The Crevasse, by Edgar Mueller
Today your mission is…
To start doing citizen science with us!
Ready, Set, Go
Review the list of signs or precursors that point to developing dislike and avoidance of mathematics. If these signs describe your child, don’t panic, but take prophylactic measures. The rest of this course is full of activities you can do to help!
Send us stories, examples, and anecdotes that illustrate these dangers. You can talk about your own experiences, or describe your child, your student, a friend, or anyone else you know.
Why do it?
Most grown-ups have “grief stories” when it comes to mathematics. If this happened to you, what if grief comes to the next generation, too? Can you shield your children from harm?
What you see below is our first attempt at putting together a systematic, research-based, people-approved set of answers to that question. We invite you to contribute.
Ask a question if an item below seems relevant, but does not quite make sense. This is work in progress, and your questions will help.
Dangers specific to multiplication
Additive misconceptions
Frequent mistakes by using addition instead of multiplication
Mental blocks about “very multiplicative” ideas, such as proportions, ratios, division of fractions, or combinatorics
“Everything is linear” misconception: can’t understand diminishing returns or exponential growth, thinks (a+b)^2=a^2+b^2
Multiplicative misconceptions
Frequent mistakes by using multiplication instead of addition
Mental blocks about sequential ideas, such as skip-counting
“Everything is nonlinear” misconception
Mnemonic “complexes” set in stone
Can’t recall a fact without reciting the full sequence of facts leading up to it
Still recalls visual mnemonics or rhymes every time a fact appears in a problem
Has trouble seeing patterns that involve multiple facts
You can help by only using mnemonics for names and other pattern-free information, never for facts that you can compute, or for anything with a conceptual explanations
Collapse of scales
Deep down, thinks that a billion is “just a bit more” than a million
Can’t understand how a Solar System, an atom, or other structures with a large spread of sizes would look, to scale
You can help by training with exponential systems. More at: http://io9.com/how-to-comprehend-incomprehensibly-large-numbers-1531604757
Psychological and social dangers in all math topics
Stereotype threats
Identifies with a demographic that is perceived to be less successful in math
Feels vulnerable about the group identity
Performs worse in mixed groups, or when reminded of the demographic identity
You can help by separating math activities from social discussions, and by affirming your individual values. More at: http://anniemurphypaul.com/2013/04/from-the-brilliant-report-the-power-of-affirming-your-values/
Ability attributions
Believes that success is largely due to abilities (rather than effort and support)
Thinks that some people are born to be good in math and some can’t do it
Easily loses heart over mistakes or feeling stuck on a problem, believing this to prove low ability
You can help by praising work, explaining success by effort, and giving feedback about specific features of math processes (rigor, perseverance, etc.). Examples at: http://www.washingtonpost.com/local/education/in-schools-self-esteem-boosting-is-losing-favor-to-rigor-finer-tuned-praise/2012/01/11/gIQAXFnF1P_story.html
Math anxiety
Panics about timed work, tests, word problems, or “everything math”
“Eyes glaze over” if a problem has new notation or is from a topic not practiced before
Is reluctant to show or share math work, feels shame around mathematics
STEM careers seem out of limits
You can help by providing a learning environment that supports well-being, increasing free play, celebrating mistakes, reducing coercion around math, and learning psychological coping techniques.
Loss of creativity, divergent thinking, problem-solving
Can’t see connections between math and other endeavors
Can’t model, draw, or explain mathematical concepts
Gives up early on any non-standard problems
Has difficulty finding extreme cases, examples, counter-examples, or non-examples
You can help by posing creative and divergent tasks, learning brainstorming, lateral thinking, and problem-solving techniques. Example: http://www.math.utah.edu/~pa/math/polya.html
Do you have examples of any of that happening? Stories of how you coped? More items we should add to this list of dangers?
