Open Middle Math

In my previous post, I shared some of the principles that guided Marc and me when creating a series of math videos for parents (Mathematics 6 & 7; 8–10): make it visual, make it conceptual, and make it inviting. In this way, we also set out to make these videos representative of math class. It was our hope that they presented parents with a view into their child’s classroom (“window”). Further, we hoped that Surrey teachers saw their classrooms in what was reflected (“mirror”). In that spirit, several videos in this summer’s collection included an open-middle problem.

In Open Middle Math, Robert Kaplinsky describes what makes a math problem an open-middle problem:

[M]ost math problems begin with everyone having the same problem and working toward the same answer. As a result, the beginning and ending are closed. What varies is the middle. Sometimes a problem’s instructions tell students to complete a problem using a specific method (a closed middle). Other times, there are possibly many ways to solve the problem (an open middle). Problems with open middles tend to be much more interesting and lead to richer conversations.

Robert Kaplinsky
Visual depiction of problems with a closed beginning, open middle, and closed ending from Dan Meyer

This use of open-middle to describe problems has always irked me. There, I said it. To me, open- vs. closed-middle is not a characteristic of a problem itself. Robert argues that a problem’s instructions can close a problem’s middle. Agreed! But I go a step further. There are other ways through which students are told to use a specific method. It’s us. For example, consider a boilerplate best-buy problem. The middle is wide open! Doubling, scaling up, common multiples, unit rates — dollars per can or cans per dollar — and marginal rates are all viable strategies. However, we close the middle when we give this task after demonstrating how to use unit prices to determine best deals (i.e., “now-you-try-one” pedagogy). If students — and teachers! — believe that mathematics is about plucking numbers to place into accepted procedures then they are unlikely to experience the rich “open-middleness” of this task, regardless of its instructions. It’s no accident that the book’s introduction is titled “What Does an Open Middle Classroom Look Like?”

Most of the problems posted on the open middle site involve number — or digit — tiles. But I get why Robert didn’t go with “Number Tile Math.” The boxes in 25 × 32 = ⬚⬚⬚ and 63 − 28 = ⬚⬚ give each a fill-in-the-blanks answer-getting feel. The routine nature misses the problem-solving mark (despite their open middles). So, “open-middle” as an adjective for problems it is. Besides, math class could use more openness, which needn’t come at the end for problems to be interesting and conversations to be rich.

When I look at an Open Middle problem from the site, the mathematical goal of the teacher who created the problem isn’t always clear to me. (The same is true, by the way, of wodb.ca.) What is the deep conceptual understanding that they anticipate their students will develop by working on the problem? What ideas will emerge? What misconceptions might be addressed? Throughout Open Middle Math, Robert describes how Open Middle problems can give us X-ray vision into our students’ mathematical understanding. Similarly, he provides readers with X-ray vision into his thinking during the process of creating these problems. Below, I’ll share a few of the open-middle problems from our video series (plus some that ended up on the cutting room floor) as well as a peek behind the curtain into my thinking.

Polynomials

Towards the end of the Math 10 Factoring Polynomials video, I present two open-middle/number-tile problems. Teachers will recognize these as familiar “find k” problems: For what value(s) of k is x2 + kx − 8 factorable? x2xk? See the answer animations below.

x2 + kx − 8 (solution)
x2xk (solutions)

I think that the number tiles add an element of play to these problems. The tiles are forgiving. Make a mistake? No biggie, just move ’em around. (The decision to show an initial misstep in the first animation above was deliberate.) This upholds our third principle: make it inviting.

These two sample tasks above highlight the role of students’ prior knowledge in solving open-middle problems. My assumption here is that teachers have not “proceduralized” these problems — that students have not been provided with predetermined solution pathways (e.g., “First, list all the factors of the constant term c. Then, …”). Note the open end of the second problem. The intent of my animation is to convey that there are infinitely many solutions. The problem presents students with an opportunity to generalize.

Each of these problems can be classified as Depth of Knowledge Level 2 (Skill/Concept). In both, students need a conceptual understanding of factoring x2 + bx + c where b and c are given. The second requires pattern-sniffing (or logic). I created a third problem that asks students to think about these two equations — and a third — simultaneously.

set of equations

Note that x − 4 could be a factor of each trinomial. However, students need to determine where to put 4 so that the other digits can be placed in the remaining boxes. This twist might not be enough to raise it to DOK 3 (Strategic Thinking). Roughly speaking, Robert’s DOK 2 problems involve making statements true. Sometimes it’s satisfying an equation, sometimes it’s satisfying a condition (e.g., a system of equations having no solution). Robert’s DOK 3 problems call for optimizing a result — least, greatest, closest to.

