Again, what follows are largely notes and excerpts from What’s Math Got to Do With It?: How Teachers and Parents Can Transform Mathematics Learning and Inspire Success (2015 revised and updated) by Jo Boaler.
Yesterday I posted about how Boaler argues that math instruction in the United States promotes a kind of passive learning that does disservice to the actual nature of real mathematics. Efforts to provide reforms or alternatives, including Common Core, have roundly been met with resistance, often organized and political.
Putting that aside, what’s a parent to do? Boaler herself helped create a summer camp designed to give a sort of Jedi training in math to math-averse students.
One of the goals… was to give students opportunities to use mathematics flexibly and to learn to decompose and recompose numbers. We also wanted students to learn to ask mathematical questions, to explore patterns and relationships, and to think, generalize, and problem solve…. We decided to focus… on algebraic thinking, as we thought that would be most helpful to the students in future years, and to focus upon critical ways of working including asking questions, using mathematics flexibly, reasoning, and representing ideas. (146)
As you can see from what I’ve put in bold, Boaler suggests a couple of major objectives that will make our kids stronger mathematicians: (1) getting the confidence and curiosity to ask good math questions, (2) playing with numbers and using concepts flexibly, (3) being able to articulate, discuss, and defend their reasoning and problem-solving, (4) having multiple ways to mathematically represent ideas and processes.
What are some things that can be done at home, with or without school?
There are many books filled with great mathematical problems for children to do, but it is my belief that the best sort of encouragement that can be given at home does not involve sitting children down and giving them extra math work, or even buying them mathematical books to work on. It is about providing settings in which children’s own mathematical ideas and questions can emerge and in which children’s mathematical thinking is validated and encouraged. (169)
Any sort of play with building blocks, interlocking cubes, or kits for making objects is fantastically helpful in the development of spatial reasoning, which is fundamental to mathematical understanding.
In addition to building blocks, other puzzles that encourage spatial awareness include jigsaw puzzles, tangrams, Rubik’s Cubes, and anything else that involves moving objects around, fitting objects together, or rotating objects. Mathematical settings need not be sets of objects. They can be simple arrangements of patterns and numbers in the world around us. If you take a walk with your child, you will stumble upon all sorts of items that can be mathematically interesting, from house numbers to gate posts. (170-171)
Research on learning tells us that when children are working on their own ideas, their work is enriched with cognitive complexity and enhanced by greater motivation. Duckworth proposes that having wonderful ideas is the “essence of intellectual development” and that the very best teaching that a parent or teacher can do is to provide settings in which children have their most wonderful ideas. All children start their lives motivated to come up with their own ideas—about mathematics and other things—and one of the most important things a parent can do is to nurture this motivation. (172)
❑ Buy more Legos.
[Sarah] Flannery says: “…almost without our knowing we’ve been getting help since we were very young—out-of-the-ordinary help of a subtle and playful kind which I think has made us self-confident in problem solving. Ever since I can remember, my father has given us little problems and puzzles…. These puzzles challenged us and encouraged our curiosity, and many of them made math interesting and tangible. More fundamentally they taught us how to reason and think for ourselves. This is how puzzles have been far more beneficial to me than years of learning formulae and ‘proofs.'” (173)
❑ Tell dinner time stumpers.
No matter how outrageous a student’s contribution or question, he could respond: “Oh, I see what you are thinking. You’re looking at it as if…” This is a very important act in mathematics teaching because it is true that unless a child has taken a wild guess, then there will be some sense in what they are thinking—the role of the teacher is to find out what it is that makes sense and build from there. (176)
❑ Try to figure out what the heck your kid is doing.
Math conversations should be relaxed and free from pressure. Fear and pressure impede learning, and children should always feel comfortable when offering their ideas in math…. When students know that I am not judging them harshly and that I genuinely value errors, they are able to think more productively and learn more. (178)
❑ Don’t talk about grades.
One of the very best methods I know for encouraging number flexibility is that of number talks…. The aim of number talks is to get children to think of all the different ways that numbers can be calculated, decomposing and recomposing as they work. (178)
Some good prompts to use while you are working with children are:
* How did you think about the problem?
* What was the first step?
* What did you do next?
* Why did you do it that way?
* Can you think of a different way to do the problem?
* How do the two ways relate?
* What could you change about the problem to make it easier or simpler?
❑ Talk about numbers.
I’m being facetious in an impoverished attempt at levity, but I actually think Boaler’s advice is excellent. In the ninth chapter of her book she has a list summarizing her practical takeaways for parents. She uses a lot of absolutes in this list, each of which I can think of numerous exceptions, but the gist of it is good.
- Never praise children by telling them they are smart. This may seem like it is encouraging but it is a fixed-ability message that is damaging.
- Never share stories of math failure or even dislike.
- Always praise mistakes and say that you are really pleased that your child is making them.
- Encourage children to work on problems that are challenging. We know that it is really important for students to take risks, engage in “productive struggle,” and make mistakes.
- When you help students, do not lead them through work step-by-step. Doing this takes away important learning opportunities for them.
- Encourage drawing whenever you can…. Both drawing and restating problems help children understand what questions are asking and how the mathematics fits within them.
- Encourage students to make sense of mathematics at all times…. If children seem to be guessing, say, “Is that a guess? Because this is something we can make sense of and do not need to guess about.” Mathematics is a conceptual subject, and students should be thinking conceptually at all times.
- Encourage students to think flexibly about numbers.
- Never time children or encourage faster work. Don’t use flash cards or timed tests.
- When children answer questions and get them wrong, try and find the logic in their answers—as they have usually used some logical thinking.
- Give children mathematics puzzles.
- Play games, which are similarly helpful for children’s mathematical development.
Next: What I do, as a homeschooler