Answer by fcogan · Apr 11, 2014 at 02:46 PM
I see that many students in high school "give up" on math and science because they believe it is "too difficult" and therefore they won't get marks high enough to allow entrance to university or college. I see many cases of students who enter university hoping to get into the sciences or medicine but give up after a year or less because they cannot tolerate the mathematics courses. This is a shame. I still think that the best way to set a foundation in multiplication is to learn the basic multiplication tables from 1 x 1 to 9 x 9 and the rest of multiplication is basically derived from there. Yesterday, my 4 year old grandson was having a sandwich which was cut in halves and he asked if it could be cut in "fours". This was a perfect opportunity to learn the concepts of 2 halves, 4 quarters make a whole, 2 quarters make a half, etc.. He really enjoyed that and the sandwich too! I think too much stress is put on students with testing, especially in math. Students have to be able to experiment and make mistakes in order to truly learn the concepts and applications of mutliplication. Testing and the expectation of perfection is very stressful for students, especially in math, where it is generally expected that there are only right or wrong answers.
Yay for tasty math! What a great example of spontaneous multiplication fun.
Here are a few more ideas from our Eat Your Math collection.
Answer by annboyd · Apr 11, 2014 at 01:06 PM
We've been using "xtramath.org" to do some math drills on basic addition facts. I thought it would help my daughter if I walked away while she worked on it so she wouldn't feel the pressure of me watching her every move. But I found that she actually performed worse this way. So instead, I decided to try "math with love," meaning that I sat next to her and smiled and encouraged her while she practiced her drills. Her performance improved a bit, but even better was her attitude about the practice sessions.
Interesting, annboyd! I once interviewed kids and parents from a math circle about well-being during math. Everybody said YES to having a dog or a cat nearby. But the group split very neatly in half on the question of relatives. Half of the people said that a family member in the room made them feel loved and supported. But other people strongly preferred not to be with family - or at least, not with this or that particular family member they named. I think your focus on your daughter's well-being, rather than just math values, made the difference. Math with love!
Here's a quote about sports that seems relevant:
College athletes were asked what their parents said that made them feel great, that amplified their joy during and after a ballgame. Their overwhelming response: “I love to watch you play.”
Answer by Eogruen · Apr 11, 2014 at 01:31 PM
Uh-oh, I am totally busted. I still, at the age of 38, have to go through large portions of the multiplication tables to answer something like 7x 8. Not sure exactly what is meant by "not using mnemonics" to learn them though, as I always thought most people learned them by essentially "skip counting" their way through. Hope my girls can learn them another way., maybe more securely? I am re learning conceptual math with my girls and find, to my surprise that I do like puzzling over mathematical concepts once I get started, after a lifetime of hating to do equations on paper.
Skip-counting is not mnemonics, it's more a calculation technique. Do you skip-count by 7s or 8s? That's very impressive!
But if you memorize the skip sequence as a song, that's another story. For example, you can sing "four eight twelve sixteen twenty" to the tune of "Old MacDonald had a farm" - and it will help to memorize the skip sequence, but only as a whole. It may interfere with calculations and with algebra in some cases, since you will always need the whole song to get to 20.
Mnemonics are non-mathematical (non-calculation) memory techniques, for example:
"Five, six, seven, eight - fifty-six is seven times eight!" This is non-mathematical, because the fact that 56=7*8 has consecutive numbers is a coincidence.
Here is a visual mnemonic for 9*9 from Times Tales. The images have no connection with quantities or numbers or the mathematics of multiplication 9*9=81. So they don't support number sense or algebra.
Thanks for the clarification, that helps a lot. Yes, I can skip count by sevens and eights but not very far, hence my perennial inability to answer 7x 8 etc. without serious thought. Am planning to finally do this right when I do it with my kids!
If you like to calculate, you can do 8th by doubling three times. It's easier than skip-counting seven times! There is a whole neuroscience-based explanation why 7*8 is the hardest multiplication fact in Dehaene's "The Number Sense." The short version is that both numbers are outside of the subitizing range, outside of the 10- and the 9-patterns, and it's not a full square (which are easier to remember).
7*8 has always been one of the hardest for me to remember. I minored in math and still don't have my multiplication tables completely memorized. What are your thoughts on mnemonic devices like the Times Tales? I've always thought that mnemonic devices like the skip counting songs were good, but have also regarded ones like the 9*9=81 story with suspicion. Would you consider them to be helpful? Dangerous? Harmless?
