The Perniciousness of Negative Numbers: Are Our Children Safe?

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.

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List 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.

Here's how I've sorted websites (sometimes arbitrarily):

Math Websites: Specific Lessons
Math Websites: Classroom Culture & Mathematical Practices
Math Websites: Professional Development and Teaching Ideas
Math Websites: Meaty Problems
(Not Just) Math Websites: Games & Puzzles
Math Websites: Educational Technology

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:

Math You Can See: Art, Nature, Patterns, Society, and More

(Not Just) Math Websites: Games & Puzzles

Some math games & puzzles are also included in my Meaty Problems list.

• Yahtzee score sheet
• http://www.hasbro.com/common/instruct/yahtzee.pdf
  Yahtzee rules.
• MathDice Basic Rules
  For an easier version, the target number can be the sum of the two 12-sided dice rolls.
• Set Game Routine for Class | Michael Fenton
  Technically the puzzles come from online, but they're displayed to the class separately from a network connection.
• Pose a Puzzle | Solve My Maths
  Collection of great math & logic puzzles from around the web, including Set, KenKen, and many, many more. VERY FUN.
• Steve Miller's Math Riddles
• Mathigon | Resources for Teachers
• Puzzles Kingdom | Conceptis via oregonlive.com (Oregonian)
  Ad-heavy site where lots of fun puzzles are accessible. Most are logic-oriented.
• KenKen Puzzles
  Number/logic puzzles, like Sudoku with some number operations and properties. Lots of difficulty levels available. Ad-heavy site.
• Lure of the Labyrinth | MIT
  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.
• Lewis Carroll Symbolic Logic Puzzles
  "Purposefully inane" statements by Lewis Carroll (Alice in Wonderland author) that can be logically analyzed. Fairly complicated notation used on this page.

Math Websites: Specific Lessons

GOOD SOURCES FOR MULTIPLE LESSONS

• Illustrative Mathematics
  Website that includes shared lessons to illustrate teaching with Common Core math standards. Started by Dr. Bill McCallum, a Common Core author.
• Robert Kaplinsky's Math Lessons
  Some very cool real-world stuff here. Math lessons are sorted by grade.
• OpenCurriculum Mathematics
  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.
• Common Core Problem Based Curriculum Maps | Geoff Krall (emergent math)
  Sequences of great CC online math lessons.
• Math Teacher Resources from I Speak Math
  Big list of math teaching resources, with pointers to lesson plans, problems, professional development...
• Teacher Resources on Line (UK)

RATIO AND PROPORTIONAL RELATIONSHIP LESSON

• Proportional Reasoning: Hamburger Nutrition
  Graphite "App Flow" describing use of various apps to do free Mathalicious rates/ratios lesson.

NUMBER SYSTEM LESSONS

• Fraction Division via Rectangles | Fawn Nguyen
• Drill Bit Fractions | Yummy Math
  Drill bits are ordered by size, and the denominators of the fractions (up to sixteenths) are given. Find the numerators.
• Long Division from Strip Model | Graham Fletcher
  Really aimed at 4th/5th grade division, but can help make sense of standard algorithm.
• Life on the Number Line - board game for real numbers | Cheesemonkeysf
• sqrt(2) is irrational: Tennenbaum’s proof | Division by Zero
  Proof by contradiction; pretty pictures.

EXPRESSIONS AND EQUATIONS (ALGEBRA) LESSONS

• Central Park: Making Sense of Variables
  A Dan Meyer & Christopher Danielson lesson for early algebra. Students solve one problem, then variations, then move to variables for power and sense-making.
• Charge! | Michael Fenton
  Model the time it takes a cell phone to charge with a linear function... then find out there's a catch!
• Scattergories Style Describing Graphs Game | Sarah Hagan (Math = Love)
  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.
• Linear Equation Card Sort | Nora Oswald
  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").
• Substitution with stars | cheesemonkeysf
  "Training wheels" to help with substitution: write x (or y) on one side of a star and whatever it equals on the other.

