**By Zachary
Champagne, posted June 5, 2017 —
**

Recently, I’ve been thinking about and sharing some of my
ideas on becoming a better
listener. In a profession where we are envisioned as sharers of
information, it was tough for me to swallow that I should be listening more and
talking less. However, I’ve come to believe that it’s the single most important
thing we can learn to do better as teachers.

One of the ideas I’ve been exploring is how providing less
structure can allow us to elicit more student thinking from our students’
written work. (Full disclosure: I was also recently inspired by my friend and
colleague Brian Bushart and his NCTM Ignite talk “Do Less, Get More”).

Consider this
problem that Mike Flynn
wrote for *Teaching Children Mathematics *(“Problem
Solvers: Problem: The Cycling Shop,” vol. 23, no. 1, August 2016, pp. 10–13):

Imagine you work at a cycling shop building unicycles,
bicycles, and tricycles for customers. One day, you receive a shipment of 8
wheels. Presuming that each cycle uses the same type and size of wheel, what
are all the combinations of cycles you can make using all 8 wheels?

This problem is rich with various entry points for students.
They can start in a variety of ways. Additionally, and perhaps more important, the
problem is also rich in the variety of structures that different students may
rely on to keep track of their work. I’ve seen students use pictures, charts,
tables, tally marks, equations, and more when they begin this problem.

But it is only when we provide students with the problem as written above, and nothing else other than a pencil, that we can begin to see the power of the blank page. As teachers, we easily slip into trying to be too helpful. With good intentions, we take a problem like the Cycling Shop problem, and many times we give students something like this table:

In my opinion, providing this structure before students get
started limits what we can find out about how our students are mathematizing
this situation. We are stuck with learning if they can accurately fill out a
table that we created. I think we are better suited to *first* let students be the problem solvers we know they can be, by
letting them solve this problem using whatever structure (if any) makes sense
to them! If students are struggling to get started or make sense of the
problem, then we can help them think of ways that they might use structure to
solve the problem. This “less is more” strategy provides us with much more
information about students and does not require them to use a structure that we
created.

I encourage you to respect the power of the blank page and
give students *more* space to solve
problems and *less* structure in the
beginning.

**Zachary Champagne**
is an assistant in research at the Florida Center for Research in Science,
Technology, Engineering, and Mathematics (FCR-STEM) at Florida State
University. He previously taught for thirteen years as an elementary school
teacher with a specialization in math and science. During this time, he
received many state and national awards for excellence in teaching,
including the Presidential Award for Excellence in Mathematics and Science
Teaching (PAEMST), Duval County Teacher of the Year, and Finalist for Macy’s
Florida Teacher of the Year. Zak is the current president of the Florida
Council of Teachers of Mathematics (FCTM) and is currently interested in
learning how young students think about mathematics and how to help them
understand that mathematics makes sense. He tweets at @zakchamp.

## Leave Comment

## All Comments

Sabrina Kuss- 9/2/2019 4:19:03 PMFenster, Wintergarten und Türen nach Maß, direkt ab Werk, passgenau und ohne Umwege

Fenster Kaufen

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Mark Ludwig- 8/30/2019 7:55:00 AMBusy z Polski do Anglii

przewóz osób w systemie door to door

Busy do Anglii

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Carolina DeLucio- 11/3/2017 8:47:40 PMHi, I am Carolina. I am a college student observing my mentor teacher every Wednesdays this Fall semester. I am going to use this problem for my second-grade class for my math observation lesson plan. I want my students to use collaboration, strategical thinking, and pictures, models, and/or manipulatives to solve the problem. I decided to write the problem as:

“Imagine you work at a pet shop feeding lizards, frogs, and spiders. One day, you receive a shipment of 12 flies. Presuming that each lizard eats 3 flies, each frog eats 2 flies, and each spider eats 1 fly, what are all the combinations of feeding the lizards, frogs, and spiders you can make using all 12 flies?”I think my students will be challenged because as of now the students do not know their multiplication. The challenge would be to visualize and calculate to form a number sentence. I believe my students will use addition and subtraction. I am excited to see all the strategies used. Thank you.## Reply processing please wait...

Amy Zimmer- 7/17/2017 11:48:59 PMIf Zak, you are saying to offer less structural prompts, I am with you. As a high school math teacher, I get so much push back from my colleagues, the most often I hear is about time. "I don't want the students spending all their time creating a table or creating the graph." So the students get scaffolding upon scaffolding. And they wait until you start because they know they will wear you down. I don't want to be mean, or unhelpful, and I don't throw the students under the bus with NO skill set. I do want them to do something, anything, ON THEIR OWN. A way that I get more is to have students start with VNPS ala Alex Overwijk, http://slamdunkmath.blogspot.com/

There is something wonderul that every student group contributes whether it is spot on or off the ramp. I admit that a lot of my students hate the lack of structure I sometimes offer, and I believe someone has to offer this space.

