Any kid who has memorized their times tables can tell you what 5x3 equals. But pose the same question in the form of a story, and the same kids will be far less sure of their answers.
So what’s going on? Why is the same problem presented in a different form so much harder? After all, if kids understand what mathematical symbols mean, shouldn’t the written story problem be just as easy to solve as the problem expressed in mathematical notation?
In this article I’m going to analyze why word problems are hard for kids, describe a promising solution called Cognitively Guided Instruction, and outline my recommendations for how to solve this crisis.
In a followup paper I will describe in detail games I am developing to teach problem solving.
Challenge 1: Kids are trained to not care about understanding mathematics
The brutal truth is that the way mathematics is taught in school, kids are rewarded get the right answer as quickly as possible, NOT to understand or make sense out of mathematical symbols. Math educator Phil Daro labels this approach to teaching math as “answer-getting”. See Phil Daro Against Answer Getting for his devastating takedown of this widely accepted but massively misguided approach to math education.
Kids are drilled to manipulate symbols according to abstract rules, without pausing to reflect on what the symbols really mean. Telling kids to “show their work” is supposed to fix this problem, but in truth it’s an annoying bit of busywork that fails to motivate true comprehension.
Here’s an analogy. You can learn the rules of four-voice counterpoint purely as a set of rules, and be able to write syntactically correct chorales without having any idea of what they sound like…or even knowing that that there is such a thing as enjoying listening to music. And so it is with mathematics. Most kids learn to dutifully follow rules without hearing the music of mathematics, or even knowing there is such a thing as enjoying understanding mathematics.
Teachers know this, but at the end of the day teachers are are under so much pressure to just get through the material on a fixed schedule that they don’t have time to make sure that every student truly understands what they are doing. And thus, over time, almost all students gradually accumulate holes in their understanding — holes that eventually sink them. Even students who are able to get good grades by following rules often don’t understand what they are doing — and don’t realize that they don’t understand it.
Furthermore, kids are drilled on a very narrow range of problems written in highly predictable forms. The result is what I called “brittle understanding”, that crumbles the moment a problem is stated in a slightly different form. My personal pet peeve is that kids see problems in the form “3+4=_” so often that if I show them the problem “3+4=_+1” they will literally ignore the “+1”, because they have been trained to think that the equals sign means “the answer is”, rather than “the expressions on the two sides have the same value”.
So when kids are given word problems, what do they do? Kids commonly scan the story for numbers, and try to determine what operation is embedded in the problem, then carry out that operation. They will literally not read most of the problem. That’s not because kids are stupid. It’s because they are smart, and have learned to play the game we have set up for them.
Challenge 2: Translating between words and mathematical symbols IS hard
Mathematical notation has a conciseness and clarity that human language does not. It takes more effort to read “Lyra had 4 Hot Wheels cars, and then her mom gave her 3 more”, than it does “4+3”. You have to decide which parts of the story are significant, and understand that “3 more” means “3 more Hot Wheels cars”. And you have to remember that all while the rest of the story is being read.
As adults we take this level of cognitive processing for granted, but for children who are learning to read, this is not easy. And I’m not talking about understanding what the individual words mean, or whether English is your native language. Even if you understand individual words, comprehending a story IS a complex task that requires years of practice to master.
CGI to the rescue
So what can we do about this? Current Common Core math standards give lip service to the importance of teaching problem solving as part of math class. But teachers are left stranded with NO curriculum that actually teaches this.
Well, not quite. Several years ago I watched talented kindergarten teacher Allison Krogmann-Jordan teach her students basic mathematical problem solving using a method called Cognitively Guided Instruction (a bland name for a good technique). CGI is an excellent first step toward developing a method to teaching problem solving, that works for both students and teachers. It is starting be adopted by many schools, and centers around the country are helping to spread it.
CGI reminds me strongly of the excellent Reader’s and Writer’s Workshop programs for teaching language skills, and not coincidentally, CGI books are published by Heineman, the same company that publishes the Reader’s and Writer’s workshop books.
Here’s how it works
- A teacher presents a simple arithmetic problem in story form. Here are a couple typical problems at the K-1 level. Note that the second problem is a division problem, which isn’t usually taught till a later grade, but when it is presented in this form it is well within the reach of first graders to solve.
- All kids are asked to draw pictures that show their understanding of the problem.
- Members of the class take turns showing their drawings, and the class as a whole discusses and appreciates how different kids have drawn their understanding.
- Crucially, the discussion does not stop when one child gets the right answer.
CGI is based on several key ideas that run counter to common math education practices.
- Concepts are presented in multiple representations, including spoken and written language, hand-drawn diagrams, and sometimes physical manipulatives.
- Multiple ways of thinking about a problem are acknowledged and celebrated.
- Class discussion focuses on understanding the problem, and not exclusively on getting the right answer
- By having kids draw their understanding, the teacher and other kids can see how kids are understanding (or misunderstanding) the problem.
Taking CGI further
CGI is a decisive step in the right direction. But it is only the first step in a journey of a thousand miles. Here are the problems I see in CGI, and my proposed solutions.
Problem: Drawings are great, but they are more abstract than physical manipulatives.
Solution: Kids need to start by acting out math stories with physical objects or manipulatives before they draw diagrams. The #1 rule of manipulatives that I learned from manipulatives pioneer Mary Laycock is that understanding is built by first working with manipulatives, then diagrams, then language, then mathematical notation. You don’t really understand an idea unless you can climb both up and down the ladder of abstraction.
Problem: The cognitive load of understanding stories is still too high, especially for younger learners.
Solution: I suggest we scaffold the understanding of language by making manipulatives that represent language. Working with manipulatives that represent the objects being talked about in the story — Hot Wheels cars or seashells — definitely helps, but it is also possible to turn the words and sentences themselves into manipulatives, which again reduces cognitive load.
Problem: The problems are boring, artificial and out of context. If we taught English class this way, we would analyze individual sentences without ever reading whole stories, or even being aware that meaningful books exist. And it would be just as dull and meaningless as math class.
Solution: Tell good stories with real stakes that make kids want to know what happens next. The key is to remove all the scaffolding. Don’t tell kids in advance how to solve a problem. Let them struggle, and be intrigued. How? See this inspiring talked by Math Ed thought leader Dan Meyer: Math Class needs a makeover.
Problem: There is no room for creativity and self-expression.
Solution: Let kids make up problems, not just solve other people’s problems. This is the soapbox I proudly stand on. As a puzzle designer, I love asking new questions and inventing new problems. It’s a natural extension of what every child does, which is to ask lots of questions. Good questions, silly questions, hard questions, questions that stump adults. Lots and lots of questions. Just as kids in English class are encouraged to write original stories based on their own experiences, so kids in math class should be given the opportunity to create original math problems based on their own creative exploration. I realize this is a tall order for teachers trained to deliver rote math curriculum, but it is the necessary spark that will ignite genuine interest in problem solving.
In part 2 of this paper I’ll describe in detail games I’ve developed to teach problem solving, based on the ideas I’ve laid out in this paper.
