[BC’s Curriculum] “Know-Do-Understand” Model

This year, BC teachers (K-9) implement a new curriculum. For the past two years, much of my focus has been on helping teachers–in all subjects–make sense of the framework of this “concept-based, competency-driven” curriculum. This will be the topic of these next few posts.

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.

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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 oversimplifies 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-for-blog

kdu-proportional-reasoning.pdf

I wonder if the KDU model overcomplicates 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.

 

On Pace

Act 1

Steph Curry On Pace Headline Retouch

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.

history-making historically dominant

Act 2

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:

Steph Curry On Pace Graph 1:2I cropped the infographic because it resolves an extension (see it from the waist down below). And because it’s too damn long.

Act 3

Steph Curry On Pace Headline

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?”

Steph Curry On Pace Graph 2:2

Source: http://www.cbssports.com/nba/eye-on-basketball/25386027/steph-curry-3-point-tracker-on-pace-for-404-makes-in-2015-16

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)

 

Fair Share Pair

A couple weeks ago, I was discussing ratio tasks, including Sharing Costs: Travelling to School from MARS, with a colleague who reminded me of a numeracy task from Peter Liljedahl. Here’s my take on Peter’s Payless problem:

Three friends, Chris, Jeff, and Marc, go shopping for shoes. The store is having a buy two pairs, get one pair free sale.

 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.

Sharing Pairs.pdf

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.

Sharing PairsWhen 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

Sharing Pairs - Sample Student Response A

Sharing Pairs – Sample Student Response A

Sharing Pairs - Sample Student Response B

Sharing Pairs – Sample Student Response B

Sharing Pairs - Sample Student Response C

Sharing Pairs – Sample Student Response C

Sharing Pairs - Sample Student Response D

Sharing Pairs – Sample Student Response D

Sharing Pairs - Sample Student Response E

Sharing Pairs – Sample Student Response E

Cola Comparison

Coke is now sold in 20, not 24, packs!

Coke 20 (2)Coke 12 (2)Pepsi 24 (2)So to determine the best buy, I couldn’t just double. I use that strategy all the time; it’s my Frank’s RedHot. The exclamation point is there because I think that 20 leads to more strategies than 24. (Some of) these strategies are listed in my 5 Practices monitoring tool below. I’m curious if you think that I have anticipated likely student responses correctly. What incorrect strategy could I have anticipated? I wonder how you’d purposefully sequence these responses during the discussion.

0001ar

More than SWBAT solve problems using unit rates, I want my students to recognize that there are many ways to solve rate problems and understand that we can easily compare rates with one term the same. This big ideas connects the strategies. In the fourth strategy above, we can think of 24 cans as a unit. Call it a “two-four” (Is that just a Canadian convention?) or a “flat” (Are we cool with calling the Pepsi cube a flat?). In fact, Save-On-Foods wants us to think of 24 as one; we’re encouraged to buy two packs of 12, a composed unit. For this task, I’d prefer that they didn’t, so I went back to the store and found this:

Cola 12

Comparing 20 packs with 15 packs is more likely to lead to common multiples than comparing 20 packs with 24 packs as above. Numbers matter. There’s this, but it doesn’t get us a clear winner:

Pepsi 15

Recommended: Dan knocks motivating unit rates out of the park; Christopher asks “What is one?” 

May 13, 2016: de La Cruz, Jessica and Sandra Garney. 2016. “Saving Money Using Proportional Reasoning.” Mathematics Teaching in the Middle School 21 (9): 552-561.

World’s Worst Person In Sports

Last week, Keith Olbermann named the Canucks’ Tom Sestito “World’s Worst Person In Sports.” In a game against the Kings, Sestito racked up 27 penalty minutes. His total ice time for the night? One second.

27:00 to 0:01 is an impressive stat. It’s hard to imagine this being surpassed. Sure, twenty-seven minutes can be topped. Randy Holt holds the NHL record for most penalty minutes in one game (67). The NHL record for most penalties in one game (10) belongs to Chris Nilan. But to do so in one second?! Inconceivable.

“I’d describe [Sestito] as a hockey player except he’s not,” Olbermann says. To make this point, he goes on to compare Sestito to Gretzky. That’s right: “The Great One” is his hockey player/”boxing hobo on skates” referent. In 101 games, Sestito had scored 9 goals, 885 shy of Gretzky’s record. Olbermann notes that Sestito would have to play about 10 000 games, or 123 seasons, to break the NHL record. Well, yeah, assuming he can keep up this pace.

I considered giving this the three-act treatment and bleeping Olbermann. But “When will Sestito break Gretzky’s record?” is not the first question that comes to your mind, is it? A more natural question re: Sestito might be “How many seasons would Sestito have to play to break Dave “Tiger” Williams’ record of 3966 career PIMs?” Apples to apples.

Olbermann, 54, followed this up by feuding with Tom Sestito’s sister, 13, on Twitter. Nice use of a unit rate by the kid: