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Nat Banting is a mathematics teacher working in Saskatoon, Saskatchewan, Canada. He currently teaches grades nine through twelve at Marion M. Graham Collegiate.
Much of what appears in mathematics textbooks is what I like to call, downstream thinking. Downstream thinking usually involves two features that set the stage for learners. First, it provides a context (however doctored or engineered–often referred to as “pseudo-context”). Second, the problem provides a pre-packaged algebraic model that is assumed to have arisen from that context.
from “Pre-calculus 11,” McGraw-Hill Ryerson, 2011
I am imagining the reasons for this are three-fold:
Instructional time. Developing these models is messy and takes time, especially with class sizes in the mid-to-high 30s.
Accessibility. Developing these models isn’t always possible (i.e. can’t run trials and collect data on every context).
Scaffolding. Glancing through the assignment pages of our textbook, it follows this trajectory: explicitly give model and describe it, give context but ignore it, allow for students to build a sterilized version1. The textbook is trying to walk students (hand-in-hand) toward abstraction.
One might assume that the provision of a context would mean that operating within it would prove useful for the resolution of the task, but students quickly learn that the context is only important for as long as it takes to read it. After that, the pre-packaged model is all that matters. This phenomenon is worsened by the lack of student involvement in building the model. These sorts of activities do all the important modelling work upstream, leaving only the completed model downstream for students to interact with. In doing so, we are missing the important work where students interact and experiment with the intricacies of the model.
What’s further, I do not think that all important modelling work needs to take place in a context. What’s important in the direction of the work: that it asks students to build a model to specifications before turning around and employing the model to solve for unknowns. This is a theme I tackled at a session at NorthWest Math Conference hosted by the BCAMT. However, we ran out of time at the session before we could really sink our teeth into one of my favourite activities: the Quadratics Menu.
The idea is a mash-up of two ideas that I had rattling around in my brain. The first is an assessment structure called a “menu.” Essentially, the assessment is made up of a series of questions (exactly like a traditional quiz), and each is assigned a value. The one my colleague showed me had questions ranging from 1-point to 3-points. The student is required to complete “10 points” of questions. By infusing this extra layer of choice, the assessment increases metacognitive demand on the student. They now must think about the topics they know well, questions they are most comfortable with, and where they are going to get the most bang for their buck–so to speak.
The second idea was one I’ve written about previously on this blog: Building custom parabolas. I love this activity, and it has become a staple in my classroom. The activity asks students to think upstream by building models to specifications. Notice, they do this without a whiff of context. That is, the models satisfy the constraints, but do not claim to map to some experience a student might encounter in their daily life. Not all context is bad (and mathematical modelling is important work). As mentioned previously, I am interested in the direction of the thinking: from specifications to model.
So these two ideas combine to form a new activity. The goal is no longer to design 10 different quadratics to fit 10 different sets of specifications, but to add a layer of decision making by asking students to satisfy 10 specifications (in whatever combinations they desire) by using as few functions as possible.
The task moves from:
Build a parabola that has two x-intercepts and an axis of symmetry of x=1.
to
The menu requires students to think upstream. Not only are they building models from specifications, they are asked to add a layer of decision making to the process by choosing which constraints to combine into specific functions.
Give this to your students and ask three guiding questions:
Which pair nicely?
Which cannot be paired?
Is it possible to solve in 2, 3, or 4 parabolas?
If you choose to work on the task yourself, I am directly the same three question your way.
Exactly one month ago, fellow Saskatchewan mathematics teacher Ilona Vashchyshyn tweeted about an area task that she used in her class. Long story short, it captured the imagination of Math Ed Twitter like elegant tasks have a tendency of doing.
The challenge: Write your name so that it covers an area of exactly 100 cm squared.
Two weeks later, my grade 9s and I were beginning a unit on surface area and this seemed like a perfect entry point. I thought it would provide a perfect window into what geometric knowledge they were bringing with them into the unit. Because my crew would be dealing with the surface area of a cylinder, I altered the task slightly:
Write your name so that it covers an area between 99 and 100 cm squared.
*you must include at least one rectangle, one triangle, and one circle in your design.
Student work:
The results were as creative as expected. The task seems to encourage a commitment to symmetry with many students calculating a sort of square budget for each letter of their names or making all letters uniform in height. Also, their attention was immediately drawn to the shape of most resistance–the circle. There was almost an immediate recognition that rectangles and triangles live nicely on a grid, but circles do not. This became an important talking point throughout their work.
