Ovechkin Contract Extension Ruins Teacher’s Mathematical Modelling Problem

The puck drops next week on the 2021-22 NHL season. As a hockey fan and mathematical modelling aficionado, I’m looking forward to watching Alex Ovechkin chase Wayne Gretzky’s goal record–a record I used to think was unbreakable. I was in good company:

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TSN posted the above back in November of last year. Since then, Ovi went on to do two things:

score 24 more goals (in a COVID-shortened season)
sign a five-year, $47.5 million contract extension with the Caps

The question is less “Will ‘The Great Eight’ catch ‘The Great One’?” and more “When will he do it?” Still fun to watch but not as interesting as “Will he or won’t he?” It was uncertainties–since made certain by the offseason signing–that made it interesting, made it open to debate. There’s a takeaway for math class here.

Permit me to time travel. It’s November, 2020. Our known knowns, at that time:

Wayne Gretzky is the leading goal scorer in NHL history
Gretzky played in 1487 games (20 seasons)
Gretzky scored 894 goals in his career
Gretzky retired in ’99; he was 38
Alex Ovechkin has played in 1152 games (15 seasons)
Ovechkin sits at 706 goals
Ovi’s contract will expire at the end of the 2020-21 season; he will be 35

Known unknowns:

Ovechkin’s future goal-scoring pace
years remaining in Ovi’s career
where those years will be spent (i.e., NHL or KHL)
Ovi’s future health and fitne ss

In BC, teachers–in all areas of learning!–are being asked to embed numeracy tasks in their classrooms. Numeracy tasks are solved using five processes–Interpret, Apply (Mathematize), Solve, Analyze, and Communicate. Outside of BC, this is called the mathematical modelling cycle.

Interpret

Here’s how BC defines Interpret:

Students are able to read and decode a range of situational contexts by identifying the real-world problems to be solved. Given insufficient or excess information, students will need to decide what information is relevant to solve the problem. This process is about students making decisions. Situational contexts may require students to identify constraints and ambiguities, and decide on next steps.
https://curriculum.gov.bc.ca/sites/curriculum.gov.bc.ca/files/pdf/assessment/gna-10-specifications.pdf

At this stage, students must understand the scenario and determine what the important variables are. Students must also make–and state–assumptions. Here, they are required to make assumptions about my unknowns above. Unfortunately–for teachers not hockey fans–Ovechkin’s new contract resolves some of these ambiguities.

Apply (Mathematize)

Students are able to identify and activate their mathematical understanding by translating real-world problems into mathematical problems (mathematizing). This process involves choosing a mathematical tool, determining how to organize the information, and creating relationship(s) in order to represent the real-world problem. (Students will need to flexibly use mathematical tools for a host of real- world problems.)

Here, that mathematical tool is likely averages/unit rates. “Likely” because students make the call about what calculations or methods to use. They may decide to use linear functions, which gives a clearer picture of the problem:

Solve

Students are able to solve mathematical problems through a variety of approaches and representations. Students may also need to check mathematical solutions to determine if their solutions make mathematical sense.

Ovechkin trails Gretzky by 188 goals (894 – 706). Again, talking November, 2020. We can play with possibilities. For example, if Ovi goes on to play four more seasons, then he’d need to score more than 47 goals per season. Or, if he continues to score at about 0.6 goals per game, then he’d need to play in a little over 300 more games.

(For readability, I’m jumbling Solve and Analyze. Solve looks more like $\frac{894-706}{4}$ or $188 = 0.6x$ . Math minus context.)

Analyze

Students are able to interpret mathematical solutions in context, such that the solutions are reasonable within the situational contexts. Students may need to assess the practicality and possible limitations of solutions, identify possible improvements to an approach, or identify other situations to which solutions can be applied. In doing so, students consider how contextual factors may affect the results. For example, students may reflect on their solutions to assess risks and address social, ethical, and environmental implications.

Students do not need to be hockey fans to check if their mathematical solutions make sense in this context. A search on NHL dot com and some familiarity with scoring being part of sportsball will suffice. You don’t have to be Chi-Chi Rodríguez to approach Fawn’s putt-putt problem.

Communicate

Students are able to clearly and precisely construct valid logical arguments to defend their decisions and assumptions, explain the tools and approaches they used, and present their solutions in context. This may require students to make recommendations and use a variety of ways (e.g., tables, graphs, diagrams, equations, symbols) to visibly represent their thinking and solution.

Some students might recognize the flaws in considering averages, especially for a skater approaching 40. For example, suppose Ovechkin plays six more seasons and averages 32 goals per season. At first glance, this feels reasonable. But it hides that he will need to outperform the average at the beginning to offset a decline at the end. It might be better to consider–and discuss–the set {42, 40, 35, 30, 25, 20}.

* * ** *** ***** ******** *************

“Ruins” in this post’s title might be a tad rough. Taking away students’ control of an important variable–the number of seasons Ovechkin will play–might push this from numeracy/modelling task to word problem (if one skate wasn’t hanging the line before). It limits how they can tinker with relationships between quantities. Still, there’s space for multiple strategies and justifications (e.g., goals per season or goals per game); there’s an opportunity for students to develop mathematical autonomy.

Alex Ovechkin is already the greatest goal-scorer in NHL history. Attaining the record is beside the point. This argument demands a different mathematical model: era-adjusted goals. But that’s for another time…

[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.

November 25, 2015September 5, 2020

On Pace

Act 1

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.

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:

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

Act 3

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?”

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)

This was the 3rd time in 4 years that Steph Curry broke the record for threes in a season pic.twitter.com/8LGr0cDUJc

— ESPN Stats & Info (@ESPNStatsInfo) May 10, 2016

October 27, 2014September 20, 2020

Cola Comparison

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

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.

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:

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:

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.

January 20, 2014

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:

https://twitter.com/vsestito32/statuses/423291409937235968

Tag: rates