At that moment, three students jumped to their feet and cheered. High fives may have even been shared. I asked them what was up. They asked if we could talk about it later. (Never press here, by the way. Rookie mistake. If kids give you an out, take it.) So we did. Each student had estimated how many times I would say “Okay, so, um…” during the lesson. Their earlier excitement? I hit the highest of the three estimates.

I had completely forgotten about this episode until last spring when Canucks rookie Brock Boeser’s first ever NHL postgame interview made it into my Twitter timeline. At that time, I was helping teachers make sense of the Ministry of Education’s (MoE’s) “Process for Solving Numeracy Tasks” (a/k/a a mathematical modelling cycle). This post is a collision between the two.

The *Interpret* process in this mathematical modelling cycle involves reading contextualized situations in order to identify real-world problems.

In this task, we can start with the following clip and ask “What do you notice?”

I noticed the sports clichés (NSFW). Brock Boeser’s “I just want to come here and help the team get a win” is damn close to “Nuke” Laloosh’s “I’m just happy to be here, hope I can help the ball club.” I also noticed that Boeser says “you know.” A lot. I wasn’t alone.

From here, we can develop a real-world problem by asking “What do you wonder?” or “What’s the first question that comes to mind?” My question: How many times does Brock Boeser say “you know” in the postgame interview?

Note: the starting point — in the diagram and in the video — is a situation, not a problem.

The next process involves identifying and activating mathematical understanding in order to translate real-world problems into mathematical problems. The MoE calls this *Apply*, a misused and abused term in mathematics education. Thankfully, *Mathematize* immediately follows in brackets throughout the documents.

We can ask “What information would be helpful to know here?” Students might want to know:

- the number of times that Boeser says “you know” in the clip (12)
- the length, in seconds, of the clip (44)
- the length of the entire interview (2:58)
- the rate at which Boeser says “you know” (?)
- the fraction of the time in which Boeser is speaking (?)

This process also involves — among other things — creating relationships to represent the real-world problems. Here, a proportional relationship. A simple approach might involve setting up 12/44 = *x*/178. A *math* problem.

At first glance, this looks trivial: simply cross-multiply and divide. But the *Solve* process involves using a variety of approaches and representations. For example, students might use scale factors or unit rates; bar models or ratio tables. Or, not proportions, but linear relations. Tables, equations, graphs. Does the solution make *mathematical* sense?

Does the mathematical solution (*x* = 48.545454…) make sense within the contextualized situation? The *Analyze* process involves identifying possible limitations and improvements. Brock Boeser says “you know” 12 times in the 44 second Act 2 video. But he reaches this count at 33 seconds and finishes answering the reporter’s question at 40 seconds. Does any of this matter? Is my simple proportional approach still useful?

Students communicate throughout the *Interpret*, *Mathematize*, *Solve*, and *Analyze* processes. This communication happens *within* their groups. The *Communicate* process in this mathematical modelling cycle involves clearly and logically defending, explaining, and presenting their thinking and solutions *outside* of their groups.

There are better tasks that I could have picked to illustrate this mathematical modelling cycle. In fact, last year — in the absence of sample numeracy tasks from the MoE — my go-to here was Michael Fenton’s Charge. BC’s Graduation Numeracy Assessment aside, mathematical modelling with three-act math tasks (and the pedagogy around these tasks) has played an important role in my work with Surrey math teachers for several years. The MoE did release a sample numeracy assessment in late September; I am now able to include a *Reasoned Estimates*, *Plan and Design*, *Fair Share*, and *Model* task in these conversations with colleagues. For more numeracy tasks, see Peter Liljedahl’s site.

Okay, so, um, if I didn’t pick this Brock Boeser task because it, you know, epitomizes the mathematical modelling cycle, then why did I share it? Coming full circle to the story of my three students at the beginning of this post, there’s a missing piece. Yeah, we shared a laugh and I was more self-conscious of my verbal fillers for the rest of the year (2005 ± 3). But the most embarrassing part is that I have no idea how my students came up with their estimates. Because I didn’t ask. I mean, three girls spontaneously engaged in mathematical modelling — I promise there was more mathematical thinking here than in the task at hand — and not a single question from their math teacher! In my defence, it would be several years before mathematical modelling was on my radar — an unknown unknown. Still, what a complete lack of curiosity!

