MariaD’s blog

Some mathematical definitions seem ridiculous to the general public. Consider a set operation called Inclusion Map. Here is one silly-looking definition!

Definition: Inclusion Map
Given a subset B of a set A,
the injection f: B –> A
defined by f(b)=b
for every b that is a member of B.

This definition seems to describe a function that does not do anything whatsoever, in fancy terms. What is it for?!

March is the Disability Awareness Month, which I did not know until the good people in CVS announced their Caremark program and blogging contest. The way a society treats every member, in all the diversity of styles and abilities, is the reflection of its humanistic advancement.

Math is often used inhumanely. Many people see math as an abstract torture tool, because it repeatedly causes them mental anguish. No less frequently, math is used as a gatekeeper, a way to prevent large groups of people from access to careers or programs that often have nothing to do with the entrance test’s mathematics. This is not right, and we need to solve these problems.

Here are the 60 ways to make your math activities inclusive, humane, and meaningful. It is my modest contribution to the cause of the month.

1. Help everybody create their own math: definitions, formulas, examples, pictures, problems and other math entities. Every person will make something accessible and meaningful for him or her, and the group will be enriched by the wide variety.
2. Give the gift of time! Make it OK to take as much time as everybody needs to figure out their mathematics. Encourage contemplation, careful slow drawing, using objects as models. Math is not a sprint.
3. Snack as you do math. Vigorous mathematics burns as much blood sugar as vigorous exercise. In one study, the number one correlation with math success was the fact that kids had breakfasts.
4. Use multiple representations for your math: words, formulas, gestures, tables, diagrams, pictures, manipulatives, modeling software, graphs, mind maps, movies, sounds. You never know which of them will “click” with different people, or different math concepts for the same person. Use at least three representations for every math idea you encounter.
5. Mind your breathing! Mathematical concentration makes breathing too shallow for some people, and then their brains don’t get enough oxygen for successful math. If this happens, make frequent micro-breaks for several deep, cleansing breaths every few minutes.
6. Make peace with mistakes. Play with mistakes and try to find out if they can make some sort of sense. Can you change axioms of mathematics to accommodate a mistake? Try to make as many mistakes in one problem as you can. Can you repeat the same type of a mistake consistently, noticing and controlling its pattern? Analyzing mistakes playfully is a great way to gain power over mistakes, including their avoidance - and to reduce anxiety.
7. Consider listening to music, moving around or rocking, or doing mild tactile stimulation (e.g. playing with modeling clay) while doing mathematics. These background activities help many types of people concentrate better and sustain concentration longer.
   
   “Headphones and Math” by Zach Hale
8. Give mathematical compliments. Learn what makes mathematics beautiful and meaningful, and notice these features in other people’s work. Originality, efficiency, references to work of others, connections among topics, reuse of good algorithms, nice ways to represent your math are just a few of the mathematical features you can notice and compliment.
   
   “Chrome dripping heart” by ViaMoi
9. Have quality math tools easily available at all times. There should be plenty of graph paper, colored pens and pencils, good rulers, protractors, compasses, scales, and modeling software appropriate for your topics. Montessori had a lot to say on the subject of good tools.
10. Capture and display personal history of math work. This is especially important for personal math creations - pictures you draw to illustrate concepts, definitions you author, applets you program, problems you pose. Use photographs, videos and audio records to capture work in different ways, together accessible for everybody. Reggio Emilia approach has good documenting ideas.
11. Celebrate not knowing as an opportunity to learn. “Take chances, make mistakes, get messy” - Ms. Frizzle from The Magic School Bus.
12. Do math with friends. Run Math Clubs, create study groups, solve problems together, blog and chat about math, create group collages of math representations. Mathematics is a social endeavor.
13. Create or find math holidays. Pi day is 03/14, one of better known math holidays. Celebrate birthdays of your math heroes, dates that make some sort of math sense, but more importantly - your own math accomplishments: “The day I created my first theorem,” for example.
    
    “Pi Day” by Jose Kevo
14. Explore Humanistic Mathematics - art, music, dance, poetry related to mathematics. Approach mathematics from storytelling, drawing, history perspectives. Internet in particular makes it easier to be Renaissance People where mathematics is concerned.
15. Play games. Recreational mathematics has something to offer for everyone, because play is an innate quality of the human animal.
16. Love one another in the context of mathematics. Support members of your math communities in caring for one another, helping one another, providing appropriate math support and inspiration for one another. Nothing says “love” like a nice, fun math problem created especially for you!
17. Find mathematics everywhere you go. Help others find mathematics in their everyday lives.
18. Create math-rich habitats by decorating with math posters and math sculptures, pinning math comics to your wall, or making math toys and games available in your home, your class rooms, or your office space. Google offices have math-rich playspaces to promote creativity and productivity - why not the rest of us?
19. Meet math practitioners and help kids meet a lot of math practitioners, chat with them, make friends, visit them at work. Role models you know personally is one of the best ways to identify with math success.
20. Learn about lives and times of cool math people from the past, and some of the current math celebrities. It may make people feel more confident than many mathematicians and scientists had various disabilities. Academia may be more accepting of disabled people than the rest of the world.

