Okay, So, Um, Mathematical Modelling, You Know

“Okay, so, um, square both sides of…”

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.

Interpret

Mathematical Modeling.002

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.

Apply (Mathematize)

Mathematical Modeling.003

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.

Solve

Mathematical Modeling.004

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?

Analyze

Mathematical Modeling.005

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?

Communicate

Mathematical Modeling.006

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 DesignFair 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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Alike & Different: Which One Doesn’t Belong? & More

I have no idea what I was going for here:

WODB? Cuisenaire Rods

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

WODB? Hundreds 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:

Alike&Different.006

Another that elicits equivalent fractions and place value:

Alike&Different.007

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.

Alike&Different.009

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:

Alike&Different.013

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:

Alike&Different.015

This time, the first and second involve a quotative (or measurement) interpretation of division: groups of (−3) or 3x, not (−3) or 3x 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.

Alike&Different.017

Alike&Different.023

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:

.pdf

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.

 

Paint Splatter Arrays

This isn’t Splat!

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…

Number Talks -- Dot Cards

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:

Paint Splatter Arrays.004

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:

Paint Splatter Arrays.010

4 × 6

Paint Splatter Arrays.028

6 × 6

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

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3 × 7

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

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How confident are you?

Paint Splatter Arrays.033

<shrinks> 6 × 8

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.

Dividing by Decimals & Fractions: Ham & Ribs

I bought a ham. It was touch-and-go there for awhile. As I was picking up and putting down hams of various sizes, I was calculating baking times. My essential question was, can I have this on the table by six? Simultaneously, I was trying to remember if this was partitive or quotative division.

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:

Ham

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.

How Much Does a Ladybug Weigh? Snails &amp; Starling

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?).

How Much Does a Ladybug Weigh? Data

[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:

RibRacks

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.)

The New York Times’ “What’s Going On in This Graph?”

The “Numeracy Helping Teacher” part of my job immerses me in things related to the teaching and learning of mathematics. I’m comfortable in that space. The “Curriculum, Innovation, and Priority Practices Helping Teacher (Guildford/Fleetwood)” part of my job? Not always.

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.

20121009POD-slide-LF4J-superJumboLater, 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.

Flushed (Stripped)

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:

WGOITGraph? Sushi (Stripped)

“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.

Halving & Doubling: Very Fun to Play With

On last week’s Last Week Tonight with John Oliver, John Oliver used the mental math/computation strategy of halving and doubling as a punchline to a news story on nuclear waste.

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:

Number Talks (pdf)

 

Dollars to Donuts

A week and a half ago, I was at The Fair and noticed this:

FunDunkersFunDunkers1thoselittleDONUTSthoselittleDONUTS1Two 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…

FunDunkers2

FunDunkers

thoselittleDONUTS2

those little DONUTS

… 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.)