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

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 *x*^{2} + *kx* − 8 factorable? *x*^{2} − *x* − *k*? See the answer animations below.

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 *x*^{2} + *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*.

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 3x ^{2} + 7x + 6. What could they be?* Here’s a problem, adapted from

*Open Middle Math*, that also tackles 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 *(3x ^{2}) + (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 3x*~~+ 7x~~ + 6. What could they be?) bridges this gap.

^{2}## Integers

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

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

- e.g., (+9) − (+3) is equal to

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.

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…

… and another that utilizes tenths.

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:

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”!

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.)

*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.

This has been so much fun to read through, Chris. I am so honored. I valued you sharing what resonated with you as well as your push back on whether it’s the problem or the teacher that opens/closes the middle. I appreciate you unpacking all of this. Truly, thank you.

Thanks for the comment, Robert. And for all your work in raising teachers’ awareness of open middle math. We’ve been sharing problems from your site with our colleagues for a few years now and we know that many of them have been using Open Middle in their classrooms. So, it was a pleasure to be able to include similar tasks in our video series. (I do think that some problems lend themselves better to multiple solution pathways. Topics, too, as proportional reasoning is admittedly low-hanging fruit.)

Chris

So thoughtful! I really enjoyed reading your comments.

Thank you, Marian, for the kind words… and for introducing me to open questions… and for influencing me to think more about which big mathematical ideas they are designed to elicit!