I consider all non-mathematical mnemonic devices dangerous, when it comes to patters, conjectures, results of computations - anything that has meaning to it. You can use a mnemonic to remember which one is sine and which one cosine, because these are rather random names. But 9*9=81 is not random. It is a part of several neat patterns, and it can be computed, so it's better for our minds to focus on these structures. To memorize times tables, if memory is desired, I would use a spaced repetition tool like Anki. You want your tool to serve you individual facts, straight up (no mnemonics), and only those you actually have trouble remembering. We'll work more on that stuff in Week 4.
My mum used to tell me a story about a girl who named her doll 'fifty-six' to help her remember that difficult fact!
Answer by babyhclimber · Apr 11, 2014 at 03:50 PM
There is a danger in our society of not valuing math and science. The various fields/careers that involve math and science are often portrayed as geeky, nerdy, boring, etc. Politicians and parts of the media bashing science and math don't help. Anti-intellectualism doesn't help. Our country needs more scientists. But first we need more teachers who understand math & science to help pass on the love of math & science. And our society as a whole need to realize how much math is everywhere instead of the dissing of math with the "I don't need to know that" or "when will I ever need that." Combine with the culture attitudes with a student struggling in middle school or high school and then quitting on math is a problem. Sadly schools are so rigid that they don't realize if they changed the presentation or math or looked more globally at the problem, the student might be able to get the concept. Also there is a problem of holding kids back. If a kid can grasp higher level math, let them move on. Sometimes the repeated drills ending up killing the love or curiousness of math.
Ways to change: expose kids to algebra thinking earlier on, discussing how natural math can be, showing how math is all around us, getting & keeping kids curious, and connecting math with art & music.
That danger of not valuing science (as a society) should be added to our list. Some groups of people don't have formal math and science, historically - and some have lost their scientific renaissances to social pressures. I lived through the event where a country lost 30-40% of its STEM publications within some 5 years, due to emigration and social disruptions of academic life...
With these comments I wanted to share
Answer by Hascoorats · Apr 11, 2014 at 07:45 PM
WOW! Can't wait to scour through the links :) My DD is 7 and was told by her grade 1 teacher that she wasn't allowed to count on her fingers. After that moment, she asked to go to the bathroom every time they did math and stayed there until the recess bell rang. We homeschool now and she still has an aeration to math... But when I play subitization games, cards, dice, pattern play, she is quite bright and enjoys *math* as long as it doesn't look, feel, smell like *real* math...
One wonders what real math is, and who decides!
I am copying my comment about embodied math:
It's okay to count on fingers. Some people who think a lot with their bodies - dancers, mechanical engineers, crafters - need to move and touch to think. Other people need to speak (maybe in their heads), others to doodle...
Answer by dnamkrane · Apr 11, 2014 at 11:22 PM
When I was seven years old I remember crying in frustration as I did my subtraction homework. The concepts of carrying and borrowing didn't make sense to me. By the time I was in the third grade though I was one of the "good in math" kids though. I didn't feel math anxiety after that until I hit calculus. After two months of scratching my head, I finally opened my text book and started at the very beginning until I figured out where I got tripped up. Not surprisingly, it's the momentary suspension of disbelief: "imagine you *could* do something an infinite number of times". Once I realized what it was, I was back in the saddle :-) My son with math anxiety is a perfectionist. If he can't get it immediately, he doesn't know how to cope because he gets so much very quickly. He doesn't use mnemonics, but he still has to count on his fingers and "reinvent the wheel" every time he needs to do simple multiplication. I'm torn between letting him work this way and feeling like he *should* have certain facts at the ready.
Some perfectionist and/or quick kids like to work with reference tables, handy at all times. Like the Pythagorean table for multiplication:
It's different from using a calculator, because you navigate a chart by patterns. People usually end up color-coding, or otherwise tweaking their charts.
