tag:blogger.com,1999:blog-67049878529979396422017-07-23T02:59:47.453-07:00Sad Armadillo: Julie Wright's Math Teaching BlogJulie Wrightnoreply@blogger.comBlogger21125tag:blogger.com,1999:blog-6704987852997939642.post-44548842985936396312017-03-31T10:31:00.000-07:002017-03-31T10:31:06.323-07:00Hidden FiguresLast Friday, every student at the <a href="http://www.pps.net/davinci" target="_blank">arts-focused public middle school</a> where I teach math walked to a nearby theater to see Hidden Figures, the <a href="http://www.foxmovies.com/movies/hidden-figures">2016 movie about African-American women who were essential to the success of the early US space program</a>, based on the <a href="http://margotleeshetterly.com/hidden-figures-nasas-african-american-computers/">book by Margot Lee Shetterly</a>. We were accompanied by most of the school staff, including our principal and assistant principal, as well as dozens of family chaperone volunteers, a crowd totaling more than 400 people. Seeing the film together and talking about it afterwards with one of my classes were some of the most meaningful and fulfilling experiences I've had as a teacher, and I feel so much love and pride for my whole school community.<br /><br />A lot of people made that field trip happen, but I was the main force behind it and grew almost obsessed with it, and I want to write about why. As a math teacher, I've become more and more focused on equity in math education and passionate about inclusion and empowerment, and this event sprang from that passion. I love helping my students love math and become confident and competent mathematicians and problem solvers, but <a href="http://sadarmadillo.blogspot.com/2014/11/school-math-and-me.html" target="_blank">I've never lost the memory of what it feels like to know others think you don't belong in math or science and to wonder if maybe they are right</a>. I know that having good intentions and promoting logical thought isn't enough by itself to counteract the fog of racism we all live in, and that <a href="https://www.susanjfowler.com/blog/2017/2/19/reflecting-on-one-very-strange-year-at-uber" target="_blank">STEM fields are still unstable territory</a> for girls and women in our culture. <a href="https://docs.google.com/document/d/1r69Fd16fX6Tuzxyz-eTTFFRWQH2Z6Y4If8Q8pIgyp0I/edit" target="_blank">As a math teacher, I have the responsibility and the privilege to help make things better. </a>Sometimes I manage it, and sometimes I still fall short of where I would like to be, but I have to believe that if I keep caring and educating myself, I will be more and more effective at raising everyone up as mathematicians, students, and citizens. I'm fortunate to teach in a time when we can <a href="http://tinyurl.com/mathandsocialjustice" target="_blank">collect and share resources to do that.</a><br /><br />As soon as I started seeing information about Hidden Figures last fall, I was hooked. It sounded almost too good to be true — a fun, feel-good, PG film that brings STEM and civil rights achievements by Black women in the early 1960s into the light, shows the excitement of the intellectual accomplishments behind the space program, features stellar actors, and puts a mathematician center stage as a heroine — wow! I read a blog post by a fellow member of the "MathTwitterBlogosphere" (<a href="https://twitter.com/#MTBoS" target="_blank">#MTBoS</a>), Delaware physics and math teacher John Burk, with the subject "<a href="https://quantumprogress.wordpress.com/2016/12/11/lets-start-a-movement-for-hidden-figures/" target="_blank">Let's Start a Movement for Hidden Figures</a>," and thought a school field trip would be a fantastic idea if the movie turned out to be as terrific as it looked. Teachers in my professional learning community at school were intrigued with the idea and interested in working on it, and the school administrators, the school Site Council, the Equity and Climate and Culture Committees, and every other teacher who heard about it were positive and supportive.<br /><br />As more previews were posted and early rave reviews came out in the media and <a href="http://rafranzdavis.com/why-seeing-hidden-figures-is-important/" target="_blank">from STEM educators</a>, I got more and more eager to have my students see this story of courage and accomplishment. I made my screensaver into the picture of Dorothy Vaughan (Octavia Spencer) and her coworkers marching down the hall, posted about it on Twitter, and <a href="https://twitter.com/TequillaKiki/status/815729031106723842" target="_blank">got an "I LOVE THIS!!" retweet from one of the actors in the picture (fun!!)</a>. The week Hidden Figures came out in local theaters, I showed all my math classes previews and <a href="http://www.makers.com/katherine-g-johnson" target="_blank">a brief clip of the real mathematician Katherine Johnson</a> (now 98) and told them about my own excitement. One African-American girl had seen it before the regular release date, and had watched the previews enough that she was <i>mouthing along to all the words</i>... about women doing math and engineering! A Black mom of another girl later told me the family went to see the movie and thanked me, saying, "I think it’s great that she’s being exposed to careers that require a good math knowledge." Several Black and multiracial girls told me about doing their Social Studies current event assignments on the real-life figures from the movie. White students and boys also went to see it and told me how much they loved it. After a snow-related delay, I finally saw it myself with my own teenage son and we both found it incredibly moving and thought-provoking (and the whole, full theater cheered at the end). As I had expected to be, I was struck by the main characters' heroism as individuals, but I was also impressed with how well the movie showed teamwork and support, how accurately it portrayed math as problem solving, and by the complicated story line about Dorothy Vaughan promoting her whole group's skills and successes and anticipating how the IBM computer would change their workplace.<br /><br />It was around this point that my enthusiasm for a whole-school field trip moved to determination, especially when our principal went to see Hidden Figures and became one of the strongest proponents of a whole-school field trip, and when our school counselor encouraged me to follow through with planning and generously shared materials, tips, and work from her experience on similar trips. The whole thing took a ton of planning and support: honestly, I might have hesitated more if I had realized how much I was asking from our secretaries and counselor, especially. But I could not have asked for a more positive, can-do attitude from the whole community, and it made the whole thing stay fun and inspiring. Just about the entire staff worked on organizing permission forms and payments and sponsorships, even the school nurse, who wouldn't even be at our site that day. For our professional learning community (PLC) one afternoon, a dance teacher, a social studies teacher, the principal and I had one of my favorite school meetings ever, as we brainstormed about curriculum for teachers to use the afternoon of the movie viewing. Staff, students, and parents thanked me so many times for leading this effort that I lost count, and it was so encouraging to know they were happy about it. Even the National Council of Teachers of Mathematics president, <a href="http://www.nctm.org/News-and-Calendar/Messages-from-the-President/Archive/Matt-Larson/Do-You-See-and-Engage-Your-Hidden-Figures_/" target="_blank">Matt Larson, was talking about Hidden Figures on his NCTM blog</a> the same week as our trip!<br /><br />The day of the trip, I was a little edgy as our large crowd walked to the theater, wondering how it would go, but the kids were in excellent spirits and great fun, and I don't think I heard any complaints on the 3/4 mile walk, even though it rained all the way (our native Portlanders were completely unphased by that). We actually split across two theaters, and it was pretty awe-inspiring to see how many people were there in each. I loved seeing the movie just as much the second time. The kids seemed totally absorbed during the movie, though in our theater they were quieter and more solemn than I expected (except with Octavia Spencer's line to the white supervisor, "I'm sure you believe that."... that got a vocal reaction!). I was delighted and a little relieved when they clapped as the credits rolled. It seemed fitting that the sun unexpectedly came out for our walk back. Everybody seemed happy to have this experience together and I heard such a positive response to the movie from those who had seen it for the first time, both students and adults.<br /><br />After lunch, we split up into our sixth period classes, and using our PLC's discussion guide which was a lot like <a href="https://docs.google.com/presentation/d/13n_j_yaADl4ZTIWBdUQgEtYiBdhNWtN_8IqcnYgj_U0/edit#slide=id.p" target="_blank">this one</a> (in which I edited out a few questions that originated elsewhere), we talked through these questions:<br /><ul><li>What did you think of the movie? Did you enjoy it? </li><li>What questions or feelings did it leave you with?</li><li>Why do you think we spent school time seeing this movie?</li><li>What does the term “Hidden Figures” refer to?</li><li>In the movie, we saw many times characters were treated unfairly because they were Black. Which three examples stood out most to you?</li><li>What character attributes and/or actions did you admire most about <a href="https://www.nasa.gov/content/katherine-johnson-biography">Katherine Johnson</a> (the mathematician), <a href="https://www.nasa.gov/content/dorothy-vaughan-biography/">Dorothy Vaughan</a> (the manager who taught herself how to program the computer), and/or <a href="https://www.nasa.gov/content/mary-jackson-biography">Mary Jackson</a> (the engineer)? [I also ended up asking them which was their favorite; as I expected, Mary, played by the glamorous and fun Janelle Monáe, was in the lead, but I was surprised that at least a third of them picked serious, ultra-competent Dorothy Vaughan.]</li><li>In the film, Space Task group director Al Harrison (Kevin Costner) was depicted as a heroic breaker of boundaries when he smashed down the “Colored Women” restroom sign. <a href="https://news.vice.com/story/oscar-nominated-hidden-figures-was-whitewashed-but-it-didnt-have-to-be">Unlike many of the dramatic moments in the movie, this incident was entirely made up.</a> In real life, Katherine Johnson herself chose to use the restroom white women used. Why do you think the screenwriter and filmmakers added this incident? Do you think adding this incident improved the movie? Why or why not?</li><li>What was most striking to you about how men treated women in this movie, and how women treated each other? What aspects of how women were treated do you think seem similar to our times, and what seemed different?</li></ul><div><div>Just two hours before spring break, almost all of my sixth grade class members were intent on considering weighty questions and issues about history and justice and racism and sexism. One girl wondered, "Why were these women 'hidden'? Why didn't we all know this history?" These eleven- and twelve-year-old students were so insightful, intelligent, and empathetic that I couldn't possibly do justice to the whole discussion here. I was so incredibly proud of them. I did write down their answers to the question, "What examples of characters being treated unfairly because they were Black stood out to you?" so you can get an idea of how observant and thoughtful they were. This list was generated not just by a few kids, but from a majority of kids in the room:<br /><div><div style="-webkit-text-stroke-width: 0px; color: black; font-family: Times; font-size: medium; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: normal; letter-spacing: normal; margin: 0px; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></div><ul><li style="font-style: normal;">Dorothy Vaughan and her kids were chased out of the library by the security guard (and she pays taxes for libraries!).</li><li style="font-style: normal;">Katherine Johnson's coworkers set up a "Colored" coffee pot for her... and it was empty.</li><li style="font-style: normal;">The bathrooms and drinking fountains were segregated, and the courthouse and bus had "colored seats in back." Separate was NOT equal: the things labeled "colored" were dirtier and cheaper. White people acted like this segregation was right, using words like "your kind". </li><li style="font-style: normal;">Mary Jackson couldn't access the class she needed to because of the segregated night school.</li><li style="font-style: normal;">Katherine Johnson's new coworkers assumed she was the custodian.</li><li>Paul Stafford kept telling Katherine Johnson that computers can't be authors of technical papers. <i>[Actually, I think the way she was restricted to being a computer was more about her being a woman, but it wasn't completely clear.]</i></li><li style="font-style: normal;">Mrs. Mitchell claimed to Dorothy Vaughan that "I don't have anything against you all," which was obviously false (and even the way she talked about black women as if they weren't people like her was insulting).</li><li style="font-style: normal;">Dorothy Vaughan was doing a supervisor's work, but without the credit or pay.</li><li style="font-style: normal;">When the police officer approached the women by their broken-down car, he assumed they were doing something wrong, and they were scared of what he would do.</li></ul><br />We had been afraid that the hour would drag by as we teachers tried to convince kids to have a serious conversation when their minds would be on the Talent Show that followed it and, of course, on the start of spring break right after the show. Instead, we had a fascinating and vigorous discussion that I was sorry to cut short. Kids even asked if we could continue it the Monday after spring break!<br /><br />I went to the Talent Show full of fondness for my own class around me, the wonderful performers who included my present and past students, and the adults that support them in their arts and in their studies. Da Vinci teachers regularly go so far beyond classroom teaching. Our drama, dance, visual arts, music, and writing teachers do incredible work to bring our students' art to the wider community and bring professionals to the students; a math teacher is one of the Talent Show organizers and a mentor to the student rock band; a language arts teacher organizes a yearly trip to the Oregon Shakespeare Festival in southern Oregon; our science teachers have kids work on community projects; our social studies teachers have them engage with the broader world as citizens in so many ways; our librarian just spent many lunchtimes and a Saturday with kids doing Oregon Battle of the the Books; the list just goes on and on. I volunteer to assist with some of these events, but usually I feel like my community contributions are limited to my classroom role and General Adult Help. It was inspirational to have my own idea and to be supported and given the freedom to carry it through, which was an experience that makes me appreciate my school and its administration and staff even more. And although it's nothing new for me to get resources and ideas from my online math teacher community, <a href="https://sites.google.com/site/mathandsocialjustice/curriculum-resources/hidden-figures-resources" target="_blank">this was yet another time that they helped me create something better than I ever could have on my own</a>.<br /><br />I was already elated with the entire day, but it got even better when the assistant principal thanked me from stage during the Talent Show for organizing it, and the whole school cheered — what an incredible feeling! From what I heard from teachers at lunch and after school, they also loved the whole event, both the movie and the discussion afterwards in their classrooms. As if all that weren't enough, one of those teachers sent the most amazing email the next day to me and the counselor and the secretaries. Here's the part that brought tears to my eyes: "Of everything I've been involved in at dV, this was the most rewarding and was true community building from the heart. [...] Overall, what amazing kiddos we have!! Thank you guys again.....it was so meaningful."<br /><br />So I started spring break with a full heart, in the best way. It felt good to contribute to community building, good to do a little bit to move society forward, and absolutely fantastic to have 389 kids see this movie and love it. I am so proud of our kids and our community, and grateful to Hidden Figures that we are all more educated on this story than we were last year. What will happen next with our community discourse on race, gender, workplaces, schools, justice, film, math, science, technology, engineering, heck, the whole world? I don't know, but I can't wait to see where we go.<br /><br /></div><div><br /></div></div></div>Julie Wrighthttps://plus.google.com/105628344902704593026noreply@blogger.com1tag:blogger.com,1999:blog-6704987852997939642.post-74382130369195453282016-05-07T21:43:00.001-07:002016-05-08T09:25:32.716-07:00How a Farflung Internet Conversation Helped Me Make Sense of My Students’ Sense-Making<i>A shorter version of this article originally appeared in <a href="http://www.octm.org/publications.html">The Oregon Mathematics Teacher (TOMT)</a> in May/June 2016, and is <a href="https://drive.google.com/file/d/0B6ZGw2bZPNA1YlBXV05hYUY4c0E/view?usp=sharing">reprinted here</a> by kind permission of TOMT and the Oregon Council of Teachers of Mathematics. The writing of this article was supported by the Writers’ Retreat facilitated by the editors of TOMT and funded by the <a href="http://octm.org/">Oregon Council of Teachers of Mathematics</a>.</i><br /><br />Math educators almost universally place high value on reasoning, sense-making, and logical thinking. The first two <a href="http://www.corestandards.org/Math/Practice/">Common Core Standards for Mathematical Practice</a> begin “Make sense of problems” and “Reason.” We promote math education as a means for students to learn how to analyze, understand, and improve the world around them. But what are we to conclude when students respond to a mathematical absurdity not with skepticism and reason, but with hapless and nonsensical attempts to impose classroom math upon it? <br /><br />An <a href="https://tjzager.wordpress.com/2014/10/18/making-sense/" target="_blank">online blog article</a> by math educator and researcher <a href="http://www.twitter.com/TracyZager" target="_blank">Tracy Zager</a> titled “Making Sense”raises this question with a fascinating example, a video filmed by <a href="https://twitter.com/robertkaplinsky" target="_blank">Robert Kaplinsky</a>, a district math specialist in California, who also <a href="http://robertkaplinsky.com/how-old-is-the-shepherd/" target="_blank">wrote about it at his website</a>. In the video, eighth graders were presented with <a href="http://hub.mspnet.org/index.cfm/9217" target="_blank">a question popularized by Katherine Merseth</a>: “There are 125 sheep and 5 dogs in a flock. How old is the shepherd?” Astonishingly, 75% of the students performed operations with the numbers in order to come up with a numerical answer. Robert added that a group of sixth graders had done even worse, universally coming up with numerical answers rather than pointing out that the problem does not make sense as written. <br /><br />I’m sure I wasn’t the only person to watch this video and immediately question whether it was truly representative of American students. Maybe, I thought, Robert’s group of students was particularly intimidated, or unusually bad at math. But further reading showed his results were actually quite typical. The original researcher whom Merseth wrote about, Kurt Reusser, also found in 1986 that three fourths of schoolchildren did not make sense of the question. More recently, Robert’s comment section had comments from several teachers who tried the experiment with their students, and Tracy wrote about elementary school teacher colleagues who tried similar corresponding problems with their younger students. In many different groups, with varied teachers, a majority of students failed to make sense of the problem or object to it. <a href="https://tjzager.wordpress.com/2014/10/18/making-sense/#comment-78" target="_blank">Robert added in a comment on Tracy Zager’s post</a> that when he reran the problem with a whole class, even after thorough group discussion of all the individual answers, many students still chose the nonsense answer. <a href="https://tjzager.wordpress.com/2014/10/18/making-sense/#comment-83" target="_blank">Tracy admitted that she had thought younger children would do better</a>: “I had assumed kids went with nonsense by middle school because of what we’ve taught them about math. Now that I have preschoolers and 1st graders going with nonsense too, I have to revisit my assumptions.” <br /><br />What is going on? “America has produced a generation of students who engage in problem solving without regard for common sense or the context of the problem,” concluded Merseth in 1993, before blaming societal attitudes about math, limited curricula in which “stress is on computation and procedure, not understanding and sense making,” and poor teacher preparation. <a href="https://twitter.com/joboaler" target="_blank">Jo Boaler</a>, author and professor of mathematics education at Stanford University, has also criticized our culture’s emphasis on computation and procedure. In <a href="https://www.youcubed.org/why-we-need-common-core-math/" target="_blank">her video “Why We Need Common Core Math,”</a> she states, “If you ask most students what they think their role is in maths classrooms, they’ll tell you that it’s to answer questions correctly. They don’t even think that they’re in maths classrooms to learn or to explore the rich set of connections that make up mathematics. They think they’re there to get questions right.” <br /><br />On Robert’s site and in Tracy’s blog’s comment section, teachers struggled to make sense of why students weren’t making sense. Here were some of their thoughts: <br /><ul><li>Some believed the interaction between the teacher and student was key: “The student’s main objective is not to make sense of the question, or even to answer it correctly, but to give an answer that will satisfy the teacher,” said a commenter named Dave on Robert’s site. “Students believe that an honest answer (‘I don’t understand the question’) would not satisfy the teacher, so they make random guesses instead.” Commenter Pierre added, “Students are conditioned to assume that the teacher is teaching something sensible.” “It does appear that some students are overriding their common sense… and my hope for this video is that it will help us all reflect on why that is,” Robert Kaplinsky responded to Dave.