How Many Do You See (Part 2 of 2)

A few weeks ago I shared with you a quick blog post showing a simple worksheet at the grade 2 level – the kind of  simple worksheet that is common to many classrooms.  If you haven’t seen the image, here it is again:


As you can see, the task asks students to correctly count the number of each shape they notice.  In my first post (Part 1) I asked us a few questions to start a conversation:

  1. Pick one shape (or more if you’re adventurous)
  2. Think about what you believe the teacher’s edition would say
  3. Count how many you see
  4. Share the 3 points above as a comment here or on Twitter

I was quite happy with where some of the conversations led…

Some of the conversations revolved around the issue many have with resources perpetuating stereotypical definitions of shapes:


If we look, there are exactly 4 shapes that resemble the diagram at the top of the page labelled as “rectangle”, however, there are several different sized squares as well (a square are a special case of a rectangle).

Other conversations revolved around actually counting the number of each item:


trapezoids 21

What interests me here is that we, as a group of math teachers, have answered this grade 2 worksheet with various answers.  Which brings about 2 important conversations:

  1. What are we looking for when students complete a worksheet or textbook questions?
  2. Are we aiming for convergent or divergent thinking?  Which of these is more helpful for our students?

What are we looking for?

Given the conversations I have had with math teachers about the worksheet being shared here, it seems like there are a few different beliefs.  Some teachers believe the activity is aimed at helping students recognize traditional shapes and identify them on the page.  Other teachers believe that this activity could potentially lead to discussions about definitions of shapes (i.e., What is a rectangle?  What is a hexagon?…) if we listen to and notice our students’ thinking about each of the shapes, then bring students together to have rich discussions.

It’s probably worth noting that the Teacher’s Edition for this worksheet includes precise answers.  If a typical teacher were to collect the students’ work and begin marking the assignment using the “answers” from the teacher’s guide, some of the students would have the “correct” answer of 8 trapezoids, but many others would likely have noticed several of the other trapezoids on the page.  If we are looking / listening for students to find the correct answer, we are likely missing out on any opportunity to learn about our students, or offer any opportunity for our students to learn themselves!

I would hope that an activity like this would provide us opportunities for our students to show what they understand, and move beyond getting answers into the territory of developing mathematical reasoning.

Convergent vs Divergent Thinking

Again, many of the teachers I have discussed this activity with have shared their interest in finding the other possible versions of each shape.  However, what we would actually do with this activity seems to be quite different for each educator.  It seems like the decisions different teachers might be making here relate to their interest in students either having convergent thinking, or divergent thinking.  Let’s take a look at a few possible scenarios:

Teacher 1:

Before students start working on the activity, the teacher explains that their job is to find shapes that look exactly like the image in the picture at the top of the page.

Teacher 2:

Before students start working on the activity, the teacher tells the students exactly how many of each shape they found, then asks students to find them.

Teacher 3:

Before students start working on the activity, the teacher explains that their job is to find as many shapes as possible.  Then further explains that there might be ones that are not traditional looking.  Then, together with students, defines criteria for each shape they are about to look for.

Teacher 4:

Before students start working on the activity, the teacher explains that their job is to find as many shapes as possible.  As students are working, they challenge students to continue to think about other possibilities.

In the above scenarios, the teachers’ goals are quite different.  Teacher 1 expects their students to spend time looking at common versions of each shape, then spot them on the page.  Teacher 2’s aim is for students to be able to think deeper about what each shape really means, hoping that they are curious about where the rest of the shapes could possibly be leading their students to challenge themselves.  Teacher 3 believes that in order for students to be successful here, that they need to provide all of the potential pieces before their students get started.  Their goal in the end is for students to use the definitions they create together in the activity.  Finally, teacher 4’s goal is for students to access the mathematics before any terms or definitions are shared.  They believe that they can continue to push students to think by using effective questioning.  The development of reasoning is this teacher’s goal.

Looking back at these 4 teachers’ goals, I notice that 2 basic things differ:

  1. How much scaffolding is provided; and
  2. When scaffolding is provided

Teachers that provide lots of scaffolding prior to a problem typically aim for students to have convergent thinking.  They provide definitions and prompts, they model and tell, they hope that everyone will be able to get the same answers.

Teachers that withhold scaffolding and expect students to do more of the thinking along the way typically aim for divergent thinking.  That is, they hope that students will have different ideas in the hopes for students to share their thinking to create more thinking in others.

Whether you believe that convergent thinking or divergent thinking is best in math, I would really like you to consider how tasks that promote divergent thinking can actually help the group come to a consensus in the end.  If I were to provide this lesson to grade 2s, I would be aiming for students to be thinking as much as possible, to push students to continue to think outside-the-box as much as possible, then make sure that in my lesson close, that we ALL understood what makes a shape a shape.