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At that time, I was creating Which One Doesn’t Belong? sets. Cuisenaire rods didn’t make the cut. Nor did hundreds/hundredths grids:

I probably painted myself into a corner. Adding a fourth shape/graph/number/etc. to a set often knocks down the reason why one of the other three doesn’t belong. Not all two-by-two arrays make good WODB? sets (i.e., a mathematical property that sets each element apart).

Still, there are similarities and differences among the four numbers above that are worth talking about. For example, the top right and bottom right are close to 100 (or 1); the top left and bottom right are greater than 100 (or 1); top left and top right have seven parts, or *rods*, of tens (or tenths); all involve seven parts in some way. There is an assumed answer to the question, “Which one is 1?,” in these noticings — a flat is 100 if we’re talking whole numbers and 1 if we’re talking decimals. But what if 1 is a flat in the top left and a rod in the bottom left? Now both represent 1.7. (This flexibility was front and centre in my mind when I created this set. The ten-frame sets, too.)

Last spring, Marc and I offered a series of workshops on *instructional* routines. “Alike and Different: Which One Doesn’t Belong? and More” was one of them. WODB? was a big part of this but the bigger theme was same and different (and justifying, communicating, arguing, etc.).

So rather than scrap the hundreds/hundredths grids, I can simplify them:

Another that elicits equivalent fractions and place value:

For more, see Brian Bushart’s Same or Different?, another single-serving #MTBoS (“Math-Twitter-Blog-o-Sphere”) site.

Another question that I like — from Marian Small — is “Which two __________ are most alike?” I like it because the focus is on sameness and, like WODB?, students must make and defend a decision. Also, this “solves” my painted-into-a-corner problem; there are three, not six, relationships between elements to consider.

The numbers in the left and right images are less than 100 (if a dot is 1); the numbers in the centre and right can be expressed with 3 in the tens place; the left and centre image can both represent 43, depending on how we define 1.

At the 2017 Northwest Mathematics Conference in Portland, my session was on operations across the grades. The big idea that ran through the workshop:

“The operations of addition, subtraction, multiplication, and division hold the same fundamental meanings no matter the domain in which they are applied.”

– Marian Small

That big idea underlies the following slide:

At first glance, the second and third are most alike: because decimals. But the quotient in both the first and second is 20; in fact, if we multiply both 6 and 0.3 by 10 in the second, we get the first. The first and third involve a *partitive* (or *sharing*) interpretation of division¹: 3 groups, not groups of 3.

Similar connections can be made here:

This time, the first and second involve a *quotative* (or *measurement*) interpretation of division: groups of (−3) or 3*x*, not (−3) or 3*x* groups. (What’s the reason for the second and third? Maybe this isn’t a good “Which two are most alike?”?)

I created a few more of these in the style of Brian’s Same or Different?, including several variations on 5 − 2.

Note: this doesn’t work in classrooms where the focus is on “just invert and multiply” (or butterflies or “keep-change-change” or…).

And I still have no idea what I was going for with the Cuisenaire rods.

The slides:

¹Likely. Context can determine meaning. My claim here is that for each of these two purposefully crafted combinations of naked numbers, division as sharing is the more intuitive meaning.

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Paint Splatter Arrays (.key)

Paint Splatter Arrays (.pdf)

In Steve Wyborney’s *Splat!*, the total number of dots is given and the number of dots under each splat is unknown. In my *Paint Splatter Arrays*, the total number of dots is unknown. My paint splatters do cover some dots but how many is beside the point. Also, Steve’s dots are scattered; mine are arranged in arrays. (More on that below.) Steve’s splats splat. My splatters are there from the get-go. See? Not the same.

h/t Andrew Stadel

Here’s why I created this activity…

T: “How many do you see?”

S: “Twenty-five.”

T: “How do you see them?”

S: “Two, four, six, …”

Every. Single. Time.

Not all students. Most students do see and use groups or arrays to figure out how many. Those strategies are described in this post. But some students don’t seem to make sense of others’ ideas. That’s a greater challenge than I’ll tackle here. (Recommended: *Intentional Talk* by Elham Kazemi and Allison Hintz.) Instead, I designed the activity above to (gently) shove students towards looking for and making use of arrays.

The first three are softballs. For example, the second:

Students can still see each dot and count all by ones or twos. But a more efficient strategy is to see 3 × 5 (3 rows, 5 columns).

