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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In this series on curriculum, I’ll do my best not to use *curriculum*. There is no agreed upon definition. I imagine that if any educator in the “MathTwitterBlogoSphere” (#MTBoS) followed the link above, she’d be shouting “Those are *standards*, not curriculum!” Similarly, when #MTBoS folks talk about adopting curriculum, I’m shouting “That’s a *resource*, not curriculum!”

My union makes the following distinction: “Pedagogy is how we teach. Curriculum is what we teach.” *Curriculum* as *standards*. For the most part, this jibes with how *curriculum* is used in conversations with colleagues and is echoed in this Ministry of Education document. But Dylan Wiliam doesn’t make this distinction: “Because the real curriculum – sometimes called the ‘enacted’ or ‘achieved’ curriculum – is the lived daily experience of young people in classrooms, curriculum *is* pedagogy.” *Curriculum* as *experiences*. Or pedagogy.

Rather than *curriculum*, I’ll try to stick with *learning standards*, *learning resources*, or *learning experiences*.

Three elements–Content, Curricular Competencies, and Big Ideas–make up the “what” in each subject and at each grade level. Last summer, the Ministry of Education simplified this as the “Know-Do-Understand” (“KDU”) model. The video below describes how content (what students will know), curricular competencies (what students will do), and big ideas (what students will understand) can be combined to direct the design of learning activities in the classroom.

I imagined planning a proportional reasoning unit in Mathematics 8 using the KDU model and shared my thinking throughout this process.

Click to view slideshow.Teachers can start with any of the three elements; I started by identifying content. (It’s a math teacher thing.) Then, I paired this content with a big idea. In English Language Arts and Social Studies, it makes sense to talk about you, as the teacher, making decisions about these combinations. In Mathematics and Science, this mapping is straightforward: algebra content pairs with a big idea in algebra, not statistics; biology content pairs with a big idea in biology, not Earth sciences. (BC math teachers may notice that the big idea above is different than the one currently posted on the Ministry of Education website. It may reflect a big idea from a previous draft. I can’t bring myself to make that change.)

Identifying curricular competencies to combine with content and big ideas is where it gets interesting. Here, my rationale for choosing these two curricular competencies was simple: problems involving ratios, rates, and percent lend themselves to multiple strategies… we should talk about them. The video makes the point that I could go in the opposite direction; if I had started with “use multiple strategies,” I likely would have landed at proportional reasoning. Of course, other curricular competencies will come into play, but they won’t be a focus of this unit. This raises questions about assessment. (More on assessment in an upcoming post.)

Note that “represent” is missing from my chosen curricular competencies. Why is that? My informed decision? Professional autonomy for the win? Or my blindspot? A teacher who sees proportional reasoning as “cross-multiply and divide,” who is unfamiliar with bar models, or double number lines, or ratio tables, or who sees graphs as belonging to a separate and disconnected linear relations chapter wouldn’t think of connecting this content to “represent.” Making connections between these representations is an important part of making sense of proportional reasoning. Will this build-a-standard approach mean missed learning opportunities for students? This speaks to the importance of collaboration, coaching, and curriculum, er, I mean quality *learning resources*.

In early talks, having these three elements fit on one page was seen as a crucial design feature. Imagine an elementary school teacher being able to view–all at once!–the standards for nine different subjects, spread out across her desk. As a consequence, the learning standards are brief. Some embraced the openness; others railed at the vagueness. In some circles, previous prescribed learning outcomes are described using the pejorative “checklist”; in others, there is a clamouring for “limiting examples.” (Math teachers, compare these content standards with similar Common Core content standards.)

I wonder if the KDU model *over*simplifies things. If you believe that there is a difference between *to know* and *to understand*, then you probably want your students to *understand* ratios, rates, proportions, and percent. For a “concept-based” curriculum, it’s light on concepts. Under content, a (check)list of topics. To that end, I fleshed out each of the three elements (below). But I have the standards I have, not the standards I wish I had. (Free advice if you give this a try: don’t lose the *that* in that stem below.)

kdu-proportional-reasoning.pdf

I wonder if the KDU model over*complicates* things. Again, U is for what students will *understand*. But “understanding” is one of the headers within the D, what students will *do*.

Despite this, I have found the KDU model to be helpful. In particular, it’s been helpful when discussing what it means to *do* mathematics. The math verbs that we’re talking about are *visualize*, *model*, *justify*, *problem-solve*, etc., not *factor*, *graph*, *simplify*, or *solveforx*. Similar discussions take place around *doing* science (scientific inquiry) and social studies (historical thinking).