Beyond the products:
As the students worked, three important themes came out of our conversations. These pieces of intel gave me a sense of where we were collectively strong and where more connections were needed.
1. Unearthing of vocabulary
The task provided plenty of opportunity to talk about the vocabulary associated with area. Words like dimension, formula, area, radius, diameter, base, height, and squared were all discussed. I really like introductory tasks that provide opportunities to acclimatize to the verbiage while content demands remain relatively low.
2. Connecting counting to calculating
The grid encourages students to count squares in order to tally their total square area. While this is a nice connection between the idea of shaded squares and square units of area, I was interested if the students could also assign dimensions to shapes and calculate their area with appropriate formulae. Requiring them to incorporate a circle made calculation a necessity, but also provided the opportunity to connect this back to counting.
After having several conversations with students about the link between counting and calculating, I asked some students to put their work under the document camera. My set-up is an iPevo VZ-R, which allows me to clearly project and zoom in on sections of student work. The camera has all kinds of functionality, but I have found it most useful in my classroom as a tool for sharing ideas. (I often “pause” activity half way to discuss strategies that are emerging in the classroom).
As a class, we focused on the ways that students had incorporated the circles. We then calculated their area with the formula and compared with a counting strategy to check their correlation. This exercise helped students add meaning to the calculations that (I knew) would dominate much of their activity in the unit.
3. Practicing sub-sectioning and calculating
We started outlining the dimensions of the shapes we were viewing under the document camera. It didn’t take long for students to see the compound calculations in different ways. For instance, the “L” in “Laurier” can be divided into a 6×1 rectangle and a 1×3 rectangle. Alternatively, it can be dissected into a 5×1 rectangle and a 1×4 rectangle. This slicing of compound shapes into more familiar ones is a critical skill when calculating surface area. One student suggested that we could calculate the area of the “A” in “Laurier” by tracing a 6×5 rectangle around the entire letter and then subtracting the triangles created “on either roof”. In the end, the class decided that this resulted in more work, but zooming in and tracing out the idea provided us the chance to discuss the possibility of multiple correct dissections.
___
I think the immediate appeal of the task (as evidenced by the sheer volume of social media activity) was in the beautiful creations of students. However, its value goes beyond what appears on the paper; the simple prompt is a powder keg for critical conversations about area.
One of the great parts of my job as a split classroom teacher and division consultant is that I get to spend time in classrooms from grades six to twelve. This means I often need to be in one head space to teach my own Grade 12s and then switch gears to act with younger mathematicians. It also means that the classroom experiences are sporadic and involve teachers working in several different places in several different curricula.
On this particular occasion, I was working with a 7/8 split class that had just finished a unit on perfect squares and divisibility rules, and we wanted an activity that could serve as a sort of review of divisibility rules but also reveal something cool about perfect squares. I thought about the locker problem, but it doesn’t require students to factor in order to see the pattern. Instead, I took some of the underlying mathematical principles (namely: that perfect squares have an odd number of factors) and wrapped it in a structure suited for a Friday afternoon.
The task:
While the students were at recess, the classroom teacher and I sectioned the board into 16 spaces. In each space we wrote, “1 Factor,” “2 Factors,” “3 Factors,” etc. This continued until there was a section for every number of factors from one to sixteen. Then we printed out a hundreds chart and put it in a dry erase sleeve. When the students came in, they sat in groups and we reviewed two types of factoring: listing prime factors and listing all factors. We made it clear that today were were focused on finding all factors. We then introduced the activity.
Each group was to nominate a reporter. Their job would be to get a number from me and then bring back to their group. The group was tasked with listing all the factors of the number and counting them up. When they were confident in their total, the reporter would then record the number in the appropriate section on the board and see me for another number. As an example, we counted the factors of 30: {1, 2, 3, 5, 6, 10, 15, 20} and placed the 30 in the section entitled “8 Factors“.
The goal was to crowd source the work for the numbers 1-100. Every time a reporter asked for a new number, I would give them the next available on the hundreds chart and cross it off. The system worked really well! We took a break after the first fifty numbers and gave the students a chance to challenge any of the placements. We also strategically challenged perfect square numbers that were placed incorrectly.
The fallout:
One student immediately worried that we might get a number with more than 16 factors. I told him that we would happily add a section on the board if that were to occur. Not long into the activity, I heard him comment to his group that there was no way that 16 factors would happen. It didn’t take long for learners to get the impression that factors are tough to come by.