In my Math 9 Polynomials video, I pose the following open question in the style of Marian Small: Two trinomials add to 3x2 + 7x + 6. What could they be? Here’s a problem, adapted from Open Middle Math, that also tackles adding polynomials:

adding polynomials

Both tasks can help reveal students’ understanding of combining like terms and manipulating coefficients and exponents. (In Task 1, I specify that the two polynomials be trinomials. This rules out responses such as (3x2) + (7x + 6) that sidestep like terms.) Task 2 is much more likely to show what students know about additive inverses, although a small tweak to Task 1 (e.g., Two trinomials add to 3x2 + 7x + 6. What could they be?) bridges this gap.

Integers

I include one open-middle problem in the application section of each of my Math 7 integer videos:

adding and subtracting integers
order of operations with integers

Note that the first is DOK 2 whereas the second is DOK 3. Maybe. I don’t want to quibble. What matters more than the differences between DOK 2 and 3 is that these questions require a deeper understanding than DOK 1 problems such as Evaluate: (−9) + (+3) or Evaluate: (+3) − [(−5) + (−4)] × (+5).

In the first video, I ask “How might finding one solution help you find more? How are some of the solutions related to one another?” These questions aren’t answered in the video — an exercise left to the viewer. Here are just some of the ideas that I would anticipate to emerge in an Open Middle classroom discussion:

  • addition and subtraction facts are related
    • e.g., (+6) + (+3) = +9 and (+9) − (+3) = +6 belong to the same “fact family”
  • this relationship extends from whole numbers to integers
    • e.g., (−6) + (−3) = −9 and (−9) − (−3) = −6 also form a fact family
  • subtracting a number can be thought of as adding it’s opposite
    • e.g., if (+6) + (+3) = +9 makes the equation true then so, too, does (+6) − (−3) = +9
  • swapping the number being subtracted (subtrahend) and the number it is subtracted from (minuend) changes the sign of the result (difference)
    • e.g., (+9) − (+3) is equal to positive 6 whereas (+3) − (+9) is equal to negative 6

Order of operations is a natural fit for optimization problems. In the second video, the intent of my answer animation is to communicate my mathematical reasoning. Once more, note that I show a couple of slight missteps and revisions to my thinking.

order of operations with integers (solution)

In addition to performing the operations in the correct order, students must think about how to maximize sums and minimize products. They must consider how subtracting a number increases the result. See one of Marc’s Math 7 decimal videos for another open-middle order of operations example.

Percents

I did not include an Open Middle problem in my Math 7 Percents video. Rather, I chose to present a percents number talk: Estimate 78% of 49. Note that I show two strategies: one that makes use of quarters…

75% of 48

… and another that utilizes tenths.

80% of 50

Because there are many ways to solve this problem, it can be thought of as a having an open middle despite it not having number tiles. A third, interesting, solution pathway can be taken. Instead of 78% of 49, we can estimate 49% of 78: 50% — or one-half — of 78 is 39. The idea that x% of y is equal to y% of x should emerge from the following:

commutative percents

It’s for this reason that I did not add the constraint Each number can be used only once. You can always add it later, which should bring about doubling and halving — and tripling and “thirding”!

x% of y is equal to 2x% of y/2; x% of y is equal to kx% of y/k

I like that the double number line problem below incorporates estimation; both 20 and 25 per cent are perfectly reasonable estimates. Also, it embraces our first principle — make it visual — which is largely missing from my other open-middle examples.

I wrestle a bit with whether or not to include the “only once” constraint. Does it enhance the problem above? I guess that it necessitates more practice; disqualifying 25% of 64 is 16 does compel students to seek out 25% of 68 is 17 or 25% of 76 is 19. But concentrating on unique tens and units digits of parts and wholes is irrelevant to percents. Again, you can always add this constraint later. (Update: Check out this slick Desmos activity from @TimBrzezinski!)

What might a DOK 3 Open Middle percent problem look like? Below is a possibility — or three! — that uses Robert’s optimization approach. (I haven’t played with the dollar value amounts so treat them as placeholders.)

percent discounts

Open Middle Math is a must read that will help you implement these engaging tasks. Whether you’re new to Open Middle problems or think you know all about ’em, you’ll love the glimpse into how Robert designs opportunities for students to persevere in problem solving and for teachers to gain insights into what students really understand.