Answer by PruSmith · Apr 12, 2014 at 03:32 AM
Man, some of the stories you hear about math trauma...makes you wonder why so many unhappy people decide to teach math. Anyway, I don't really know math instruction (I was "unschooled" as a kid) but if it's anything like art, just sloppy language on the part of a well meaning person can stunt creativity and stifle genius. For example, be sure you REALLY know something about color before you try to teach a bunch of preschoolers their colors. Take a poor or incomplete understanding, couple it with authority and you've got a recipe for creating some tremendous road blocks in their brains. I can only imagine what us numerically challenged people must do to kids with our imprecise and fumbling talk about mathematical concepts!
This is a complex discussion: the danger of mentors not knowing the content. Overall, I am cautiously optimistic about "numerically challenged people" if they take steps to learn - as you do. I've seen very nice results, especially with support! For example, the leader of Living Math Julie Brennan and my co-leader at Natural Math Yelena McManaman describe their journeys from rocky relationships with math to reading algebra and calculus for fun, with and for their kids. Many parents get their second chances at math with children! Leading big math projects is not for everyone, but helping kids and friends learn math well is doable and frequent among parents with past math challenges. So I am happy to see you here in the course!
Answer by yileinei · Apr 14, 2014 at 09:19 PM
I've always been good at maths but one of my biggest problems is related with the way I learnt the multiplication tables (I learnt them by heart). When I try to recall, for example, 8x7 I have to recall the full table to get there and sometimes I have to change the order of numbers, that is I can remember 6x8 but not 8x6 and it makes me feel kind of stupid! I'd love my daughters to learn this in any other way.
Answer by classicalmama · Apr 17, 2014 at 11:11 PM
I have say homeschooling has opened my eyes to education.(even as a preschool teacher I was blind to things) I could pass a math class and I had math through 12th grade pre or calculus. I wanted my kids to really understand math. I still struggle with the multiplication tables.
Answer by lisa.koops · Apr 21, 2014 at 05:03 PM
Thanks for this list - separating it out makes it easier to think through! My 7-y-o daughter said last week she was tired of getting 100% on her math tests because the other kids made (unkind) comments about it. I wanted to shout "NO! Shine on!" or something because this was such a familiar story to my middle school and early high school years (it's not cool to be good at math). amightygirl.com has some great resources on this as well.
Answer by nancy · Apr 11, 2014 at 06:03 PM
I was usually fine with multiplication...memorized the tables by third grade. My challenges came when the concepts became more abstract. I identify with my students who find it difficult to represent or explain their process for solving a math problem.
I prefer to start "high concepts" early - even with toddlers. Arithmetic is not mathematics, and exclusive exposure to arithmetic only leads to conceptual shocks later on!
I would have thought that arithmetic is rather a part of mathematics. How do you, then, define arithmetic in relation to mathematics? (I do get and appreciate what you say, ". . . exclusive exposure to arithmetic only leads to conceptual shock later on!" TY!
Answer by Sblair · Apr 11, 2014 at 07:18 PM
Since I started this class, I have been stretching my brain to see ordinary activities in a math sense. My children and I are probably torn between Loss of creativity and misconceptions. I am wondering what the (^) means in the algebraic equation in the misconception portion?
My first F in the 6th grade was in math. From then on I thought I didn't have geometric brain function. I had just coasted by and am pretty anxious about teaching my children the upper level math classes for credit. Although I have confidence now, only because I feel I have to just let it go and face my fear, to do the math that is needed to show completion of work I want to STOP COUNTING ON MY FINGERS! It rolls off my tongue when the kids are doing worksheets or a timed drill.
As I literally look around me, I am starting to visualize the math concepts around me. I feel a fuller concept of natural view ahead.
The world looks different through math goggles! Thank you for sharing this feeling!
^ stands for power. So (a+b)^2 reads "a plus b to the second power." Many parents and teachers say the fear of mistakes (or misconceptions) stops their creativity... So I would focus on the natural view and creativity first.
It's okay to count on fingers. Some people who think a lot with their bodies - dancers, mechanical engineers, crafters - need to move and touch to think. Other people need to speak (maybe in their heads), others to doodle...
