Like my dad did for me, I shower my son with Mathematician Love. That’s when the person helping you believes in you so much that he holds you hostage until you show you’ve learned it. He is willing to wait as long as it takes for you to do it, because he knows you can. You could be nine years old and it’s already past your bedtime and you’re crying because you can’t figure out which one of the sixteen three-digit multiplication problems you just spent a whole hour doing is wrong. And even at that young age, you can clearly envision nine more years of mandatory school and its accompanying Math Hell. But deep down, (and many years later), you know that someone believes in you.
I do this to my son because I want him to know that he is loved. OK, maybe not the crying-desperate-nine-year-old bit. But when I can, I try to enforce these ideas:
1) Learn it now, learn it well. It is important to learn skills well when they first come up, because playing catch-up is hard.
2) No matter how long it takes, I know you will learn it. I will not give up on you.
These two ideas still work really well for a lot of things, particularly skill-based and fact-based work, for example, practicing the steps of long division, or learning multiplication facts.
But there is more to being a good problem-solver than being good at skills and facts. Simply knowing skills and facts does not automatically lead to developing the art of problem solving: creativity, insight, inspiration, and perseverance. We know it when we see it. It's when a kid can pick up a problem, and even without knowing what exact procedure she is going to use, she starts working on it. She tries something. If that doesn't work, she tries something else. She is able to tell that it didn't work, because she is paying attention to whether or not her thinking makes sense. Sometimes, by trying something, she discovers something new about the problem, getting closer to the answer. She knows when she is done. She double-checks her work. She knows why the answer is complete and reasonable, and can explain it if asked. She knows that she can set it aside and come back to it later.
This is what some call "Doing Math"--the ability to pick up an new problem, use concepts and strategies to discover new things about the problem, and eventually come up with a reasonable solution. And over the past few decades, educators have worked to identify those specific behaviors that comprise Doing Math. Those behaviors have evolved to what is now the eight Common Core Mathematical Practices (MP), which is a list of mental habits that allow kids to (and this is oversimplifying it): make sense of problems and keep trying until they get a reasonable answer.
Promoting Mathematical Practices (MP)
If you’ve ever known a kid, you know how long it takes to get him to change behaviors. And I don’t mean change behavior right now. I mean change behavior forever. Without being reminded. You can’t teach behaviors in a single, sleepless lesson. Not when just learning simple behaviors like remembering not to wipe snot on your pants can takes a full three years (like it did with my son).
In trying to help my son develop soft Mathematical Practices such as “perseverance” and “attending to precision” and “choosing tools strategically,” I, as a parent who was all about “learn it now, learn it well,” had to adopt an additional skill: patience.
Practices are a part of one’s personality. I can’t apply Mathematician Love with the same intensity as if I’m asking him to learn a two-step procedure. I can’t demand that, instead of throwing his pencil across the table when he can’t think of what strategy to use, that I can say “Persevere! Dammit.”—and expect that this will solve the problem. I can’t say, “Just think of something else” if he can’t think of something else. It is hard to think creatively when you’re already upset
What's great about being a parent is, unlike my kid's teacher, I do not have to come up with the lessons to promote all eight Mathematical Practices. However, just by doing a few things when helping with homework, I can provide a lot of opportunities for him to engage in Mathematical Practices.
When the kid gets the right answer. When my kid gets the right answer, I promote problem-solving strategies by asking him to explain his answer. The requirement to explain answers or show work has recently gotten a bad rap, mostly because it is associated with the Common Core Learning Standards. It is often thought of as "extra" work at best, and at worst, a punishment for kids who are good at math but bad writers.
But the reason it is important is because the ability to explain your thinking (MP.3, Construct viable arguments and critique the reasoning of others) is closely linked to having complete, coherent thoughts. The fact that my kid can explain his thinking and show how he is able to get from the question to the answer shows me that he has solved the problem authentically. He hasn't taken a random guess at the answer, and he hasn't taken a random guesses in the middle of solving the problem.
Additionally, the act of explaining a problem can deepen a child's understanding, because it often forces the child to link ideas and pay attention to cause-and-effect in reasoning.
When the kid gets the wrong answer. When my kid gets a problem wrong, I resist the urge to immediately point out his mistake, re-teach the process, or redirect his thinking. Instead, I start by asking him to explain his answer. I try to say it in the same tone as when he gets it right so that he can't tell from my voice whether his answer is right or wrong. Much of the time, this act of explaining helps him to find his own mistakes (MP.6, Attend to precision).
If it doesn't help him see his mistake, his explanation helps me understand what he was thinking, and what exactly he is confused about.
When the kid gets stuck. A lot of times, when our kids get stuck or don't know how to solve a problem, we jump right in to help. We might just tell them what to do, or ask a series of bite-sized questions, preventing them from engaging deeply with the problem.
Instead, when my kid gets stuck, I ask him to read the problem again and tell me what the question is asking, in his own words (MP.1, Make sense of problems and persevere in solving them). If he can't tell me what is happening in the question or what it is asking for, I know what my next step is: to help him understand what the problem says, including the mathematical vocabulary. If he doesn't know the math words, I tell him what they mean. Sometimes I ask him to read each sentence carefully and explain it to me, draw it, show me what it means with objects, act it out (MP.5, Use appropriate tools strategically). Ninety percent of the time, he figures out what to do once he understands what the question is asking.
If he doesn't figure it out on his own, I may act as a thought-partner. I pretend I don't know the answer either and model the way I handle not knowing, such trying various approaches such as writing an number sentence, equation, or expression) that describes the situation (MP.4 Model with mathematics), seeing if I can find any patterns (MP.7 Look for and make use of structure), or see if I there are any rules that describe what is happening (MP.8 Look for and express regularity in repeated reasoning).