The next several slides completely cover at least one dot, so students can’t count all by counting what they can see. In each, at least one complete row and one complete column is visible. For example:

I had some fun with the last two. In the next-to-last one, the middle column is completely concealed.

In the last one, most of the dots are hidden. A bit of estimation. How many?

How confident are you?

What about now?

I test-drove these on my daughters. (Keira likes *Booger Math!* over *Paint Splatter Arrays*, by the way. It *is* catchier.) I’m looking forward to trying this out in Surrey classrooms. Feedback welcome!

And mine goes ding ding ding di di ding ding DING ding ding ding di di ding ding.

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In partitive division problems, a.k.a division as (fair) sharing, the number of groups is known. This type of problem asks how many are in each group. In quotative division problems, a.k.a. division as measurement, the number in each group is known. This type of problem asks how many groups. For example: 6 ÷ 3 = 2 (partitive) means ; 6 ÷ 3 = 2 (quotative) means . This distinction isn’t limited to collections of objects. Consider 6 ÷ 3 as cutting a 6 m rope into 3 parts (sharing) vs. cutting lengths of 3 m (measurement). Nor are these meanings limited to whole numbers. Which brings me back to my ham…

The directions read “bake approximately 15 minutes per pound (0.454 kg) or until internal temperature reaches whatever.” But here’s the thing:

Kilograms, not pounds. I could have converted from kilograms to pounds by doubling then adding ten percent of that. Instead, I divided 1.214 by 0.454. I know, I know, this still gives me the weight of my ham in pounds. But at the time, I interpreted 2.67 as the number of repeated additions of 15 minutes in my baking time. Either way, I determined how many 0.454s there are in 1.214. Quotative division. By a decimal.

As a math task, this is clunky. The picture book *How Much Does a Ladybug Weigh?* by Alison Limentani is a more promising jumping off point for quotative division in the classroom. On each page, the weight of one animal is expressed in terms of a smaller animal.

Using the data at the back of the book, we have 3.2 ÷ 0.53 = 6. We could ask children to make other comparisons (e.g., how many grasshoppers weigh the same as one garden snail?).

[Insert link to Marc‘s First Peoples beaded necklace task here]

In the past, I have struggled with partitive division by decimals (or fractions). But I found the following example at The Fair this summer:

It’s not intuitive–at least to me–to think of 1/3 in 12 ÷ 1/3 as the number of groups. Take a step back and think about 26 ÷ 1 = 26. The cost, $26, is shared between 1 rack of ribs; the quotient represents the unit price, $26/rack, if the unit is a rack. This result should be… underwhelming.

Before we think about dividing by a fraction here, let’s imagine dividing by a whole number (not equal to one). What if I paid $72 for 3 racks? (Don’t look for these numbers in the photo above–I’m making them up.) In 72 ÷ 3 = 24, the cost, $72, is shared between the number of racks, 3; again, the quotient represents the unit price, $24/rack. Partitive division.

So what about 12 ÷ 1/3? The cost is still distributed across the number of racks; once again, the quotient represents the unit price, $36/(full) rack. The underlying relationship between dividend, divisor, and quotient hasn’t changed because of a fraction; the fundamental meaning (partitive division) remains the same.

We could have solved this problem by asking a parallel question, how many 1/3s in 12? And this quotative interpretation makes sense with naked numbers. But it falls apart in this context–how many 1/3 *racks* in 12 *dollars*? Units, man! If dollars were racks, a quotative interpretation would make sense–how many 1/3 racks in 12 full racks?

As a math task, this, too, is clunky. My favourite math tasks for partitive division by fractions are still Andrew Stadel’s estimation jams.

(Looking for a quotative division problem that involves whole numbers? See Graham Fletcher’s Seesaw three-act math task. For partitive, there’s *Bean Thirteen*.)

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It can be challenging to plan experiences that will be meaningful to English teachers and physical education teachers, to science teachers and French teachers, etc. One approach is to have teachers participate in a math activity, then wave my hands–magic!–and say “Of course, this strategy translates to your social studies classroom.” In fact, swap the two subjects in the previous sentence and you will accurately sum up much of my professional development experiences from early in my career. Turnabout is fair play?

A more promising approach is to find a colleague from another subject and talk pedagogy. Discuss similarities. Discuss differences. Several years ago, my literacy colleague hipped me to *The New York Times*‘ “What’s Going On in This Picture?” feature. An evocative image is stripped of its caption and students discuss/write about what they see.

