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:

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

  1. score 24 more goals (in a COVID-shortened season)
  2. 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:

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:

desmos.com/calculator/mok03hi6oe

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…

Wolf, N. B. (2015). Modeling with mathematics: Authentic problem solving in middle school. Heinemann.

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!