Math Instruction Shouldn't Kill Thinking
Mathematics has been stripped of its context and purpose
by Gerald Aungst
“How many of you would like to visit the Museum of Ice Cream?” I asked a group of fourth graders in Ms. Farnsworth’s class at Cheltenham Elementary School. Two dozen hands shot up and there was an immediate buzz of excitement. We had just watched a short promotional video for the pop-up museum in Los Angeles which highlighted its swimming pool filled with candy sprinkles.
But we weren’t planning a field trip to the west coast. “I wish I could go too. Unfortunately,” I told them, “Tickets are already sold out for the whole summer.” I paused for the predictable groan. “But did you notice that cool swimming pool? How many sprinkles do you think it takes to fill it up?” We spent the next couple of minutes making guesses and talking about how we might make better guesses about the number.
This was the start of a multi-day math lesson. It wasn’t going to look like a traditional math class. There would be no worksheets full of computation exercises. No bland and arbitrary word problems about buying seventeen candy bars for 61 cents each. (Where can you buy 61-cent candy bars? And why would anyone buy seventeen of them at once?) And definitely no teacher saying, “Just follow these steps exactly as I do and you’ll get the right answer.” All of those techniques just kill any thinking that students might have been doing.
The Purpose of Math
I recently asked a group of mathematically talented third graders why we learn math.
“So we can add and subtract.”
“So we’ll be ready for fourth grade.”
“To get good grades.”
I reframed the question: “But what’s the purpose? What do we use math for?” Silence and befuddlement. It was as if they hadn’t even considered there was any use for it beyond the endless exercises we put them through. Basically, to all of these students who were already very good at math class, the only point of math class was to keep getting better at math class.
Youth soccer is huge in suburban Philadelphia where I live. Every night of the week teams practice in any available open space, and some kids spend two straight days every weekend on the soccer field playing multiple games for multiple teams. They completely understand that the whole point of the weeknight practices is to perform on the field on Saturday and Sunday.
Imagine if we taught soccer the same way we teach math in K-12 schools. Kids would spend years practicing drills and exercises and artificially-constructed scenarios. And when we asked them the point of all of these experiences, they would say it’s to get better at the drills and exercises and scenarios. They wouldn’t even know that a sport called “soccer” exists, or if they did, they would perceive it as something only adults do and that most of them would never get to try.
In school we have stripped mathematics of its context and purpose and reduced it to arranging digits and signs in a specific way.
But mathematics is not about mindless and mechanical manipulation of symbols. Math is a beautiful language we use for solving problems and fostering innovative thinking, and students shouldn’t have to wait for years to experience its meaning and purpose. This isn’t just a matter of tweaks to the curriculum or substituting a few new teaching tricks, though. What we need is a different kind of classroom culture: one that promotes thinking instead of killing it.
5 Principles of the Modern Math Classroom
Creating a problem-solving orientation in the mathematics classroom comes down to five core principles that give students a framework for understanding the context and purpose of the math we teach: Conjecture, Communication, Collaboration, Chaos, and Celebration. Alone, each principle can greatly enhance student learning. Together, they form an integrated framework that can transform the way students think and learn.
If I’d written this article for Buzzfeed, it probably would have been titled “5 Jaw-Dropping Things That Will Make Kids Go Nuts for Math Class (#4 will absolutely BLOW YOUR MIND).” And if the link showed up in your Twitter feed, you probably would have clicked on it, even though you know it’s hyperbole and nothing in it is actually likely to drop your jaw, let alone blow your mind.
So why do even the most intelligent among us still click on clickbait? Because we’re curious. When we think there’s a gap in our knowledge, we want to fill it. When we encounter a mystery, we want to solve it. It’s built into our neurons: humans are wired to wonder. Curiosity is associated with dopamine and opioid production in the brain, which relate to motivation and pleasure.
Conjecture is about activating that curiosity. One simple strategy to promote Conjecture is to never end with the answer. A student’s answer to a question or problem should almost always be followed by another question. Here are a few questions that you can use this way:
- Why do you think so?
- How do you know?
- Are there any other ways to answer it?
- What was hard about solving that problem?
- What did you use to help you solve this? How did it help?
Mathematics is not just a method for doing computations. It’s the language of problem solving. And like learning a language, full understanding requires we use it in context. If you were taking a French class and all you did was learn vocabulary words and grammar rules but never had a conversation with someone, you’d be missing the whole point of learning French.
