In discussions of progressive and constructivist teaching practices, math is often the odd subject out. Teachers and schools that are capable of creating real-world, contextualized, project-based learning activities in every other area of school often struggle to do the same for mathematics, even as prospective employers and universities put more emphasis on its importance.

This struggle may come from a fundamental misunderstanding about the discipline and how it should be taught.

That’s the stance David Wees has arrived at after more than 20 years of teaching at many different kinds of schools all over the world. It has taken a long time, but Wees has stopped labeling student work with the word “mistake” and has started paying attention to what he can learn about how students are thinking, based on the work (right or wrong) they produce.

“I want to know the ways that they are thinking rather than the ways they are making mistakes,” said Wees, who now works as a formative assessment specialist in mathematics for New Visions for Public Schools, an organization supporting public school teachers in New York City. “My interpretation that they’re making a mistake is a judgment and usually ends my thinking about what they are doing.”

In that situation, it’s extremely tempting to tell the student where he or she went “wrong” and move on. But what does the student learn in that scenario? Not much, beyond how to memorize computational formulas, said Wees.

“My goal is for them to become the truthmakers,” Wees said. “I’m trying to build a mathematical community where something is true when everyone agrees it’s true.” To do that, he asks students to talk through mathematical ideas, struggle with them and give one another feedback. “A major goal of math classrooms should be to develop people who look for evidence and try to prove that things are true or not true,” Wees said. “You can do that at any age”

Fundamentally, Wees wants to increase the amount of thinking “at the edge of their knowledge” that students do. “There’s lots of evidence that what we think about is what we know later,” he said. “I want to increase the amount of thinking going on in math class.”

Wees points out that while practice is important, students are repeating an action with which they are at least a little familiar.

He wants students to struggle in the zone of proximal development, where they don’t quite understand yet but aren’t frustrated. When working in New York public schools, Wees found if he gave students problems to solve that allowed for different points of entry, all students could struggle together. One student might be more advanced than another, but if each could access some element of the problem, they discussed it together and either relearned core concepts or were exposed to more advanced ones.

For example, Wees asked his students to solve the Seven Bridges of Konigsberg problem. It goes like this: A river flows through the middle of Konigsberg, forming an island in the middle and then separating into two branches. The citizens of Konigsberg have built seven bridges to get from place to place. The people wondered if they could walk around the city in such a way that they would cross each bridge once and only once.

Konigsberg
Visualization of the Seven Bridges of Konigsberg problem. (Math Forum)

“The kids understood the problem and virtually all attacked it,” Wees said. “Some kids worked on it for weeks.” Wees posted it in the hallway and at one point almost all the ninth-graders were working on the problem. Students got tired of carefully drawing the bridges, river and city over and over, so they naturally began to abstract the map into something that looked like a graph.

No student solved the problem — in fact, the mathematician Leonhard Euler proved it was impossible. Wees showed his students Euler’s proof, and pointed out how similar their graphing was to his. Wees said kids were a little mad when they discovered there was no answer, but they enjoyed the experience and along the way realized that learning is about the process.

“Over time I tended to embed projects of various kinds because at the time I was thinking I needed to get them interested,” Wees said. “They weren’t interested directly in the mathematics itself because they’d experienced so much failure, so I was trying to get them excited.”

Slowly throughout his career, Wees began to see that projects could be more than just excitement builders — they could be the vehicle for teaching content and the assessment. And the range of mathematical ideas was much broader than he thought if he used his imagination.

“The range of mathematical ideas the kids struggled with were pretty wide,” Wees said. After working in inner-city schools, Canadian schools and international schools for expat kids in London and Bangkok, Wees has come to the conclusion that all kids make the same kinds of mistakes.

“It was clear to me that the mistakes in some cases were a function of the mathematics and the way kids think about the math, rather than whether the kid is rich or poor,” he said.