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Zachary Champagne- 7/9/2017 11:49:05 PMHi everyone,

Thanks for the feedback! I'm so glad to know these ideas are resonating and starting some discussions. I want to start by saying that I truly believe in the power that these tasks hold and that it's important to not overly-scaffold for students from the beginning. On a related note, my friend and colleague, Andrew Gael, wrote some ideas about open problems and special education students on this very blog. Check them out here and here. And as I wrote in the post, "I think we are better suited to

let students be the problem solvers we know they can be, by letting them solve this problem using whatever structure (if any) makes sense to them." But, I want to be very clear here. When our students struggle to get started, or need some support, it is our responsibility as educators to provide those supports. Robert hit the nail on the head, this is about formative assessment. We won't know what our kids already know if we give them the support and scaffolds before they show that they need them. *And thanks for sharing this on facebook and getting some conversation started over there - I too hope some more of it will come this way.firstBut all of that brings up the THE point. Which only partially addresses Kim's poignant thoughts. This work can't be done in a classroom where the norms and expectations are not set to support this work. It's a completely separate post (that others are way more suited to write than me) on how to set up classrooms where we truly allow for problem solvers and not problem performers. This work can't happen instantly in a classroom where students only are challenged to solve sets of problems that are all similar in nature. You know the page I'm talking about. Those students will need lots of experiences being problem solvers before they can move into this kind of work.

Now, to try and address Kim's more larger (very valid) concerns. I'm not convinced that these types of problems, including the variations you mentioned, don't have an important place at the table in a mathematics classroom. They certainly aren't the ONLY type of problems that students need to be doing. But, I'd like to argue that we, as a math ed. community, need to get better at how to implement and work with students on these types of problems. Even if they are a bit less "real" or "practical." Ultimately, I'm not convinced that a "real" problem that doesn't promote problem solving is any better than a "fake" problem that does encourage problem solving. In fact, I’d love it if the arguments you make about the market and which wheel combinations are more suited for a sale came up during this work. That helps kids see how math and the real world interact. Much more than any problem context ever could.

I do wish I had more hours in my day to continue to respond to all your points Kim – because they are important and I truly appreciate your skepticism and the push back. I value that in our friendship very much. Thanks for taking the time to read and respond to this. I hope it continues to spawn more conversation and discussion around these ideas. –Zak

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Drew Polly- 7/8/2017 1:44:48 PMGood points here, Zak! There definitely is power in open-ended or rigorous tasks without too much scaffolding or over guidance at first. Definitely a reason why there is strength in curricular resources that do not overly guide and direct students to a specific procedure or process!

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Kimberly Morrow Leong- 7/7/2017 3:25:51 PMZak,

I am going to push back on this, because I think that you would appreciate it, Zak. Ben Sinwell makes a good point when he asks how students are prepared to be offered "less help." I have seen that happen in my role as a coach - students are set loose to solve an open task and have precisely zero idea how to start because they don't have adequate structure-making skills under their belt to access the task.

My objection, however, is a bit different, and is directed at the task. When I read the task, my initial solution strategy was to draw bicycles and I completely missed the references to the other types of cycles. In the past I might have chided myself for "missing the point" of the task. But now I recognize that my approach is far more practical, seeing the task as an economic task. Why use the wheels to make anything other than bicycles, because that's where the market is? The intention of this task is to find all combinations of unicycles, bicycles, tricycles (and quadricycles?). By design it is traditionally a logical task, and is "best" solved by creating some sort of organizing structure (a table, a list, a tree diagram, etc.). As such it calls on a far different set of skills than the problem I solved, which is a simpler factors problem.

So I return to the task design. It is an open task, but while we might seek "creative and out-of-the-box" thinking in theory, it's really not what

wewant.Webeing teachers, mathematicians, assessment creators, textbook writers standard writers, and other powers that be. How do we know this? Each year I am involved in a lesson study project, and each year I am interested in what solution strategies teachers generate in anticipation of teaching the lesson (as in the 5 Practices). Rarely are we able to generate "out-of-the-box" possibilities, yet each lesson yields at least one that makes us shake our heads trying to figure out the student's thinking. This isn't evidence, per se, but it is a common experience. I believe this is because the open tasks we select can address radically different mathematics topics depending on the student's strategy. My interpretation of the wheel problem is a factors problem, while other students' strategies might address tree diagrams, numerical expressions, and of course using an organized list. While this sounds great, it isn't practical when the task-based lesson refers to a certain standard or falls on a particular day in the pacing guide, which is a reality of teaching.Inherently this problem is a code for a particular solution strategy, which is a result of the problem's structure. There are many of those: the Handshake Problem, the Mango Problem (something involving a series of creatures indulging in a fraction of a bowl of some fruit), the Locker Problem, etc. You know the ones. How do you know a task is like this? In PD settings it is hard to encourage teachers of older grades to solve it differently than the "typical" way! I am not really so much being critical of the tasks, but of the assumptions we make when we give these tasks. The way students mathematize a real situation is so unpredictable, just as their interpretations of literature are. I think mathematical modeling offers a lens that is more open to teaching students to freely mathematicize, which gives us the freedom to teach a task like the wheel task as a structure lesson, rather than pretending it is actually "open."

Thanks for making me think, Zak!

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Robert Berry- 7/7/2017 2:35:22 PMI posted this piece on my Facebook page and a wonderful conversation emerged. Below is my response on FB to questions concening support for students' in special education and concerns about student being left out to be lost without supports.

"I took Zak's piece as a form of pedagogical support after the norms and nurturing are in place. In my mind this is a way to support problem solving. That is, "giving less" can be use for formative assessment and to provide opportunities to understand students' depth in understanding. It is not meant for teachers to give students tasks without the necessary supports. It would be irresponsible for teachers to not provide the necessary instructional supports or to meet students' needs. That is, use direct instruction where necessary but also allow space for students to use mathematics in creative ways for problem solving etc...I also think we should follow Kimberly Morrow Leong's advice and move this to the NCTM page to promote conversation."

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