Tuesday, December 30, 2014

Diversity and the MathTwitterBlogoSphere (MTBoS)

I'm catching up on my MTBoS reading after the holiday rush and the post-holiday sluggishness. Today I read Michael Pershan's Year in Review blog post, in which he referred back to a sort of self-reflection 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 one comment I wanted to highlight and write about:

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.

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.

It bugs me that people treat this like it's some grand puzzle to be cracked. It's just hard work.

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.

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.

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 posts on feedback. 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:

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 noticed and cared that including me would change it from an all-male conversation.

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.

Saturday, November 1, 2014

School Math and Me

Did 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, "MAA Calculus Study: Women Are Different," made me dig up some parts.

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".

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.

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.

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.


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.

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).

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.

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.

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.

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.

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.

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.

Saturday, September 13, 2014

Circles, Percents, and Butter Cake

I was reminded of my grandmother's butter cake recipe today after seeing a math-related scenario (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).

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 too much batter for 2 9" pans." 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.)

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?

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.

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.

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!

Why Do (Some) Students Hate Math?, or How I Learned To Love a Concept Map

This past summer, I took an incredibly good online course called How to Learn Math by Professor Jo Boaler at Stanford University. Its focus is on research on math learning and student mindsets that can transform students' experiences with math.

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.

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:

Thanks partly to wonderful math education classes with Kasi Allen at Lewis & Clark, 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.

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.

Sunday, September 7, 2014

Typed, Targeted Feedback on Student Papers: How To Do It

As a grizzled veteran of the MathTwitterBlogosphere of, let's see, about 5 weeks, I've already seen several really inspiring conversations among teachers (including , , , , and @TRegPhysics) who are searching for a good way to achieve all of these things at once:
  • 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)
  • avoid repeatedly handwriting the same comments
  • have the comments personalized and right beside the work being commented upon
  • provide each student (and family?) with an online, commented document instead of or in addition to a physical paper
  • do all this without installing expensive software (sometimes, without third-party software at all)
  • do all this in a reasonable amount of time and without losing one's sanity
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. 
  1. 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. (idea from Trevor Register, whose blog post was passed on by John Burk
  2. 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'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!)
  3. If you have a Mac, open the pdf file using Preview. (If you have a PC, you'll need some other procedure for steps 3 & 4; I'll add it if someone suggests one.) 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. 
  4. 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.
    Here's an example 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!!
  5. Now you can split the pdf with everyone's papers into individual papers. You can use the free website splitpdf.com for this (found by John Burk). 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. (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. John also found PDFsam, which might be even more powerful but needs to be installed and uses Java.)
  6. 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, as I did for my split, commented sample test
  7. 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?
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.) However, 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).

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.