Math is supposed to be cut and dried, right? Do the work, find the one correct answer, and… you're done.
We must retire this way of thinking. The Common Core State Standards (CCSS) in mathematics call for shifts in depth, focus, and rigor.
In my last Tch Blog post, I examined the first CCSS practice standard. Today I'd like to take a look at practice standard three, which calls on students in grades K-12 to "construct viable arguments and critique the reasoning of others."
Here are three ways to bring argument construction and critique into your classroom, strategies that will lead to deeper understanding of key math concepts:
1. Be wrong more often.
Consider intentionally presenting incorrect solutions to students and asking them to analyze the errors. Ask questions such as: "What did I do wrong here?" or "How could I have come to this answer?" or "Does this answer make sense?" This can lead to rich discussion and deeper thinking.
When you do this, you're teaching students to be mathematical critics. This widens their focus from only the answer to the process as well. In turn, it embeds the habit of paying close attention to work and learning from mistakes.
Errors are a part of math. Everyone makes them…even teachers! Practicing error analysis in this manner can help embed the importance of mathematical critique and free students from the pressure of finding the right answer.
Here's a problem that was recently giving trouble to a group of fifth graders I was working with: "Mr. Jones bakes 6 loaves of bread. He cuts each loaf into equal slices that are ¼ of a loaf each. How many slices of bread does he have?"
In solving this problem for them on the board, I intentionally came up with an incorrect answer of 6/4 (six-fourths), an answer that was showing up on a lot of students' papers.
After seeing my answer and thinking about it for a moment, students discovered that an answer of 6/4 slices simply doesn't make sense in this scenario.
Together we reached the conclusion that I had multiplied when I should have divided. This turned into a great opportunity to stress the importance of thinking about whether your answer makes sense or not: a key, yet often overlooked, component of the problem solving process.
2. Examine student work.
One way I have attempted to build problem solving skills is through the use of Problems of the Week (POWs), weekly multi-step problems that push students to work through challenging mathematical situations.
After we complete a POW, I project a few student examples with my document camera—some with incorrect solutions, some with correct solutions. Together, we take a close look at the students' processes.
We zoom in on each step and ask questions like "Why did she choose addition?" or "What might he have been thinking here?" The class attempts to get inside the mind of the problem solver.
This may seem like a potentially uncomfortable proposition. However, by making many public errors of my own, I've built a classroom culture in which mistakes are embraced as learning opportunities.
So it's just natural to ask, "What could have solved this problem in fewer steps?" or "Where did he go wrong—and why?" Students embrace opportunities to see how others approached the problem and to examine their own and others' thinking processes together. There's no stigma.
3. Ask "Why?"
How often do we take the time to ask students why they chose an operation or a strategy? Doing so can prompt them to "construct viable arguments." Let's turn to part two of the bread-baking question:
"Mr. Jones sells 2/3 of the slices he made. How many slices are left?"
This was a little tricky for my students. Many answered "16," the number of slices sold, rather than the number left. But when I asked them to defend their thinking, to prove to me that their answer was correct, the error became clear.
We've all seen students rush through a problem and make a careless mistake. We can help change the way they think about their work if we slow them down more often and ask them, "Why?"
Consistent questioning leads to better arguments and reasoning, as well as a more thoughtful approach to their work. Math, after all, is about thinking, not just about right and wrong.
Common Core implementation calls upon us to teach math students to look critically at their work and the work of others… just as closely as we ask them to look at a piece of literature or a poem.
If we can teach them to pay attention to detail, and to analyze and critique the steps they take to solve a problem, we will boost their work in all areas of mathematics.
Center for Teaching Quality is writing a series of blogs in partnership with Teaching Channel. CTQ is transforming the teaching profession through the bold ideas and expert practices of teacher leaders.
Ben Curran is a K-5 instructional coach at a charter school in Detroit, Michigan. He is also co-founder of Engaging Educators, co-author of Learning in the 21st Century and a member of the Center for Teaching Quality’s Collaboratory. Follow him on Twitter @engaginged.