Later, the photo’s caption and story are revealed.

A child jumps on the waste products that are used to make poultry feed as she plays in a tannery at Hazaribagh in Dhaka, Bangladesh on Oct. 9, 2012. Luxury leather goods sold across the world are produced in a slum area of Bangladesh’s capital where workers, including children, are exposed to hazardous chemicals and often injured in horrific accidents, according to a study released on Oct. 9. None of the tanneries, packed cheek-by-jowl into Dhaka’s Hazaribagh neighborhood, treat their waste water, which contains animal flesh, sulphuric acid, chromium and lead, leaving it to spew into open gutters and eventually the city’s main river.

I connected this strategy to my teaching with three-act math tasks. In both, subtracting information adds perplexity. Central to this strategy, in English language arts or mathematics classrooms, was the shared belief that our students were, above all, *curious*. At a workshop for department heads from every department, we invited teachers to look closely at and interpret several photos that we selected from *The Times*‘ series. We invited them to notice and wonder. This activity was invaluable in tackling the challenge of talking big ideas and inquiry across all subjects. So I was very excited to see the announcement of a new monthly *NYT* feature: “What’s Going On in This Graph?”

I needed the activity before the first “WGOITGraph?” would be published so I created my own. I modelled the strategy using Dan Meyer’s “Canada Flushed” graph. Only I stripped it *naked*.

I asked “What do you notice?” and recorded their noticings:

- the blue line is more variable than the green
- there are two lines: blue and green
- there are large peaks and valleys

“What else?”:

- the lines are more similar at the ends
- the maximum points are evenly spaced
*(“periodic” even!)* - the minimum values are decreasing
- there are small spikes between the dips
- the blue line is thicker than the green line

I asked “What do you wonder?” and recorded their wonderings:

- ECG? Stock market?
- is the scale of the
*x*-axis seconds, minutes, hours, days?*(“… because the*x*-axis is always time.” — Geoff Krall)* - are the lines related?
*(Ha! Spurious correlations?)* - what’s the
*y*-axis? - what’s causing the peaks and valleys?

I gradually provided answers to some of their questions. “The lines *are* related. The green and blue lines represent two successive days, February 27 and 28. The horizontal axis is time of day from noon to six.”

I added a critical piece of information: “February… *2010*.” That was enough for some to blurt out “Olympics!” or “Hockey!” I gave them time to talk at their tables. There was a natural transition to “What’s going on?” before I asked the question (just as there’s a natural transition from noticings to wonderings). You can read the graph’s story here.

In the second half, I invited one teacher at each table to try out the strategy. I provided these facilitators with a rough script (anticipated noticings and wonderings, what to reveal and when, the complete graph with all its bits and pieces). They shared the following stripped graph with their colleagues:

“What do you notice?”

- junk foods bottom left, healthy foods top right
- there’s a diagonal line (or
*y*=*x*) - some foods are farther away from this line than others
- there are arrows pointing to two foods

It is key that participants notice the line as it implies two variables (that are largely similar). If what’s going on is just a measure of the healthiness of foods, a continuum (a.k.a. a number line) is all that is needed. A coordinate system is overkill. There were some related false starts here (e.g., fat, sugar). I encouraged facilitators to let that play out. Groups self-corrected. The arrows indicate where text boxes have been removed. Learners were told that they could choose to revisit the “stories” of these foods later.

“What do you wonder?”

Facilitators gradually provided answers to their group’s questions. For example, if participants were told that the *y*-axis represents the percent of *nutritionists* saying a food is healthy, they reasoned that the *x*-axis represents another group: the public (or all Americans).

“What’s going on in this graph?”

Groups revisited granola bars and quinoa. They discussed possible reasons for differences in opinions between nutritionists and the public. They noticed and wondered some more.

When planning, I considered slowly providing this information in a series of slides. To learn more about this approach, see Brian’s post.

On Monday, I was invited to Fraser Heights Secondary to be part of their professional development day. I chose to lead teachers through this activity. (Last week, I test drove it with Surrey Math Department Heads.) Teachers–from all departments–were engaged in the activity. Not because of the “real-world” context but because “we *had* to figure out what was going on.” A *math* activity.