I also noticed that the divisibility rules came into play quicker than I expected. I imagine this was due to the recency with which they had studied them. Also, I realized that I really hate the divisibility rules. I saw several students default to mechanically employing these rules instead of using numerical reasoning. When questioned, they were able to reason with a variety of strategies like finding a close multiple (i.e. 3 divides 84 because 3 divides 81) or decomposing and dividing (i.e. 4 divides 68 because 4 divides 60 and 4 divides 8). However, these strategies took a back seat to rules that (in my observation) contained little connection to the structure of numbers. Rant over. Our standards include the divisibility rules, so we provided them as an option alongside these other strategies.
Probably the most interesting action came from two students who, when they asked for a number, would ask for a switch when they were given a large, odd number. For instance, I offered a student “71” (because it was the next on the list), and she requested “72” instead. Then another student asked for “90” instead of “89”. When I pushed for the reason behind the trend, they said that they didn’t want to get “stuck” with a large number that didn’t have any factors. This struck me as interesting, and we discussed this sort of intuition. What features of a number make it seem like it will have few factors? Students talked very openly about this prime number inkling.
After the numbers 1-100 were categorized, we took a deep breath and asked the class what they noticed about the board. Below are some of their observations:
Only “1” has a single factor
Lots of numbers have only 2 factors (which a student identified as the set of prime numbers)
All the 2-factor numbers are odd (which instigated a discussion about “2”)
It is rare to have a number with an odd number of factors
Numbers don’t have as many factors as we thought
Large numbers might have a few factors
Even numbers cannot have less than 3 factors (again, the conversation of “2” came up)
All the numbers in the odd spaces are perfect squares
This last noticing was the place where we wanted to live for a while. We wrote out the factors for “36” and noticed what was different than the previous trial with “30”. Very quickly, students saw that not every number had a “partner,” and this was causing the odd number of factors. Before we cleaned up, we left the class with a challenge:
Build a number with exactly 11 factors.
One student immediately asked, “So just build a random number?” to which another student immediately replied, “Well, we know it must be odd”.
Every time I teach a unit on fractions, there are many students who insist that they’d rather use decimals, and I don’t blame them. The obvious parallels to the whole numbers make decimals a “friendly” extension from the integers into the rational numbers.1 Many of the things school math asks kids to do with rational numbers can be easily transferred into decimals with minimal stress on the algorithms. Such is not the case with fractions. Take addition for example.
When we add whole numbers, we combine collections of equal denomination until it is possible to collect, bind together, and count them alongside collections of a higher denomination (a process often called “carrying” when the process is abstracted to an algorithm, but I am still unsure of what, exactly, is being “carried”. I think I’d prefer to call this process, “re-grouping”). This process of “carrying” or “re-grouping” is preserved when working with rational numbers in decimal form. However, when asked to add fractions, students encounter problems above and beyond that of re-grouping collections. All of a sudden the process involves common denominators. Students quickly realize that they can always work with decimals at face value2, but need to massage fractions into form before their addition behaves nicely.
The same goes for ordering rational numbers. The bulk of the opportunities I provide to my students revolve around ordering fractions, because they always arrive in my room well-versed in ordering decimals. Most of our work with this involves a “dynamic number line“3. Usually, I check in with their understanding of ordering decimals quickly and then move on, but this semester I decided to provide a wrinkle to the usual question about decimals.
I wanted to accomplish two things:
I wanted students to show they understand magnitude of decimals by placing them on the number line.
I wanted to push their algorithmic thinking about what makes a decimal large or small.
The lesson went like this:
I asked each student to take out a blank sheet of paper and copy down the following structure:
I showed the class a 10-sided die with the digits 0-9 on the faces. The instructions were straightforward: I was going to roll the die four times. After each roll, they needed to choose a blank to place the result. They had to choose after each roll, and they were not permitted to switch locations later on. In the first round, students were tasked with assembling the largest possible decimal. In the second round, they were tasked with assembling the smallest decimal possible. As they placed the numbers, I asked questions to the class:
“Who dared to put that 6 in the hundreds place?”
“Does it matter where you put a zero?”
“If the next roll was a 9, would that be good news”
Often times, the last roll was met with equal parts pain and exhilaration, as half the class was hoping for a large number and the other half for a small number. All of these sounds signalled t0 me that the class had an understanding of what they wanted, and therefore an understanding of what made decimals large or small. After the two rounds, I asked if they could definitely create the largest possible decimal if I gave them the four digits beforehand. They guaranteed me, I rolled for four more digits, and each student came up with the same result. We were ready.
I asked them to write the following structure on their pages. (I added a spot because I wanted to have a lot of variety in the class).