More broadly, the model has been helpful in making sense of the framework of our new curriculum, or *standards*. It’s a useful exercise to have to think about specific combinations–far more useful than:

Q: “Which competencies did we engage in?”

A: “All of ’em!”

We’re still some distance from “the lived daily experience of young people in classrooms” but it isn’t difficult to imagine learning experiences in which this specific combination of the three elements come together.

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The retouched headline is designed to have students ask “How many 3-pointers will Stephen Curry make this season?” There are related questions: “At what pace (*rate*) is Curry making 3-pointers? What makes this pace historically ridiculous? What’s the difference between a historically ridiculous pace and a ridiculously historic pace?”

Here’s the thing about historic paces: historically, they happen weekly.

I retouched the first sentence in the article to open things up a bit. Pre-edit: “We’re nearly through *20 percent* of the 2015-16 season…” Only the number of 3-pointers made to date (74) is needed. We don’t need to know the number of games played to date (15) or the number of games played in an NBA season (82). That’s the point of percent: fanatical comparison to 100. (I wonder if students would ask for this superfluous information anyway.) Post-edit, this information might, in fact, be useful to know. And help draw out multiple strategies. Perhaps students will ask for a fraction, rather than a percent, to fill in the blank. Games played and 3-pointers made to date can be determined from the following graph:

I cropped the infographic because it resolves an extension (see it from the waist down below). And because it’s too damn long.

The article suggests two possible extensions: “How many 3-pointers does Steph Curry need per game *remaining* to reach 300? How many games will this take?”

**April 7, 2016:** Steph Curry Is On Pace To Hit 102 Home Runs

**May 11, 2016:** 3-Point Tracker — 2015-16 Season

**May 11, 2016:** Misleading y-axis (h/t Geoff Krall)

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If you *do* know the four digits, how many combinations¹ could there be?

Students may ask to see the four digits.

Remember to ask later if this information matters. That the digits are 1, 3, 4, 5 doesn’t; that there are four different digits — no repetition — does.

My hope is that this resolution feels sort of anticlimactic — that Raymond Reddington’s “Now there’s only twenty-four combinations” on the screen doesn’t measure up to students’ shared strategies in the classroom.

Elizabeth Keen’s “Could be thousands of combinations” prior to Red’s sand trick could be an extension. At first viewing, it seems far-fetched that the character — an FBI profiler — doesn’t understand that there are exactly ten thousand four-digit possibilities (0000, 0001, 0002, …, 9999). But has Liz assumed that the digits cannot repeat? If so, how many combinations could there be? Students can no longer answer this question by systematically listing and counting each possibility.

I imagine this task as an introduction to, not an application of, permutuations. It provides a context for students to develop — not practice! — methods of counting without counting. Don’t bother if you’re anticipating a lot of knee-jerk 4!s from your students.

¹I know, I know… *permutations*.

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Three friends, Chris, Jeff, and Marc, go shopping for shoes. The store is having a

buy two pairs, get one pair freesale.Chris opts for a pair of high tops for $75, Jeff picks out a pair of low tops for $60, and Marc settles on a pair of slip-ons for $45.

The cashier rings them up; the bill is $135.

How much should each friend pay? Try to find the fairest way possible. Justify your reasoning.

I had a chance to test drive this task in a Math 9 class. I asked students to solve the problem in small groups and record their possible solutions on large whiteboards. Later, each student recorded his or her fairest share of them all on a piece of paper. If you’re more interested in sample student responses than my reflections, scroll down.

The most common initial approach was to divide the bill by three; each person pays $45. What’s more fair than *same*? I poked holes in their reasoning: “Is it fair for Marc to pay the same as Chris? Why? Why not?” Students notice that Chris is getting more shoe for his buck. Also, Marc is being cheated of any discount, as described by Student A. (This wasn’t a happy accident; it’s the reason why I chose the ratio 5:4:3.)

Next, most groups landed on $60-$45-$30. Some, like Student A, shifted from equal shares of the cost to equal shares of the discount; from ($180 − $45)/3 to $45/3. Others, like Students B, C, and D, arrived there via a common difference; in both $75, $60, $45 and $60, $45, $30, the amounts differ by $15. This approach surprised me. Additive, rather than multiplicative, thinking.