</li><li>Commenter MG on Robert’s post blamed overtesting, in part: “When we base schooling around testing, students learn to look like they’re smart in order to fool the test […] without understanding.”</li><li><a href="http://www.twitter.com/MFAnnie" target="_blank">Annie Fetter</a> of the <a href="http://mathforum.org/" target="_blank">Math Forum</a> <a href="https://tjzager.wordpress.com/2014/10/18/making-sense/#comment-80" target="_blank">wondered in Tracy’s comment section if drawing a picture</a> would help kids “step back and do the sense-making that they won’t do if they think it’s math.”</li><li>Commenter Joshua (of <a href="http://3jlearneng.blogspot.com/">3jlearneng.blogspot.com</a> ) on Tracy’s post <a href="https://tjzager.wordpress.com/2014/10/18/making-sense/#comment-106" target="_blank">theorized that </a>“students seem to be triggered to think they are answering a ‘math-class’ question (MCQ). MCQs have […] exactly one single correct answer, […] can be answered using the tools you have been taught, […] require a mathematical object as an answer […], are asked of students, not by students, [and] they require entering into an idealized world where you only have the information given to you within the question, no more and no less.” Joshua’s ideas seemed to correspond with some shared by <a href="http://www.twitter.com/ddmeyer" target="_blank">Dan Meyer</a> in his well-known <a href="http://www.ted.com/talks/dan_meyer_math_curriculum_makeover" target="_blank">TED talk “Math class needs a makeover”</a>: “[W]hat problem have you solved, ever, that was worth solving where you knew all of the given information in advance; where you didn't have a surplus of information and you had to filter it out, or you didn't have sufficient information and had to go find some.”</li></ul><br /><b><i>I Wonder What Would Happen In My Classroom?</i></b><br /><br />If you’re like me, as you read about this problem and how students reacted to it, you started itching to go ask your own students how old the shepherd is! After all this online discussion, I decided to try to investigate my students’ sense-making with a nonsense question. These were my goals: <br /><ul><li>Compare the performance of my sixth graders at an arts focus middle school (<a href="http://www.pps.net/schools/davinci" target="_blank">da Vinci, Portland Public Schools</a>) with that of other groups of students. Although math is not generally the main interest or passion of my students, they often seem to me to be more willing to be original thinkers and to question authority than many sixth graders, so I thought perhaps they would be less likely to comply with a nonsense answer against their intuition.</li><li>Give a nonsense prompt that seemed very likely to generate multiple attempted numerical answers, in order to help ensure that no individual felt exposed or stupid when the answer was revealed in a whole-class discussion. I decided to change the question to one that I felt was somewhat less obviously nonsensical, to be cautious on this point. I chose a fraction problem because we had recently finished a fraction unit.</li><li>Generate a prompt which was close to one that could be solved, to see if students would make suggestions of what missing piece of information would help.</li><li>Examine the role of drawing pictures, as suggested by Annie Fetter.</li><li>Debrief in phases: sharing anonymous individual answers, discussing in pairs, then having a whole-class discussion. Ultimately, I wanted students to leave class understanding the actual answer (not enough information), knowing what motivated me to ask the question, and becoming aware that giving a numerical answer was not at all unusual. I also wanted to elicit their thoughts on why it is so common to try to answer a messed-up question with a specific answer, and how teachers can build a space where students feel safe and confident critiquing the question. </li></ul><br />With these goals, I prepared this Google slide for one class (Group A) and another just like it except without the cue to draw a picture for two other classes (Groups B and C): <br /><br /><a href="https://4.bp.blogspot.com/-QVXhDOGIjEQ/Vy6o29vXyiI/AAAAAAAAAFM/TINO-BdpDJwUeLm04gKDb0FmhWqzmT2cACLcB/s1600/HOWOLD.cocoaatjoes.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="263" src="https://4.bp.blogspot.com/-QVXhDOGIjEQ/Vy6o29vXyiI/AAAAAAAAAFM/TINO-BdpDJwUeLm04gKDb0FmhWqzmT2cACLcB/s640/HOWOLD.cocoaatjoes.jpg" width="480" /></a><br /><br />and provided it to my students to “solve” on individual whiteboards, after giving them two key verbal instructions: that it was essential they work silently and individually, and that if they had questions for me, they should write them down as part of their work, but that I would not answer questions while they worked individually. Each class (Groups A, B, and C) had 27, 23, and 21 students present, respectively, sitting at tables of three or four students. <br /><br />A followup slide for a “pair share,” which happened after I collected the individual whiteboards, included the question (“It takes…”) preceded by the prompt: “Now discuss the problem with the person next to you (or the people across from you if you’re at a table of 3).” <br /><br />Following the pair share, we had a group discussion, during which students shared out their table’s thinking, then I showed them the Robert Kaplinsky video and summarized the research results. After our class discussion, I made sure to obtain the first two groups’ agreement not to spoil the surprise by discussing the problem with other classes (and I believe they did not). <br /><br /><b><i>What Actually Did Happen In My Classroom</i></b><br /><br /><i>Initial Student Reaction</i><br /><br />Our school has a large emphasis on performance arts, and my students tend to be extroverts, so I expected that watching their faces as they processed the question would be entertaining, and I was not disappointed. In Group A, many students looked worried, confused, and doubtful, and a few looked indignant, within a minute or so, before they picked up the pens to draw or write. Several insisted on calling me over to whisper to me (I responded as described above). To my surprise, the expressions on the students’ faces were very different for Group B as they absorbed the question: they looked much less worried and more confident, and many looked amused. Fewer called me over to question the problem, and most were quicker to write down their answers. Group C’s students’ confidence seemed between Group A’s and Group B’s, and the few students who called me over seemed, by their body language, more ready to tell me off. <br /><br /><i>Effect of the Pictures</i><br /><br />For Group A, the students who were prompted to draw a picture, students were generally willing to do this, but I saw little evidence the pictures helped them realize the problem was nonsense: in fact, if anything, my impression was that my insistence on a picture made some think it must be a problem I thought they could solve. <br /><br /><i>Overall Results for Individual Answers</i><br /><br />Unfortunately, these were not perfect research conditions: I don’t have a background as an educational researcher, and with back-to-back classes it was difficult to keep perfect records. Nevertheless, the results I was able to record were quite interesting and suggestive to me. <br /><br />First of all, a significantly higher proportion of da Vinci students identified the “Cocoa at Joe’s” question as a nonsense question than students in previous formal and informal studies tended to do for the “How old is the shepherd” question. In Group A, more than half clearly stated in some form or another that it was a nonsense question, and why. (My notes seem to say it was 15 out of 27, they but are not completely clear.) In Group B, a few students looked worried about it, but virtually everyone ended up answering that the problem did not make sense or there was not enough information. In Group C, 12 out of 21 students present clearly indicated that the problem could not be answered and/or was nonsense. Here are some examples of ways students expressed this thought on their whiteboards:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-dd_mQm_ZrX0/Vy6p9ymTqdI/AAAAAAAAAFY/UICbope4Ib4hzZjeCamBw5AIeVWxz0HAgCLcB/s1600/idk.pic.p4.JPG" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="174" src="https://4.bp.blogspot.com/-dd_mQm_ZrX0/Vy6p9ymTqdI/AAAAAAAAAFY/UICbope4Ib4hzZjeCamBw5AIeVWxz0HAgCLcB/s320/idk.pic.p4.JPG" width="250" /></a><a href="https://1.bp.blogspot.com/-7fMFCy3aKgY/Vy6puiYjl8I/AAAAAAAAAFU/sPGwzIc0y1As5s6uVWTQ6nUgaUXdobadgCLcB/s1600/depends.pic.p4.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="170" src="https://1.bp.blogspot.com/-7fMFCy3aKgY/Vy6puiYjl8I/AAAAAAAAAFU/sPGwzIc0y1As5s6uVWTQ6nUgaUXdobadgCLcB/s320/depends.pic.p4.JPG" width="250" /></a></div><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-5FOdouXi68w/Vy69VFA1_bI/AAAAAAAAAF0/4Jd5nSgm_sQN-QuZiTmmIM1wX7Tl8KMOwCLcB/s1600/notcomplete.p4.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="177" src="https://4.bp.blogspot.com/-5FOdouXi68w/Vy69VFA1_bI/AAAAAAAAAF0/4Jd5nSgm_sQN-QuZiTmmIM1wX7Tl8KMOwCLcB/s320/notcomplete.p4.JPG" width="250" /></a><a href="https://1.bp.blogspot.com/-2F6k9MBcs3U/Vy69QDh2f4I/AAAAAAAAAFw/vOTuXi_JwOsC2lBmUoYqMzg6eN_CVRN5QCLcB/s1600/moarinfosad.p6.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="180" src="https://1.bp.blogspot.com/-2F6k9MBcs3U/Vy69QDh2f4I/AAAAAAAAAFw/vOTuXi_JwOsC2lBmUoYqMzg6eN_CVRN5QCLcB/s320/moarinfosad.p6.jpg" width="250" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-l7se4ysGxA4/Vy6-40BN14I/AAAAAAAAAGA/zPsCBjD7GoMQIsKjt56DW0zNaXGWKNhlQCLcB/s1600/notfunny.p6.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="175" src="https://3.bp.blogspot.com/-l7se4ysGxA4/Vy6-40BN14I/AAAAAAAAAGA/zPsCBjD7GoMQIsKjt56DW0zNaXGWKNhlQCLcB/s320/notfunny.p6.JPG" width="250" /></a><a href="https://4.bp.blogspot.com/-3gkZ3ZhO6Kk/Vy6-9fqcE2I/AAAAAAAAAGE/rOi4fWeh8Bs_L1COhaib8E002tzHwCZ4QCLcB/s1600/infiniteamts.p4.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="162" src="https://4.bp.blogspot.com/-3gkZ3ZhO6Kk/Vy6-9fqcE2I/AAAAAAAAAGE/rOi4fWeh8Bs_L1COhaib8E002tzHwCZ4QCLcB/s320/infiniteamts.p4.JPG" width="250" /></a></div><br /><a href="https://drive.google.com/open?id=0B6ZGw2bZPNA1SGtldm54YXU4ejA" target="_blank"><i>(More photos of answers of this type are available here in the "Not Enough Info" folder.)</i></a><br /><br />A few other students in Groups A and B, as well as three in Group C, were clearly uneasy with the question and refused to commit to a real answer, writing “IDK” (I don’t know) or similar statements. <br /><br />There seemed to be no clear correlation between sense-making and general math achievement; if there was any trend, the students who seemed most confident it was a nonsense question tended to be from both the high and low ends of the class, grade-wise, more than from the middle. <br /><br /><i>Details of Individual Answers</i><br /><br />As I expected, many of the students who gave a numerical answer answered 1 ¼ (the sum of 1 and ¼, the two numbers given) or 4 (the number of ¼’s in 1; the quotient, although they probably didn’t think of it that way). I was surprised by how many people answered 1, including one girl who said she thought it was a riddle: perhaps in their attempts to make sense of the question, they decided only one serving was described, so it must be one person. Here are some pictures of work from people who tried to explain their numerical answer: <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-Qfhxx5NBWYo/Vy7ANe9U1FI/AAAAAAAAAGY/_v15Fs0GO6cy-QKWf1SOMsxokEA48-WbQCLcB/s1600/1.25.dubious.pic.p4.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="194" src="https://2.bp.blogspot.com/-Qfhxx5NBWYo/Vy7ANe9U1FI/AAAAAAAAAGY/_v15Fs0GO6cy-QKWf1SOMsxokEA48-WbQCLcB/s320/1.25.dubious.pic.p4.jpg" width="150" /></a><a href="https://3.bp.blogspot.com/-eYmS3ufiCC0/Vy7ANadVg9I/AAAAAAAAAGc/EZDw0GvLJN8kDFyY8BnFhT1ixnjWaCmqQCLcB/s1600/1.25.pic.p5.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="166" src="https://3.bp.blogspot.com/-eYmS3ufiCC0/Vy7ANadVg9I/AAAAAAAAAGc/EZDw0GvLJN8kDFyY8BnFhT1ixnjWaCmqQCLcB/s320/1.25.pic.p5.JPG" width="170" /></a><a href="https://1.bp.blogspot.com/-AOzC_iZHm-Q/Vy7ARHoxb7I/AAAAAAAAAGg/3Z40FWpWT7Qu4QOkWYg1bjiVIX6juhFqACLcB/s1600/1person.pic.p5.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="180" src="https://1.bp.blogspot.com/-AOzC_iZHm-Q/Vy7ARHoxb7I/AAAAAAAAAGg/3Z40FWpWT7Qu4QOkWYg1bjiVIX6juhFqACLcB/s320/1person.pic.p5.jpg" width="150" /></a></div><br /><a href="https://drive.google.com/open?id=0B6ZGw2bZPNA1SGtldm54YXU4ejA" target="_blank"><i>(More photos of answers of this type are available here in the "Numerical or No Answers" folder.)</i></a><br /><br />I suspect several of the people who avoided writing down a specific answer or line of reasoning (despite my encouragement) didn’t think the problem made sense, but were too confused and underconfident to write that down. <br /><br /><i>Results for Answers After Table Discussions</i><br /><br />In all three groups, during table discussions, every single student became convinced it was a nonsense question. Group B seemed to spend the most time talking about why they thought I asked a nonsense question. In Group C, I was interested to hear several students loudly and confidently referring to algebra (this idea may have spread from one table to another as they overheard it, or maybe several people came up with it). “It’s algebra!”, “n,” and “random number” were some sentence fragments I wrote down. <br /><br />After discussion at their tables, many students volunteered to summarize their tables’ thoughts. The people who spoke for the tables were mostly medium- to high-achievers in math class, but many others seemed intent on listening to their answers and clearly agreed. A few typically high-achieving speakers had begun with a numerical answer and seemed eager to share why they were now confident it was a nonsense question. Here was what the groups said when they reported back (comments recorded in chronological order): <br /><br /><i>Group A:</i><br />Our table agreed that if you knew how much they used in one day, then you could answer, but [that was missing].<br />You were trying to make us look at the question more than the other information. <br />It seems like they took out a sentence. It seems like algebra – looking for missing things, and how they fit together. Some of the things we need to answer the question aren’t there. <br />I changed my mind from the discussion. It says “how many people ordered,” and you only have information about one cup. <br />It doesn’t give you that little teaser! Any number could be right. There’s not enough information. <br />The question doesn’t make sense! <br /><br /><i>Group B:</i><br />You can’t do the problem. There’s no information about how much milk or cocoa. You wanted us to think of how equations can be wrong or [how there can be] not enough information. <br />You’d need to know how much was used. At first I thought it was a riddle. <br />I thought it was n. You have the information but not all of it, so it’s as many numbers as it is. [Another student from that table interjected: It could also be n x ¼ and n x 1 and add together.] It reminds me of <a href="http://www.ted.com/talks/dan_meyer_math_curriculum_makeover" target="_blank">[Dan Meyer’s] TED talk</a> – you give them partial information and make them ask for more. [My mind was blown by this, since I wasn’t thinking of the TED talk at all, and wasn’t even certain which periods I’d shown it to. I asked, “Who thought of the TED talk while you were working on this?” and four people around the room raised their hands.] <br />There’s not enough information. At the end it doesn’t say who ordered. <br />We all agreed there was not enough information. [We thought] “Maybe she wanted to test us.” <br />We could not do it! Somebody got some numbers, but I said, how could you do it?? <br /><br /><i>Group C:</i><br />I wrote one person, but I realized there’s no way to answer, so now I say x. <br />We all could have realized it wasn’t multiplying fractions or anything. [There’s] no answer, so we could call it x. <br />At first we looked for an answer but [… it] could be any number. <br />I think it’s also a letter because we’ve been studying algebra. <br /><br />I asked Group C, “Why do you think I gave you this problem?” Four new students answered (this time across the achievement spectrum): <br /><br />To show sometimes problems have algebra letters. <br />In algebra, there’s not always a specific number answer. <br />Maybe the problem was written wrong. [Interestingly, though, this answer was from the only student in this class who reported that an elementary school teacher had given his class a similar (nonsense) problem.] <br />Maybe whoever wrote this wanted us to think about algebra. <br /><br />I believe I asked Group B a similar question, although unfortunately I didn’t write down my question(s), just their answers. One student stated firmly that people “lack common sense.” Another said that people “talking among themselves helps make sense.” <br /><br /><i>How They Felt</i><br /><br />I asked students how they felt about this question when they were working on it on their own, and being middle schoolers, they were eager to share. <br /><br />In Group A, a student reported feeling “annoyed,” and about three-fourths of the students agreed. Another student reported thinking it was funny; three or four people agreed. One added, “I thought it was one of the funniest things a math teacher’s ever done.” <br /><br />In Group B, when asked to describe how they felt while working on this problem, one student replied, “That was amazing!” Students volunteered words to describe other ways they felt while they were working on the problem by themselves, and we got a count for each adjective of how many people felt that way. Out of 23 students, five felt “frustrated/GRRR,” nine felt confused, four felt content (which puzzles me!), six said “funny,” two felt surprised, and eight felt stupid (“because I couldn’t get the answer”). <br /><br />In Group C, I solicited a list of words from students to describe ways they felt while they were working on the problem by themselves, and we got a count for each adjective of how many people felt that way (out of 21). Eleven people reported feeling stupid, seven felt confused, five felt indignant or annoyed, three felt determined, and two felt amused. <br /><br /><b><i>Take-Aways and Conclusion</i></b><br /><br />I probably question and second-guess every instructional decision I make as a math teacher. Do problems that are challenging or that do not have a particular right answer help students develop as critical thinkers, or are they just discouraging “trick questions”? Does group work help students communicate ideas and come to a deeper understanding of math, or is it just pointless off-topic chatter or copying of work without comprehension? Do my students need more time for practice of procedures and less for conceptual understanding? Should they respect my authority more, or continue to be allowed to question me and the tasks they are given? <br /><br />I’m going to continue asking those questions of myself, and I am sure the answers will fluctuate depending on students’ needs. Nevertheless, this activity and the results from it, especially compared to the results obtained in other settings with similar problems, gave me a renewed appreciation for the role of non-“math class questions,” group work, conceptual understanding, and student questioning in developing mathematical reasoning and sense-making. I work hard at promoting critical thinking and conceptual understanding, and believe elementary school teachers in my district do too. I think this emphasis contributed to how well the da Vinci students did at identifying the nonsense question. Students also clearly progressed in their thinking in the group discussion, partly because they were comfortable with this format, and they developed sophisticated ideas together of what the activity was about. Finally, I allow and sometimes even encourage students to criticize textbook problems that don’t seem to match real life (though often we solve them anyway). This used to feel indulgent, but now I feel it is important for my classroom culture and their critical thinking. <br /><br />I also realized that our students need us to convey that mathematical reasoning and sense-making is <b><i>not</i></b> about getting procedural answers as fast as possible. When students use math outside of the classroom, they must make decisions about what types of problems they can solve and what information they will need. We do them a disservice if we only give them “plug-and-chug” problems. For Groups B and C, I recorded that 19 out of 44 students (43%) felt stupid when they could not solve the nonsense problem, including many who had correctly answered that there was not enough information. I hope they finished class realizing that the same critical thinking that made them so uneasy that they were “doing it wrong” was actually a strength. <br /><br />As <a href="http://www.ted.com/talks/dan_meyer_math_curriculum_makeover" target="_blank">Dan Meyer says</a>, “Math makes sense of the world. Math is the vocabulary for your own intuition.” Our students all deserve to have that experience.Julie Wrighthttps://plus.google.com/105628344902704593026noreply@blogger.com0tag:blogger.com,1999:blog-6704987852997939642.post-16422626868703119362016-05-04T22:04:00.001-07:002016-05-04T22:33:13.766-07:00Online Networking for Math Educators (or: How to MTBoS)<div class="MsoNormal" style="margin-bottom: 12.0pt; mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;"><i style="mso-bidi-font-style: normal;"><span style="font-family: "futura"; font-size: 11.0pt;">This article originally appeared in <a href="http://www.octm.org/publications.html" target="_blank">The Oregon Mathematics Teacher (TOMT)</a> in March/April 2016, and is reprinted (with minor edits) by kind permission of TOMT and the <a href="http://octm.org/" target="_blank">Oregon Council of Teachers of Mathematics</a>. The writing of this article was supported by the Writers’ Retreat facilitated by the editors of TOMT and funded by the Oregon Council of Teachers of Mathematics. </span></i><i style="mso-bidi-font-style: normal;"><span style="font-family: "times"; mso-bidi-font-family: Times;"><o:p></o:p></span></i></div><div class="MsoNormal" style="line-height: 150%; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-indent: .5in;">Over the course of just four days last August, my math teacher colleagues and I had lively, fun conversations in which we exchanged curriculum and lesson ideas, played math games, shared examples of math in the news, recommended teaching materials, and discussed classroom culture in math class. Several of us talked about ways to teach order of operations and the distributive property. I learned a <a href="http://marilynburnsmathblog.com/wordpress/four-strikes-and-youre-out/" target="_blank">math version of hangman</a>, and we had a great time playing several rounds. I heard about a <a href="https://www.theguardian.com/science/alexs-adventures-in-numberland/2015/aug/10/attack-on-the-pentagon-results-in-discovery-of-new-mathematical-tile" target="_blank">fascinating article about the discovery of a fifteenth type of pentagon tiling</a> that covers the plane and a neat <a href="http://www.slate.com/articles/life/culturebox/2014/10/population_map_use_our_interactive_map_to_figure_out_how_many_flyover_states.html" target="_blank">online “population mapper”</a> that shows different areas of the United States which have equivalent populations. A teacher shared a <a href="https://illuminations.nctm.org/Lesson.aspx?id=3170" target="_blank">National Council of Teachers of Mathematics free lesson on considering percentages and nutrition</a> while picking out food from a McDonald’s menu, and another recommended an excellent (and cheap) whiteboard cleaner. Finally, several of us talked with some experts in <a href="http://twitter.com/@math4justice" target="_blank">math trauma</a> and <a href="http://www.twitter.com/tchmathculture" target="_blank">math classroom culture</a> to try to analyze what steps we could take to improve our students’ experience with and confidence in math.