Answer by Elizabeth02 · Apr 11, 2014 at 08:48 PM
This is a tricky one. The only time I felt math anxiety is when I got moved to the more difficult math halfway through the year and I thought I would not be able to catch up. My son's biggest problems with math occured when he was still in the classroom and not progressing at all in math. He was so bored that he was really starting to hate math time. There was also a large emphasis on memorization and fact recall, as opposed to playing with concepts, which was disheartening in that they were only 6 year olds. Since he has been homeschooled he has really bloomed and I haven't seen any signs of math anxiety. He abhors memorization and solves problems on his own. Since math was very sterile when I went to school I really want to help him see connections more, so we are all working on that together.
Memorization can be meaningful, if you memorize something that is beautiful or useful TO YOU. I am afraid the danger you name is deep and insidious: it's not about the overall amount of memorization, but about forcing everyone to memorize the same content, whether they find it meaningful or not.
Absolutely! Everyone figures out things on their own and ends up memorizing certain facts that stand out to them for whatever reason. Why do we force everyone to do it the same way though, as if there is one right way? I really try to just teach the concepts, and practice. Everyone will develop their own personal connections to the material and thus learn it, hold on to it, and hopefully apply it their own way.
Answer by James · Apr 11, 2014 at 11:21 PM
I still remember when I was in kindergarden and wanted to move to first grade early. The class was in the middle of learning subtraction, so the teacher gave me a long worksheet and quickly went over the two practice problems with me so I'd be good to get the rest. It just so happened the practice problems were both something minus zero, so I though subtraction was just picking the bigger of the two numbers. I got every problem on the worksheet wrong except when it was something minus zero, and felt stupid. My daughter will be unlikely to experience a similar problem as I'll be guiding her throughout her education, giving her tons of attention. If she makes a silly mistake, we can laugh together over it.
This should be added to the list of dangers: a mistake that gets repeated a lot of times because of lack of feedback! It's okay if problems are self-checking or connected to other problems, because mistakes have consequences kids can notice. In worksheets... it's a danger.
Answer by TraceySeier · Apr 11, 2014 at 11:59 PM
Haven't had any of these things happen to me... except the eyes glaze over part... that has happened with anything past Calculus ... ie Fourier series. But it is helping me to understand what has happened to so many people around me, who seem to struggle with math. I just hadn't thought of all the ways in which they don't get it, or how/when they fell off the boat. I was lucky: I was always so far ahead of my (working-class town) school mates, that I just taught myself from books for fun, and so I never had pressure to perform, and I could always take a playful approach. Maybe the reason I struggled with the "harder topics" was that I ran into them in a high- pressure context... in college.
This is a subtler danger for more successful kids: never running into math difficulties until something like calculus hits like a ton of bricks! I will add it to the list. Julia Brodsky of The Art of Inquiry talks about an algebra student she helped simply by explaining that some problems have more than one step and require taking notes for intermediate steps. Up to that poor kid's sixths grade, all the exercises were so easy he could leap to the answer in his head. When algebra happened, he thought he was missing some math recipes for solving those problems in one step, as before...
Answer by ramandada · Apr 12, 2014 at 11:33 AM
In 5th grade I was moved back up to the advanced math group because the middling group was too slow for me, according to the teachers. The first homework I was given in the advanced group were word problems where the teacher explicitly emphasized how we were to "write out" the problems completely. I spent the entire evening, weeping, late into the night, copying out longhand the entire question in my notebook before attempting to answer it. Math association with meaningless drudgery. Much later, in geometry, I invented new proofs for some problem that made sense given what we knew at the time. The teacher informed me that it would not work given what I did not yet know. Oh. The unknown will defeat you, there is no point trying. Unlike many subjects where I had an idea of the grade I would earn on an exam, I could never predict with any accuracy the grade I would get on a math exam, through college. It felt like the rules of math were held tight by some exclusive club of people who always knew the right answer. And there was *only* one answer.
How would you name the danger of what happened to you? People usurping and then abusing power over your learning?
