I Got it Right, So Why Do I Have to Show You What I Did?

This was, I thought, a fairly legitimate response to a prompt for students to "show their working", which is rapidly becoming an essential component of math assessment for students at all levels. My student did indeed draw a picture to show what she did. But she had no sense of explaining how she arrived at her answer, nor could she visually represent how she thought about the problem. She was so used to the standard algorithm, that she simply provided me with a beautiful illustration of her doing just that. Artistry aside, it became very clear that I needed to do a much better job of teaching my students to record their thinking. I thoroughly support the principle of encouraging students to explain their reasoning, but I am not convinced that math curriculum as it stands, or the culture of mathematics classrooms, equip students with the skills to do this.

I have several reasons for thinking this. So often it is the case that students are given no room to write down their thinking, no blank space to work problem through on paper, and so they find it unnatural when they are asked to do so. I believe that this is very well demonstrated by Tracy Zager, author of the upcoming book "Becoming the Math Teacher You Wish You'd Had," in her blog post "An ode to blank paper". She includes the following workbook page, brought home by her daughter, lamenting the fact that "students have no room to work, but the publisher left plenty of room for cutesie drawings." And rightly so! This multiple choice format, with large, many-limbed bugs obscuring any space to record thinking, suggests somehow that there is no place for working through the problem on paper. Forget partial sums. This page, to me, communicates to students must be able to compute this in their heads. Just look at the smug smiles on that grasshopper and ladybug!

Providing blank space to think through a problem is clearly not enough, and this is illustrated nicely by the following gem of an answer that I found on Reddit.

Students have to believe in the efficacy of explaining their thinking, rather than seeing it as a chore. A self-professed math-hater in my student days, I can clearly remember saving any "show your work" questions until last, when I would begrudgingly provide some sort of calculaton that may or may not have reflected what I actually did. As far as I was concerned, if I'd already got the right answer, that was pretty bloody impressive. What did it matter to anyone how I had got there?

With this memory in mind, I was particularly taken with David Ginsburg's blog post for Education Week, in which he advocates for giving students the answers and having them explain how this answer was achieved. Mixing it up, giving students the answers, and saying "prove to me that this is right" seems to me to be a great way to engage students. I love this idea because it gives students a purpose for showing their work, and also lays a foundation for mathematical debate in the classroom. When students become accustomed to having mathematical conversations, I believe that explaining how they arrived at a particular answer will become natural and even, dare I say it, useful to them.

I will freely admit that I feel more at ease teaching language arts, but this does not mean that I don't relish the challenge of math! That said, I am inclined to approach math instruction from a literary angle. As such, I want to teach my students, as I do in writing, to "show" not "tell" me how they solved a problem. Donald Murray, author of “A Writer Teaches Writing," wrote the following,

“You may tell us it was hot. Or you may show us by writing” My shirt stuck to my back. Sweat ran into my glasses.” In the first sentence, we had to take your word for the heat. In the other two sentences, we were there. We too have the same stickings and sweatings. Showing makes it possible for the reader to identify with the writer.”

I feel that this is equally relevant to math. If you tell me that the answer is 26, I have to take your word for it. But if you show me how you got there, I can identify with you, share in your thought process, and perhaps even learn a new approach to the problem. Showing your working is the window into your mind that lets others connect. It seems more natural to do so with writing, but I think that this comparison might help students to make sense of its importance in mathematics.

Tracy Zager wrote a highly entertaining and thought-provoking blog about an experiment that her colleage, Debbie Nichols carried out in which she asked her first and second graders the following question: “There are 25 sheep and 5 dogs in the flock. How old is the shepherd?” Concerned about vocabulary, and added a second question, “There are 25 kids and 5 dogs in the classroom. How old is the painter?” The range of responses included the following:

"30, because I added them up."

"25. You said 25 sheep."

"8. I counted by 9s."

And my personal favorite:

"69? I got it out of my brain and my brain is made of pink worms."

These responses all suggest that students believe that all math problems have to be answered with a number, that it is normal for problems to make no sense, and that every problem is solvable. And this is exactly why when asked how she arrived at a particular answer, my student simply wrote out the standard algorithm for multiple-digit addition for a second time. Even a how question, to her, simply because it was a math question, meant that answer had to be in numbers. Words had no place in her conception of answering a math problem. And this is exactly why it is so important talk about math.