Student C noticed that this discount of $15 represented different fractions of the original prices; $15/$75 = 1/5, $15/$60 = 1/4, $15/$45 = 1/3. He applied a discount of 1/4 to all three because “it’s the middle fraction.” Likely, this is a misconception that didn’t get in the way of a reasonable solution.

Student D presented similar amounts. Note the interplay of additive and multiplicative thinking. She wants to keep a common difference, but changes it to $10 to better match the friends’ discounts *as percents*.

Student E applies each friend’s percent of the original price to the sale price. This approach came closest to my intended learning outcome: “Solve problems that involve rates, ratios and proportional reasoning.”

In spite of not reaching my learning goal, I think that this lesson was a success. The task was accessible yet challenging, allowed students to make and justify decisions, and promoted mathematical discourse.

Still, to increase the future likelihood that students solve this problem using ratios, I’m wondering about changes I could make. Multiples of 20 ($100-$80-$60) rather than 15 ($75-$60-$45)? Different ratios, like 4:3:2 or 5:3:2, might help; the doubles/halves could kickstart multiplicative thinking. (Also, 5:3:2 breaks that arithmetic sequence.)

Or, I could make changes to my questioning.

When I asked “What do you notice?” students said:

- the prices of the shoes are different
- Chris’ shoes are the most expensive
- Marc’s shoes are the cheapest
- Chris’ shoes are $15 more than Jeff’s, which are $15 more than Marc’s
- Jeff’s shoes are the fugliest

Maybe I could ask “What else could you say about the prices of Chris’ shoes compared to Marc’s?” etc. to prompt comparisons involving ratios. If that fails, I’m more comfortable connecting ratios to the approaches taken by students themselves than I am forcing it.

BTW, “buy one, get one 50% off” vs. “buy two, get one free” would make a decent “Would you rather?” math task.

h/t Cam Joyce, Carley Brockway

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As a K-12 Numeracy Helping Teacher, I have the opportunity to teach in elementary and secondary classrooms alike. Often, I’m struck by how pedagogical similarities overwhelm any differences. Malcolm Swan’s ordering decimals lesson — a current favourite of mine — and my sinusoidal sort illustrate this.

In Grade 5, students are asked to put the decimal cards in order of size, smallest to largest.

The most common response is 0.4, 0.8, 0.04, 0.25, 0.75, 0.125, 0.375. This can be explained by how students compare whole numbers. Misconceptions are revealed, but not corrected. (The rest of the activity will take care of that.)

In Grade 12, students are asked to put the equation cards in piles.

The most common response is to make two piles: sine and cosine. This can be explained by how students were introduced to *y* = sin *x* and *y* = cos *x*. This isn’t a misconception; there is no “right” sort. But it is unsophisticated. And it goes unchallenged. (The rest of the activity will take care of that.)

In Grade 5, students match hundred grids and number lines to their decimals. They explain how they know that the cards make a set, building connections between decimals and their understanding of fractions and place value. They argue. When there’s consensus, students are again asked to put the cards in order. I note *their* strategies that emerge (benchmarks, place value, equivalent decimals) for a class discussion.

In Grade 12, students match graphs and characteristics to their equations. They make conjectures. They explain how they know that the cards make a set, building connections between transformations of trigonometric functions and their understanding of transformations of other functions. When there’s consensus, students are again asked to put the cards in piles. I note *their* sorting rules that emerge (amplitude, period, phase shift, vertical displacement, range, maximum/minimum values) for a class discussion.

Elementary and secondary lessons need not be as closely aligned as above. It’s not about matching card matching activities. Or parallel “Which one doesn’t belong?” prompts. Or three-act math tasks for K-5. More generally, exploring and discussing ideas, working collaboratively in pairs/small groups, problem solving (and posing!), … you could make a case for students needing these experiences in elementary school because… wait for it… they’ll need it for high school. We want them to get used to it. I hope.

A nagging thought… I’m not convinced that pedagogy preparation is ever a satisfactory response to “So, what will *they* need?” Even if that pedagogy is positive. If we’re going to follow that pathway, then this question demands an answer not in terms of teaching methods but in terms of the mathematical habits of mind (or processes or practices or competencies) that these teaching methods aim to promote. (See my “Affective Domain” concerns in Part 1.)

That’s not to say that I wasted my time with this post. For me, it’s always helpful to think about and present examples of secondary mathematics education as something other than passive. Maybe that — reframing those pedagogical “TNIFHS” conversations with colleagues into something more promising — is the real value here.

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