I explained a new set of rules. They would get all five digits at the onset, and would still be required to place one in each spot. However, this time they were trying to create a decimal that would fall in the exact middle of the pack of created decimals. Notice that I did not ask them to create the middlemost decimal; I asked them to create a decimal where half of their classmate’s creations would fall above and half of the classmate’s creations would fall below.
Each student kept their answer to themselves and wrote it on a small, folded piece of paper we call a “tent”. (They put their name on the inside of the tent). One by one, they were asked to place their values on the clothesline across the front of the class.4 This allowed for conversations about magnitude to become public. I was able to assess if students understood not only the magnitude of the decimals but also the relative magnitude as they analyzed relationships between the created decimals.
After they all had gone, we pulled off the tents one by one to reveal the exact middle decimal.
To accomplish goal number 2, I asked the following question:
“How could we find out if this decimal is the “real” middle number?”
The conversation turned algorithmic quickly. To build the largest number, you place the largest digit first. To build the smallest number, you place the smallest digit first. So doesn’t it make sense that to build the middle number, you place the middle-est number first? The conversation was awesome!
Some questions I used to guide their thinking:
What if we tried it with three spots?
What if we forgot about the decimal point? Does that matter?
What if there were repeat digits?
We never arrived at a distinct algorithm on how to build the middle. We did, however, interrogate the algorithms for largest and smallest, and talk a lot about rational numbers, relative magnitude, and the base-10 number system in the process. In my book, these are all wins that resulted from a subtle shift in focus.
We (as teachers) act like we want students to ask questions; however, there are plenty of implicit messages about teaching that tell us that good teachers don’t need students to ask questions. One of the oldest pillars of teaching tells us to provide adequate wait time for students to formulate and ask questions, but there is a sense of relief when time passes without the need for clarification. This feeling essentially equates clarity with quality. Wait time becomes an emergency procedure to be used when we feel an awkward imbalance in the room.
These messages are also felt by students. Students might want to ask questions but school sends implicit messages that good learners don’t need to ask questions. In the same way that teachers can feel like questions reflect poorly on their ability to teach, students interpret their desire for clarification as an indictment on their ability to learn. Sayings like, “there are no bad questions” or “if you are thinking it, so is someone else” feel fake. Questions signal a breakdown in the smooth communication between teacher and student. What’s weird is that both sides blame themselves. In an attempt to shift this culture, I set out this year to stop entertaining questions and start expecting them.
Two focus going into year:
1) (a pedagogical structure shift) Lag assessments with immediate student-teacher interviews on every “work day”.
2) (a pedagogical stance shift) Don’t pause and ask if there are questions. Instead say, “I want someone to ask a clarifying question”
Others have reported a similar focus, and it seems to lead back to the initiative of Howie Hua. To remind myself, I printed out a page that said, “I want someone to ask a clarifying question” and posted it in plain sight in my classroom.
Here, I describe the positive initial experiences with this simple shift in verbiage. As an illustration, I’ll use a task that appeared in my Twitter feed thanks to Ed Southall.
I projected the image, and talked through the example of 8, 5, and 3. I then chose two other numbers (5 and 11), and then asked which number is then banished from the set (6). I then clearly re-stated what would signal a resolution to the task: they are done when they successfully place each number into exactly one of the three sets.
I paused for a brief moment and then said, “I want someone to ask a clarifying question”. One student got the ball rolling:
“Do we only place each number once?”
I did my best to address the question, and then was sure to add, “That’s an important question”.
“What if we get a negative number?”
“Do all sets have to be the same size?”
Again, I answered and then added, “Thanks. That’s a great question”.
I found that after a while it not only changed the culture about students asking questions, but it also encouraged a new type of question. Soon the questions became a mix of clarification and idea generation. They began to hint at their imagined strategy, and the time morphed into a sort of group brainstorm.
“Can we leave a set empty?”
I responded and added, “I never thought of that. Interesting question”.
“What if the difference between two numbers is one of the two numbers?”
(He meant like the case of 2 and 4 having a difference of 2 or 3 and 6 having a difference of 3)
This signalled (to me) a distinct shift away from clarification and toward strategy. I responded with, “That sounds like a question for your group. Go ahead”.
This little shift has had a huge impact on how I launch tasks in my room. Expecting questions opens up a platform for students to voice and re-voice their initial interactions with the constraints of a task. I feel like it builds a culture of mutual responsibility. Rather than the teacher being solely responsible for a clear introduction, I now share this responsibility with students, and they are taking the opportunity to think deeply about the intricacies that await them in each task. This represents one step in a new message to students: I both want and need their questions.