Math Learning for Kids Who Have a Tough Time

This is an interesting panel discussion. In addition to useful background explanation, the discussion had three specific bits I found revealing. Let me transcribe those parts. Then I will expand on what Michele Mazzocco says. 

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Lindsey Jones: ... there is a beautiful study by Aaron Maloney and Sian Beilock showing that when parents have math anxiety and they do homework with their children, then their children end up being even more math anxious and performing even worse in school. 

Lindsey Jones:  But there are things we can do as well. So there is research showing that suggests that for example if you engage students in what is called "expressive writing" exercises, before they take, for example, a high-stakes math exam—they simply write down how they feel about the upcoming test, how they feel about the upcoming exam—and if you do that, you lessen the effects of the anxiety on the subsequent test performance. 

Michele Mazzocco:  In our longitudinal study, we looked back at kindergarten, first, second graders' descriptions of math and reading. And, again, looking at the different groups of children,  those who had consistently shown difficulties with math throughout the study, we looked back at how they had defined math and reading in first and second grade. They were significantly more likely to use negative language in describing math compared to reading, more likely to describe it as difficult compared to their typically-achieving peers.

And this gets back to the chicken-or-egg problem that Daniel was talking about: Is it that they disliked math and therefore they avoided it and therefore didn't get better at math? Or is it that they disliked math because it was difficult and math would have continued to be problematic for them, regardless of their disposition towards math?

And I think that this speaks to the broader issue of math disposition. Math anxiety is a very real phenomenon; it's something that we should not ignore. But there's also the part of the math disposition that reflects how children feel about the importance of math—it's usefulness—and the sentiment that "Hey, you know this might be one of those things that's hard. It might be something where I need a recipe guide or I need some help. I need some scaffolds.  But that's OK, because we all need some scaffolding for different kinds of tasks, and putting forth effort actually leads to some success." So if we have this healthy disposition towards math and recognize that effort is required, that effort doesn't mean necessarily that I'm bad at it—it just means that I need to work harder. 

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After Noah Smith's and my column "There's One Key Difference Between Kids Who Excel at Math and Those Who Don't" went viral (and was translated into Spanish here), many people shared their own personal stories of ultimate success at learning math even when they went through a period of thinking they were bad at math. Many other people shared wonderful ideas for how to help people learn math. I tried to distil those ideas and some of my own into the advice in "How to Turn Every Child into a 'Math Person'" One of the key ideas is that it is OK to learn math slowly:

• Cathy O'Neil on Slow-Cooked Math
• Mary O'Keeffe on Slow-Cooked Math
• Fields Medal Winner Maryam Mirzakhani's Slow-Cooked Math
• Jenny Anderson: The best way to learn math is to learn how to fail productively

A second related idea is that for children, math needs to be made as fun, relevant and unthreatening as possible:

• Laura Overdeck: Math for Pleasure
• Laura Overdeck: Street Math
• Jing Liu: Show Kids that Solving Math Problems is Like Being a Detective
• Examining the Statistics in “Math at Home Adds Up to Achievement in School” by Talia Berkowitz, Marjorie Schaeffer, Erin Maloney, Lori Peterson, Courtney Gregor, Susan Levine and Sian Beilock

In "How to Turn Every Child into a 'Math Person'," I pointed to leading a math club for children as a wonderful choice for those willing to volunteer to help kids. 

I have also been interested in how to teach math effectively in more formal settings:

• Barbara Oakley: How We Should Be Teaching Math
• Jessica Lahey: Teaching Math to People Who Think They Hate It
• Math Camp in a Barn
• Jethra Spector: Using Miles and Noah's Math Column in the Classroom
• Kevin Remisoski on Teaching and Learning Math
• A Mathematician Has Created a Teaching Method That’s Proving There’s No Such Thing as a Bad Math Student
• Elizabeth Cleland: How I get all my students to be good at math
• My Advice to Qatar: Make Math Education a Research Grand Challenge
• The Coming Transformation of Education: Degrees Won’t Matter Anymore, Skills Will
• Karthik Muralidharan, Abhijeet Singh, and Alejandro J. Ganimian: Disrupting Education? Experimental Evidence on Technology-Aided Instruction in India

More recently, I have run across discussions of stereotypes for particular demographic groups that can get in the way of learning math:

• You, Too, Are a Math Person; When Race Comes Into the Picture, That Has to Be Reiterated
• Calculus is Hard. Women Are More Likely to Think That Means They’re Not Smart Enough for Science, Technology, Engineering and Math
• Kate Owino: Kenyan Women Can Love Math Too

For those who need a little extra confidence boost to counteract a tendency to underestimate themselves, I also discuss "Travis Bradberry: Ten Guaranteed Ways To Appear Smarter Than You Are." 

At a high level, the confidence that one can learn math, as well as the idea of "slow math" is important in the advice Noah and I give in "The Complete Guide to Getting into an Economics PhD Program."

By the way, I dug out these links simply by typing "math" into my blog search box. There are more, especially guest posts people telling personal stories of overcoming the belief that they were bad at math.  

The most important principle is the key not only to learning math but to succeeding at almost anything: knowing that you can get smarter by working hard at getting smarter. Here is that message, generalized:

• Miles's Recipe for Success
• Effort vs. Innate Ability: What I Learned from Being in the Spelling Bee
• How the Idea that Intelligence is Genetic Distorted My Life—Even Though I Worked Hard Trying to Get Smarter Anyway

Math Post: Isomorphismes on Transforming a Problem Into Something Easier to Understand

Isomorphismes is one of my favorite Tumblogs. Here is a math post I liked. 

Going the long way: What does it mean when mathematicians talk about a bijection or homomorphism?

Imagine you want to get from X to X′ but you don’t know how. Then you find a “different way of looking at the same thing” using ƒ. (Map the stuff with ƒ to another space Y, then do something else over in image ƒ, then take a journey over there, and then return back with ƒ ⁻¹.)

The fact that a bijection can show you something in a new way that suddenly makes the answer to the question so obvious, is the basis of the jokes on www.theproofistrivial.com.

In a given category the homomorphisms Hom ∋ ƒ preserve all the interesting properties. Linear maps, for example (except when det=0) barely change anything—like if your government suddenly added another zero to the end of all currency denominations, just a rescaling—so they preserve most interesting properties and therefore any linear mapping to another domain could be inverted back so anything you discover over in the new domain (image of ƒ) can be used on the original problem.

All of these fancy-sounding maps are linear:

Fourier transform
Laplace transform
taking the derivative
Box-Müller

They sound fancy because whilst they leave things technically equivalent in an objective sense, the result looks very different to people. So then we get to use intuition or insight that only works in say the spectral domain, and still technically be working on the same original problem.

Pipe the problem somewhere else, look at it from another angle, solve it there, unpipe your answer back to the original viewpoint/space.

“Going the long way” can be easier than trying to solve a problem directly.