<s><o:p></o:p></s></div><div class="MsoNormal" style="line-height: 150%; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-indent: .5in;"><br /></div><div class="MsoNormal" style="line-height: 150%; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-indent: .5in;">Wouldn’t you want to have discussions like these with your fellow teachers? You can! Every conversation mentioned in the previous paragraph happened over a few days on Twitter. The network called “MathTwitterBlogosphere” (or MTBoS, sometimes pronounced “mitt-boss”) is open to any teacher who wants to join the online conversation or just listen in. The site <a href="http://exploremtbos.wordpress.com/">exploremtbos.wordpress.com</a>, set up by teacher volunteers, has good advice on how to search for people and resources that interest you. The MathTwitterBlogosphere is a positive, welcoming community of creative math teachers from all over the world who share activities, resources, ideas, feedback, and encouragement. As one participant, Anna Blinstein (<a href="http://www.twitter.com/@borschtwithanna" target="_blank">@borschtwithanna</a>), summarized, “It's the place for connections, rich discussion & exchange of ideas. People here support, question, and challenge me to be better.” You can find discussions at times that suit you (Tuesday at 9 pm? No problem!), and connect with specialists who have expert knowledge, with teachers who have experience with the same ages, classes, lessons or teaching ideas that you do, or with people who teach math to students older or younger than yours. You can find and solve puzzles and problems that remind you why you love math so much. If more of us in Oregon join the MTBoS, we can use it to share Oregon-based lessons and tasks with each other and coordinate in-person gatherings to share ideas, expertise, support, and professional development.</div><div class="MsoNormal" style="line-height: 150%; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-indent: .5in;"><br /></div><div class="MsoNormal" style="line-height: 150%; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-indent: .5in;">Among all the social media sites out there, <a href="http://www.twitter.com/" target="_blank">Twitter</a> is uniquely suited to act as both a discussion platform and a set of links to other websites with valuable math teaching content. You can read anyone’s tweets by going to <a href="mailto:twitter.com/@username">twitter.com/@username</a>(for example, <a href="mailto:twitter.com/@OregonMath">twitter.com/@OregonMath</a>for OCTM’s account), but setting up your own free Twitter account will be far more convenient for browsing, even if you never tweet anything yourself… although I hope you’ll be inspired to do that, too, before long. I recommend setting up an account just for interacting with math educators. If you have or want other Twitter accounts, keep in mind that it will be far easier if each account has a unique email address associated with it (other people on Twitter won’t see it). Most MTBoS participants recommend picking a user name that corresponds somehow to your real name (as you’d introduce yourself to teachers at a conference, for instance, <a href="http://www.twitter.com/tracyzager" target="_blank">@TracyZager</a>) or is otherwise memorably connected to your personality (like <a href="http://www.twitter.com/@Veganmathbeagle" target="_blank">@Veganmathbeagle</a>). Lots of names are taken, so you might need to use a middle initial or add something like “math,” but choose a shorter and easier to say user name when possible, and you may want to avoid special characters like “_” that are harder to access from phones. For your profile (brief autobiographical information), it’s helpful to add an image and identify yourself as a math teacher; consider adding your grade level and geographical location, too. You can change any of these items later. </div><div class="MsoNormal" style="line-height: 150%; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-indent: .5in;"><br /></div><div class="MsoNormal" style="line-height: 150%; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-indent: .5in;">After you have a Twitter account, you can set up your “timeline” (the tweets you’ll see on your Twitter home page) by following other accounts. To do this, search for a person’s or organization’s name or Twitter user name, click to go to their Twitter page, and click “Follow”. I’d suggest following OCTM (<a href="http://www.twitter.com/@Oregonmath" target="_blank">@OregonMath</a>), NCTM (<a href="http://www.twitter.com/@NCTM" target="_blank">@NCTM</a>), and Explore MTBoS (<a href="http://www.twitter.com/@ExploreMTBoS" target="_blank">@ExploreMTBoS</a>) to start. As you follow more accounts, Twitter’s recommendations of other accounts to follow will be increasingly useful. </div><div class="MsoNormal" style="line-height: 150%; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-indent: .5in;"><br /></div><div class="MsoNormal" style="line-height: 150%; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-indent: .5in;">When you feel ready, retweet some tweets you like, tweet your own ideas and questions, or reply to a tweet in your timeline by clicking on the little left-pointing arrow at the bottom. Accounts mentioned in your tweets or retweets will get notifications, although they won’t always respond. If the first character of a tweet is “@” (as it is for most replies, the tweet shows up in the timelines of only accounts mentioned in the tweet and accounts following both the author <i style="mso-bidi-font-style: normal;">and</i>a mentioned account. These tweets are not private, though – in fact, you can see any public account’s replies by clicking on “Tweets & replies” from their Twitter page – and in the MTBoS, at least, it’s considered perfectly fine to join a conversation whether you’re mentioned or not. You can also click on any tweet to see more of the conversation. If someone wants to tell you they especially liked what you had to say, you might get a notification that they “favorited” your tweet (by clicking on the heart). </div><div class="MsoNormal" style="line-height: 150%; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-indent: .5in;"><br /></div><div class="MsoNormal" style="line-height: 150%; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-indent: .5in;">The last big piece of Twitter know-how you need concerns hashtags (words or phrases starting with “#”). Hashtags are just markers that let people search for tweets about a particular topic or event. They can be used to flag people’s attention: if you add “#mtbos” to a tweet, for instance, people may find it even if they didn’t previously know about your account. At math conferences, people may add a hashtag like “#NCTMAnnual” (for NCTM’s annual conference) to their tweets, so that other people can find them. </div><div class="MsoNormal" style="line-height: 150%; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-indent: .5in;"><br /></div><div class="MsoNormal" style="line-height: 150%; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-indent: .5in;">Hashtags are also used to mark tweets that are part of “chats,” which are scheduled (and occasionally unscheduled) conversations open to anyone who wants to join. You can read tweets in a chat by entering the hashtag (for instance, #elemchat) as a search term in Twitter. You must click “Live” to see all tweets. Scheduled Twitter chats about math focused on a particular grade level range or subject area include elemmathchat, msmathchat, alg1chat, geomchat, alg2chat, statschat, precalcchat, and calcchat.<span style="mso-spacerun: yes;"> </span>Some more general chats are probchat (about complicated problem solving; currently on hiatus), slowmathchat (various topics, with tweets spread out over a week or so; also on hiatus), spedmath (Special Education), edtechmath (educational technology), and the British chat mathstlp (Twitter Lesson Planning). About the last site, co-organizer Jo Morgan (<a href="http://www.twitter.com/@mathsjem" target="_blank">@mathsjem</a>) reports, “It's bloody brilliant. We're working together for the sake of our students. I cannot think of a more effective use of social media.” This comment could really apply to any of the chats listed here! </div><div class="MsoNormal" style="line-height: 150%; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-indent: .5in;"><br /></div><div class="MsoNormal" style="line-height: 150%; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-indent: .5in;">Chats are an excellent way to introduce yourself to the online math community while learning a lot, but busy chats can be confusing to follow at first. Sometimes people use Storify (<a href="http://storify.com/">storify.com</a>) to save a series of tweets in an organized way, so that people can go back and read the “story” later. Two examples I especially like are <a href="https://storify.com/ColeGailus/scifri-does-math-matter-live-chat">storify.com/ColeGailus/scifri-does-math-matter-live-chat</a>and <a href="https://storify.com/TracyZager/multiplying-fractions">storify.com/TracyZager/multiplying-fractions</a>. You need a free Storify account to make one, but anyone can read a story at a storify link.<br /><div class="MsoNormal" style="text-indent: 0.5in;"><br /></div><div class="MsoNormal" style="text-indent: 0.5in;"><span style="line-height: 150%; text-indent: 0.5in;">If you’re feeling a little panicky at the idea of doing all the things in this article, please remember, take your time and move at the pace that works for you! I’m pretty certain none of the math teachers on Twitter participate because they’re expected to be there or get paid for it; we just do it when we can because it’s fun and valuable. If you do get to the point that you feel comfortable sharing your online contact information, let others know you’re online by adding yourself to the </span><a href="http://sites.google.com/site/mtbosdirectory" style="line-height: 150%; text-indent: 0.5in;" target="_blank">MathTwitterBlogosphere (MTBoS) directory</a><span style="line-height: 150%; text-indent: 0.5in;"> at sites.google.com/site/mtbosdirectory. Let’s get more pins on that map and find each other! I hope to see you there and on Twitter soon!</span></div></div><div class="MsoNormal" style="line-height: 150%; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-indent: .5in;"><a href="https://www.blogger.com/null" name="_GoBack"></a></div>Julie Wrighthttps://plus.google.com/105628344902704593026noreply@blogger.com0tag:blogger.com,1999:blog-6704987852997939642.post-27679183156934222242015-09-21T17:05:00.000-07:002015-09-21T17:05:32.127-07:00"What three words come to mind when you think of school math?"<i>This post is a <a href="http://juliewright.weebly.com/class-activities-blog/what-three-words-come-to-mind-when-you-think-of-school-math" target="_blank">duplicate of a post I made to my classroom blog</a>.</i><br /><i><br /></i>In a survey at the beginning of the year, I asked my 125 Math 6 students to tell me what three words came to mind when they thought of school math. Almost everyone answered, and this Wordle, which I also shared at Back to School Night, shows all the words that were listed by two or more people. The larger a word, the more students listed it. (The colors are random.)<br /><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-S5qgK6eaD-M/VgCaA2-tirI/AAAAAAAAAEs/NLyZhgOQJnU/s1600/3wordsaboutmath.sept2015.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="460" src="http://4.bp.blogspot.com/-S5qgK6eaD-M/VgCaA2-tirI/AAAAAAAAAEs/NLyZhgOQJnU/s400/3wordsaboutmath.sept2015.png" width="533" /></a></div><div><br /></div><div>To my teacher mind, some of these words describe associations I'm pleased to see incoming sixth graders having with math, some are merely neutral or factual, and some are associations I hope will change. I'd class them like this (listing the words in each category in order of frequency):<br /><br style="background-color: white; color: #868686; font-family: Tahoma, Geneva, sans-serif; font-size: 14px; line-height: 21px;" /><b>Positive</b> (8): fun, challenging, interesting, learning, challenge, exciting, yay, smart<br /><b>Neutral/Unclear</b> (22): hard, multiplication, numbers, difficult, homework, easy, complicated, addition, subtraction, division, fractions, shapes, adding, equations, algebra, practice, math, complex, school, long, work, ok<br /><b>Negative</b> (9): boring, confusing, ugh, scary, irritating, annoying, weird, meh, stressful<br /><br style="background-color: white; color: #868686; font-family: Tahoma, Geneva, sans-serif; font-size: 14px; line-height: 21px;" />I think it's great for students to find math challenging, as long as they're not discouraged, so I put "challenging" and "challenge" into the positive category. But I'm actually uncertain whether to classify "hard" and "difficult" as neutral or negative. Hard work can be intensely satisfying and can lead to great learning, but when "hard" is one of the three <i>primary</i> associations eleven- and twelve-year-olds have with math, well, I worry that they're feeling overwhelmed. At the other extreme, "easy" can be OK if it is a word used by a happy and confident student, but it might also be contributing to why "boring" appears so many times, so it reminds me I need to provide challenges in math class for all kids. <br /><br />By the end of the year, I'm hoping "fun" and "challenging" replace both "hard" and "easy," and that the rest of the negative words are wiped out! <br /><br />As for the students' description of the subject matter of math, arithmetic looms large, which is not unexpected for students coming out of elementary school. I do find it interesting that many mathematicians describe math as the study of patterns, yet not one student listed that word, even though they've undoubtedly looked many times at patterns in math class. Could it be that they believe "real" math is the symbols and arithmetic, not the patterns and relationships? If so, I'd like to change that so their view of math is more expansive.<br /><br />Finally, I would love to see the word USEFUL showing up by the end of the year. Middle school math is arguably the <b>most</b> useful math students learn, but I hope they will realize how powerful it is <b>now</b>, not just later in life.</div>Julie Wrighthttps://plus.google.com/105628344902704593026noreply@blogger.com3tag:blogger.com,1999:blog-6704987852997939642.post-8168902972613642092015-07-18T00:21:00.001-07:002015-07-18T00:21:50.641-07:00The Perniciousness of Negative Numbers: Are Our Children Safe?<i><b>What follows is a paper I just wrote for a class called History of Mathematics for Middle School Teachers. The actual name of the paper was "The History of Negative Numbers in Mathematics and Education," but that's not exactly click bait, so I livened it up here. Most of the paper is about how mathematicians sort of pretended they were ignoring negative numbers for a while, then publicly freaked out about how ridiculous they were, then finally came to love them. I'm sure many of us have had an in-law relationship like that, so maybe you can relate. The end of the paper is an appendix with some thoughts about activities for students that reinforce the methods and philosophies described in the rest of the paper.</b></i><br /><hr /><br /><a name='more'></a>Why do we believe –5 is a number? To adult math teachers, the question seems silly or startling. We were told about negative numbers before our teenage years, and we’ve seen them used in budgets or economic reports, weather reports of temperatures, elevations below sea level, and, of course, math textbooks. We’ve added, subtracted, multiplied, and divided them just as we have with positive numbers. Why wouldn’t they be numbers? To our middle school students, however, understanding negative numbers and performing arithmetic with them are often far from natural, and in their confusion and skepticism they have much in common with mathematicians from previous centuries, who believed negative numbers were ridiculous or impossible. Considering the history of how mathematicians came to accept and embrace negative numbers provides some guidance for how we can help our students to understand them and work confidently with them.<br /><br />In past centuries, many mathematicians, like our students, accepted the idea of unitless or abstract positive numbers, although with some limited exceptions, they avoided or openly scorned negative numbers in their methods and solutions. Philip E. B. Jourdain described abstract numbers in The Nature of Mathematics in 1913 [Newman p. 24]:<br /><div class="MsoNormal" style="margin: 12pt 0.25in;"></div><blockquote class="tr_bq"><i style="line-height: 150%;">In arithmetic we use symbols of number. A symbol is any sign for a quantity, which is not the quantity itself. […] When we shake off all idea of “1,” “2,” &c., meaning one, two, &c., of anything in particular […] then the numbers are called abstract numbers.</i></blockquote><div class="MsoNormal" style="margin: 12pt 0in;">Early Egyptian, Mesopotamian, and Greek mathematicians came to use abstract positive numbers to varying degrees, but they did not use negative numbers. In about 300 A.D., Diophantus, says Herbert Westren Turnbull in The Great Mathematicians, referred to “the impossible solution of the absurd equation 4 = 4x + 20” [Newman p. 115], although he accepted fractional solutions without difficulty. </div><div class="MsoNormal" style="margin: 12pt 0in;">Early Chinese and Indian mathematicians began using negative numbers in a limited way, often as tools for intermediate steps of problem solving. Early Chinese mathematicians used negative numbers as coefficients in intermediate steps of solving systems of equations. [Berlinghoff p. 81-82] In the 600s, Brahmagupta of India used and calculated with negative numbers to represent debts, and other Indian mathematicians later continued to use them and develop arithmetic rules for them. In the twelfth century, in his text the Lilavati, Bhaskara II found a negative distance for the position of a triangle’s altitude from a vertex, and he correctly interpreted it as “in the contrary direction,” producing an obtuse triangle. [Mumford, p. 126-127] Still, the Indian mathematicians were somewhat dubious about negative numbers, rejecting them as solutions to quadratic equations, for instance. [Berlinghoff p. 82]</div><div class="MsoNormal" style="margin: 12pt 0in;">In any case, Chinese and Indian mathematicians’ work with negative numbers did not end up being passed to other cultures. Arabic mathematicians, like Muhammad Ibn Musa Al-Khwarizmi in the 800s and Umar Al-Khayammi (Omar Khayyam) in about 1100 AD, avoided negative numbers in their work on complicated algebraic equations. Al-Khwarizmi expressed algebraic problems in words; Al-Khayammi placed each term on the side of the equation for which the coefficient would be positive. The limitations in how they wrote equations hindered them from seeing all quadratics or cubics as particular examples of a single type of equation. [Berlinghoff p. 82, 109-110] </div><div class="MsoNormal" style="margin: 12pt 0in;">In Europe from the time of the Renaissance onward, many types of mathematics flourished, including methods to solve increasingly complicated algebraic equations and the development of the coordinate plane. The focus on higher-order equations seems to have been part of why negative numbers seemed so unreasonable to European mathematicians, because if they accepted that negative numbers existed, they also felt obliged to accept taking their roots, and complex and imaginary numbers seemed completely over the top to them. (As David Mumford puts it, “It was because of negatives that square roots had a problem, so maybe it was best to consider them both as second class citizens of the world of numbers. […] The fate of –1 and i were inseparable.” [Mumford p. 140 & 142]) European mathematicians were finding negative numbers increasingly useful, yet they still shied away from embracing them fully. In his work <i>Arithmetica integra</i> in 1544, Michael Stifel represented all quadratic equations as a general form by using negative coefficients, but called them <i>numeri absurdi</i> and would not accept them as solutions. [Hollingdale p. 109] Turnbull says that in the 1500s, Girolamo Cardan (sometimes known as Cardano), “surmised the need” for negative, imaginary, or complex roots to cubic and quartic equations in accord with his ideas of how many roots these equations should have [Newman p. 119], but he seems to have avoided such solutions in his publications. </div><div class="MsoNormal" style="margin: 12pt 0in;">Before the late 1600s, many mathematicians were also confused about the size of negative numbers and their physical meaning. Mumford attributes European discomfort with negative numbers largely to “the overwhelming importance of Euclid,” with his focus on geometry and positive quantities, in the development of math in Europe. [Mumford p. 140-143] Antoine Arnauld felt, not unreasonably, that if -1 were truly smaller than 1, then it was not reasonable for their ratio to be the same no matter which came first (since this would never happen with positive numbers of different sizes). John Wallis believed that dividing by a negative number was essentially an even more extreme case of dividing by zero, and that therefore all such quotients would yield infinity. [Berlinghoff p. 84] Wallis was, however, a pioneer in the 1600s in using negative coordinates in the coordinate plane, which had been originally developed by Descartes with only positive coordinates. [Berlinghoff p. 140] This alteration allowed Europeans to think about positions of points more flexibly, in a way similar to that used by Bhaskara II centuries earlier in his writing about the obtuse triangle.