    
    “Professor Stephen Hawking in Cambridge” by doug88888

21. Use cues to manage your attention and to help others. For example, a pleasant sound can go off every so often to remind you to check your work or just to stay on task. A group of people can help each other stay on task if they regularly ask questions of each other, share work, or otherwise collaborate.
22. Model problems, act them out, roleplay with toys and objects. Make math real and meaningful in any form that makes sense to you.
23. If your mathematics involves measurements (angles, weights, heights) either carry them out with appropriate measurement tools, or visualize the situation with familiar objects. Is “a mile” more or less than the distance from your house to your supermarket?
24. When you use graphs and diagrams, make them clear, uncluttered, and beautiful. Encourage others to use graph paper, colored pencils, rulers and other tools to create clear and beautiful mathematical representations. Use software, such as spreadsheets or concept mapping programs, and hands-on tools to assist with clarity.
25. Work with reference sources such as Wikipedia or math dictionaries to make sense of math words and expand your math vocabulary. Create definitions of the math entities you use in your own words, pictures, or other representations. Understanding words and having power over definitions helps people be included in a field.
26. Recitation is a time-honored practice that can feel relaxing and meditative. Recite important and relatively complex math truths, alone or in groups, for mild, pleasant changes in your state of consciousness and for a huge math memory boost. Again, reserve recitation for important math statements and procedures, not for long lists of minor facts.
27. How do you eat an elephant? One bite at a time. Use good task management practices for your math work. Plan your steps, take little breaks, track your progress visually and so on. These practices will help people with any sort of math, cognitive, and even emotional difficulties to accomplish their math tasks and feel better in the process.
28. Use calculators and computers appropriately: to remove drudgery from your math reasoning. Computing devices can help boost the number sense by making number patterns more obvious, more visible, and easier to manipulate. Do not replace understanding with calculation, whether by hand or by computers, and do not send humans to do machine’s jobs.
29. Learn to use summary tools, such as diagrams, mind maps, graphic organizers, highlights and text outlines. Concise summaries reach people who have trouble with large chunks of math materials.
30. Use mnemonics after you understand the concepts - never instead of understanding. Do not use mnemonics for massive groups of facts, such as multiplication tables, because they will mess up topics based on multiplication. Mnemonics are good for stand-alone multistep procedures. They are especially helpful for people with attention issues.
31. Collect your own math, and collect works of everybody in a group. For example, everybody can keep adding examples to a topic’s bulletin board. You can keep a math journal, blog, or notebook. Use “math frames” to make everybody’s work uniform in style while creating a group collection - for example, a shape of a function machine. With collections, each person can feel a part of some bigger project.
32. Base your math activities on popular games. I am explaining the obvious here, but if the game is popular, it automatically works well for a lot of people. Inventing games is harder than it looks.
33. Make sufficient pauses after asking questions, and help people around you to learn the skill. Because math conversation is more contemplative than the average banter, silently count to at least 100 before barging in with comments. In a group, mandate pauses after each question, so that everybody has a chance to think quietly. People who answer slower will feel more included this way, and everybody will gain in more depth and variety of answers.
34. Use at least three examples of every idea introduced. Make your own examples or search for them. Say, if you introduce the Arabic numerals for quantities to your three year olds, research other ways humans devised for the purpose (Roman numerals, Mayan symbols and so on). The goal is to be able to create your own examples of everything you meet in math. Studying commutative operations, can you make up an operation and convince others it’s commutative?
35. Run an activity where everybody makes up problems with too much information, irrelevant information, or contradicting information. “If my brother is three years older than me, where do I live?” Then invite the group to “fix” problems. Silliness of this sort helps people relax, but also leads to very helpful analysis, support and discussions of problem structures. Since everybody owns some problems, they are likely to want to pay attention.
36. Run an activity where everybody draws random pictures and then everybody searches for math in them. People who are stronger in math will probably find more ideas, but everybody’s picture will be a part of the activity. Searching the web for relevant terms and ideas as they are found in pictures helps to make the activity more interesting.
37. In general, run activities where math order emerges from a mess of examples - stories, pictures, problems, paradoxes, and so on. Stronger people can work on the emergent math, but everybody can contribute the examples and learn from the process.
38. Consider using some visual programming language, like Scratch from MIT, or Geogebra. Working with hands-on algorithms is accessible to everybody, very empowering, and very effective for promoting mathematical thinking.
39. Use physical estimates. Constantly estimate measures (size, distance, speed) and compare your estimates to reality. Number operations can be based on physical estimates, and made more concrete and accessible this way. For example, decimals make much more sense if people connect them with money.
40. Appreciate the difference between math exercises and problem solving. Find individual balance between the two to reach everybody.
41. Use qualitative examples and analogies together with number examples. Say, a function machine that turns baby animals into adult animals (kitten-cat, foal-horse, duckling-duck) can be used to explore many function concepts (input, output, domain, range, inverse) before any numbers are introduced. The order of dressing (underpants, then shorts, not the other way around) can be used to think about non-commutative operations. Qualitative examples make math ideas accessible more broadly.
42. Use “fill in the blanks” tasks to set up the frame of an activity. Create your own “fill in the blanks” MathLibs. This way, people who can’t set up a problem from scratch can participate.
43. Color code your math. Color parts of equations, different graphs, chart details. Color-blind people can use different thicknesses of pens, wavy vs. straight lines and other coding systems. Any coding or highlighting system of their own choice makes people “own” mathematics much more, and have more power over it.
44. Find references to mathematics in books, movies and computer games. Invite everybody to search their favorites for references, or head for internet collections of math references in the popular culture.
45. Explore ethnomathematics. Many ethnic groups contributed their wonderful ideas, representations, and traditions to the common mathematical culture. Even if people do not know much about that part of their heritage, it is a good way to learn and to connect more to the roots. Moreover, “folk math” part of ethnomathematics tends to be hands-on, robust, interesting and more accessible than “classroom math.”
46. Work with historical representations and computational devices, like Napier’s bones or the abacus. They may reach some people not digging modern conventional methods. Also, these devices are usually beautiful and pleasant to the touch.
    