My plan was to connect this activity to core and curricular competencies. Also, I wanted to ask teachers to consider where–in *their* discipline, in *their* practice–they could remove (and gradually provide) information. I wanted to ask teachers to consider where they could ask their students “What do you notice? What do you wonder? What’s going on?” I rushed that. “Never skip the close!” I hope that challenge didn’t come across as hand-waving.

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The graphics nicely–and quickly!–illustrate why this strategy works. Starting with 1 × 20 (one football field twenty feet tall), if we double the first factor (area in football fields) and halve the second factor (height in feet), the product (volume in piles of nuclear waste), expressed as 2 × 10, remains the same. Similarly, we can halve and double to visualize that 1 × 20 is equivalent to ½ × 40. (Oliver also throws in the commutative property at the end–twenty football fields one foot tall.)

This reminded me of a video clip from Sherry Parrish’s *Number Talks*. In it, the teacher poses the problem 16 × 35. The fifth graders share several strategies: partial products (10 × 30 + 10 × 5 + 6 × 30 + 6 × 5); making friendly numbers (20 × 35 − 4 × 35); halving and doubling (8 × 70); and prime factors (ultimately unhelpful here).

I’ve probably shared this video in about a dozen workshops. There are some predictable responses from attendees. Often “not *my* kids” is the first reaction. I remind teachers that the teacher in this video has implemented this routine three to five times a week in her classroom. This isn’t her kids’ first number talk. Pose 16 × 35 in your fifth–or ninth!–grade classroom tomorrow and, yeah, the conversation will probably fall flat. Also, this teacher is part of a schoolwide effort (seen in other videos shared at these workshops).

Teachers are always amazed by Molly’s halving and doubling strategy. Every. Single. Time. I ask attendees to anticipate strategies but they don’t see this one coming. I note that doubling and halving wasn’t introduced through 16 × 35. I would introduce this through a string of computation problems (e.g., 1 × 12, 2 × 6, 4 × 3). “What do you notice? What patterns do you see? Does it *always* work? Why?” We can answer this by calling on the associative property: 16 × 35 = (8 × 2) × 35 = 8 × (2 × 35) = 8 × 70 above. Better yet, having students play with cutting and rearranging arrays provides another (connected) explanation.

Rather than playing with virtual piles of nuclear waste, I had fun with arrays of candy buttons:

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Two different mini donut vendors, two different sets of prices. I wondered “What’s the best deal?” As much as I love asking students “What do you notice? What do you wonder?” when introducing problems–see this introduction to “I Notice, I Wonder” from The Math Forum— I’m thinking about skipping this routine here and just presenting the context and problem (using these photos). Let me explain that decision later in this post.

I love best buy problems because they lend themselves to multiple strategies. *From students*, not “let *me* show you six different ways to solve these.” For example, I anticipate that many–most?–students will determine and compare unit rates. It’s an intuitive thing to do. (Or not. See Robert Kaplinsky’s discussion of his Carnival Tickets task.) At *FunDunkers*, it’s $12 for 3 bags, or $4 per bag; at *those little DONUTS*, it’s $10 for 2 bags, or $5 per bag. Winner: *FunDunkers*. Students may also think common multiples (or scale up). At *Fundunkers*, it’s $12 for 3 bags, so $24 for 6 bags; at *those little DONUTS*, it’s $10 for 2 bags, or $20 for 4 bags, or $30 for 6 bags. We can easily compare ratios or rates when one term is the same, be it one bag or six.

After having some students present their solutions, I’d display these photos…

… and ask students if they’d like to revise their solutions. Now, students will likely determine and compare new unit rates. “One” has changed: dollars per one *donut *instead of dollars per one *bag* (#unitchat). Here some students may also consider one *dollar* to be the unit (and avoid fractions or decimals in doing so). At *FunDunkers*, it’s 36 donuts for $12, so 3 donuts per dollar; at *those little DONUTS*, it’s 45 donuts for $15, so 3 donuts per dollar. A tie. Less interesting than a reversal but, hey, “real world” numbers.

I like the teacher move of gradually providing students with new information and asking them if they’d like to revise their thinking. (It’s a strategy I used with Sinusoidal Sort and “Selfiest” Cities.) Not all the time. But in this task, if students wonder how many donuts are in each bag, then you kinda have to provide this up front. This means that we might not get the dollars per bag idea on the table at all–a missed opportunity to compare and connect strategies.

(Anyone else notice my donut hole-like tunnel vision in that last *FunDunkers* photo? One step back and maybe there’s a math task involving souvenir cups and pop refills.)