A Star Wars Tomatometer Story

I wrote this post two years ago but decided against hitting publish. With the final film in the “Skywalker saga” opening next week, now is as good a time as any. And yes, we have tickets!

Like The Force Awakens and Rogue One, my daughters and I saw The Last Jedi on opening night. It’s become a bit of a Hunter holiday tradition. Gwyneth loves the stories; Keira loves the Porgs. As much as the movies themselves, Gwyneth loves watching and discussing YouTubers’ takes on them — reactions, explanations, theories. She shared this one from New Rockstars with me, which begins with this:

“… Star Wars: The Last Jedi is the most polarizing film of the year, with one of the biggest gaps between critics ratings and audience scores for a major film ever. What the hell is going on here? Why are some people so annoyed with it, saying it ruined what made the original trilogy and The Force Awakens so good? Why are others fanboy crushing so hard over it, calling it the best Star Wars film ever made?”

New Rockstars

This reminded me of another passion of mine: the fundamental meanings of the operations. More specifically, subtraction as difference/comparison rather than take away/removal.

Here are the Rotten Tomatoes scores for The Force Awakens:

Tomatometer -- Episode VIII

Episode I | II | III | IV | V | VI | VII | Rogue One | What is the Tomatometer?

What’s the meaning, in context, of 50 – 90?

We’re measuring the gap between the percentage of professional critics (“Tomatometer rating”) and Rotten Tomatoes users (“Audience Score”) who rate the movie positively. We’re talking comparison, not removal. There’s a difference of 40%. Moreover, the difference here is negative (albeit my minuend/subtrahend decision is kinda arbitrary). This means that The Last Jedi is far less favourable among moviegoers as a group than among professional movie critics. We can compare this gap with that of others in the Star Wars franchise:

Episode IV: A New Hope (1977) → 96 – 93 = +3
Episode V: The Empire Strikes Back (1980) → 97 – 94 = +3
Episode VI: Return of the Jedi (1983) → 94 – 80 = +14
Episode I: The Phantom Menace (1999) → 59 – 55 = +4
Episode II: Attack of the Clones (2002) → 57 – 65 = -8
Episode III: Revenge of the Sith (2005) → 65 – 79 = -14
Episode VII: The Force Awakens (2015) → 88 – 93 = -5
Rogue One: A Star Wars Story (2016) → 85 – 87 = -2
Episode VIII: The Last Jedi (2017) → 50 – 90 = -40

Some patterns emerge. For example, all three films in the original trilogy received positive reviews from critics and audiences alike; all three are Certified Fresh. A greater percentage of Rotten Tomato users than critics liked A New Hope, The Empire Strikes Back, and Return of the Jedi: Audience Score – Tomatometer rating > 0. The Force Awakens and Rogue One received similar positive reviews, again from critics and audiences alike. However, these recent movies rated a little lower among audiences than among critics: Audience Score – Tomatometer rating < 0.

We can use absolute value to measure agreement between the two groups. For A New Hope, The Empire Strikes Back, The Phantom Menace, The Force Awakens, and Rogue One, |Audience Score − Tomatometer rating| ≤ 5. Rotten or fresh, there’s consensus. For The Return of the Jedi and Revenge of the Sith, |Audience Score − Tomatometer rating| = 14. Still, a relatively small difference of opinions.

The Last Jedi breaks this trend. Professional critics place it alongside fellow Disney films The Force Awakens and Rogue One. RT users score it lower than the prequels. Below Binks!

Movies may be more engaging than the usual contexts for integers — a diversion from temperatures and bank balances. Thinking about this data graphically may have more potential.

https://www.desmos.com/calculator/zzplnokx7f
https://www.desmos.com/calculator/zzplnokx7f

It’s very similar to my take on the food graph, with movie critics in place of nutritionists in the role of expert. Gwyneth played along as I asked “What’s going on in this graph?”. We predicted where some of our favourite movies would land. We explained our reasoning. We compared our predictions with Rotten Tomato data. And then we shut down the laptop and rewatched The Empire Strikes Back.

The data for eight of these nine movies hasn’t changed much in two years. The outlier? Yep, The Last Jedi. The difference is now up to — or down to? — negative forty-eight (Audience Score: 43; Tomatometer rating: 91).