My favorite way to do this is to use a method coined by fellow blogger Brian Stockus called "Numberless Word Problems" such as the following,

"Some girls entered a school art competition. Fewer boys than girls entered the competition.

Automatically, students are compelled to consider the situation, and to find the math in the problem for themselves, rather than pulling out the numbers somewhat mindlessly, adding, multiplying, or even "counting by nines" . Just as literacy is making meaning from print, so too should math be, and yet somehow this often gets lost. Numbers are as ubiquitous as the written word, and so math conversations can be created out of virtualy anything. I am relatively new to the twittersphere, but from some cursory dabbling, I discovered the wonderful hashtag: #tmwyk - Talking Math With Your Kids! If you have any doubts about how to weave math into conversations about everything from hot chocolate to Dora the Explorer underpants, you won't after you give this a read!

The main takeaway for me was to subtly weave numbers into general conversation- I recently snuck some steeplechase math into my classroom. I told students that the race was seven and a half laps, with 4 barriers and a water pit each lap. And then I posed the question: "how many barriers does Miss Mel have to jump over in a steeplechase race?" before promptly dismissing the class for recess. It's pretty cool to hear your kids debating a math problem as they wait in line for the monkey bars.

It is a delicate balance that must be maintained between teaching students to value process and make sure that they have memorized the number facts to implement procedures effectively. Steven Strogatz, a professor of Mathematics at Cornell, uses the analogy of teaching music,explaining, "you need to have technique before you can create a composition of your own. But if all we do is teach technique, no one will want to play music at all." I love this comparison, because I believe that music is the perfect combination of mathematics and artistic expression, but without basic knowledge, be it of numbers or notes, or of foundational skills in either music or math, students cannot be creative.

I know that this is a contentious issue, but as far as I am concerned creativity very much has a place in the math classroom. I want to reward student-generated strategies, regardless of whether they are more time-consuming or involve more steps than the standard algorithm. Students can learn so much from finding their own way to solve a problem! But to give students a chance to show creativity, you need to ask the right questions. SanGiovanni and O'Connell, authors of the great book "Putting the Practices into Action" call these "rich" math problems, giving the example of the "poor" 4 x 5 = versus the rich "how many hours of TV would you have watched if you live to be one hundred years old?"

This second problem requires students to perform mutliple calculations, to estimate, and to really think, plus it is embedded in a real-world context. Better still, it invites a wide range of possible strategies for solving it, and, best of all, there is no one right answer! Students can debate, discuss and collaborate, learning from the ways that their peers approached and solved to problem, and gaining skills in "showing work" by explaining their own reasoning to others- double win! As such, I can deemphasize speed, focus on only a few problems per lesson, and reward students for innovative approaches, rather than problems completed, or number of correct anwers.

Not everyone agrees. I was somewhat shocked to read the following from John Hopkins Math Professor, W. Stephen Wilson, in an interview for Education Next about the Common Core State Math Standards,

"There will always be people who think that you must be able to solve problems in multiple ways. This is probably similar to thinking that it is important to teach creativity in mathematics in elementary school, as if such a thing were possible. Forget creativity; the truly rare student is the one who can solve straightforward problems in a straightforward way".

I feel that the word :straightforward" has little place in the reality of the elementary classroom. Children are not "straightforward" people, and that is what makes them so wonderful. They approach problems from all different angles, they ask questions that I would never have thought of, and they are innately creative. I believe that we should teach in a way that is repsonsive to these assets. And this is why I want to make sure that my students are consistently engaged in mathematical discussions and discoveries. I think students should have room to play with manipulatives, explore shapes, and choose their favorite representation of a particular concept or problem. And that is why I am so taken with Steven Strogatz's idea, tried and tested by Jessica Lahey in her piece for The Atlantic, "Teaching Math to People Who Think They Hate It", in which she challenged my son and husband to create their own scalene triangle origami. The premise is simple: give students a scalene triangle, and ask them to fold it into two equal halves. And, with that, students will set about making their own math, discovering for themselves the properties of this triangle. She calls it "making math together", and I love this idea of a discovery-based learning approach to math, redefining the subject as one in which creativity and collaboration are integral.

There will be those who object, but know that my classroom is unabashedly full of scalene triangle origami, geometric shape turkeys and good serving of math chat.