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.

Principles of Math Videos

This summer, Marc and I made several videos designed to help parents understand what mathematics their children are learning. As well, we wanted to give parents a feel for how their children are learning in their math classes. We added Mathematics 6 & 7 videos to the previous summer’s 8 & 9 collection. The work of producing videos for Foundations of Mathematics and Pre-calculus 10 is well underway; I expect to add two more videos–Solving Systems of Linear Equations Graphically & Algebraically–this week. Although intended for parents, we believe that this series could be a helpful resource for teachers, especially those having to teach in an online or blended environment due to COVID-19.

In this post, I’ll share some of the principles that guided us when creating the videos. This’ll be a peek behind the curtain of interest more to educators than to parents.

Make it visual.

Math is visual. Videos are visual. So math videos should be visual. It is disappointing how often makers of digital content fail to take full advantage of visual aspects available to them. The animation of symbolic representations–line-by-line equation solving or drawing little arrows to show the distributive property–should not be the extent to which content is presented visually. It’s ballsy to sell this to educators as visual if that’s all you’ve got. By “make it visual,” I mean include images and animations that help viewers make sense of the mathematics at hand or the context in which the mathematics is situated. For example, we show that 2:3 is equivalent to 8:12 by repeatedly extending a black-red-red-red-black pattern of beads; we don’t describe two candles with different heights and different rates at which they burn, we show it–so long as we can figure out how to do it in Keynote.

Mathematics 6: Ratios & Percents (1/2)
Foundations of Mathematics & Pre-calculus 10: Systems of Linear Equations (1/2)

In our videos, we make use of virtual manipulatives–or virtual virtual manipulatives?–like pattern blocks, colour tiles, counters, multi-link cubes, base ten blocks, algebra tiles, tangram-like puzzles, Solo cups and paper clips, etc. We employ other pictorial representations such as hundred charts, decimal/percent grids, number lines, double number lines, factor trees, factor rainbows, tables of equivalent ratios, graphs, etc.

Make it conceptual.

I once watched a short video to fix an issue with my dishwasher. I carefully followed the directions, pausing the video at each step along the way. But no one is ever going to mistake me for a handyman! I don’t really understand how dishwashers work. I couldn’t connect the problem to any knowledge of the machine’s mechanical or electrical systems. If the solution shared online didn’t work, I was hooped. And even though I was successful, my procedure for fixing my dishwasher was useless for fixing my washing machine, let alone a different make and model of dishwasher. This skill–long-forgotten, by the way–didn’t transfer from one household appliance to another.

But it didn’t matter. I set out that morning to make one small repair, not become an appliance repair technician. Mathematics is different. The emphasis in math class must be on sense-making, not answer-getting. The same should be true of math videos. In our videos, we attempt to always address the why.

For example, we answer “Why is a negative divided by a negative a positive?” by revisiting what it means to divide whole numbers and then applying these two fundamental meanings to dividing integers. Later, a thermometer example reinforces dividing as measuring.

Conceptual understanding means seeing mathematics as a coherent whole rather than isolated procedures. Digital content can support students in developing conceptual understanding by creating opportunities for them to connect models and representations. For example, we ask learners to connect multiplying binomials to what they already know about multiplying two-digit numbers (i.e., an area model, partial products, the distributive property).

Of course, procedural fluency is important. Effective mathematics teaching focuses on the development of both conceptual understanding and procedural fluency. However, procedural fluency depends and builds on a foundation of conceptual understanding.

For example, percents are presented as fanatical comparisons to 100. No part-whole-percent triangles or is-over-of-equals-percent-over-100 cross-products here. Relating percents to decimals does not appear until grade 7. So, in our Mathematics 6 video, the emphasis is on equivalency and benchmark percents (i.e., 50%, 25%, 75%; 10%, 20%, … , 90%).

Procedural fluency includes the ability to apply procedures flexibly. Throughout each video, multiple strategies are discussed. For example, see the strategies–and representations–used in this proportional pizza problem.

To bridge conceptual understanding and procedural fluency, we try to build on learners’ own mathematical ideas. For example, before the elimination method for solving systems of linear equations is introduced, viewers are first encouraged to solve a puzzle using their intuition. Then, they are presented with a pictorial representation of a solution to a problem. It’s at this time that algebraic symbols and notation appear:

(Note: Each row and column is a sum. None of that fries times Coke nonsense!)
Foundations of Mathematics & Pre-calculus 10: Systems of Linear Equations (2/2)

This process plays out when solving equations in one variable (boxes of doughnuts; algebra tiles) in Math 7 and when solving systems using substitution (scale puzzle; two types of tickets problem) in Math 10.

Of course, we can build on viewers’ own ideas only to the extent to which they engage with and participate in the learning experiences that we design. This segues into our last principle…

Make it inviting.

I think that this is the right adjective. To me, inviting goes beyond accessible.

One way in which we make an effort to invite parents to “do the math” is to use open questions. Sometimes, this means open-ended. For example, Show me one-quarter in as many ways as you can, What could the numbers be? and What comparisons can you make? (and later What is being compared in the ratio 1:2?) are open-ended; they allow for many correct answers and signal that a range of responses are valued.

Other times, we used open middle problems. An open middle problem may have one correct answer but multiple ways of getting it. For example, there is an arrangement–or two–of number tiles that maximizes the expression below. The animated placement of the number tiles is meant to model one strategy and includes me making missteps and backtracking as I went along.

(Here’s a number tile factoring task from Math 10 that has both an open end and an open middle.)

In addition to openness, we try to hold off on introducing formal symbols and notation early. For example, Two numbers add to 12. What could they be? comes before Solve the system x + y = 12 & y = 2x. (See also Burgers, Fries, and Cokes and Tees and Hoodies above.)

Sometimes, when a task is not a soft place to start, we may still present it up front and then return to it later, after we’ve built up some knowledge. In this way, we hope to “make it inviting” by piquing the curiosity of viewers. For example, asking parents to pick two numbers that differ by two and multiply them is accessible whereas asking them to explain why this product is one less than the square of the number between them is not. A similar approach was taken with a gas vs. electric vehicle application of linear systems; it serves as a hook in the video’s introduction.

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

We believe that learners of mathematics should be active participants rather than passive spectators. There’s a tension between this belief and video. In our videos, we put a lot of trust in parents pressing pause when prompted. It’s in these moments that they “do” math, that they play, notice and wonder, solve problems, visualize, look for patterns, make conjectures, generalize, reason, explain, connect ideas, take risks, etc. We were limited by the medium–or our tech skills within this medium.

In a mathematics classroom–be it face-to-face or remote–this tension can be resolved. And this is one reason why we’re just as, if not more, excited about teachers using these videos. At the moments when we ask viewers to pause, students could be placed in visibly random groups or breakout rooms. Teachers are not limited by our prompts–or these moments. They can observe and adapt to what’s happening with their learners in the moment and ask How else might you have solved the problem/represented your thinking? What does this remind you of? How are these the same? How are they different? What would happen if… ? etc. There’s no need to “fake it” coming out of a pause as we had to do (e.g., “You might have noticed that…”).

These videos were intended to capture the big ideas or enduring understandings or key concepts of a topic–a whole chapter or unit. Although each clocks in at about twenty minutes, it would be inappropriate to have students experience an entire video in one sitting. Instead, a task or two clipped from a downloaded video could make up one day’s learning experience.

If you find this video series helpful, we’d love to hear from you. Drop a comment, question, or complaint in the comments.