GEOMETRY LESSONS

• Stained Glass Circle/A/P/Modeling Problem | Illustrative Mathematics
  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.
• Rigid Transformations (Grade 8) | Fawn Nguyen
  Ideas for student-created procedures for transformations (create, then do someone else's).
• Box of Clay (Volume and Units) | Joe Schwartz, Exit 10A
  Fantastic post about learning issues and teaching choices involved in a fifth grade lesson about volume and units.
• 3 Act Volume | Tap Into Teen Minds + Lisa Bejarano
  3-act task around predicting relationships between volumes of various 3-D prisms, pyramids, cones, etc.
• Arguing about Shapes (Kate Nowak)
  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.
• Proofs of the Pythagorean Theorem | Mathematics Assessment Project (map.mathshell.org)
• Constructions Castles | Cheesemonkey
  Creativity with high-school-level Geometry constructions.
• The Secret of Flowchart Proofs #officesupplies | cheesemonkeysf
  Flow-chart proofs with post-it notes.

STATISTICS AND PROBABILITY LESSONS

• Census at School (US) | American Statistical Association
  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.
• Scatterplots Idea: "Are We Movie Compatible?" | Cathy Yenca
  Fantastic idea for introducing scatterplots and correlation: students rank 10 things in order (here, movies), then compare rankings via scatterplot.
• Are you reflected in the new Congress? | The Guardian
  Enter demographic info (sex, age, etc.) and find out how many members of Congress share it.

LOGIC/PROOF LESSON

• "Words" Logic/Proof Handout | Justin Lanier (ichoosemath)
  Use "rules" to change letter strings, then move to inventing your own rules. Not specifically geometry.

Math Websites: Meaty Problems

See also my Games and Puzzles list.

GENERAL PROBLEMS AND PUZZLES (ordered very roughly by likely age of interest, youngest to oldest)

• Numeracy Tasks (All Grade Bands) | Peter Liljedahl
• NRICH enriching mathematics
  British site with lots of cool problems.
• 1001 Math Problems | Sian Zelbo
  Fun, fairly complicated visual and logic math problems that don't require more than elementary school math. Same author has very fun site 1001visualpuzzles.blogspot.com, but that's now removed from public view.
• Play With Your Math
  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.
• Math Tasks | youcubed
• Brain Teasers | NCTM Illuminations
  Various cool problems aimed at different age levels. Good to browse!
• 30 Math Starters (Quick Puzzles)
  Recommended on a UK site. Quick but interesting puzzle problems. Look mainly geared for upper elementary.
• Inside Mathematics
  Includes Common Core resources, problems of the month, performance tasks and other assessments, etc.
• Open Middle: Challenging math problems worth solving
  Problems with a defined beginning and end, but an "open middle": different solution paths and problem solving techniques could be used.
• Would You Rather?
  Asks students to choose their own path and justify it.
• Problems for Developing Mathematical Habits of Mind | NRICH
  Problems designed to encourage certain mathematical habits of mind: being curious, collaborative, thoughtful, or determined.
• Math Explorers' Club | Cornell Dept. of Mathematics
  Advanced math "modules" for middle school & high school students. Wow!!
• Five Triangles
  Terrific, complicated, fun problems, especially for geometry and for math-lovers.
• Brilliant (Math & Science Problems)
  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.
• Math Teacher Resources from I Speak Math
  Big list of math teaching resources, with pointers to lesson plans, problems, professional development...
• Sunday Puzzle | io9
  Aimed at adults, but likely to have good kid puzzles sometimes too.

PROBLEMS AND PUZZLES FOR PARTICULAR TOPICS AND SKILLS (ordered roughly by age, youngest to oldest)

• Proof (Elem. & up): Numbers of Squares Making a Square | artofmathstudio
• Factor Craze | NCTM Problems to Ponder
  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?"
• Car Talk Gone Fishing Puzzler | A Recursive Process
• Stained Glass Circle/Area/Perim/Modeling Problem | Illustrative Mathematics
  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).
• Conway’s Rational Tangles (of Rope) | Fawn Nguyen
  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.
• Great Circle/Square Problem with Multiple Solutions | Chris Harrow, NCTM
  Multiple solution methods for a @Five_Triangles geometry problem.
• World's Hardest Easy Geometry Problem | Keith Enevoldsen
  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!

Math Websites: Educational Technology

ONLINE MATH TOOLS STUDENTS CAN USE

• CUEthink Math Problems App Recommendation | Carpe Ed Tech
  Word problems (many from The Math Forum) and prompts for students to break them into 4 clear phases (based on Polya's Problem Solving Techniques): Understand, Plan, Solve and Review. Students' work can be shared within a class.
• Learn Desmos | Michael Fenton
  Includes a series of challenges.
• GeoGebra at TMC14 | John Golden
  On Twitter, #ggbchat can help.
• Google Sheets: Create Pixel Art | Alice Keeler, Teacher Tech
  Using Google Sheets to make pixel art. Teaches lots about spreadsheets; could be used to make visual patterns or do something with proportions or...
• Box Plot Generator
  Plotly and Geogebra may also be good for this, say Twitter users.
• MathDice Online Archive
  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).
• Common Core Standards Explorer | Graphite
  Reviews of free and paid software and apps related to different Common Core standards.