</div><div class="MsoNormal" style="margin: 12pt 0in;">In his <u>Treatise on Algebra</u> in 1685, Wallis developed clear explanations of the meaning and laws of arithmetic of negative numbers, despite his confusion about division with them. He explained that multiplying by a negative number could mean taking away that many times. He also described multiplying a negative number by a positive number, and even explained the produce of two negative numbers: “[T]here may well be a Double Deﬁcit as a Double Magnitude; and−2A is as much the Double of –A as+2A is the Double of A. . . But to Multiply –A by −2 is twice to take away a Defect or Negative. Now to take away a Defect is the same as to supply it; and twice to take away the Defect of A is the same as twice to add A or to put 2A .” [Mumford p. 137]<br /><br />In another breakthrough, Wallis explained and illustrated a number line with negatives on the left and positives on the right, explaining that</div><blockquote class="tr_bq" style="line-height: 150%; margin-bottom: 12.0pt; margin-left: .25in; margin-right: .25in; margin-top: 12.0pt;"><i style="mso-bidi-font-style: normal;"><span style="mso-spacerun: yes;"> </span>[W]hen it comes to a Physical Application, [a negative number] denotes as Real a Quantity as if the Sign were +; but to be interpreted in a contrary sense. […] [If a man] having Advanced 5 Yards […] thence retreat 8 Yards […] and it then be asked, How much is he Advanced […]: I say –3 Yards […] That is to say, he is advanced 3 Yards less than nothing […] (Which) is but what we should say (in ordinary form of Speech), he is Retreated 3 Yards […] [Mumford p. 138]</i></blockquote>Very soon thereafter, Isaac Newton was thinking about negative numbers in a similar way:<br /><blockquote class="tr_bq" style="line-height: 150%; margin-bottom: 12.0pt; margin-left: .25in; margin-right: .25in; margin-top: 12.0pt;"><i style="mso-bidi-font-style: normal;">[I]n local motion, progression may be called affirmative motion, and regression negative motion; because the first augments, and the other diminishes the length of the way made. And after the same manner in geometry, if a line drawn in a certain way be reckoned for affirmative, then a line drawn the contrary way may be taken for negative. [Mumford p. 138]</i></blockquote><div class="MsoNormal" style="margin: 12pt 0in;">It is easy to picture that Newton’s acceptance of negative numbers helped pave the way for his tremendous advances in calculus and physics.</div><div class="MsoNormal" style="margin: 12pt 0in;">Other scientists of the Enlightenment era in Europe were probably less comfortable than Newton was with negative numbers. The development of temperature scales gives some interesting evidence of a pronounced desire to avoid them. Early scales of the 1700s, including Fahrenheit’s, set 0° temperatures low enough that lab scientists of the time could describe lab temperatures with non-negative numbers; cold weather temperatures were not yet an area of interest. Anders Celsius developed a scale using the endpoints that are still familiar to us, of water’s freezing and boiling temperatures, which he determined precisely through experiments. Negative temperatures on the Celsius scale are, of course, much more likely to occur than with the Fahrenheit and related scales; for instance, the freezing temperature of brine used to set 0° Fahrenheit is negative on the modern Celsius scale. But Celsius decided to avoid negative temperatures for lab scientists by reversing the direction of the scale, making his 100° temperature water’s freezing point, and 0° the boiling point! Scientists used this reversed direction for a few decades before settling on the modern Celsius scale in the mid-1700s. [Beckman]<br /><br />As for the post-Enlightenment European mathematicians, even after advances like Wallis’s and Newton’s, many remained skeptical of negative numbers for more than a century. In 1843, Augustus De Morgan wrote in his article Negative and Impossible Quantities, “These creations of algebra retained their existence, in the face of the obvious deficiency of rational explanation which characterized every attempt at their theory.” [Mumford p. 113] Philip Jourdain, the twentieth century mathematician mentioned earlier, perceived that negative numbers were useful; he acknowledged that “‘generalisations of number’ and transference of methods to analogous cases” were useful tricks of the trade that had led mathematicians to have “arrived at the truth by a sort of instinct.” Nevertheless, he did not feel that the generalization of numbers to negative numbers had been on a sound logical footing historically:</div><blockquote class="tr_bq" style="line-height: 150%; margin-bottom: 12.0pt; margin-left: .25in; margin-right: .25in; margin-top: 12.0pt;"><i style="mso-bidi-font-style: normal;">For centuries mathematicians used “negative” and “positive” numbers […] without any scruple, just as they used fractionary and “irrational” numbers. And when logically-minded men objected to these wrong statements, mathematicians simply ignored them or said: “Go on; faith will come to you.” And the mathematicians were right, and merely could not give correct reasons—or at least always gave wrong ones—for what they did. [Newman p. 25-27]</i></blockquote><div align="center" class="MsoNormal" style="margin-bottom: 12.0pt; margin-left: 0in; margin-right: 0in; margin-top: 12.0pt; text-align: center;"><!--[if gte mso 9]><xml> <o:DocumentProperties> <o:Revision>0</o:Revision> <o:TotalTime>0</o:TotalTime> <o:Pages>1</o:Pages> <o:Words>1802</o:Words> <o:Characters>10276</o:Characters> <o:Company>Portland Public Schools</o:Company> <o:Lines>85</o:Lines> <o:Paragraphs>24</o:Paragraphs> <o:CharactersWithSpaces>12054</o:CharactersWithSpaces> <o:Version>14.0</o:Version> </o:DocumentProperties> <o:OfficeDocumentSettings> <o:AllowPNG/> </o:OfficeDocumentSettings></xml><![endif]--> <!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w:TrackMoves/> <w:TrackFormatting/> <w:PunctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:DoNotPromoteQF/> <w:LidThemeOther>EN-US</w:LidThemeOther> <w:LidThemeAsian>JA</w:LidThemeAsian> <w:LidThemeComplexScript>X-NONE</w:LidThemeComplexScript> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w:DontGrowAutofit/> <w:SplitPgBreakAndParaMark/> <w:EnableOpenTypeKerning/> <w:DontFlipMirrorIndents/> <w:OverrideTableStyleHps/> <w:UseFELayout/> </w:Compatibility> <m:mathPr> <m:mathFont m:val="Cambria Math"/> <m:brkBin m:val="before"/> <m:brkBinSub m:val="--"/> <m:smallFrac m:val="off"/> <m:dispDef/> <m:lMargin m:val="0"/> <m:rMargin m:val="0"/> <m:defJc m:val="centerGroup"/> <m:wrapIndent m:val="1440"/> <m:intLim m:val="subSup"/> <m:naryLim m:val="undOvr"/> </m:mathPr></w:WordDocument></xml><![endif]--><!--[if gte mso 9]><xml> <w:LatentStyles DefLockedState="false" DefUnhideWhenUsed="true" DefSemiHidden="true" DefQFormat="false" DefPriority="99" LatentStyleCount="276"> <w:LsdException Locked="false" Priority="0" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Normal"/> <w:LsdException Locked="false" Priority="9" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="heading 1"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 2"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 3"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 4"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 5"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 6"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 7"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 8"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 9"/> <w:LsdException Locked="false" Priority="39" Name="toc 1"/> <w:LsdException Locked="false" Priority="39" Name="toc 2"/> <w:LsdException Locked="false" Priority="39" Name="toc 3"/> <w:LsdException Locked="false" Priority="39" Name="toc 4"/> <w:LsdException Locked="false" Priority="39" Name="toc 5"/> <w:LsdException Locked="false" Priority="39" Name="toc 6"/> <w:LsdException Locked="false" Priority="39" Name="toc 7"/> <w:LsdException Locked="false" Priority="39" Name="toc 8"/> <w:LsdException Locked="false" Priority="39" Name="toc 9"/> <w:LsdException Locked="false" Priority="35" QFormat="true" Name="caption"/> <w:LsdException Locked="false" Priority="10" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Title"/> <w:LsdException Locked="false" Priority="1" Name="Default Paragraph Font"/> <w:LsdException Locked="false" Priority="11" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Subtitle"/> <w:LsdException Locked="false" Priority="22" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Strong"/> <w:LsdException Locked="false" Priority="20" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Emphasis"/> <w:LsdException Locked="false" Priority="59" SemiHidden="false" UnhideWhenUsed="false" Name="Table Grid"/> <w:LsdException Locked="false" UnhideWhenUsed="false" Name="Placeholder Text"/> <w:LsdException Locked="false" Priority="1" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="No Spacing"/> <w:LsdException Locked="false" Priority="60" SemiHidden="false" UnhideWhenUsed="false" Name="Light Shading"/> <w:LsdException Locked="false" Priority="61" SemiHidden="false" UnhideWhenUsed="false" Name="Light List"/> <w:LsdException Locked="false" Priority="62" SemiHidden="false" UnhideWhenUsed="false" Name="Light Grid"/> <w:LsdException Locked="false" Priority="63" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 1"/> <w:LsdException Locked="false" Priority="64" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 2"/> <w:LsdException Locked="false" Priority="65" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 1"/> <w:LsdException Locked="false" Priority="66" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 2"/> <w:LsdException Locked="false" Priority="67" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 1"/> <w:LsdException Locked="false" Priority="68" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 2"/> <w:LsdException Locked="false" Priority="69" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 3"/> <w:LsdException Locked="false" Priority="70" SemiHidden="false" UnhideWhenUsed="false" Name="Dark List"/> <w:LsdException Locked="false" Priority="71" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Shading"/> <w:LsdException Locked="false" Priority="72" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful List"/> <w:LsdException Locked="false" Priority="73" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Grid"/> <w:LsdException Locked="false" Priority="60" SemiHidden="false" UnhideWhenUsed="false" Name="Light Shading Accent 1"/> <w:LsdException Locked="false" Priority="61" SemiHidden="false" UnhideWhenUsed="false" Name="Light List Accent 1"/> <w:LsdException Locked="false" Priority="62" SemiHidden="false" UnhideWhenUsed="false" Name="Light Grid Accent 1"/> <w:LsdException Locked="false" Priority="63" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 1 Accent 1"/> <w:LsdException Locked="false" Priority="64" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 2 Accent 1"/> <w:LsdException Locked="false" Priority="65" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 1 Accent 1"/> <w:LsdException Locked="false" UnhideWhenUsed="false" Name="Revision"/> <w:LsdException Locked="false" Priority="34" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="List Paragraph"/> <w:LsdException Locked="false" Priority="29" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Quote"/> <w:LsdException Locked="false" Priority="30" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Intense Quote"/> <w:LsdException Locked="false" Priority="66" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 2 Accent 1"/> <w:LsdException Locked="false" Priority="67" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 1 Accent 1"/> <w:LsdException Locked="false" Priority="68" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 2 Accent 1"/> <w:LsdException Locked="false" Priority="69" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 3 Accent 1"/> <w:LsdException Locked="false" Priority="70" SemiHidden="false" UnhideWhenUsed="false" Name="Dark List Accent 1"/> <w:LsdException Locked="false" Priority="71" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Shading Accent 1"/> <w:LsdException Locked="false" Priority="72" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful List Accent 1"/> <w:LsdException Locked="false" Priority="73" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Grid Accent 1"/> <w:LsdException Locked="false" Priority="60" SemiHidden="false" UnhideWhenUsed="false" Name="Light Shading Accent 2"/> <w:LsdException Locked="false" Priority="61" SemiHidden="false" UnhideWhenUsed="false" Name="Light List Accent 2"/> <w:LsdException Locked="false" Priority="62" SemiHidden="false" UnhideWhenUsed="false" Name="Light Grid Accent 2"/> <w:LsdException Locked="false" Priority="63" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 1 Accent 2"/> <w:LsdException Locked="false" Priority="64" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 2 Accent 2"/> <w:LsdException Locked="false" Priority="65" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 1 Accent 2"/> <w:LsdException Locked="false" Priority="66" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 2 Accent 2"/> <w:LsdException Locked="false" Priority="67" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 1 Accent 2"/> <w:LsdException Locked="false" Priority="68" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 2 Accent 2"/> <w:LsdException Locked="false" Priority="69" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 3 Accent 2"/> <w:LsdException Locked="false" Priority="70" SemiHidden="false" UnhideWhenUsed="false" Name="Dark List Accent 2"/> <w:LsdException Locked="false" Priority="71" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Shading Accent 2"/> <w:LsdException Locked="false" Priority="72" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful List Accent 2"/> <w:LsdException Locked="false" Priority="73" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Grid Accent 2"/> <w:LsdException Locked="false" Priority="60" SemiHidden="false" UnhideWhenUsed="false" Name="Light Shading Accent 3"/> <w:LsdException Locked="false" Priority="61" SemiHidden="false" UnhideWhenUsed="false" Name="Light List Accent 3"/> <w:LsdException Locked="false" Priority="62" SemiHidden="false" UnhideWhenUsed="false" Name="Light Grid Accent 3"/> <w:LsdException Locked="false" Priority="63" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 1 Accent 3"/> <w:LsdException Locked="false" Priority="64" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 2 Accent 3"/> <w:LsdException Locked="false" Priority="65" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 1 Accent 3"/> <w:LsdException Locked="false" Priority="66" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 2 Accent 3"/> <w:LsdException Locked="false" Priority="67" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 1 Accent 3"/> <w:LsdException Locked="false" Priority="68" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 2 Accent 3"/> <w:LsdException Locked="false" Priority="69" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 3 Accent 3"/> <w:LsdException Locked="false" Priority="70" SemiHidden="false" UnhideWhenUsed="false" Name="Dark List Accent 3"/> <w:LsdException Locked="false" Priority="71" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Shading Accent 3"/> <w:LsdException Locked="false" Priority="72" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful List Accent 3"/> <w:LsdException Locked="false" Priority="73" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Grid Accent 3"/> <w:LsdException Locked="false" Priority="60" SemiHidden="false" UnhideWhenUsed="false" Name="Light Shading Accent 4"/> <w:LsdException Locked="false" Priority="61" SemiHidden="false" UnhideWhenUsed="false" Name="Light List Accent 4"/> <w:LsdException Locked="false" Priority="62" SemiHidden="false" UnhideWhenUsed="false" Name="Light Grid Accent 4"/> <w:LsdException Locked="false" Priority="63" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 1 Accent 4"/> <w:LsdException Locked="false" Priority="64" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 2 Accent 4"/> <w:LsdException Locked="false" Priority="65" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 1 Accent 4"/> <w:LsdException Locked="false" Priority="66" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 2 Accent 4"/> <w:LsdException Locked="false" Priority="67" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 1 Accent 4"/> <w:LsdException Locked="false" Priority="68" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 2 Accent 4"/> <w:LsdException Locked="false" Priority="69" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 3 Accent 4"/> <w:LsdException Locked="false" Priority="70" SemiHidden="false" UnhideWhenUsed="false" Name="Dark List Accent 4"/> <w:LsdException Locked="false" Priority="71" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Shading Accent 4"/> <w:LsdException Locked="false" Priority="72" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful List Accent 4"/> <w:LsdException Locked="false" Priority="73" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Grid Accent 4"/> <w:LsdException Locked="false" Priority="60" SemiHidden="false" UnhideWhenUsed="false" Name="Light Shading Accent 5"/> <w:LsdException Locked="false" Priority="61" SemiHidden="false" UnhideWhenUsed="false" Name="Light List Accent 5"/> <w:LsdException Locked="false" Priority="62" SemiHidden="false" UnhideWhenUsed="false" Name="Light Grid Accent 5"/> <w:LsdException Locked="false" Priority="63" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 1 Accent 5"/> <w:LsdException Locked="false" Priority="64" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 2 Accent 5"/> <w:LsdException Locked="false" Priority="65" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 1 Accent 5"/> <w:LsdException Locked="false" Priority="66" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 2 Accent 5"/> <w:LsdException Locked="false" Priority="67" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 1 Accent 5"/> <w:LsdException Locked="false" Priority="68" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 2 Accent 5"/> <w:LsdException Locked="false" Priority="69" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 3 Accent 5"/> <w:LsdException Locked="false" Priority="70" SemiHidden="false" UnhideWhenUsed="false" Name="Dark List Accent 5"/> <w:LsdException Locked="false" Priority="71" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Shading Accent 5"/> <w:LsdException Locked="false" Priority="72" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful List Accent 5"/> <w:LsdException Locked="false" Priority="73" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Grid Accent 5"/> <w:LsdException Locked="false" Priority="60" SemiHidden="false" UnhideWhenUsed="false" Name="Light Shading Accent 6"/> <w:LsdException Locked="false" Priority="61" SemiHidden="false" UnhideWhenUsed="false" Name="Light List Accent 6"/> <w:LsdException Locked="false" Priority="62" SemiHidden="false" UnhideWhenUsed="false" Name="Light Grid Accent 6"/> <w:LsdException Locked="false" Priority="63" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 1 Accent 6"/> <w:LsdException Locked="false" Priority="64" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 2 Accent 6"/> <w:LsdException Locked="false" Priority="65" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 1 Accent 6"/> <w:LsdException Locked="false" Priority="66" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 2 Accent 6"/> <w:LsdException Locked="false" Priority="67" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 1 Accent 6"/> <w:LsdException Locked="false" Priority="68" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 2 Accent 6"/> <w:LsdException Locked="false" Priority="69" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 3 Accent 6"/> <w:LsdException Locked="false" Priority="70" SemiHidden="false" UnhideWhenUsed="false" Name="Dark List Accent 6"/> <w:LsdException Locked="false" Priority="71" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Shading Accent 6"/> <w:LsdException Locked="false" Priority="72" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful List Accent 6"/> <w:LsdException Locked="false" Priority="73" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Grid Accent 6"/> <w:LsdException Locked="false" Priority="19" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Subtle Emphasis"/> <w:LsdException Locked="false" Priority="21" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Intense Emphasis"/> <w:LsdException Locked="false" Priority="31" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Subtle Reference"/> <w:LsdException Locked="false" Priority="32" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Intense Reference"/> <w:LsdException Locked="false" Priority="33" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Book Title"/> <w:LsdException Locked="false" Priority="37" Name="Bibliography"/> <w:LsdException Locked="false" Priority="39" QFormat="true" Name="TOC Heading"/> </w:LatentStyles></xml><![endif]--> <!--[if gte mso 10]><style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:Cambria; mso-ascii-font-family:Cambria; mso-ascii-theme-font:minor-latin; mso-hansi-font-family:Cambria; mso-hansi-theme-font:minor-latin; mso-fareast-language:JA;} </style><![endif]--> <!--StartFragment--> <!