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I have to reach waaay back to late August/early September.

One day, she came home from Michael’s with several new bottles of paint.

“They were two for a buck fifty, so I got twelve. Twelve!”

I asked her how much she paid.

“Twelve bucks.”

I shot her a look. No poker face.

“Okay, okay,” she said and began to figure it out, spreading out the bottles in pairs on the lawn.

Then, she joined two pairs. “Two dollars and… three.”

She made two groups of four from the remaining four groups of two.

“Three, six, nine. Nine dollars!”

I’ve tagged this post “#tmwyk” (“talking math with your kids”). That’s generous, I know. I took “less helpful” to heart. Bordering on no help at all. But hey, it was summer. “I’m off the clock, kid.”

As a result, Keira developed a strategy. It’s *hers*. In joining two pairs of bottles of paint, she dealt with dollars before considering cents; she started with the part of the quantity that’s more important–the larger part. After joining two pairs, she knew that $3 was easier to work with than $1.50. Friendlier. She can skip-count by threes. If I got a do-over? More paint. Would Keira have skip-counted to $12 (16 bottles) or would she have doubled $6 (8 bottles)? In about five years, she’ll be expected to set up a proportion–and set aside her intuition!–or calculate a unit price to practice similar textbook exercises.

At the time of this conversation, I was also reading up on proportional reasoning. I noticed that Keira was “attending to and coordinating two quantities”: the number of bottles of paint and the amount of money.

Previously, I shared my thinking about planning a proportional reasoning unit using the KDU model. In that post, I came up with some elaborations to flesh out the MoE’s open/vague content standards. I missed the “attending to and coordinating two quantities” thing. Later in the year, I attempted Dan Meyer‘s Nana’s Paint Mixup math task in a Mathematics 8 classroom. It flopped. For several reasons. For one, I had taken for granted that students had understood that the problem involved two quantities. Paint was being added willy-nilly. I could have asked “What quantities can be counted/measured?” I didn’t.

Fast-forward one week…

This surprised me. I mean, here she is days before proudly wearing a t-shirt that she picked out for back to school.

Keira identifies as a mathematician. And author, and artist, and athlete, and engineer. When I created the Which One Doesn’t Belong? sets here, Keira was my go-to. She loved the challenge of identifying (at least) one reason why each image in a set didn’t belong. It didn’t matter that the content came from high school.

(John Stevens tells a similar story in his new book, *Table Talk Math*. Highly recommended!)

“I’m nervous about new Grade (Math)” didn’t add up, so I asked Keira about it. “You have to add and subtract big numbers,” she said. This is a kid who wrote several stories over the summer, such as The Magical adventures of the Fruitimals and the Food Fight, in which the plot could be summarized as strings of two-digit subtraction problems. For *fun*.

Sometimes, there can be a disconnect between mathematics at school and mathematics at home. This is not one of those times. Keira described *schoolmath* in terms of calculations. Her description of *homemath*–at least here–wouldn’t be markedly different. We’re talking about arithmetic, not ideas about infinity.

For the record, I don’t believe that Keira was nervous about Grade 3 math. Rather, she has picked up on peoples’ perceptions of mathematics: math is something that is okay to be nervous about.

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A few years ago, I completed a questionnaire to determine my personal operating style. I’m green. Creativity. At first, I questioned the validity of the assessment. I didn’t see myself as creative. I’m not… *artsy*.

But taking a closer look, the results made sense. I scored very highly in the four strategies that made up creative thinking in this system: brainstorm ideas, challenge assumptions, reframe problems into opportunities, and envision possibilities. To be clear, this was an assessment of *preferences*, not *proficiencies*.* *Also, there are trade-offs; to choose one thing is to reject another. For example, my 98 in *reframe* and 91 in *envision* meant zero — zero! — in *tune-in to feelings* and ten in *empathize with others*. These results did not suggest that I *can’t* tune-in and empathize; they did suggest that I don’t *want* to. Preferences, not proficiencies.

More important to this post, this assessment tool offered a different definition of creativity: “the generation of a wide variety of options, ideas, alternatives and fresh ways of approaching difficult situations and everyday challenges.” BC’s Ministry of Education defines creative thinking, one of the core competencies, as “the generation of new ideas and concepts that have value to the individual or others, and the development of these ideas and concepts from thought to reality.” There are similarities between these two definitions: both talk of the generation of novel ideas; neither talk of art.