    “Abacus abstract” by aussiegall
47. Let math simmer for a while. Make pauses after learning an involved concept. A spiral approach, returning to a topic once a week or once a month, may be more efficient than storming the topic hour after hour. Likewise, read a challenging text at night and return to it in the morning. This approach is especially good for math anxiety sufferers.
48. Consider math unit studies, exploring many connections of a topic to different areas of life. The more areas you can connect, the higher the chances of different people relating to the topic.
49. Pay attention to emotions, since math is notorious for bringing on strong emotions, from elation to angst. Use relaxation techniques, pauses and time-outs, massage, food, tactile stimulation, good Feng Shui, meditation and prayer or whatever else helps to manage emotions while doing mathematics. Learning to control the many emotional states of math practice is huge help for everybody, but especially for people with disorders that make emotion management harder.
    
50. Monitor the rhythms of the day and do math at opportune times. Body rhythms are very individual. Sleepy, relaxed time may be great for reviewing old ideas, and the time when you bounce off the walls with energy can be wonderful for brainstorming or problem solving. You can somewhat control the rhythms with food, exercise and other activities. Controlling biorhythms is somewhat easier in a group, because rhythms are “contagious.”
51. Think of multiple roles in group projects. For example, one person can be the principal investigator of a math exploration project and another can be the group’s photographer, contributing and participating in more central or more peripheral (apprenticeship) roles and learning as the group goes along. Use good balance between individual projects and group projects, though.
52. Practice and help others learn Active Listening techniques. Seek goodness and sense, rather than mistakes, in what others create and express mathematically. This is especially important for working with kids. Do not focus on minor issues such as reversing letters or even arithmetic mistakes, but on the meaning of what the person is trying to express mathematically.
53. Use adaptations for physical limitations. Computer software and hardware, in particular, can be adapted to a wide range of restrictions through eye tracking, touch screens, read-aloud programs and so on. When designing software, mind the accessibility standards!
54. Size matters. Large-square graph paper may do wonders for younger kids or dyspraxic people. Consider using stairs or outdoor chalk drawings instead of number lines on paper. Even for healthy people, using gross motor skills for their mathematics feels very good; for some disabled people, it is the only way to go.
    
    n(n+1) by Jal Tik
55. Think of different ways to “input” your math. For example, depending on body limitations and learning styles, people can dictate, type or write their texts. Graphs can be done on paper, using keyboard with computers, or by special voice activated add-ons for graphing software.
56. In group activities, take turns in everything you do, for example, pose questions or invent problems. Let people respond in a variety of ways when their turn comes - by words, pictures, or web references as needed. Use a “conch shell” or a “talking stick” if people speak out of turn.
57. Use prompts and lists for sequential directions. Following many math procedures and making sense of problems depends on sequences of steps. Individual short term memory may not be sufficient to hold all the steps, and some disabilities may disrupt sequencing. Visual organizers and lists will reduce sequencing problems.
58. Do not raise tall, narrow towers of one math concept building on another building on another and so on. If proportions are based on multiplication, which is based on repeated addition, which is based on counting, the idea is too shaky for many people. Such structures are unstable, easy to forget and inaccessible to some people. Instead, ground mathematics, even advanced concepts, in concrete, everyday, intuitive experiences. Ground proportions directly in stretching or sharing, for example, in addition to exploring purely mathematical links and connections.
59. Use “think-pair-share” and “buddy system” strategies to rehearse and contemplate math individually before presenting to a group, or participating in group activities. Some people may need more individual help within groups to succeed.
60. Support strong, healthy, well-connected, inclusive communities and networks involving mathematics.

And now, as a bonus, the explanation of the Inclusion Map definition that hopefully makes sense. Here it is again, with a translation into our situation:

Given a subset B of a set A
Let’s say we have a community of people, call it A, and a subset of disabled people, call it B

the injection f: B –> A
we will make B a part of A in such a way that everybody will preserve their individuality (that’s what “injection” means)

defined by f(b)=b
for every b that is a member of B
and every disabled person, b, will maintain who they are, but be considered a member of the community A from now on.