--EndFragment--></div>Jourdain reconciled himself to the logic of negative numbers by concluding that a negative number is the “operation which is the fulfillment of the order: ‘Subtract,’” adding, “Mathematicians call it a ‘number’ […] because of analogy: the same rules for calculation hold […] when ‘addition,’ ‘subtraction,’ &c., are redefined for these operations.” [p. 27-28] He goes on to describe negative numbers on a number line in a way that would be familiar to any of us in 2015, including the idea of +a and –a being placed symmetrically about 0. Later, he summarizes “the essence of algebra” as describing generally the “exceedingly complicated relations in which abstract things stand to one another. The motive for studying such relations,” he continues, “was originally, and still is in many cases, the close analogy of relations between certain things we see, hear, and touch in the world of actuality round us,” but in general, he notes approvingly, “we have reduced the definitions of all ‘numbers’ to logical terms” [p. 65], which is obviously a relief to him. Modern algebraists took similar ideas much further, defining our number system as a ring in ways that require negative numbers to be included as inverse elements for addition. At long last, we have come to a time when mathematicians consider negative numbers on an equal footing with their positive counterparts, and teaching children about them is not only accepted, but encouraged. <br /><hr /><h2>Activities for Students</h2><div class="MsoNormal" style="margin: 12pt 0in;">As mathematicians came to accept negative numbers, they were used more freely in various contexts, so they are more familiar to today’s middle school students than they would have been to math students before the 1900s. Debts are still frequently described as a negative amount of money. Now that outdoor temperatures are frequently measured, negative temperatures are common contexts. Negative elevations (locations below sea level) are rare for land, but make sense to students as a context for problems. Students might be familiar with scales that go from a negative to a positive number (for instance, rating mood on a scale of -10 to 10). Finally, and perhaps most importantly for their flexible use of math as they grow up, students are now exposed to a variety of changes described with negative numbers, such as stock market drops or weight loss, and in middle school science they start describing physical changes (in position, for example) in terms of negative numbers as well. These examples provide a rich source of contexts for teachers to help students make sense of negative numbers, and our cultural acceptance and familiarity with negative numbers give our students advantages in learning about them that mathematicians in past centuries did not have. Seeing the contrast between historical and current use of negative numbers in “regular life” helps me better appreciate how essential it is to build upon this familiarity in order to help my students achieve a greater degree of comfort with negative numbers than mathematicians of the past ever experienced. The Illustrative Mathematics tasks “<a href="https://www.illustrativemathematics.org/content-standards/tasks/288">Above and below sea level</a>,” “<a href="https://www.illustrativemathematics.org/content-standards/tasks/285">Comparing temperatures</a>,” and “<a href="https://www.illustrativemathematics.org/content-standards/tasks/1475">Bookstore account</a>” would each help students explore negative numbers in a meaningful and familiar context.</div><div class="MsoNormal" style="margin: 12pt 0in;">Nevertheless, negative numbers will always represent another level of abstraction for our students beyond what they have experienced with positive numbers or even zero, which are easier to “see”. Mathematical advances of the past can guide how we help students with this abstract thinking. Multiplication and division, in particular, often are difficult to understand, and don’t have a clear meaning for many contexts, such as temperatures in degrees Fahrenheit or Celsius. Reading about historical explanations of negative numbers and how mathematicians made sense of them has given me a renewed appreciation for the number line developed by Wallis and currently promoted within the Common Core Mathematical Content Standards. The Illustrative Mathematics tasks “<a href="https://www.illustrativemathematics.org/content-standards/tasks/283">Integers on the number line 1</a>,” “<a href="https://www.illustrativemathematics.org/content-standards/tasks/2009">Integers on the number line 2</a>,” “<a href="https://www.illustrativemathematics.org/content-standards/tasks/284">Fractions on the number line</a>,” and “<a href="https://www.illustrativemathematics.org/content-standards/tasks/591">Distances between homes</a>” are representative student work I will consider to help them understand and use number lines.<br /><br /><hr /><h2>References</h2><div class="MsoListParagraphCxSpFirst" style="margin: 12pt 0in 12pt 0.25in;"></div><ol><li>Olaf Beckman, “<a href="http://www.astro.uu.se/history/celsius_scale.html" target="_blank">History of the Celsius temperature scale</a>,” 2001.</li><li>William P. Berlinghoff and Fernando Q. Gouvea, <u>Math Through the Ages: A Gentle History for Teachers and Others</u>, Oxton House Publishers, Farmington, Maine, 2002 [1st edition].</li><li><a href="http://www.corestandards.org/Math/" target="_blank">Common Core State Standards Initiative: Mathematics Standards.</a> </li><li>Stuart Hollingdale, <u>Makers of Mathematics</u>, Penguin Books, London, 1989.</li><li>David Mumford, “<a href="http://www.dam.brown.edu/people/mumford/beyond/papers/2010b--Negatives-PrfShts.pdf" target="_blank">What’s So Baffling About Negative Numbers? — a Cross-Cultural Comparison</a>.”</li><li>James R. Newman, <u>The World of Mathematics</u> (vol. 1), Simon & Schuster, New York, 1956.</li></ol><br /><br /></div>Julie Wrighthttps://plus.google.com/105628344902704593026noreply@blogger.com0tag:blogger.com,1999:blog-6704987852997939642.post-46827486052625682002015-05-25T17:40:00.000-07:002015-05-26T19:54:41.482-07:00List of List of Websites for Math Teaching Resources In less than a year on the "MathTwitterBlogosphere," I've collected tons of great resources in Diigo lists, which I've now exported to blog posts here. Some ideas and lessons on these lists are ones I've used already in my middle school math classes; others are on my summer research list.<br /><br />Here's how I've sorted websites (sometimes arbitrarily):<br /><br /><a href="http://sadarmadillo.blogspot.com/2015/05/math-websites-specific-lessons.html" target="_blank">Math Websites: Specific Lessons</a><br /><a href="http://sadarmadillo.blogspot.com/2015/05/math-websites-classroom-culture.html" target="_blank">Math Websites: Classroom Culture & Mathematical Practices</a><br /><a href="http://sadarmadillo.blogspot.com/2015/05/math-websites-professional-development.html" target="_blank">Math Websites: Professional Development and Teaching Ideas</a><br /><a href="http://sadarmadillo.blogspot.com/2015/05/math-websites-meaty-problems.html" target="_blank">Math Websites: Meaty Problems</a><br /><a href="http://sadarmadillo.blogspot.com/2015/05/not-just-math-websites-games-puzzles.html" target="_blank">(Not Just) Math Websites: Games & Puzzles</a><br /><a href="http://sadarmadillo.blogspot.com/2015/05/math-websites-educational-technology.html" target="_blank">Math Websites: Educational Technology</a><br /><br />And here's one more I forgot to add originally! This is a list of sites I made for my students so they could investigate math in different contexts that interested them:<br /><br /><a href="http://msjwright2.blogspot.com/2015/05/math-you-can-see-art-nature-patterns.html" target="_blank">Math You Can See: Art, Nature, Patterns, Society, and More</a>Julie Wrighthttps://plus.google.com/105628344902704593026noreply@blogger.com0tag:blogger.com,1999:blog-6704987852997939642.post-62343905138395246082015-05-25T17:21:00.002-07:002015-05-25T17:21:42.036-07:00(Not Just) Math Websites: Games & Puzzles<i>Some math games & puzzles are also included in my <a href="http://sadarmadillo.blogspot.com/2015/05/math-websites-meaty-problems.html" target="_blank">Meaty Problems</a> list.</i><br /><ul><li><span class="content"><a href="http://www.eecis.udel.edu/~davis/yahtzee.pdf" target="_blank"> Yahtzee score sheet </a></span></li><li><span class="content"><a href="http://www.hasbro.com/common/instruct/yahtzee.pdf" target="_blank"> http://www.hasbro.com/common/instruct/yahtzee.pdf </a></span><br /><span class="description">Yahtzee rules.</span></li><li><span class="content"><a href="https://www.diigo.com/outliner/diigo_items/92215/8223492/231453342?key=rpgewp4pw4" target="_blank"> MathDice Basic Rules </a></span><br /><span class="description">For an easier version, the target number can be the sum of the two 12-sided dice rolls.</span></li><li><span class="content"><a href="http://reasonandwonder.com/?p=1630" target="_blank"> Set Game Routine for Class | Michael Fenton </a></span><br /><span class="description">Technically the puzzles come from online, but they're displayed to the class separately from a network connection.</span></li><li><span class="content"><a href="http://solvemymaths.com/2015/01/11/pose-a-puzzle/" target="_blank"> Pose a Puzzle | Solve My Maths </a></span><br /><span class="description">Collection of great math & logic puzzles from around the web, including Set, KenKen, and many, many more. VERY FUN.</span></li><li><span class="content"><a href="http://mathriddles.williams.edu/" target="_blank"> Steve Miller's Math Riddles </a></span></li><li><span class="content"><a href="http://mathigon.org/teachers" target="_blank"> Mathigon | Resources for Teachers </a></span></li><li><span class="content"><a href="http://www.oregonlive.com/puzzles-kingdom/" target="_blank"> Puzzles Kingdom | Conceptis via oregonlive.com (Oregonian) </a></span><br /><span class="description">Ad-heavy site where lots of fun puzzles are accessible. Most are logic-oriented.</span></li><li><span class="content"><a href="http://www.kenkenpuzzle.com/" target="_blank"> KenKen Puzzles </a></span><br /><span class="description">Number/logic puzzles, like Sudoku with some number operations and properties. Lots of difficulty levels available. Ad-heavy site.</span></li><li><span class="content"><a href="http://labyrinth.thinkport.org/www/" target="_blank"> Lure of the Labyrinth | MIT </a></span><br /><span class="description">Computer game designed for pre-algebra middle schoolers. Rescue a lost pet from monsters in a labyrinth by solving complicated math puzzles. A bit of a pain to set up; some fun, though time-consuming, puzzles once you do. </span></li><li><span class="content"><a href="http://www.math.hawaii.edu/~hile/math100/logice.htm" target="_blank"> Lewis Carroll Symbolic Logic Puzzles </a></span><br /><span class="description">"Purposefully inane" statements by Lewis Carroll (Alice in Wonderland author) that can be logically analyzed. Fairly complicated notation used on this page.</span></li></ul>Julie Wrighthttps://plus.google.com/105628344902704593026noreply@blogger.com0tag:blogger.com,1999:blog-6704987852997939642.post-65295843776356348752015-05-25T17:18:00.000-07:002015-05-25T17:18:02.630-07:00Math Websites: Specific Lessons<h3>GOOD SOURCES FOR MULTIPLE LESSONS</h3><ul><li><span class="content"><a href="https://www.illustrativemathematics.org/" target="_blank"> Illustrative Mathematics </a></span><br /><span class="description">Website that includes shared lessons to illustrate teaching with Common Core math standards. Started by Dr. Bill McCallum, a Common Core author. </span></li><li><span class="content"><a href="http://robertkaplinsky.com/lessons/" target="_blank"> Robert Kaplinsky's Math Lessons </a></span><br /><span class="description">Some very cool real-world stuff here. Math lessons are sorted by grade.</span></li><li><span class="content"><a href="http://opencurriculum.org/browse/common-core/mathematics/" target="_blank"> OpenCurriculum Mathematics </a></span><br /><span class="description">Lessons, projects, activities, videos, worksheets & tests that are open-source (from Illustrative Mathematics, Engage NY, Khan Academy, and others). Quality varies widely, but ability to search by CC standards is VERY helpful.</span></li><li><span class="content"><a href="http://emergentmath.com/my-problem-based-curriculum-maps/" target="_blank"> Common Core Problem Based Curriculum Maps | Geoff Krall (emergent math) </a></span><br /><span class="description">Sequences of great CC online math lessons.</span></li><li><span class="content"><a href="http://ispeakmath.org/math-teacher-resources/" target="_blank"> Math Teacher Resources from I Speak Math </a></span><br /><span class="description">Big list of math teaching resources, with pointers to lesson plans, problems, professional development...</span></li><li><span class="content"><a href="http://www.cleavebooks.co.uk/trol/index.htm" target="_blank"> Teacher Resources on Line (UK)</a></span></li></ul><h3><span class="content">RATIO AND PROPORTIONAL RELATIONSHIP LESSON</span></h3><ul><li><span class="content"><a href="http://www.graphite.org/app-flows/applying-proportional-reasoning?utm_source=twitter&utm_medium=social&utm_term=&utm_content=8302014&utm_campaign=reviews" target="_blank"> Proportional Reasoning: Hamburger Nutrition </a></span><br /><span class="description">Graphite "App Flow" describing use of various apps to do free Mathalicious rates/ratios lesson.</span></li></ul><h3><span class="content">NUMBER SYSTEM LESSONS</span></h3><ul><li><span class="content"><a href="http://fawnnguyen.com/fraction-division-via-rectangles/" target="_blank"> Fraction Division via Rectangles | Fawn Nguyen </a></span></li><li><span class="content"><a href="http://www.yummymath.com/2014/drill-bit-fractions/" target="_blank"> Drill Bit Fractions | Yummy Math </a></span><br /><span class="description">Drill bits are ordered by size, and the denominators of the fractions (up to sixteenths) are given. Find the numerators.</span></li><li><span class="content"><a href="https://gfletchy.files.wordpress.com/2014/09/scaffolding-division-through-strip-model-diagramming.pdf" target="_blank"> Long Division from Strip Model | Graham Fletcher </a></span><br /><span class="description">Really aimed at 4th/5th grade division, but can help make sense of standard algorithm.</span></li><li><span class="content"><a href="http://cheesemonkeysf.blogspot.com/2012/08/life-on-number-line-board-game-for-real.html" target="_blank"> Life on the Number Line - board game for real numbers | Cheesemonkeysf </a></span></li><li><span class="content"><a href="http://divisbyzero.com/2009/10/06/tennenbaums-proof-of-the-irrationality-of-the-square-root-of-2/" target="_blank"> sqrt(2) is irrational: Tennenbaum’s proof | Division by Zero </a></span><br /><span class="description">Proof by contradiction; pretty pictures.</span></li></ul><h3><span class="content">EXPRESSIONS AND EQUATIONS (ALGEBRA) LESSONS</span></h3><ul><li><span class="content"><a href="http://blog.mrmeyer.com/2014/makeover-central-park-these-tragic-write-an-expression-problems/#comments" target="_blank"> Central Park: Making Sense of Variables </a></span><br /><span class="description">A Dan Meyer & Christopher Danielson lesson for early algebra. Students solve one problem, then variations, then move to variables for power and sense-making.</span></li><li><span class="content"><a href="http://reasonandwonder.com/charge/" target="_blank"> Charge! | Michael Fenton </a></span><br /><span class="description">Model the time it takes a cell phone to charge with a linear function... then find out there's a catch!</span></li><li><span class="content"><a href="http://mathequalslove.blogspot.com/2013/06/scattergories-style-describing-graphs.html" target="_blank"> Scattergories Style Describing Graphs Game | Sarah Hagan (Math = Love) </a></span><br /><span class="description">Good practice at identifying features of graphs and describing them with correct terminology. Competitive group game in which teams win for having the highest number of correct and unique descriptions.</span></li><li><span class="content"><a href="http://simplifyingradicals2.blogspot.com/2015/02/linear-equation-card-sort-day-1.html" target="_blank"> Linear Equation Card Sort | Nora Oswald </a></span><br /><span class="description">Match cards for y=mx+b, std form, slope-int, graph, table. Editable too!! At this time, card 2 is a bit confusing (needs fractional coefficient filled in, but it reads as "-x").</span></li><li><span class="content"><a href="http://cheesemonkeysf.blogspot.com/2013/05/substitution-with-stars.html" target="_blank"> Substitution with stars | cheesemonkeysf </a></span><br /><span class="description">"Training wheels" to help with substitution: write x (or y) on one side of a star and whatever it equals on the other.</span></li></ul><h3><span class="content">GEOMETRY LESSONS</span></h3><ul><li><span class="content"><a href="https://www.illustrativemathematics.org/illustrations/1513" target="_blank"> Stained Glass Circle/A/P/Modeling Problem | Illustrative Mathematics </a></span><br /><span class="description">Very cool and realistic scenario in which students need to choose a reasonable model and use knowledge of area (especially of circles), perimeter, and numerical calculations. In PPS, can be used in Math 6.</span></li><li><span class="content"><a href="http://fawnnguyen.com/rigid-transformations/#comment-19936" target="_blank"> Rigid Transformations (Grade 8) | Fawn Nguyen </a></span><br /><span class="description">Ideas for student-created procedures for transformations (create, then do someone else's).</span></li><li><span class="content"><a href="http://exit10a.blogspot.com/2015/05/box-of-clay.html" target="_blank"> Box of Clay (Volume and Units) | Joe Schwartz, Exit 10A </a></span><br /><span class="description">Fantastic post about learning issues and teaching choices involved in a fifth grade lesson about volume and units.</span></li><li><span class="content"><a href="https://docs.google.com/document/d/1YYpEP_aT6yHRD0N5Hr0Gd_BY8czQ8m17hpgHoaM_NFE/edit" target="_blank"> 3 Act Volume | Tap Into Teen Minds + Lisa Bejarano </a></span><br /><span class="description">3-act task around predicting relationships between volumes of various 3-D prisms, pyramids, cones, etc.</span></li><li><span class="content"><a href="http://function-of-time.blogspot.com/2014/08/arguing-about-shapes.html?m=1" target="_blank"> Arguing about Shapes (Kate Nowak) </a></span><br /><span class="description">Great intro geometry lesson (could be used in 7th or 8th grade math also). Stresses finding patterns, sense-making, communication and critique of reasoning, math vocabulary... awesome.</span></li><li><span class="content"><a href="http://map.mathshell.org/materials/download.php?fileid=1231" target="_blank"> Proofs of the Pythagorean Theorem | Mathematics Assessment Project (map.mathshell.org) </a></span></li><li><span class="content"><a href="http://cheesemonkeysf.blogspot.com/2014/10/constructions-castles.html" target="_blank"> Constructions Castles | Cheesemonkey </a></span><br /><span class="description">Creativity with high-school-level Geometry constructions.</span></li><li><span class="content"><a href="http://cheesemonkeysf.blogspot.com/2014/10/the-secret-of-flowchart-proofs.html" target="_blank"> The Secret of Flowchart Proofs #officesupplies | cheesemonkeysf </a></span><br /><span class="description">Flow-chart proofs with post-it notes.</span></li></ul><h3><span class="content">STATISTICS AND PROBABILITY LESSONS</span></h3><ul><li><span class="content"><a href="http://www.amstat.org/censusatschool/about.cfm" target="_blank"> Census at School (US) | American Statistical Association </a></span><br /><span class="description">Census questions for classes to take, with answer statistics to be compared against other national & international groups. Requires part of a class period for online activities.</span></li><li><span class="content"><a href="https://twitter.com/mathycathy/status/523172651850682368" target="_blank"> Scatterplots Idea: "Are We Movie Compatible?" | Cathy Yenca </a></span><br /><span class="description">Fantastic idea for introducing scatterplots and correlation: students rank 10 things in order (here, movies), then compare rankings via scatterplot.</span></li><li><span class="content"><a href="http://www.theguardian.com/us-news/ng-interactive/2014/nov/06/-sp-congress-diversity-women-race-lgbt-are-you-represented" target="_blank"> Are you reflected in the new Congress? | The Guardian </a></span><br /><span class="description">Enter demographic info (sex, age, etc.) and find out how many members of Congress share it.</span></li></ul><h3><span class="content">LOGIC/PROOF LESSON</span></h3><ul><li><span class="content"><a href="http://ichoosemath.files.wordpress.com/2011/11/words-sheets.pdf" target="_blank"> "Words" Logic/Proof Handout | Justin Lanier (ichoosemath) </a></span><br /><span class="description">Use "rules" to change letter strings, then move to inventing your own rules. Not specifically geometry.</span></li></ul>Julie Wrighthttps://plus.google.com/105628344902704593026noreply@blogger.com1tag:blogger.com,1999:blog-6704987852997939642.post-62833516294866972392015-05-25T17:13:00.000-07:002015-05-25T17:22:30.047-07:00Math Websites: Meaty Problems<i>See also my <a href="http://sadarmadillo.blogspot.com/2015/05/not-just-math-websites-games-puzzles.html" target="_blank">Games and Puzzles list</a>.</i><br /><i><br /></i><br /><h3>GENERAL PROBLEMS AND PUZZLES (ordered very roughly by likely age of interest, youngest to oldest)</h3><ul><li><span class="content"><a href="http://www.peterliljedahl.com/teachers/numeracy-tasks" target="_blank"> Numeracy Tasks (All Grade Bands) | Peter Liljedahl </a></span></li><li><span class="content"><a href="http://nrich.maths.org/frontpage" target="_blank"> NRICH enriching mathematics </a></span><br /><span class="description">British site with lots of cool problems.</span></li><li><span class="content"><a href="http://1001mathproblems.blogspot.com/" target="_blank"> 1001 Math Problems | Sian Zelbo </a></span><br /><span class="description">Fun, fairly complicated visual and logic math problems that don't require more than elementary school math. Same author has very fun site <a class="link" data-auto="true" href="http://1001visualpuzzles.blogspot.com/" rel="nofollow" target="_blank">1001visualpuzzles.blogspot.com</a>, but that's now removed from public view.</span></li><li><span class="content"><a href="http://www.playwithyourmath.com/" target="_blank"> Play With Your Math </a></span><br /><span class="description">Most of these puzzles require only elementary-level math content knowledge, but they are not easy to solve! Fun challenges! Many are classics. Recommended by NCTM's Teaching Children Mathematics.</span></li><li><span class="content"><a href="http://www.youcubed.org/tasks/" target="_blank"> Math Tasks | youcubed </a></span></li><li><span class="content"><a href="http://illuminations.nctm.org/BrainTeasers.aspx" target="_blank"> Brain Teasers | NCTM Illuminations </a></span><br /><span class="description">Various cool problems aimed at different age levels. Good to browse!</span></li><li><span class="content"><a href="http://www.statisticsonline.org/subtangent/30-maths-starters.pdf" target="_blank"> 30 Math Starters (Quick Puzzles) </a></span><br /><span class="description">Recommended on a UK site. Quick but interesting puzzle problems. Look mainly geared for upper elementary.</span></li><li><span class="content"><a href="http://www.insidemathematics.org/" target="_blank"> Inside Mathematics </a></span><br /><span class="description">Includes Common Core resources, problems of the month, performance tasks and other assessments, etc.</span></li><li><span class="content"><a href="http://www.openmiddle.com/" target="_blank"> Open Middle: Challenging math problems worth solving </a></span><br /><span class="description">Problems with a defined beginning and end, but an "open middle": different solution paths and problem solving techniques could be used.</span></li><li><span class="content"><a href="http://wyrmath.wordpress.com/" target="_blank"> Would You Rather? </a></span><br /><span class="description">Asks students to choose their own path and justify it.</span></li><li><span class="content"><a href="https://nrich.maths.org/11409" target="_blank"> Problems for Developing Mathematical Habits of Mind | NRICH </a></span><br /><span class="description">Problems designed to encourage certain <span class="text-bold">mathematical habits of mind</span>: being curious, collaborative, thoughtful, or determined.