The MoE also has this to say: “Core competencies are evident in every area of learning; *however, they manifest themselves uniquely in each discipline.*”

Over the last few years, I’ve sat through many presentations where examples of creative thinking across subject areas have been shared. The examples from mathematics almost always make me cringe. The math song is a common offender. (Usually the topic tips towards the procedural — BEDMAS, the quadratic formula, etc. — but that’s a different post.) Here, creative thinking manifests itself *outside* of mathematics. It happens *in* language/fine arts. (Maybe. Talk to a language/fine arts teacher.) You can substitute provincial capitals for divisibility rules and the nature of creative thinking within the task remains unchanged. Math is merely the context.

Worse, the message is that math is unappetizing in and of itself. Broccoli. The cheese sauce that is the math song (or poster, or skit, or diorama, or …) comes at a cost. Limited time means tension — time spent on products versus time spent solving interesting problems and having interesting conversations. Note: in my mind, the opportunity cost isn’t coverage of content; it is engaging students in the “doing” of mathematics.

Yesterday, I attended a meeting where the MoE repeated the message: “By doing the curricular competencies, students will be developing their core competencies.” The math song attempts to have students develop a core competency without doing the curricular competencies.

The connection between creativity and art is strong:

I’d like to suggest a better title:

Creative *Math* is clearly evident. Just click on one of the staff picks and look to the left. Focus not on the equations themselves, but on the thinking behind them. Not on “front mathematics,” but on “mathematics in back.” (A lovely metaphor from Reuben Hersh that I first came across in Tracy Zager’s *Becoming The Math Teacher You Wish You’d Had.)*

To most math teachers, this title makes no difference. Just me nitpicking. But it matters where teaching includes designing curriculum/learning experiences. If teachers think of creativity in terms of *art*, they may look to Pinterest when planning; if they think of creativity in terms of *ideas*, they may dive deeper into Desmos.

Last year, one of my highlights was being invited into a classroom to observe Marbleslides: Lines in action.

I observed students experimenting with new ideas by changing the variables one at a time. They asked “what if…” questions. They made — and checked — predictions. “New ideas” here means new to the students themselves. These new ideas had value, evident in cheers and high fives. “Right here, right now” value, not “real-world,” career, or “when you take Calculus” value.

(The Desmos Teaching Faculty designed the activity with students in mind who were familiar with equations for lines in slope-intercept form and the idea of domain. In the classroom that I visited, the students were not. We worried that introducing restrictions on the domain at the same time as slope-intercept form would overcomplicate things. It didn’t.)

I don’t fault my fellow educators for associating creativity with art. It’s a natural thing to do. We in mathematics education need to articulate better what creative thinking looks like in mathematics. I’ve had some success in asking teachers to sort curricular competencies by core competency. (Here they are, in random order. Venn diagrams work nicely; I let that idea come from teachers themselves.)

There’s still the leap required to go from making these connections to designing curriculum/planning learning experiences with these connections in mind. Rather than listing activities that elicit creativity, like Marbleslides above, it may be helpful to think about the attributes of these tasks.

Marbleslides is immediately accessible and highly extendable (“low floor, high ceiling”). It invites a wide range of responses (multiple *solutions*). (The teacher can view novel solutions at a glance on the dashboard.) Open questions, like *Which one doesn’t belong?*, share these attributes, as does Quarter the Cross.

A rich task can have a single solution, but invite a wide range of approaches (multiple *strategies*). To me, this has less to do with the task/problem itself and more to do with pedagogy. A curriculum that values creative thinking has pedagogical implications. Consider a typical *What’s the best deal?* task. A step-by-step-worked-examples-now-you-try-one approach to teaching leaves little room for creativity. The strategy — calculate and compare unit prices — is predetermined. What if students were presented with the problem before the strategy? The class would generate several different ways to solve the same problem. They’d see and discuss a wide range of ideas. Note: this doesn’t preclude the teacher from later bringing a particular strategy (e.g., compare unit prices) to the conversation, if necessary. Ask “Why does this strategy make sense?” or “What’s the best strategy?” and students develop another core competency: critical thinking.

In her ShadowCon talk, Tracy Zager shared a word cloud generated from the language mathematicians use to describe their work. *Creative* sticks out. And *invent*, *curiosity*, *play*, *imagination*, *wonder*, etc. The image generated from the language society/teachers use to describe math… not so much.