MATHEMATICIAN’S LAMENT

Over the course of his career, through trial and error, Wees came to see what Paul Lockhart describes in his essay, “The Mathematician’s Lament”:

By concentrating on what, and leaving out why, mathematics is reduced to an empty shell. The art is not in the “truth” but in the explanation, the argument. It is the argument itself which gives the truth its context, and determines what is really being said and meant. Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity— to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs— you deny them mathematics itself. So no, I’m not complaining about the presence of facts and formulas in our mathematics classes, I’m complaining about the lack of mathematics in our mathematics classes.

KIDS ASK THREE KINDS OF QUESTIONS

When doing his master’s in education technology and the pedagogy around it, Wees learned to categorize the three kinds of questions students ask and changed his teaching practice entirely. Kids ask questions: 1) to find out if they did the problem right; 2) because the teacher is standing near them and they can, and; 3) occasionally they ask “I wonder what if” questions, which show they are thinking about the math. Wees took to not answering the first two kinds of questions and encouraging the third.

“I went from really trying to answer questions and support them in that way, to really trying to think of questions that would support them to learn it themselves,” Wees said. He found himself often asking the same question, whether a student had gotten the problem right or wrong. He’d ask them to explain their answer or how they could check to see if they were right or wrong.

“I became better at having a poker face so I wasn’t communicating whether they were right or wrong,” Wees laughed. When students asked questions because he was nearby, he deferred them to their peers, who often explained the math quite well.

THE TIME FACTOR

Many math teachers will say a community of learners like Wees describes is a fairytale classroom with no time constraints and no standards to cover. They say their jobs depend on covering all the topics on the test and helping students correct their errors, not taking days to uncover the thinking behind that error. Wees acknowledges the limitations that many math teachers struggle with, but points out the way most people teach math now doesn’t work, so it could be considered a waste of time anyway.

“Whatever time people are putting in to teach mathematics is kind of wasted in many cases,” Wees said. “Are [students] learning anything that they can transfer, that they can use in other contexts? If they’re not doing these things, then I don’t know what they’ve learned.”

He points out students often did very well on the New York Regents test when teachers focused on teaching specific kinds of problems, but whether kids learned the full range of mathematics possible that year is another thing entirely.

Beyond time limitations, a broader problem is that many math teachers know only one way to solve the problems they teach. Even professional development often focuses on breadth instead of depth, with the result that many teachers carry the same fundamental gaps in math understanding as their students.

“We have generations of math-phobia,” said Laura Thomas, director of the Antioch Center for School Renewal. “A lot of teachers who teach math are second- and third-generation math-phobic, so our system is really calculation-based as opposed to applying in context.”

Thomas said it takes a person with deep understanding of both math and project-based pedagogy and coaching to effectively lead students through what is often a very messy process requiring students to use problem-solving skills to figure out solutions, rather than being told what skills to apply.

Wees is frustrated at how linear math learning has become. “The standards are a list of things the kids are supposed to do, not a list of things you have to teach,” Wees said.

In other words, many standards can be embedded in a problem so that students are exposed to lots of ideas in different ways. When teachers focus on clusters of standards as opposed to individual ones, “that kid who doesn’t get one idea on Thursday is going to get 10 or 12 other ways of looking at the idea in the unit,” Wees said.

For example, a teacher might give students this math problem: “I’m traveling 50 mph. How far will I have driven in 10 minutes?” This problem does not confuse students. They know what they are being asked and in discussing it they could hit many standards — multiplication, number lines, writing down possible solutions to think it through and fractions, to name a few.

“The kids get exposed to all of the standards every day in different ways,” Wees said. And more importantly, they’re having to think through the standards every day, leading to a deeper level of learning.

“You really have to understand math is a range of ideas and not individual standards,” Wees said.

When teachers are comfortable teaching in this more complex style, they are able to offer the multiple points of entry that allow for differentiation to take place — but in community, not isolation. If students are segmented out to learn only with the students “at their level,” some students will be in danger of never moving past fractions.

Seeing Struggling Math Learners as ‘Sense Makers,’ Not ‘Mistake Makers’ 4 August,2015Katrina Schwartz

  • Making it concrete with applications and projects. Encouraging kids to explain what they’re doing. And stimulating deeper thought about fewer problems. Really nice approach to teaching math.