</span></li><li><span class="content"><a href="http://www.math.cornell.edu/~mec/" target="_blank"> Math Explorers' Club | Cornell Dept. of Mathematics </a></span><br /><span class="description">Advanced math "modules" for middle school & high school students. Wow!!</span></li><li><span class="content"><a href="http://fivetriangles.blogspot.com/" target="_blank"> Five Triangles </a></span><br /><span class="description">Terrific, complicated, fun problems, especially for geometry and for math-lovers.</span></li><li><span class="content"><a href="https://brilliant.org/" target="_blank"> Brilliant (Math & Science Problems) </a></span><br /><span class="description">Free. Graphite's recommendation says Brilliant has excellent problems and adapts to students' performance, but is probably best for high-end students. Some problems have a test-prep orientation.</span></li><li><span class="content"><a href="http://ispeakmath.org/math-teacher-resources/" target="_blank"> Math Teacher Resources from I Speak Math </a></span><br /><span class="description">Big list of math teaching resources, with pointers to lesson plans, problems, professional development...</span></li><li><span class="content"><a href="http://io9.com/tag/sunday-puzzle" target="_blank"> Sunday Puzzle | io9 </a></span><br /><span class="description">Aimed at adults, but likely to have good kid puzzles sometimes too.</span></li></ul><h3><span class="content">PROBLEMS AND PUZZLES FOR PARTICULAR TOPICS AND SKILLS (ordered roughly by age, youngest to oldest)</span></h3><ul><li><span class="content"><a href="http://artofmathstudio.wordpress.com/2014/09/09/the-thrill-of-proof-in-the-early-grades/" target="_blank"> Proof (Elem. & up): Numbers of Squares Making a Square | artofmathstudio </a></span></li><li><span class="content"><a href="http://www.nctm.org/Classroom-Resources/Problems/Problems-to-Ponder_-Factor-Craze/" target="_blank"> Factor Craze | NCTM Problems to Ponder </a></span><br /><span class="description">Currently available only to NCTM members. "Which numbers have exactly three factors? Exactly four factors? Exactly five factors? Exactly six factors? Given a positive integer n, how can we tell exactly how many factors it has?"</span></li><li><span class="content"><a href="http://blog.recursiveprocess.com/2015/03/01/car-talk-gone-fishing-puzzler/" target="_blank"> Car Talk Gone Fishing Puzzler | A Recursive Process </a></span></li><li><span class="content"><a href="https://www.illustrativemathematics.org/illustrations/1513" target="_blank"> Stained Glass Circle/Area/Perim/Modeling Problem | Illustrative Mathematics </a></span><br /><span class="description">Very cool and realistic scenario in which students need to choose a reasonable model and use knowledge of area (especially of circles), perimeter, and numerical calculations. Requires knowledge of circle area (7th grade standard).</span></li><li><span class="content"><a href="http://fawnnguyen.com/students-embroiled-conways-rational-tangles/" target="_blank"> Conway’s Rational Tangles (of Rope) | Fawn Nguyen </a></span><br /><span class="description">Group activity with rope in which students need to analyze rotating & twisting mathematically. See https://hilbertshotel.wordpress.com/2015/05/06/conways-ropes/ for more on doing this.</span></li><li><span class="content"><a href="http://www.nctm.org/Publications/Mathematics-Teacher/Blog/Great-Problems-Keep-on-Giving/" target="_blank"> Great Circle/Square Problem with Multiple Solutions | Chris Harrow, NCTM </a></span><br /><span class="description">Multiple solution methods for a @Five_Triangles geometry problem.</span></li><li><span class="content"><a href="http://thinkzone.wlonk.com/MathFun/Triangle.htm" target="_blank"> World's Hardest Easy Geometry Problem | Keith Enevoldsen </a></span><br /><span class="description">Requires only elementary geometry (triangle angle sum theorem, etc.) but is very challenging. I have not solved this yet. The hints are not spoilers at all, at least for me!</span></li></ul>Julie Wrighthttps://plus.google.com/105628344902704593026noreply@blogger.com0tag:blogger.com,1999:blog-6704987852997939642.post-32617086474959362382015-05-25T17:06:00.001-07:002015-05-25T17:06:55.351-07:00Math Websites: Educational Technology<h3>ONLINE MATH TOOLS STUDENTS CAN USE</h3><ul><li><span class="content"><a href="http://carpeedtech.weebly.com/cuethink/i-think-you-think-cuethink-10-reasons-this-app-must-be-in-your-math-classroom" target="_blank"> CUEthink Math Problems App Recommendation | Carpe Ed Tech </a></span><br /><span class="description">Word problems (many from The Math Forum) and prompts for students to break them into 4 clear phases (based on <a class="link" href="https://math.berkeley.edu/~gmelvin/polya.pdf" rel="nofollow" target="_blank">Polya's Problem Solving Techniques</a>): Understand, Plan, Solve and Review. Students' work can be shared within a class.</span></li><li><span class="content"><a href="http://reasonandwonder.com/learn-desmos/?utm_campaign=coschedule&utm_source=twitter&utm_medium=mjfenton" target="_blank"> Learn Desmos | Michael Fenton </a></span><br /><span class="description">Includes a series of challenges.</span></li><li><span class="content"><a href="https://docs.google.com/document/d/18bn2zJZonywWWYWWlV2-Sr_dp-eZAQ61ohyJiXHk2Nk/edit" target="_blank"> GeoGebra at TMC14 | John Golden </a></span><br /><span class="description">On Twitter, #ggbchat can help.</span></li><li><span class="content"><a href="http://www.alicekeeler.com/teachertech/2015/05/24/google-sheets-create-pixel-art/" target="_blank"> Google Sheets: Create Pixel Art | Alice Keeler, Teacher Tech </a></span><br /><span class="description">Using Google Sheets to make pixel art. Teaches lots about spreadsheets; could be used to make visual patterns or do something with proportions or...</span></li><li><span class="content"><a href="http://www.alcula.com/calculators/statistics/box-plot/" target="_blank"> Box Plot Generator </a></span><br /><span class="description">Plotly and Geogebra may also be good for this, say Twitter users.</span></li><li><span class="content"><a href="http://www.mathdice.com/kids/archive.html" target="_blank"> MathDice Online Archive </a></span><br /><span class="description">Online version of MathDice game: make a target number from 3 other numbers ("rolled" on "dice") and any operations. Requires latest version of Flash. Instructions can be found at http://www.mathdice.com/kids/fullinstructions.html (note: I can't find the online tutorial game they mention).</span></li><li><span class="content"><a href="http://www.graphite.org/standards/common-core/math" target="_blank"> Common Core Standards Explorer | Graphite </a></span><br /><span class="description">Reviews of free and paid software and apps related to different Common Core standards.</span></li></ul><h3><span class="content">ONLINE MATH TOOLS & RESOURCES FOR TEACHER PREP</span></h3><ul><li><span class="content"><a href="http://mathandmultimedia.com/2015/02/13/create-math-expressions-in-google-forms/" target="_blank"> How to Create Math Expressions in Google Forms | Guillermo Batista </a></span></li><li><span class="content"><a href="http://www.fishing4tech.com/fishin-solo-blog/google-forms-finally-loves-math" target="_blank"> Google Forms FINALLY Loves Math | fishing4tech.com </a></span></li><li><span class="content"><a href="http://mr-stadel.blogspot.com/2014/11/video-error-analysis-anti-khan-style.html" target="_blank"> Math Videos for Error Analysis | Andrew Stadel </a></span><br /><span class="description">The particular emphasis here is on videos for error analysis (students identify mistakes), but there are great tips for making any math video.</span></li><li><span class="content"><a href="http://illuminations.nctm.org/Activity.aspx?id=3509" target="_blank"> Dynamic Paper | NCTM Illuminations </a></span><br /><span class="description">Make graph paper, number lines, nets, shapes, ...</span></li><li><span class="content"><a href="https://itunes.apple.com/us/app/myscript-mathpad-handwriting/id674996719?mt=8" target="_blank"> MyScript MathPad: Handwriting->LaTeX Generator </a></span><br /><span class="description">Saves as image or LaTeX or MathML. Plays well with Wolfram Alpha for an extra $5.</span></li></ul><h3><span class="content">ONLINE RESOURCES TO HELP STUDENTS UNDERSTAND MATH CONCEPTS</span></h3><ul><li><span class="content"><a href="http://solveme.edc.org/" target="_blank"> SolveMe Mobiles </a></span><br /><span class="description">For algebra - build or play puzzles to figure out value of various shapes balanced on a mobile. Rec'd by Tina Cardone.</span></li><li><span class="content"><a href="http://geogebratube.org/student/m42380" target="_blank"> Factorization - Visual illustration of divisor pairs | GeoGebraTube </a></span></li><li><span class="content"><a href="http://www.mathshelper.net/long_division.html" target="_blank"> British Long Division Helper | mathshelper.net </a></span><br /><span class="description">Works much like our long division, but they include all of the dividend and subtract full multiples of the divisor from it. (Longer to write but makes more sense)</span></li><li><span class="content"><a href="https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-relationships-functions/cc-8th-graphing-prop-rel/e/comparing-proportional-relationships" target="_blank"> Rates and proportional relationships | Graphing and analyzing proportional relationships | Khan Academy </a></span><br /><span class="description">Dan Meyer says: "Proportional relationships is one of KA's better sets. Many different representations. Analysis, not calculation." KA praise from DM is rare enough to be worth checking out.</span></li></ul><h3><span class="content">OTHER INTERESTING MATH STUFF ONLY AVAILABLE ONLINE</span></h3><ul><li><span class="content"><a href="http://www.ted.com/playlists/189/math_talks_to_blow_your_mind" target="_blank"> 10 TED "Math talks to blow your mind" | TED.com </a></span><br /><span class="description">Have not actually watched these yet. All men (sigh)</span></li></ul>Julie Wrighthttps://plus.google.com/105628344902704593026noreply@blogger.com0tag:blogger.com,1999:blog-6704987852997939642.post-84181095240417247222015-05-25T17:02:00.001-07:002015-05-25T17:24:52.247-07:00Math Websites: Professional Development and Teaching IdeasHere's a list (exported from Diigo) of some posts and resources I wanted to be able to come back to (often for the links within those posts as well!). The division between this list and some others I've made, especially <a href="http://sadarmadillo.blogspot.com/2015/05/math-websites-classroom-culture.html" target="_blank">Classroom Culture and Mathematical Practices</a>, is rather arbitrary.<br /><br /><h3><span class="content">FEEDBACK AND ASSESSMENT</span></h3><ul><li><span class="content"><a href="http://blog.mathed.net/2011/08/rysk-butlers-effects-on-intrinsic.html" target="_blank"> Feedback Research: Butler's Effects on Intrinsic Motivation & Performance (1986) and Task-Involving and Ego-Involving Properties of Evaluation (1987) | mathed.net </a></span></li><li><span class="content"><a href="http://map.mathshell.org/materials/lessons.php" target="_blank"> Mathematics Assessment Project | U. of Nottingham + Shell </a></span></li><li><span class="content"><a href="http://rationalexpressions.blogspot.com/2014/06/questioning-wiggins-definition-of.html" target="_blank"> Questioning Wiggins' Definition of Feedback | Michael Pershan, Rational Expressions </a></span><br /><span class="description">Things to ponder about feedback...</span></li><li><span class="content"><a href="http://www.opencolleges.edu.au/informed/features/giving-student-feedback/" target="_blank"> Giving Student Feedback: 20 Tips To Do It Right | InformED : </a></span><br /><span class="description">Very useful checklist.</span></li></ul><h3><span class="content">LEARNING ABOUT TECHNOLOGY</span></h3><ul><li><span class="content"><a href="http://reasonandwonder.com/learn-desmos/?utm_campaign=coschedule&utm_source=twitter&utm_medium=mjfenton" target="_blank"> Learn Desmos | Michael Fenton </a></span><br /><span class="description">Includes a series of challenges.</span></li></ul><h3><span class="content">SPECIFIC MATH TOPICS</span></h3><ul><li><span class="content"><a href="http://christopherdanielson.wordpress.com/2011/06/30/more-on-fraction-division-you-know-you-love-it/" target="_blank"> Fraction Division (partitive, quotative, etc.) | Christopher Danielson </a></span></li><li><span class="content"><a href="http://lanier180.wordpress.com/2013/02/20/day-93-three-eighths-of-a-pile/" target="_blank"> Fraction Divison: Three-eighths of a Pile | Justin Lanier </a></span></li><li><span class="content"><a href="http://gdaymath.com/courses/exploding-dots/" target="_blank"> Exploding Dots | G'Day Math (James Tanton) </a></span><br /><span class="description">"Exploding Dots" are about thinking of each digit in a number as the same thing as 10 of the digit to right of it, basically. The way James Tanton goes on to use this concept is VERY powerful: see, for instance, Lesson 2.2 on decimal division. Definitely considering using this in class!!</span></li><li><span class="content"><a href="https://www.youtube.com/watch?v=QjQJpRvBzD0" target="_blank"> Math Teaching & Ending Racism | Max Ray, Math Forum </a></span><br /><span class="description">Not crazy about their title, so I retitled it! Interesting talk motivating teaching proportional reasoning and statistics to help our students grow into people who can fight "fog" of racism by analyzing and rethinking perceptions.</span></li><li><span class="content"><a href="http://17goldenfish.com/2015/03/21/quadratics-mighty-square-start-by-completing-the-square/" target="_blank"> Quadratics: Mighty Square! | 17GoldenFish </a></span><br /><span class="description">Interesting description of success with quadratics by teaching completing the square first, then quadratic formula, then factoring.</span></li></ul><h3><span class="content">LESSON DESIGN</span></h3><ul><li><span class="content"><a href="http://www.educationaldesigner.org/ed/volume1/issue1/article3/" target="_blank"> A Designer Speaks | Malcolm Swan of MARS/Shell Centre </a></span><br /><span class="description">Short paper, but LOTS to unpack here: purposes for learning math, lesson design, constructivism, group work, transmission vs. discovery/collaboration, math task types that encourage concept development, misconceptions, questioning, reasoning... Wow!</span></li><li><span class="content"><a href="http://www.teachinla.com/vpss/documents/course_info/3b-support.pdf" target="_blank"> Thinking Through a (Math) Lesson Protocol </a></span><br /><span class="description">Linked by @mpershan on Twitter. Have not read through extensively yet, but looks sound.</span></li></ul><h3><span class="content">OTHER CLASSROOM TEACHING PRACTICES</span></h3><ul><li><span class="content"><a href="http://mathpractices.edc.org/" target="_blank"> Implementing the Mathematical Practice Standards </a></span><br /><span class="description">Illustrations, student dialogues, & text describing the Common Core Standards for Mathematical Practice in action.</span></li><li><span class="content"><a href="http://theconversation.com/seven-great-teaching-methods-not-backed-up-by-evidence-33647" target="_blank"> 7 'great' teaching methods not backed up by evidence | Durham U. </a></span><br /><span class="description">Authors argue evidence does not support certain teaching practices widely regarded as helpful. I probably don't agree with all this, but it's a good self-check.</span></li><li><span class="content"><a href="http://mississippimath.wordpress.com/2014/10/08/the-math-resource-primer/" target="_blank"> The Math Resource Primer | Team Mississippi Math </a></span><br /><span class="description">Good resources for lesson and unit planning, arranged by need (example: "my lessons are too boring").</span></li><li><span class="content"><a href="http://mathforum.org/blogs/max/problem-solving-strategies-and-the-common-core-practice-standards/" target="_blank"> Problem-Solving Strategies & CC Math. Practices | Max Ray (Math Forum)</a></span></li></ul><h3>OTHER/NOT YET SORTED:</h3><ul><li><span class="content"><a href="http://www.maa.org/math-competitions/teachers/curriculum-inspirations" target="_blank"> Curriculum Inspirations | Mathematical Association of America </a></span><br /><span class="description">Haven't looked through this yet but at a glance, it looks very promising for problem solving/mathematical practices.</span></li><li><span class="content"><a href="http://geekymathteacher.com/2014/08/encompass/" target="_blank"> EnCoMPASS Analysis of Student Work </a></span><br /><span class="description">Math Forum PLC (more or less) to analyze student work.</span></li><li><span class="content"><a href="http://ispeakmath.org/math-teacher-resources/" target="_blank"> Math Teacher Resources from I Speak Math </a></span><br /><span class="description">Big list of math teaching resources, with pointers to lesson plans, problems, professional development...</span></li><li><span class="content"><a href="http://www.aft.org/pdfs/americaneducator/fall1999/wu.pdf" target="_blank"> Basic Skills vs. Conceptual Understanding: Bogus Dichotomy (H. Wu) </a></span><br /><span class="description">Interesting-looking paper arguing that basic skills can be taught in a way that ties in with conceptual understanding; in particular, dividing fractions can be taught as an inverse of multiplication, not just with pictures.</span></li><li><span class="content"><a href="http://www.insidemathematics.org/" target="_blank"> Inside Mathematics </a></span><br /><span class="description">Includes Common Core resources, problems of the month, performance tasks and other assessments, etc.</span></li></ul>Julie Wrighthttps://plus.google.com/105628344902704593026noreply@blogger.com0tag:blogger.com,1999:blog-6704987852997939642.post-63824687104331082862015-05-25T16:58:00.000-07:002015-05-25T17:24:15.405-07:00Math Websites: Classroom Culture & Mathematical PracticesHere's a list (exported from Diigo) of some posts and resources I wanted to be able to come back to (often for the links within those posts as well!). The division between this list and some others I've made, especially <a href="http://sadarmadillo.blogspot.com/2015/05/math-websites-professional-development.html" target="_blank">Professional Development and Teaching Ideas</a>, is rather arbitrary.<br /><div><br /><div><h3>BUILDING CONFIDENCE, A GROWTH MINDSET, AND A SAFE CLASSROOM CULTURE</h3><ul><li><span class="content"><a href="http://teachingmathculture.wordpress.com/2014/11/13/first-do-no-harm/" target="_blank">First, Do No Harm | Ilana Horn, teaching/math/culture </a></span><br /><span class="description">A nice, though partial, summary of math teachers' responsibilities to avoid harming students' sense of competence.</span></li><li><span class="content"><a href="https://teachingmathculture.wordpress.com/2014/07/14/what-do-you-think-and-why/" target="_blank">Encouraging Student Sharing: Norms & Status | Ilana Horn, teaching/math/culture </a></span><br /><span class="description">Great list of ideas developed by reflective teachers on how to encourage democratic sharing of ideas in the math classroom.</span></li><li><span class="content"><a href="http://youcubed.org/wp-content/uploads/Positive-Classroom-Norms2.pdf" target="_blank">Positive Math Classroom Norms | Jo Boaler </a></span><br /><span class="description">Jo Boaler's positive classroom norms for math class, with ideas for supporting them. Norms: Everyone Can Learn Math to the Highest Levels; Mistakes are Valuable; Questions are Really Important; Math is about Creativity and Making Sense; Math is about Connections and Communicating; Depth is much more important than speed; Math Class is about Learning not Performing.</span></li><li><span class="content"><a href="http://youcubed.org/wp-content/uploads/The-Mathematics-of-Hope-5.pdf" target="_blank">The Mathematics of Hope | Jo Boaler </a></span><br /><span class="description">Moving from Performance to Learning</span></li><li><span class="content"><a href="http://mathmindsblog.wordpress.com/2014/09/06/week-one-talking-points-math-mindset/" target="_blank">Talking Points & Math Mindset | Math Minds </a></span><br /><span class="description">This post and the ones linked to from it (some cheesemonkeysf's), as well as https://docs.google.com/file/d/0B6W4HKOGaWhdMTM2SUViMVhBMGc/edit , have ideas on how to solicit kid ideas about what is important in doing math.</span></li><li><a href="http://bstockus.wordpress.com/2014/10/05/talking-up-talking-points/" target="_blank">Talking Up Talking Points | Brian Stockus</a></li><li><span class="content"><a href="http://educating-grace.blogspot.com/2014/08/notice-and-wonder-when-it-works-when-it.html" target="_blank">Notice-and-Wonder: When It Works, When It Doesn't </a></span><br /><span class="description">Highlighted description of notice-and-wonder. Article includes details on when it works well and when it doesn't.</span></li><li><span class="content"><a href="http://exit10a.blogspot.com/2015/05/box-of-clay.html" target="_blank">Box of Clay (Volume and Units) | Joe Schwartz, Exit 10A </a></span><br /><span class="description">Fantastic post about learning issues and teaching choices involved in a fifth grade lesson about volume and units, including "Notice-and-Wonder" strategy in practice.</span></li><li><span class="content"><a href="http://educating-grace.blogspot.com/2014/08/small-things-increasing-participation.html" target="_blank">Increasing Participation in Classroom Discussions (educating grace) </a></span><br /><span class="description">Steps novice teachers were instructed to try to increase meaningful classroom participation in math class.</span></li><li><span class="content"><a href="http://slamdunkmath.blogspot.com/2014/08/vertical-non-permanent-surfaces-and.html" target="_blank">Vertical Non-Permanent Surfaces & Visible Random Groupings </a></span><br /><span class="description">Food for thought that SlamDunkMath presented at Twitter Math Camp 2014. Strongly recommends having students work in random groups (different for each task) and at vertical non-permanent surfaces placed around the room.</span></li><li><span class="content"><a href="http://mathymcmatherson.wordpress.com/2013/10/30/something-small-positive-reinforcement/" target="_blank">Positive Reinforcement Award Cards | Mathy McMatherson </a></span><br /><span class="description">"Award" cards for students for mathematical communication, perseverance, and team work. Positive feedback for behaviors, because students will care about the things that the teacher cares about.</span></li><li><span class="content"><a href="https://banderson02.wordpress.com/2014/09/18/teacher-reflection-180-day-6/" target="_blank">Math Animals | Bryan Anderson, MathLab </a></span><br /><span class="description">Student answers to prompts "When I walk into math class I feel like a(n) _______" (which animal?) and "What animal do you think would be best for math class?" On Day 6, SPEED is seen as the main desirable quality...