But I know that there are places where *school* math is creative. In Surrey Schools (#sd36learn), in the “MathTwitterBlogoSphere” (#MTBoS), and beyond. When I wrote “We in mathematics education need to articulate better what creative thinking looks like in mathematics” above, I really meant “*I* need to articulate better…” So, I need your help. Did I get it right in this post? Artsy or not, what does *mathematical* creativity look like in your classroom?

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“You clarify and share learning intentions and success criteria. You implement rich tasks that elicit evidence of student thinking. You pose questions that cause thinking.”

I presented teachers with four sample student responses to the following question:

A store sells a box of nine 200 g bags of chips for $12. How much should the store charge for twenty-four 200 g bags?

I asked teachers to consider (1) Where is the learner going? (2) Where is the learner right now? and (3) How can the learner get to where she needs to go?

This sparked some interesting conversations. The students in the top left and top right know that a unit price is an equivalent rate where one term–number of bags in TL, dollars in TR–is one. The student in the bottom left also knows that proportion problems can be solved by looking for a scale factor–albeit an inaccurate one–between ratios. What’s going on with the student in the bottom right? What’s the learning goal in terms of *content*? What’s the learning goal in terms of *curricular competency*? This activity was preceded by a conversation about the KDU model, so teachers were thinking “use multiple strategies” and “communicate mathematical thinking.” Is it fair to consider “use multiple strategies” using this–or any single–task as evidence? (A good time to bring up triangulation–products, observations, conversations with students.) What does “good” communication look like in mathematics? Do the bottom two responses need words? Would a ratio table help answer what’s going on in the bottom right?

While this was a worthwhile exercise, this answer was “not yet meeting expectations.” One reason for this is that *assess* is often a euphemism for *evaluate*. Or *grade*. Or *report*. As a student teacher, my school associate once asked me how I planned to assess. I began to tell him about upcoming quizzes. “That’s all well and good, but that’s *evaluation*. Minute-by-minute, day-by-day, how will you know they know?” This has been helpful for me as I’ve navigated through assessment by preposition (*assessment of*, *for*, or *as learning*) and “Is this formative or summative?”

Answering with another question is probably unsatisfactory, but, to me, *what* is a much more important consideration than *how*.

The Ministry of Education released the following in the summer:

At the end of the school year or semester, Boards must provide a written summative report to parents that address the student’s progress in relation to the

learning standardsof the curriculum in all areas of learning as set out in the Required Areas of Study Ministerial Order.

(Emphasis added.)

Learning standards in BC’s curriculum are made up of curricular competencies (“what students are expected to *do*“) *and* content (“what students are expected to *know*“). (#MTBoS, think practice and content CCSS-M standards.) As late as June, some teachers were still wondering if there would be a requirement to assess–or evaluate? or report on?–the curricular competencies. To me, the MoE’s choice of “learning standards” makes this clear.

At the same time, there’s another message out there: *learning standards* and *curricular competencies* are synonymous. The gist of this idea is that content is interchangeable. And maybe that’s more true in other areas of learning. (I still take issue with “If you enjoy teaching ancient Egypt and ancient Egypt has moved, then you can still teach ancient Egypt” but social studies isn’t the hill I’ll die on.) And I’m all in favour of a greater emphasis on students *doing* mathematics. Helping teachers make this happen is my work–it’s what I (try to) do. Still, I’m baffled.

Of course, nobody argues that process and content exist without one another other. In the classroom, “I can use multiple strategies to solve problems involving ratios and rates” or “I can communicate my thinking when solving proportional problems” work as learning intentions. I can design learning experiences around these. My question is about evaluating: together or separately? Consider the student in the bottom right above. If she “fully meets expectations,” or is “proficient,” or is a “Jedi Knight,” it’s easy–the learning intentions above still work. But if she, as most agreed, isn’t, then why is that? My take is that she is proficient with respect to content (proportional reasoning)–or, at least, here’s one piece of supporting evidence–but not quite there yet with respect to competency (communicate thinking). What are some implications surrounding reassessment? And is it possible to fully meet with respect to competency without also possessing a deep level of content knowledge?

I’m beginning to enter the Land of the Gradebook, which, nine times out of ten, is at the heart of teachers’ “How do we assess it?” Standards-based grading, depth of knowledge, learning maps, rubrics, portfolios, etc. will be part of part two.

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