  • Molly Prince

    Making it relevant to their lives, interests, even passions. All of us, including our students, will invest our mental energy where we see it matters most.

  • The Hoffs

    Watching and listening to how students “see” problems rather than pushing them through a process has definitely given me a deeper understanding of math and helped me be a better teacher, not only in school, but in life. Exploring different ways to approach a problem allows all students to be engaged and isn’t the ultimate goal to teach children to be engaged as adults?

  • I have always called myself a “Why Child” and until it is explained in the terms I fully understand,….I will keep asking. As an adult, one of my biggest pet peeves is to not be acknowledged. As an undiagnosed dyslexic, I suffered immense humiliation and belittlement. I always will ask and tell you what I need. So ‘nonlogical or nonsensical” answers will get me asking you in many different ways to try and find the way I will understand. I never made this connection until reading this article. Thank you!

  • Tamera Morrison

    Just reading this fills me with anxiety. I’ve never been able to think in groups. I need quiet.

    • I’m sorry you feel anxious after reading this. I had lots of students with similar needs as yours. I did my best not to force them to work in groups and to keep the overall volume of the room down. Also, everyone needs some quiet think time, so I always included time for independent work.

  • Kim Blackhurst

    As a student teacher it is great to keep these ideas in mind when observing and teaching Math classes. I believe it is wonderful how children approach problems from different perspectives and I feel privileged to be able to watch and help them develop their confidence and understanding of Mathematics. Focus on the possibilities and not the mistakes… yes!

  • Simmons Mom

    Great article. My 5th grade daughter has always struggled in math. I think she is a visual learner and that there is a disconnect between how she learns and how it is taught. I loved your suggestion to have them teach you.

    After years of frustration trying to memorize math facts and falling behind her classmates she often bursts into tears trying to do her math. Can you give me any more suggestions on how I can help her feel more confident doing math? Or how to make those math facts stick?

    • I don’t know your daughter at all, so take these suggestions with a grain of salt.

      Some options include:

      1. JUMP Math produces a set of guided discovery curriculum that your daughter may appreciate. In particular it offers carefully sequenced ideas that may help her see the steps between more clearly than with other less explicit curriculum. It also allows for carefully structured practice over time so as to support your daughter in remembering what she is learning.

      2. Constructing arrays that represent the multiplication to help your daughter see how the idea of multiplication is related to area. Making patterns with these arrays is something my son enjoys a lot.

      3. Take the entire multiplication table and look for patterns within it that your daughter sees and can remember so that when she forgets an individual fact, she has a way to reconstruct it from the patterns. Ideally over time she needs the patterns less and less (since having to recall the patterns to construct the fact when she needs it increases her cognitive load, making it harder for her to solve problems using the fact) but her confidence may improve once she doesn’t feel as helpless every time she is faced with recalling a fact she doesn’t know.

      4. Playing games which repeatedly remind your daughter of some multiplication facts as part of game-play. For example, when my son was 5 he had his 25 times table memorized from 1 x 25 to 10 x 25 because of Plants versus Zombies. He knew very quickly how many suns it took to buy anything because of the repetition of the game although he did not use the formal language associated with multiplication.

    • Rene Kiser

      With my daughter, who just started 6th grade, I have tried to create fun and interesting ways to implement math into “everyday life” with her. I take her to the store with me and have her estimate the bill. “If this item is $3, and we are buying 7 of them, how much should our bill be?” When she is doing her homework and seems stuck, I try to use everyday examples – Sally used 1/4 of the 300 seeds she had in the garden. How many did she use? I will ask my daughter, “how much is one quarter of a dollar? And how many pennies are in a dollar? So what do you think you would need to do to get to 300?” These are things she is familiar with so it’s easier for her to make the connections without feeling so overwhelmed. Once she makes it once, her level of self-confidence begins to improve. After one or two problems, I can usually have her work at a few on her own and then show me what she did.