</span></li><li><span class="content"><a href="http://mathymcmatherson.wordpress.com/2013/06/06/sbg-the-wall-of-champions/" target="_blank">The Wall of Champions | Mathy McMatherson </a></span><br /><span class="description">Students get Post-Its (colored by class) to write their names on when they do exceptionally well on an assessment. Encourages excitement about excellence.</span></li><li><span class="content"><a href="https://www.youtube.com/watch?v=b3_lVSrPB6w" target="_blank">You're Correct Horse - YouTube </a></span><br /><span class="description">Ridiculous animation of a sax-playing horse saying "You're correct!" and "You are smart!" So over-the-top it should not undermine a culture of positivity about mistakes.</span></li><li><span class="content"><a href="http://engageinmath.blogspot.com/2014/08/tackling-group-work-part-1.html" target="_blank">Group Work Norms: Engage in Math </a></span><br /><span class="description">Group work norms/ground rules from four sources.</span></li><li><span class="content"><a href="https://www.youtube.com/watch?v=daCtIac24yU" target="_blank">Why 2 > 4 | Max Ray at NCTM Ignite </a></span><br /><span class="description">Great, brief talk on the effect of students (and teacher) listening to students.</span></li><li><span class="content"><a href="http://horizonsaftermath.blogspot.com/2014/09/bad-at-math-is-lie.html" target="_blank">Bad at Math Is a Lie | Math Horizon's Aftermath </a></span><br /><span class="description">Matt Waite, a professor and software pioneer, had to take remedial algebra to get an MBA degree at 37, and found he was not actually Bad at Math: "the truth is anyone can get math."</span></li><li><a href="http://hpl.uchicago.edu/sites/hpl.uchicago.edu/files/uploads/American%20Educator%2C%202014.pdf" target="_blank">Math Anxiety | American Educator</a></li><li><span class="content"><a href="http://ell.stanford.edu/publication/mathematics-common-core-and-language" target="_blank">Mathematics, the Common Core, and Language | Understanding Language </a></span><br /><span class="description">Paper on developing math instruction for ELL students, referenced by @MathEdnet (Raymond Johnson).</span></li><li><span class="content"><a href="https://www.youtube.com/watch?v=u086rr7SRso" target="_blank">Classroom Management Strategies To Take Control Of Noisy Students | Rob Plevin </a></span><br /><span class="description">This video gives a nice explanation of advantages of establishing positive behavior at the start of class through interactions with students outside the classroom. Considering for next year.</span></li><li><span class="content"><a href="https://jdmahlstedt.wordpress.com/2015/02/01/this-one-time-when-i-was-an-8th-grader-again/" target="_blank">When I Was an 8th Grader Again… | What is 5? </a></span><br /><span class="description">Lots of thoughtful discussion of what the student POV is like in 8th grade.</span></li></ul><h3><span class="content">REASONING, SENSE-MAKING, AND PROBLEM SOLVING</span></h3><ul><li><span class="content"><a href="http://tjzager.wordpress.com/2014/10/18/making-sense/comment-page-1/#comment-67" target="_blank"> Making Sense | Tracy J. Zager </a></span></li><li><span class="content"><a href="http://blogs.law.harvard.edu/reyes/files/2006/05/ShepMersth.pdf" target="_blank"> How Old Is the Shepherd? An Essay About Mathematics Education | Katherine Merseth </a></span><br /><span class="description">A very well-written article from 1993 on math education in the US. Some things have changed since then, some haven't. References "How old is the shepherd?" problem.</span></li><li><span class="content"><a href="http://powersfulmath.wordpress.com/2014/09/21/warm-ups-transformed-my-classroom/" target="_blank"> Warm-Ups Transformed My Classroom | powersfulmath </a></span><br /><span class="description">Transforming warmups into engaging opportunities for mathematical reasoning and real-world problem solving.</span></li><li><span class="content"><a href="https://crazymathteacherlady.wordpress.com/2015/03/20/filing-cabinet-of-warm-up-activities/" target="_blank"> Filing Cabinet of (Warm Up) Activities | Crazy Math Teacher Lady </a></span><br /><span class="description">Nice list of lots of activities from the MathTwitterBlogosphere. </span></li><li><span class="content"><a href="http://mathwithbaddrawings.com/2013/10/16/two-column-proofs-that-two-column-proofs-are-terrible/" target="_blank"> Two-Column Proofs that Two-Column Proofs are Terrible | Ben Orlin, Math with Bad Drawings</a></span></li></ul><h3><span class="content">HOMEWORK AND TEST REVIEW</span></h3><ul><li><span class="content"><a href="http://cheesemonkeysf.blogspot.com/2015/04/impromptu-twitter-master-class-on.html" target="_blank"> Homework Strategies | cheesemonkeysf </a></span><br /><span class="description">@cheesemonkeysf's homework thoughts. See #4 in particular: 2-week homework packets (students review together daily). </span></li><li><span class="content"><a href="http://mrradamsthoughts.blogspot.com/2015/04/red-yellow-green.html" target="_blank"> Red, Yellow, Green | Ryan Adams, Math Edumacation </a></span><br /><span class="description">Fantastic idea for self-assessment & setting positive & efficient classroom routines for pre-test review.</span></li></ul><h3><span class="content">FUN AND GAMES IN MATH CLASS</span></h3><ul><li><span class="content"><a href="http://mathhombre.blogspot.com/p/games.html#Review" target="_blank"> Games in Math: List of Resources | Math Hombre </a></span><br /><span class="description">This is AMAZING. A list of big and small games, for review and not, from many sources.</span></li><li><span class="content"><a href="http://teachingandlearningcommunity.blogspot.com/2011/10/free-choice-in-math-worth-our-time.html" target="_blank"> Math Free Choice (Part 1) | Amanda Northrup </a></span><br /><span class="description">Food for thought! How a (5th grade?) teacher set up 30-45 minute sessions of Math Free Choice, with activities and goals chosen by students (from logic puzzles, tangrams, games, etc.).</span></li><li><span class="content"><a href="http://teachingandlearningcommunity.blogspot.com/2012/01/math-free-time-does-it-all.html" target="_blank"> Math Free Time (Part 2): Refinements | Amanda Northrup </a></span><br /><span class="description">Refinements to Math Free Time system described in Part 1. Resource links at bottom of page.</span></li><li><span class="content"><a href="http://mathforum.org/alejandre/cooperate.html" target="_blank"> Games for Math & Social Skills (Math Forum) </a></span><br /><span class="description">Cooperative games.</span></li></ul><h3><span class="content">POSTERS</span></h3><ul><li><span class="content"><a href="http://everybodyisageniusblog.blogspot.com/2013/08/kid-friendly-mathematical-practices.html" target="_blank"> Kid Friendly Math Practices Posters (Everybody Is a Genius) </a></span><br /><span class="description">These are attractive and editable! Most are nice kid-friendly versions, but consider changing #4; to me, modeling with math is not about showing work in different ways. </span></li><li><span class="content"><a href="https://app.box.com/s/34e64500f84dccab6e3e" target="_blank"> Problem Solving Strategies Posters </a></span><br /><span class="description">pdf with Polya-esque problem solving strategies.</span></li><li><span class="content"><a href="http://mathmunch.org/classroom-posters/" target="_blank"> Math Munch Classroom Poster </a></span><br /><span class="description">Gorgeous poster that needs to be printed out. I did it at Office Depot for $15 (actually $12 with a coupon) and it looks great! Comes with puzzle.</span></li></ul><h3>OTHER/NOT SORTED YET:</h3><ul><li><span class="content"><a href="http://ispeakmath.org/math-teacher-resources/" target="_blank"> Math Teacher Resources from I Speak Math </a></span><br /><span class="description">Big list of math teaching resources, with pointers to lesson plans, problems, professional development...</span></li><li><span class="content"><a href="http://mathteachermambo.blogspot.com/2015/03/half-century.html" target="_blank"> Half Century (Comments on Teaching) | Math Teacher Mambo </a></span></li></ul></div></div>Julie Wrighthttps://plus.google.com/105628344902704593026noreply@blogger.com0tag:blogger.com,1999:blog-6704987852997939642.post-78792253710134702102015-01-01T16:12:00.001-08:002015-01-03T00:06:06.281-08:00"Value Added Model" for Teachers is Poor Modeling<a href="http://www.livingindialogue.com/duncan-brings-sham-vam-teacher-education/">Thank you, Anthony Cody, for prompting me to send a message</a> before the January 2 comments deadline to the Department of Education protesting their proposed use of so-called "Value Added Models" to evaluate teacher preparation programs. Here is what I sent, slightly edited to remove details of a specific case (the teacher in question was not me, by the way; I knew her, but not well).<br /><div><br /></div><div>To: OIRA_DOCKET@omb.eop.gov</div><div>Subject: "value-added model" for teachers is poor modeling</div><div><br /></div><div>To Whom It May Concern:</div><div><br /></div><div>I am a public school parent and have been a math and science teacher since 2009, when I attended a teacher education program after a previous career in the scientific software industry. I am writing to register my objection to the idea of using student standardized test scores to evaluate teachers or teacher training programs. Although I would be in favor of putting teachers, schools, or even education programs under closer scrutiny in cases where scores plunge or where consistent losses are documented over multiple years, the test score data on student growth is nowhere near precise enough to use fairly for these evaluations. It would be like basing the Consumer Price Index only on strawberry jam prices. I agree with many of the objections listed in this article: <a href="http://www.livingindialogue.com/duncan-brings-sham-vam-teacher-education/">http://www.livingindialogue.com/duncan-brings-sham-vam-teacher-education/</a></div><div><br /></div><div>Data on student performance also inevitably reflects far more than the learning experience the student had in the classroom. For instance, availability of computers at school could have a huge effect. For another example, [horror story redacted in which test scores plunged for some classes one new teacher had because of events the previous year, when she was not even at the school]. Should the teacher or her education program be punished for [redacted]?</div><div><br /></div><div>Furthermore, the SBAC and PARCC tests are at a "beta" stage. We all know there are going to be problems with the rollout; how could there not be, with such major changes? By putting so much pressure on teachers to "achieve" on inevitably faulty first-round tests, you risk alienating some of your best allies. Many of us want to improve our teaching with Common Core standards, and we want high-quality tests that will give us some information about what our students are learning and what we still need to improve in our instruction. Help give us a way to give feedback on the tests and improve them without the horrendous conflicts of interest you introduce when we and our education programs are punished for poor test questions or design that are likely to disproportionately impact our ELL, SpEd, and non-white-middle-class students.</div><div><br /></div><div>I do believe that over time, education programs that produce the most valued teachers (and, of course, the most value for their teacher students) will be those that have the best record of placing teachers in schools. I completely approve of efforts to increase reporting requirements for programs of education on teacher placement, and could support programs gaining an advantage if they consistently supply teachers to high-needs schools. </div><div><br /></div><div>I hope you will carefully consider all points of view and ensure that any data-driven analyses are actually using high-quality, relevant data.</div><div><br /></div><div>Sincerely,</div><div><br /></div><div>Julie Wright<br />Portland, Oregon</div>Julie Wrighthttps://plus.google.com/105628344902704593026noreply@blogger.com0tag:blogger.com,1999:blog-6704987852997939642.post-50936313287694145502014-12-30T23:58:00.000-08:002014-12-30T23:58:39.260-08:00Diversity and the MathTwitterBlogoSphere (MTBoS)I'm catching up on my MTBoS reading after the holiday rush and the post-holiday sluggishness. Today I read <a href="http://rationalexpressions.blogspot.com/" target="_blank">Michael Pershan</a>'s <a href="http://rationalexpressions.blogspot.com/2014/12/year-in-review.html" target="_blank">Year in Review</a> blog post, in which he referred back to <a href="http://rationalexpressions.blogspot.com/2014/10/an-open-comment-thread.html" target="_blank">a sort of self-reflection</a> from October which I unfortunately missed at that time. If you like Michael's writing, which I do, it's all thought-provoking and engaging, but there's <a href="http://rationalexpressions.blogspot.com/2014/10/an-open-comment-thread.html?showComment=1413751493495#c3848331650894150680" target="_blank">one comment</a> I wanted to highlight and write about:<br /><br /><div class="comment-header" id="bc_0_15M" kind="m" style="background-color: white; color: #666666; font-family: 'Trebuchet MS', Trebuchet, Verdana, sans-serif; font-size: 14px; line-height: 19.6000003814697px; margin: 0px 0px 8px;"><cite class="user blog-author" style="font-style: normal; font-weight: bold;"><a href="http://www.blogger.com/profile/17046644130957574890" rel="nofollow" style="color: #888888; text-decoration: none;">Michael Pershan</a></cite><span class="icon user blog-author" style="background-image: url(data:image/png; background-repeat: no-repeat; display: inline-block; font-weight: bold; height: 18px; margin: 0px 0px -4px 6px; width: 18px;"></span><span class="datetime secondary-text" style="margin-left: 6px;"><a href="http://rationalexpressions.blogspot.com/2014/10/an-open-comment-thread.html?showComment=1413751493495#c3848331650894150680" rel="nofollow" style="color: #888888; text-decoration: none;">October 19, 2014 at 4:44 PM</a></span></div><div class="comment-content" id="bc_0_15MC" style="background-color: white; color: #666666; font-family: 'Trebuchet MS', Trebuchet, Verdana, sans-serif; font-size: 14px; line-height: 19.6000003814697px; margin-bottom: 8px; text-align: justify;">A quick note about race: there's no real mystery about how to diversify the MTBoS or TMC. You need to seek out teachers of color and invite them to join your circles.<br /><br />This can be done bluntly or subtlety, depending on what the situation calls for. But if you're interested in twitter being more diverse, then go actually search for teachers of color to follow and then follow and chat and RT them and such.<br /><br />It bugs me that people treat this like it's some grand puzzle to be cracked. It's just hard work.</div><br />I don't especially know Michael, but I can vouch for him on this: both that he practices what he preaches, and that it made a big difference to me, a 49-year-old white woman who can be a little gun-shy about male-dominated professional spaces. I joined Twitter and started this blog this past summer. I haven't been blogging or even tweeting much lately, but I am fully intending to pick up the pace more in 2015, especially over the summer.<br /><br />The fact that I have been and will be tweeting and blogging instead of just lurking or wandering away is due more to Michael than any other single individual, and it is from just what he said here: he noticed me, paid attention to what I said, engaged in conversation with me online, was one of my first followers, and recommended others to follow me. This kind of welcome, from him and others in the MTBoS, and the active, lively, respectful conversations among women and men of diverse ages and teaching roles, kept me on Twitter at a time when I easily could have felt there was not much of a place for a 48-year-old woman in this new(ish) space whose most famous citizen (at least to me) was a young white guy. If we want to hear from all kinds of teachers, we should make sure they know they're not tweeting into a vacuum, and the best way to do that is to respond somehow.<br /><br />One conversation from my first few weeks on Twitter stands out in my memory, enough that I just went back and dug it up to make sure I had it right. It was about one of Michael's <a href="http://mathmistakes.org/what-feedback-would-you-write-for-this-quiz/" target="_blank">posts on feedback</a>. I commented there and, I think, tweeted something early on and was included among the names in the tweets for a few rounds. My name got dropped after a bit, which didn't bother me, but I kept reading the conversation because it was interesting. Then all of a sudden this happened:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-C2_xrA5MXAs/VKOQJcdR70I/AAAAAAAAAD8/MvcLKBWwDvc/s1600/mpershanjlanier.tiff" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-C2_xrA5MXAs/VKOQJcdR70I/AAAAAAAAAD8/MvcLKBWwDvc/s1600/mpershanjlanier.tiff" height="141" width="400" /></a></div><br />I was stunned, in a good way, but I don't think I answered either of those remarks (till now, obviously) because I had no idea what to say. I felt like I didn't understand Twitter culture enough (I still am not quite sure of the etiquette of barging into threads when your name's not on the list, though I do it now anyway); I felt they had nothing to apologize for about dropping me and didn't know how to say that and appreciate the thought at the same time; I worried anything I would say in response would sound patronizing (wow, you're a young white guy and you're aware of sexism? gold star for you!). All of those things are still true, but I'm going ahead and saying something now because it made a big difference to me that they <i>noticed and cared</i> that including me would change it from an all-male conversation.<br /><br />Within a few weeks, I had found plenty of women and men on Twitter to follow (and others to read frequently even if I didn't follow them), some of them followed me too, and I was hooked. I've been cautious about following too many accounts for fear of getting swamped, but after reflecting on my own experiences, I'm going to try harder next year to seek out and follow teachers who I wish I could hear more from, not just wait till their accounts are full of interesting posts... which might never happen if they are tweeting into the void.Julie Wrighthttps://plus.google.com/105628344902704593026noreply@blogger.com3tag:blogger.com,1999:blog-6704987852997939642.post-24001463789689011642014-11-01T12:59:00.000-07:002014-11-01T12:59:47.606-07:00School Math and MeDid you ever write a "mathography" or have your students write one? They're basically autobiographical accounts of your experiences with and feelings about math. I've assigned them and have written a few, and this post, "<a href="http://launchings.blogspot.com/2014/11/maa-calculus-study-women-are-different.html?utm_source=twitterfeed&utm_medium=twitter" target="_blank">MAA Calculus Study: Women Are Different,</a>" made me dig up some parts.<br /><br />The MAA calculus study looked at women and men who took calculus at the beginning of college as part of the coursework for their intended majors, and found that "any indication that they may not be up to the task is much more influential for [women] than for men.... Only 4% of the men earning an A or B were dropping calculus because they did not understand calculus well enough to continue its study, but this was true of almost a fifth of the women earning an A or B. Even more notably, not a single man earning an A or B felt that this grade was not good enough to continue the study of calculus, but this was true of 7% of the women who were switching out of the calculus sequence. [...] Strenta et al (1994) [...] found [...w]omen were much more likely to question their ability to handle the course work, and women were much more likely to feel depressed about their academic progress. They also found that women were more likely than men to leave science because they found it too competitive".<br /><br />I'm 49, so my college experience is a little dated, but these words are definitely a blast from my past. I spent much of my childhood loving math and thinking I might become a mathematician, was widely considered a "math brain" by other students and by teachers, scored extremely high (a 640) on the Math SAT in 7th grade as part of screening for what was then called the Study of Mathematically Precocious Youth, got a 790 on the Math SAT as a junior, placed out of Calculus I by taking AP Calculus in high school, was a National Merit Scholar, got into Harvard, Yale, and Princeton [note: it was a LOT easier then], and chose Swarthmore College, where I double-majored in chemistry and math and won the senior math prize for my math paper.<br /><br />I've never strung all those facts together into a brag like that before, so why am I doing it now? Because when I read it over as if I were a woman I didn't know, it seems ridiculous, astounding, even horrifying, that I WAS GOING TO QUIT MATH -- right after loving first semester Calculus II in college and getting an A in it -- BECAUSE I DIDN'T THINK I COULD CUT IT. In fact, I didn't take any math second semester freshman year. I only went back to math classwork because my chemistry major demanded it, and then I double-majored (with great trepidation and sure I'd be kicked out as a phony at any time) because my college boyfriend talked me into it.<br /><br />Now, some of this near-math-dropout status is down to me being a hot mess. My confidence in everything was low, my academic performance in college was streaky, and my work habits were inconsistent. But the rest... well, if you're still with me, read on for an example of how much difference math experiences in school can make, whether they are positive or negative. Look especially for experiences that contributed to the fixed mindset I had at the time about math -- a belief that people were simply either good at math or they weren't -- which made me vulnerable to feeling helpless, tuning out, and quitting when the going got rough.<br /><div><br /></div>*****************************************************************<br /><br />At school, math was a mixed experience for me. I’ve had years when I had nothing but praise for my math smarts, and years when I was told I wasn’t cut out for “higher math”. When I started elementary school as a first grader, I liked how predictable math was. My family had undergone a lot of upheaval: I had lived in at least seven places in three different states, my parents had gotten divorced and my mother had remarried, and she and my stepfather had started new jobs. My family was young and fairly poor at a private school where most kids’ families were rich and settled and had elegant houses and clothes and even maids. I was scared of the strict teacher and the other kids, but once I understood how to do math worksheets and get the “right answer,” I felt like I had something reliable to count on and succeed at. I liked math and worked hard at it, and my teachers had me do a lot of advanced elementary school math independently with a book in the corner. I liked concentrating on my own most of the time, although sometimes it was lonely, and it was frustrating when I got stuck.