Answer by Kris · Apr 12, 2014 at 07:21 AM
I've always found math relatively easy; however, two experiences stand out for me. 1) My brother in about grade 9 was given the task of finding the perimeter and area of three triangles (he was quite behind in math according to his teachers and has a learning disability). My mother didn't know how to help him with the questions, so she asked me to work with him. He proceeded to ask me to do it because that's what the teachers did when he didn't know. I sat there for almost an hour waiting him out. I had a very strong gut feeling that the problems were within his ability. I will never forget the look on his face when he said FINE!, put his head-down, completed the questions (only making one error), and walked out of the house to go play hockey. Fifteen years he still refers to that single experience, as one of the only times that he felt that someone believed in his academic ability and didn't let him "play" the system. 2) In grade six, my math teacher called a parent teacher interview to discuss why I was "cheating" on my math tests. In the meeting the teacher pulled out my exam and showed my mother that I had gotten 100% on the exam (long division, algebra, etc) and that I had completed the test without showing any work-his conclusion I must have cheated. I sat there confused, thinking I was getting accused of cheating because doing work in your head and not showing your work was cheating (I didn't realise that he thought I'd copied the answers). For the next exam, I "showed" my work and again a meeting was called. This time because I didn't show the correct work and again he thought I was cheating (I did long division by estimating the answer and using multiplication to find the answer). My teacher didn't understand the method I used and said it wasn't possible. At the end of that meeting my mom suggested that he give me a test right then because she believe me. I completed the test my way without "cheating".
Dangers making assumptions about a students ability or knowledge based on their performance. But I feel both these examples illustrate the protective power of believing in someone's ability.
Kris, thank you for sharing these powerful stories. I held my breath reading. We will add to the list of dangers assumptions and presumptions about abilities (and morals, as in that cheating story). Expecting kids to cheat or dislike mathematics can be very toxic.
There is a famous experiment about assuming ability, the Rosenthal-Jacobson study.
Answer by zzzeee2000 · Apr 12, 2014 at 07:03 PM
As a teenager I feel like math anxiety is something a lot of us teenagers face. Both in homeschooling and in public school.
Answer by Valerie · Apr 13, 2014 at 01:54 AM
My answer to this question really is the reason that I wanted to do this course. In primary school I loved playing with numbers and arithmetic. I had an enthusiasm for it, even up to 16 years old. Then, I lost my confidence in my ability in maths in the last two years of school when we had to tackle integration at school. My teacher would spend the lesson writing lines and lines of calculus on the blackboard and ask us to copy it down in our books without explanation. If we did ask a question, she would get very cross with us and we were too frightened to ask her questions. After that experience, I believed that I couldn’t learn maths. However, I needed maths for my biology degree and luckily my university lecturer was very different and made me start to believe that I could do it with effort and support. Now my job includes quite a lot of statistics and some maths and this wouldn’t have happened without the support of my uni lecturer. Looking back, I think much of my school teaching was focused on rote-learning methods to derive the correct answer, rather than having an understanding of what that really means. So, for example, while I had no problem learning multiplication of integers, I remember getting muddled with multiplication involving numbers with decimal points, which wouldn’t have happened if I’d understood multiplication more fully. I also think that mathematical terms can strike fear into the hearts of the under-confident, particularly if you have had a bad experience in school, so I’d like for my daughter to learn and understand mathematical concepts first without the labels. I would also like her to have a full understanding of the concepts, and not be taught primarily to learn a series of methods to get the right answer. I am grateful to you and this course for encouraging us all to find the love and enthusiasm of maths again - thank you.
Answer by AGray · Apr 13, 2014 at 04:18 PM
This is a great list! I need to study it and the suggestions. The biggest obstacle I have noticed in my kids is math anxiety. So far I have handled it by sitting with them, working problems together, modeling making mistakes and trying different approaches. They seem to think they have to know the right answer right off the bat. I'm trying to get them to see it as a process .
Week 1 Task 1: Your mathematical dreams and worries 146 Answers
Week 1 Task 2: Your child, the divergent thinking genius 94 Answers
Week 1 Task 3: Patterns and adaptations 56 Answers
Week 1 Task 4: What is multiplication? 42 Answers