I want to leave you with a few reflective questions:

  • I provided you with a specific worksheet from a specific grade, however, I want you to now think about what you teach.  How much scaffolding do you provide?  Are you providing too much too soon?
  • Do your lessons start off with convergent thinking or divergent thinking?  Why do you do this?  Is this because you believe it is best?
  • How can you delay scaffolding and convergent thinking so that we are actually promoting our students to be actively thinking?  How can you make this a priority?
  • What lesson or warm-ups or problems have you given that are examples of what we are talking about here?
  • If we do remove some of the scaffolding will some of your students sit there not learning?  Is this a sign of them not understanding the math, or a sign of them used to being spoon-fed thinking?  What do WE need to get better at if we are to delay some of this scaffolding?

I encourage you to continue to think about what it means to help set up situations for your students to actively construct understanding:

I’d love to continue the conversation.  Write a response, or send me a message on Twitter ( @markchubb3 ).

P.S.  I’m still not confident how many of each shape are actually here!


“Worst Problem Ever!”

A few years ago I was teaching a 6/7 split class.  We had been exploring surface area and volume of prisms (rectangular so far), but were quite early on in the learning cycle.  At the beginning of my math class I drew a picture of a rectangular prism and wrote the dimensions on the diagram like below:

SA and V 4

I asked everyone to calculate the Surface Area.  Really this was just a check for me to see if we understood surface area… the problem for the day was coming up.

However, as I walked around, I realized something VERY unusual.  EVERYONE got the answer of 250… but not everyone did it correctly.  Let me show you:


SA and V 3

Take a look at the 2 answers above.  Some students calculated the Surface Area correctly at 250 units squared… and others calculated 250 units cubed (they found the volume).

I apologized to the class for giving them the worst problem ever… THE ONLY POSSIBLE rectangular prism that has the same surface area as volume.  Then one student commented, “Is that the only possible prism?”  

I didn’t know the answer, so we started seeing if it was the only possible prism.

Over the next 100 minutes, every student in the class, some in pairs, some on their own, started drawing prisms and calculating the surface area and volume.

Eventually, a student told me that one of the dimensions couldn’t possibly be a 1.  I asked him to prove it… After a few minutes he showed me several examples with numbers that were small or large.  He attempted to find the limits:

SA and V

He explained to the class that if there is a 1 as any of the dimensions, the Surface Area would always be larger.   We explored why that happens.

Minutes later, another student told me that it was impossible to have a 2 as one of the dimensions.  I asked her to prove it… She showed me the work that she had completed and shared it with the class.

SA and V2

She explained that the numbers were getting close if she chose 2 of the numbers identical, but the end pieces of the shape would always add up to more than the volume would.

After lots of trial and error… guessing and refining ideas, several students started finding possible prisms that also had the same Surface Area and Volume…

The class started noticing a pattern between the dimensions and found limits between the smallest and largest possible prisms…  

By the end of class, all students had calculated dozens of surface areas and volumes… all students were making conjectures or testing out the conjectures of others.

My original problem, which I intended as a quick warm-up was not a quality engaging problem.  However, I want you to think about what made this lesson better?

WHO posed the problem that day?  Did this have something to do with the shared responsibilities that happened later in the lesson?

What if I just moved on?  would the learning have been as rich?

Think about how the students picked the shapes they were testing?  Some students worked in teams to work strategically… others made their own conjectures and followed those patterns.

In this lesson, choice was key, but the choices didn’t come from me… Students were working together to reach an ultimate goal, not in competition with each other…  Conjectures were made and tested, not because I told everyone to, but because it served our purpose!

I think about Dan Meyer’s “Real World vs Real Work” a lot.  Why were my students so engaged here?  There was NO real world connection.  That wasn’t what was motivating my students at all!


I also think we need to reflect on the level of cognitive demand we ask our students to be engaged in:


“Doing Mathematics Tasks” seems like something that is hard to do, yet, every one of my students were engaged in this problem… everyone eagerly searched for patterns, many drew pictures, used snap cubes, visualized what was happening… all in the name of better understanding the relationships between the dimensions of a rectangular prism, and its surface area and volume!

By the end of the class, my students had found exactly 10 rectangular prisms that have the same value of its surface area as its volume (using only whole numbers), and could prove that these were the only 10 possible.

I’d love to hear your thoughts about our problem… or why the students were SO engaged… or about the conditions that must have been present in the class… or how problem solving can be used as a purposeful practice of procedures…..