<br /><br />In my sixth grade year, my family moved again and I went to a larger, public, grades 1-6 school. My teacher made a big deal to the class about how I tested at 12th grade level in math (whatever that meant!). I was immediately labeled a “Brain,” in a partly friendly but still annoying way. I am not sure why he told the rest of the class about my math testing results, but I have a vague hunch that it was to demonstrate to everyone that girls could be good at math (something that was not exactly a truth universally acknowledged in 1976). <br /><br />I am 49 now, and I think a lot of my students’ parents were also raised in this time, when teachers were trying to send messages that girls could succeed in math, but unintentionally made most students feel like there were “brains” who were good at math and could do it, and then there was everybody else, who… maybe couldn’t. It makes me sad when students tell me their parents tell them they can’t do math, and I became a math teacher partly because it bothered me that my clever, hardworking adult friends sometimes felt that way too. I’ve read and heard that in other countries, people take it more for granted that anyone can put in effort and succeed at math.<br /><br />For seventh and eighth grade, after my parents’ second marriage broke up, I moved with my father back to Baltimore and the private school. Again, math was a safe and fun place for me amidst family stress and turmoil. My teacher, Mr. N-------, loved math and was really excited to work with kids who were good at it. He stretched our boundaries: he had us programming in 1978, using cassette tapes for data storage! He was very energetic and strangely charismatic, and a lot of us really wanted to do well in his class and were quite competitive with each other. I did the best, but it was an oddly uneasy role because the other best students were all male. <br /><br />In eleventh and twelfth grade, I met my math nemesis. Mr. F------, a philosophical soul of around 60, was known as a hard and eccentric teacher. I had had his wife for fourth grade and adored her. I figured I would love Mr. F------. And for a few months, for first-year calculus, I think I did. But things started to go downhill then. We noticed he would lecture only to the boys (and the boys and girls in the class gradually sat separately, unlike in our other classes). He would tell little anecdotes about one female student from a few years back who actually did really well in his class, sounding surprised. The not too subtle message was that having a girl do well in his class was a freak occurrence, possible but not likely. We all (boys and girls) picked up on this stuff, but we didn’t take it too seriously. Where was the harm? After all, he was fair… wasn’t he? The girls started to do worse in his class. But wasn’t that normal? Already there were fewer girls than boys in first-year calculus.<br /><br />By twelfth grade, Mr. F------ laid it on really thick about how THE CALCULUS was HIGHER MATH, and not everyone was cut out for it. People might have done really well at algebra or geometry, but that didn’t mean they were the ones who would do best at higher math. We all figured some of this was aimed at me. I had entered with a whiz-kid reputation, and he was letting me know it didn’t mean anything. And sure enough, by this second year with him, my performance started to fall off. His handwritten quizzes made me panic, though I still did well on printed, standardized test questions. When I got increasingly lost and my grade dropped very low, I went in to ask him for help. He listened to me with a patronizing smile, then said, "You know, you're a very pretty girl." [Yes, I am COMPLETELY sure that's what he said. And yes, I was infuriated, and I told adults and other students about it. But it was a different time and nothing happened because we all thought he "meant well".] After seeing my complete shock at that response, he said not to worry, but just that he thought I didn't seem to know that. He made it clear that he thought girls should focus on their social lives and not fret about math too much. I asked again for help on catching up and he said dismissively, "Just do the homework." I left, fuming, and of course checked out completely. Although I did fine on the AP Calculus exam, I scraped by with a C- for the class for the year, along with a vow to never take “higher math” again.<br /><br />However, I decided to major in college in chemistry, which required me to take second semester calculus. I had a sweet, smart, older gentleman as a professor: “Fast Eddie” Skeath, a former track star and a super-fast blackboard-writer. Then a miracle occurred: it was fun again. I got it again. I felt safe again. I did well again. In the end, I tentatively took more math and ended up double-majoring in math and chemistry. I did have a fairly erratic math major career, though. Sometimes I’d do really well in very hard classes, and I even won an award for a senior geometry paper, but that same semester it was a big disappointment to me that I did quite badly in the only post-elementary-school class I had in math with a female teacher, and never really did know why. I think I was just sometimes rather unfocused, and perhaps still too easily discouraged if I hit a rough spot. Overall, though, I had a great time in math, though I thought of chemistry as my “main” major because I was told it was more employable.<div><br />I have never regretted my choice of college majors. They were hard at times, because more than in other subjects, there are assessments in chemistry or math where you just flat out have the wrong answer, and it doesn’t feel great. But learning how to work past that and succeed at problems you used to do wrong is a great feeling, and gives you confidence in how you use math or science to solve real-world problems. Because fewer Americans major in these subjects even though there are more jobs related to them, I’ve always been able to use one or both of my majors to find interesting work, especially as a teacher (now) and a chemistry software company grant writer and customer support manager (earlier). I hope that any of my students leave my classes with those college and career paths open to them if they choose to take them, but regardless of their career paths, I hope all students get some of the same pleasure out of math I have, and leave my class feeling successful and knowledgable.<br /><br /><br /><br /><br /><br /><br /></div>Julie Wrighthttps://plus.google.com/105628344902704593026noreply@blogger.com0tag:blogger.com,1999:blog-6704987852997939642.post-47786294953304033052014-09-13T23:01:00.001-07:002014-09-14T00:25:42.214-07:00Circles, Percents, and Butter CakeI was reminded of my grandmother's butter cake recipe today after seeing a <a href="http://mathforum.org/blogs/pows/free-scenario-baking-blackberries-wcydwt/" target="_blank">math-related scenario</a> (sort of a prompt) on the Math Forum, posted by Annie (whom I think I remember from college decades ago, but that's another story).<br /><br />I love Gram's butter cake recipe (posted below) for several reasons. The recipe style reminds me of her (I can just hear her voice saying "Don't DUMP it in"); it's absolutely delicious; it's fun to make; and then there is this lovely warning: "This is <u>too much batter for 2 9" pans.</u>" If you're a member of my family, that is code for "You probably want to make a 9", two-layer cake, so you're just going to have to eat some of the batter so it doesn't go to waste. Oh, DARN." (I should add here that you're eating raw eggs if you choose to do this, which is not advised, not to mention the gazillion calories. So let's consider this a hypothetical scenario that you're too smart to follow.)<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-DDJATex0DTE/VBUpF2e03JI/AAAAAAAAADI/INuoC0ODxuw/s1600/buttercakerecipe.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-DDJATex0DTE/VBUpF2e03JI/AAAAAAAAADI/INuoC0ODxuw/s1600/buttercakerecipe.jpg" height="400" width="345" /></a></div><br />So the question that always occurred to me, and which I have solved several times in my life, is: just how much excess batter are we talking about? If we fill two 9" round pans to the same depth to which we would have filled three 8" pans, what percentage of the batter do the cooks need to... um... dispose of?<br /><br />I suppose if I pose this problem to students, I could come up with how many cupcakes a full recipe would make (probably about 30), then ask how many cupcakes you should make with the excess batter.<br /><br />I just remembered: the frosting I always use for it also has interesting math. This was a recipe of my other grandmother's. It says to bring 2 tablespoons milk, 3 tablespoons butter, and 4 tablespoons of brown sugar to a boil, then stir in 1 1/2-2 cups of confectioner's sugar and a pinch of salt. But there's also a parenthetical note "or 3-4-5 proportions" under the 3 ingredients you boil. I used to entertain myself figuring out how much the options shifted the share of each ingredient.<br /><br />By the way, although my frosting grandmother died about 20 years ago, Gram-of-the-butter-cake is still in fine form at age 93. If you live in Greensboro, NC, you've probably met her; it seems like everybody has!Julie Wrighthttps://plus.google.com/105628344902704593026noreply@blogger.com1tag:blogger.com,1999:blog-6704987852997939642.post-2769364322882225622014-09-13T19:51:00.000-07:002014-09-14T00:24:50.000-07:00Why Do (Some) Students Hate Math?, or How I Learned To Love a Concept MapThis past summer, I took an incredibly good online course called <a href="http://online.stanford.edu/course/how-to-learn-math-for-teachers-and-parents-s14" target="_blank">How to Learn Math</a> by <a href="http://joboaler.com/" target="_blank">Professor Jo Boaler</a> at Stanford University. Its focus is on research on math learning and student mindsets that can transform students' experiences with math.<br /><br />Near the beginning of the course, Jo Boaler and some of her students talked about some reasons why people often dislike math classes. As a response task, she asked us to summarize the reasons discussed with a concept map. I groaned, because although I am circumspect about sharing this view in education classes, I've despised concept maps ever since first trying one. I believed they must be useful for someone somewhere, so I've occasionally used them in teaching out of a sense of duty, but I consistently felt that rather than highlighting connections and promoting thought, they just resulted in a big muddled blob of words.<br /><br />Nevertheless, I can comply with educational directives, so I gave it a whirl. To my astonishment, this concept map assignment actually clarified my thinking and made me see things in a new way. But before I get to that, here's the concept map in question:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-3SxR99DiFxk/VBT280b2tOI/AAAAAAAAAC4/7hpAlrUut34/s1600/HTLM.ConceptMap1.3A.jpeg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-3SxR99DiFxk/VBT280b2tOI/AAAAAAAAAC4/7hpAlrUut34/s1600/HTLM.ConceptMap1.3A.jpeg" height="443" width="576" /></a></div><br />Thanks partly to wonderful math education classes with <a href="http://graduate.lclark.edu/live/profiles/140" target="_blank">Kasi Allen at Lewis & Clark</a>, and partly to my own observations, none of these ideas were new to me, and I think about all of them a lot already. The breakthrough for me, though, was in seeing that the "boring/irrelevant" impressions of math and the "math is not for me because..." impressions of math are really two separate strands. Improving my teaching to address only one strand would help reach some students, but would leave others with their math hatred untouched. The interconnections in each strand might lead to a positive kind of snowball effect within that strand -- for instance, if I help students not to feel ignored and excluded in math because they belong to a certain group, their math anxiety will be reduced -- but to reach all math haters, I really need to make certain I am working on both strands.<br /><br />If you want to comment, I'd love to know what you think about my concept math epiphany or what you feel is missing from the reasons students hate math. I was surprised they didn't talk about standardized tests or about the feeling that in math answers are right or wrong without any gray areas (an impression some people find reassuring but many find terrifying), but since they didn't, I left it out of my concept map assignment.Julie Wrighthttps://plus.google.com/105628344902704593026noreply@blogger.com2tag:blogger.com,1999:blog-6704987852997939642.post-80416160927804515532014-09-07T13:52:00.002-07:002014-09-14T15:26:13.060-07:00Typed, Targeted Feedback on Student Papers: How To Do ItAs a grizzled veteran of the MathTwitterBlogosphere of, let's see, about 5 weeks, I've already seen several really inspiring conversations among teachers (including <a class="twitter-atreply pretty-link" dir="ltr" href="https://twitter.com/davidwees" style="background: rgb(245, 248, 250); color: #0084b4; font-family: 'Helvetica Neue', Helvetica, Arial, sans-serif; font-size: 14.399999618530273px; line-height: 18px; text-decoration: none; white-space: pre-wrap;"><span style="color: #66b5d2;">@</span>davidwees</a>,<span style="background-color: #f5f8fa; color: #292f33; font-family: 'Helvetica Neue', Helvetica, Arial, sans-serif; font-size: 14.399999618530273px; line-height: 18px; white-space: pre-wrap;"> </span><a class="twitter-atreply pretty-link" dir="ltr" href="https://twitter.com/k8nowak" style="background: rgb(245, 248, 250); color: #0084b4; font-family: 'Helvetica Neue', Helvetica, Arial, sans-serif; font-size: 14.399999618530273px; line-height: 18px; text-decoration: none; white-space: pre-wrap;"><span style="color: #66b5d2;">@</span>k8nowak</a>, <a class="twitter-atreply pretty-link" dir="ltr" href="https://twitter.com/occam98" style="background: rgb(245, 248, 250); color: #0084b4; font-family: 'Helvetica Neue', Helvetica, Arial, sans-serif; font-size: 14.399999618530273px; line-height: 18px; text-decoration: none; white-space: pre-wrap;"><span style="color: #66b5d2;">@</span>occam98</a>, <a class="twitter-atreply pretty-link" dir="ltr" href="https://twitter.com/cheesemonkeysf" style="background: rgb(245, 248, 250); color: #0084b4; font-family: 'Helvetica Neue', Helvetica, Arial, sans-serif; font-size: 14.399999618530273px; line-height: 18px; text-decoration: none; white-space: pre-wrap;"><span style="color: #66b5d2;">@</span>cheesemonkeysf</a>,<span style="background-color: #f5f8fa; color: #292f33; font-family: 'Helvetica Neue', Helvetica, Arial, sans-serif; font-size: 14.399999618530273px; line-height: 18px; white-space: pre-wrap;"> </span><a class="twitter-atreply pretty-link" dir="ltr" href="https://twitter.com/fawnpnguyen" style="background: rgb(245, 248, 250); color: #0084b4; font-family: 'Helvetica Neue', Helvetica, Arial, sans-serif; font-size: 14.399999618530273px; line-height: 18px; text-decoration: none; white-space: pre-wrap;"><span style="color: #66b5d2;">@</span>fawnpnguyen</a>,<span style="background-color: #f5f8fa; color: #292f33; font-family: 'Helvetica Neue', Helvetica, Arial, sans-serif; font-size: 14.399999618530273px; line-height: 18px; white-space: pre-wrap;"> </span><a class="twitter-atreply pretty-link" dir="ltr" href="https://twitter.com/absvalteaching" style="background: rgb(245, 248, 250); color: #0084b4; font-family: 'Helvetica Neue', Helvetica, Arial, sans-serif; font-size: 14.399999618530273px; line-height: 18px; text-decoration: none; white-space: pre-wrap;"><span style="color: #66b5d2;">@</span>absvalteaching</a>, and <span style="background: rgb(245, 248, 250); color: #66b5d2; font-family: 'Helvetica Neue', Helvetica, Arial, sans-serif; font-size: 14.399999618530273px; line-height: 18px; white-space: pre-wrap;"><a href="http://twitter.com/TRegPhysics" target="_blank"><span style="color: #0084b4;"><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; font-size: 14.399999618530273px;">@</span></span>TRegPhysics</a></span>) who are searching for a good way to achieve all of these things at once:<br /><div><ul><li>provide high-quality written feedback on student papers (especially, feedback separate from or instead of scores and grades, since research shows students will only read the grades if both are there)</li><li>avoid repeatedly handwriting the same comments</li><li>have the comments personalized and right beside the work being commented upon</li><li>provide each student (and family?) with an online, commented document instead of or in addition to a physical paper</li><li>do all this without installing expensive software (sometimes, without third-party software at all)</li><li>do all this in a reasonable amount of time and without losing one's sanity</li></ul></div><div>So, here's the best procedure I've come up with so far. If anyone wants to comment and suggest improvements, I'll fold your ideas in, crediting you. </div><div><ol><li>Make a test that has room reserved for your comments: a column to the right or left of the problems, a box under each problem or at the bottom of the page, whatever. Make sure to instruct students not to write in that area. <i>(idea from <a href="http://twitter.com/TRegPhysics" target="_blank">Trevor Register</a>, whose <a href="http://tjregister.wordpress.com/2014/08/18/efficiency-and-writing-student-feedback/" target="_blank">blog post</a> was passed on by <a href="http://twitter.com/occam98" target="_blank">John Burk</a>) </i></li><li>After students take the test, arrange their papers in your class list order (presumably alphabetical). You may want to include blank pages for absent students. Scan the tests into a pdf file. <i>(I'm assuming most of us have access to photocopiers that can do this quickly, but sometimes teachers don't realize it can be done. Ask around if you're not sure!)</i></li><li>If you have a Mac, open the pdf file using <b>Preview</b>. <i>(If you have a PC, you'll need some other procedure for steps 3 & 4; I'll add it if someone suggests one.) </i>Add your comments using the Tools/Annotate/Text option: click and drag to make a box to type your text in, and type it in. You can change the color, font, and size for the text box; sadly, you can't do equations. </li><li>For your next comment, you don't need to select Tools/Annotate/Text again, just make a new box. If you want to reuse a comment, click on it and copy it, then just paste it in on the next student's paper. When you've finished commenting, save the pdf. <br /><i><a href="https://drive.google.com/a/apps4pps.net/?tab=jo#folders/0B6ZGw2bZPNA1bzJQWk5fNkJFQnM" target="_blank">Here's an example</a> I made using real test questions (leaving out the hardest) and real student work (copied over in my handwriting for privacy reasons). See especially how the comment for #5 is almost identical on the two papers -- no copying by hand, though!!</i></li><li>Now you can split the pdf with everyone's papers into individual papers. You can use the free website <a href="http://splitpdf.com/">splitpdf.com</a> for this <i>(found by <a href="http://twitter.com/occam98" target="_blank">John Burk</a>). </i>Select the file you are splitting, and use the page range thing (if needed; you'll also use "More+") and "Customize split files' names" to (for example) save student JW's test to jw.pdf.<i style="font-style: italic;"> (Entering individual file names is a bit painful; anyone have improvements to suggest?) (splitpdf.com also offers a free Chrome "app," but as far as I can tell that just takes you to their web page. </i><i>John also found </i><a href="http://pdfsam.org/splitpdf" style="font-style: italic;" target="_blank">PDFsam</a><i>, which might be even more powerful but needs to be installed and uses Java.)</i></li><li><i><span style="font-style: normal;">When you hit "Split!", it will make a .zip file and ask you where on your computer to put it. When you unzip the file, it will make a directory with the student papers stored in it. If you want to, you can then upload the whole directory at once to Google Drive, <a href="https://drive.google.com/?tab=jo&authuser=0#folders/0B6ZGw2bZPNA1Q3l3eDBUS19xVzg" target="_blank">as I did for my split, commented sample test</a>. </span></i></li><li><i><span style="font-style: normal;">The biggest problem I see is, now how do I share the papers out with the students? Presumably each student could have a directory that they have permissions for and others don't, but shoving each file into the appropriate directory would be a pain. Any thoughts, especially from anyone who's been working with Google Classroom?</span></i></li></ol>Conclusion so far: I would rather do this procedure than write comments on tests by hand. I'd especially like to use it to give students a chance to work more on their papers BEFORE getting the scores/grades. (I would still give highest points to students who did high-quality work in the first round, but I'd give generous partial credit for changes made after my comments.) <b>However</b>, I would like a better way to customize the file names in step 5 and (especially) to painlessly redistribute the commented student papers into appropriate directories where they could access them (step 7).<br /><br />PS Can you guys actually see the sample files mentioned in Steps 4 & 6? The permissions on the directory, and therefore the files, are supposed to be set so you can, but I'm not sure it's working.</div><div><br /></div><div><br /></div>Julie Wrighthttps://plus.google.com/105628344902704593026noreply@blogger.com5tag:blogger.com,1999:blog-6704987852997939642.post-34961206875743196042014-08-23T21:04:00.001-07:002014-08-23T21:04:03.232-07:00Math: Classroom Culture & Mathematical Practices ListThis is a test of sharing a list of bookmarks in Diigo on my Blogger blog.<br /><br /><br /><br /><a href="https://www.diigo.com/list/msjwright2/list-2014082401094260">Math: Classroom Culture & Mathematical Practices</a>Julie Wrighthttps://plus.google.com/105628344902704593026noreply@blogger.com0tag:blogger.com,1999:blog-6704987852997939642.post-13600349783802588562014-08-22T20:59:00.000-07:002014-09-13T18:00:27.632-07:00Math Facts and Drills WebsitesI decided to move this list to my classroom website. You can find it <a href="http://msjwright2.blogspot.com/2014/09/math-facts-and-drills-websites.html">here</a>.Julie Wrighthttps://plus.google.com/105628344902704593026noreply@blogger.com0tag:blogger.com,1999:blog-6704987852997939642.post-86081697121647092862014-08-22T08:25:00.001-07:002014-09-13T18:01:47.141-07:00Math Websites with Creative or Complicated GamesI decided to move this list to my classroom website. You can find it <a href="http://msjwright2.blogspot.com/2014/09/math-websites-with-creative-or.html">here</a>.Julie Wrighthttps://plus.google.com/105628344902704593026noreply@blogger.com0