ONLINE MATH TOOLS & RESOURCES FOR TEACHER PREP

• How to Create Math Expressions in Google Forms | Guillermo Batista
• Google Forms FINALLY Loves Math | fishing4tech.com
• Math Videos for Error Analysis | Andrew Stadel
  The particular emphasis here is on videos for error analysis (students identify mistakes), but there are great tips for making any math video.
• Dynamic Paper | NCTM Illuminations
  Make graph paper, number lines, nets, shapes, ...
• MyScript MathPad: Handwriting->LaTeX Generator
  Saves as image or LaTeX or MathML. Plays well with Wolfram Alpha for an extra $5.

ONLINE RESOURCES TO HELP STUDENTS UNDERSTAND MATH CONCEPTS

• SolveMe Mobiles
  For algebra - build or play puzzles to figure out value of various shapes balanced on a mobile. Rec'd by Tina Cardone.
• Factorization - Visual illustration of divisor pairs | GeoGebraTube
• British Long Division Helper | mathshelper.net
  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)
• Rates and proportional relationships | Graphing and analyzing proportional relationships | Khan Academy
  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.

OTHER INTERESTING MATH STUFF ONLY AVAILABLE ONLINE

• 10 TED "Math talks to blow your mind" | TED.com
  Have not actually watched these yet. All men (sigh)

Math Websites: Professional Development and Teaching Ideas

Here'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 Classroom Culture and Mathematical Practices, is rather arbitrary.

FEEDBACK AND ASSESSMENT

• Feedback Research: Butler's Effects on Intrinsic Motivation & Performance (1986) and Task-Involving and Ego-Involving Properties of Evaluation (1987) | mathed.net
• Mathematics Assessment Project | U. of Nottingham + Shell
• Questioning Wiggins' Definition of Feedback | Michael Pershan, Rational Expressions
  Things to ponder about feedback...
• Giving Student Feedback: 20 Tips To Do It Right | InformED :
  Very useful checklist.

LEARNING ABOUT TECHNOLOGY

• Learn Desmos | Michael Fenton
  Includes a series of challenges.

SPECIFIC MATH TOPICS

• Fraction Division (partitive, quotative, etc.) | Christopher Danielson
• Fraction Divison: Three-eighths of a Pile | Justin Lanier
• Exploding Dots | G'Day Math (James Tanton)
  "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!!
• Math Teaching & Ending Racism | Max Ray, Math Forum
  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.
• Quadratics: Mighty Square! | 17GoldenFish
  Interesting description of success with quadratics by teaching completing the square first, then quadratic formula, then factoring.

LESSON DESIGN

• A Designer Speaks | Malcolm Swan of MARS/Shell Centre
  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!
• Thinking Through a (Math) Lesson Protocol
  Linked by @mpershan on Twitter. Have not read through extensively yet, but looks sound.

OTHER CLASSROOM TEACHING PRACTICES

• Implementing the Mathematical Practice Standards
  Illustrations, student dialogues, & text describing the Common Core Standards for Mathematical Practice in action.
• 7 'great' teaching methods not backed up by evidence | Durham U.
  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.
• The Math Resource Primer | Team Mississippi Math
  Good resources for lesson and unit planning, arranged by need (example: "my lessons are too boring").
• Problem-Solving Strategies & CC Math. Practices | Max Ray (Math Forum)

OTHER/NOT YET SORTED:

• Curriculum Inspirations | Mathematical Association of America
  Haven't looked through this yet but at a glance, it looks very promising for problem solving/mathematical practices.
• EnCoMPASS Analysis of Student Work
  Math Forum PLC (more or less) to analyze student work.
• Math Teacher Resources from I Speak Math
  Big list of math teaching resources, with pointers to lesson plans, problems, professional development...
• Basic Skills vs. Conceptual Understanding: Bogus Dichotomy (H. Wu)
  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.
• Inside Mathematics
  Includes Common Core resources, problems of the month, performance tasks and other assessments, etc.