Our ability to think is dependent on our ability to use language. If we can’t use the language of math fluently, we can’t think mathematically. Students learning a new language also need to be able to translate back and forth. If you can’t explain a mathematical idea clearly in English, you don’t really understand the math very well. You may be able to flawlessly reproduce the steps in a complex calculation, but if you don’t know why they work, you won’t be able to tell when you’ve done something wrong. We see this in school all the time: students will take a rule they’ve learned, and they’ll use it incorrectly, or in the wrong place, and have no idea why they got an answer that doesn’t make sense.
To enhance students’ abilities to communicate mathematically, try a 3-stage strategy I call “Convince Me”:
Stage 1: Convince Yourself. First, students convince themselves: explaining their own reasoning in simple language. It doesn’t matter at this stage whether others understand it.
Stage 2: Convince a Friend. Students must use clearer language to communicate ideas to someone else. A friend will still give you the benefit of the doubt, though, and convincing them may not take a lot of work.
Stage 2.5: Convince a Mathematical Friend. A mathematical friend will start to insist on accurate terminology and precision. As students become more fluent in their communication, this part will become more automatic and won’t need to be a separate stage.
Stage 3: Convince an Enemy. Students who are very confident and fluent will be able to present a mathematical argument to someone who actively seeks to find flaws in logic and pick apart their reasoning.
Ultimately, students should be able to explain someone else’s work to an enemy!
Think about the last time you had to solve a complex problem. Did you do it entirely on your own? Or did you get help from others? (Hint: a Google search counts as getting help.) Real-world problem solving generally requires a team approach.
Problem solving in math class, on the other hand, is typically a solo activity. In fact, more often than not when students seek to collaborate we call it “cheating.” It will take time to teach our kids that working together on math problems is not only permitted but is preferable.
One way to promote collaboration is to flip the traditional script. Traditional skill instruction looks like this:
- I do. Teacher models an algorithm.
- We do. Everyone replicates the algorithm together under guidance.
- You do. Students practice independently to mastery.
Magdalene Lampert reverses this by giving students a problem to attack without first modeling any strategy. They then follow this procedure:
- You do. Students work on the problem alone.
- Y’all do. Students work further on it in small groups.
- We do. The whole class works collectively. When necessary, the teacher then introduces the standard algorithm or strategy which now makes sense in context.
This principle often prompts strong reactions. To be clear, I am not talking about the kind of chaos that reigned during my first attempt at teaching science to first graders. That did not end well.
Chaos in the math classroom refers instead to the problem solving process. In the real world, problem solving is messy and complicated. We take far too much of this away when we give students clean, neat exercises to work.
To leverage this principle, give students math problems engineered to be disorganized and unpredictable. Use scenarios like the one I used with Ms. Farnsworth’s fourth graders. Put them in situations and ask questions that require them to make guesses and justify their reasoning instead of just running through a series of thought-numbing algorithms:
- OK now that we’ve estimated how many candy sprinkles would fill the pool, imagine we used them all to make some banana splits.
- How much ice cream would we need?
- How long would it take to make them? Or eat them?
Math class should involve lots of celebration. Not the “everyone is great and gets a trophy we gloss over mistakes” kind of celebration, though. In fact, we need to celebrate effort, even when it results in mistakes.
Jessica, age 9, provides this wise perspective: “Mistakes lead to good places so if you make a mistake take it as a step up the learning ladder.”
One way to help students see the value of mistakes is to model it. Plan deliberate mistakes into lessons and activities and encourage students to catch them and correct them. Build a culture where pointing out and correcting mistakes is done respectfully, and students perceive it as feedback and help. Celebrate whenever anyone makes progress, even if it’s small. Make a big deal out of any learning.
From Principles to Practice
According to two researchers at Harvard and MIT, there are only three things humans can still do better than computers:
- Unstructured problem solving: knowing how to attack a problem where the rules and algorithms don’t yet exist.
- Acquiring and processing information: deciding what to pay attention to and what to filter out, for example.
- Non-routine physical work. We have robots that build cars and land on other planets. But no one has yet designed one that can give a manicure or do landscaping.
Anya Kamenetz adds a fourth:
- Being human: expressing and engaging with emotions, empathy, creativity, and humor.
How much of our curriculum, particularly in math, is focused on strengthening these things instead of the routine computations at which computers excel? Focusing on the five principles described here can go a long way towards reorienting our curriculum in this direction.
While you can begin applying the principles by implementing single strategies, it ultimately requires a shift of mindset on the part of the teacher. When the five principles are fully integrated into classroom practice in a deep way, student thinking and learning become the focus, and we will shift from preparing students for the next grade towards preparing them to be the innovative thinkers and problem solvers of the future.