  • Ari L

    Great article, it’s definitely some things I’ve noticed in my own experience

    Which brings me to wonder, is the current generation of people like myself still have a chance to learn math in the way discussed in the article?

    I’d like to begin this approach on my own studies. Id like to have a love for math, but I’m just not sure how. Any ideas? Maybe even books/courses?

    • I’d check out James Tanton’s courses (http://gdaymath.com/courses/), ideally in a small group (unless that’s really not your thing). It’s not the same experience as working with a teacher but if you are an adult and really interested, you’ll get a lot out of it. Note that James’ work includes a fair number of explanations but also a fair amount of opportunity to explore ideas. It’s also available online, for free.

  • SSINTENSE

    Math IS logic. I think application is the best way to really learn and understand it.

    As Ben Franklin once said, if you teach someone, they will remember. If you involve someone, they will learn.

  • Krista

    This is great. I see a lot of similarities with Montessori education, which follows the principle of “teach by teaching, not by correcting” and teaches an understanding of concepts before memorization and also seeks to inspire questions and then give kids the tools to find the answers.

  • Fabiana Jarrin

    Yes Maria Montessori realized that 100 years ago !!!

  • Deana

    From my own experience with math, I can agree that it’s important to try and discern where I’m thinking differently, so I can reach the right answer, and I have, in the past been very frustrated when my “why” questions were not answered. If I can’t understand why thinking about it one way instead of another gets me the wrong answer I will never consistently do it right. My father was good about trying to help me see why what I we doing wouldn’t work, and that was incredibly helpful.
    Having said that, I had a teacher in college who tried to teach geometry in a class setting using this kind of problem solving method, where he refused to answer questions directly and wouldn’t tell us where we went wrong. I have never done so poorly or been so immensely frustrated with a math class that I had wanted to do well in. The lack of clear, linear teaching made it impossible for me to work with the concepts creatively. So I would be wary of seeing this method as some key to saving future math students.

    • Not answering questions directly is not the same as not giving feedback or advice at all! I’m sorry that you had a negative experience with math. That must have been super frustrating.

      I’m clear that that the end of each lesson, students should know what they have learned and what the objective of that day was, even if the approach to reaching it is different.

  • Mark Burke

    I always try and ask two things ; why does it work, and how can we use it ? It seems to make a big difference to engagement and understanding.

  • Bernard Bagnall

    I started teaching in the 1960’s and very quickly for into an investigational approach in mathematics lessons as well as other subjects (being a primary school teacher) After many years of teaching in that style I became an advisory teacher going into schools to work in situations where the teachers wanted to take on a more investigative approach. I endorse as powerfully as possible what is said in this article. You may have to be a bit of a risk-taker but the benefits for the learners are great. I was so pleased that when my 11 year olds had to do the first Sats test (UK) they came back into the classroom and when I asked how they got on their response was ” we knew we’d be able to answer the questions somehow” and they did so well in the tests. I then moved into teacher training in Cambridge (UK) and started writing for Nrich (Nrich.maths.org.uk) and in the last 10 years have continued to work totally for Nrich which has a very similar approach to this article in very many of the 10000 activities on the site. Thank you so much for this article and I hope it inspires as well as tempts teachers to see and experience it’s great worth. Thank you David Wees may these words spread far and wide.

  • chaos123

    “My goal is for them to become the truthmakers,” Wees said. “I’m trying to build a mathematical community where something is true when everyone agrees it’s true.” To do that, he asks students to talk through mathematical ideas, struggle with them and give one another feedback. “A major goal of math classrooms should be to develop people who look for evidence and try to prove that things are true or not true,” Wees said. “You can do that at any age”

    So he turns maths into people-based consensus!

    Not a good start to the children’s maths career …

    • Why do you think children think most things in math class are true? And related … how do mathematicians know that anything they have discovered is true? What do you think the purpose of proof is?

Author

Katrina Schwartz

Katrina Schwartz is a journalist based in San Francisco. She's worked at KPCC public radio in LA and has reported on air and online for KQED since 2010. She's a staff writer for KQED's education blog MindShift.

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