What does this mean?

The other day I was working with some high school students on problems like this one:

I asked what I thought was a pretty straightforward question, "What does this mean?" [SPOILER: it wasn't as straightforward as I thought.]

One student said/asked something like, "Split the square root? Make it the square root of one over the square root of 64?" (Seriously, these were implicit questions asking, "Is this where I start?") 

STRIKE ONE!

I responded, pointing emphatically at the problem, "But what does this mean? How would you say it in words?"

The other student interjected, "The square root of 1 over 64." And then asked, "We should rationalize the denominator, right?" 

STRIKE TWO!

I tried again, "You're focusing on what you would do. I want to know what you understand about square root problems. What if it was the square root of sixty-four, then what would the problem be asking you to find?"

The first student, clearly getting frustrated, "I'd take a half of 64." 

STRIKE THREE! YOU'RE OUT, COFFEY!

I didn't see any point to continuing to frustrate them or me with more questions. So instead, I tried to model my thinking about solving problems involving radicals.

When I was a kid, I used to bring a four-function calculator with me on long road trips (like the one shown here). There was no square root button on the calculator, so I would play What's the Square Root by typing in a guess, pressing the multiplication button and then equals, and use the result to see how I might need to adjust my guess. It was hours of fun! 

If my question was, "What's the square root of two?" then I was essentially asking, "What number multiplied by itself is two?" So when I ask what this (again, emphatically point at the original problem) means, I'm thinking about playing the game and trying to determine what factor squared gives this (more pointing) product, the number inside the radical.

I went back to our simpler problem, "So this is asking me, 'What factor squared equals sixty-four." They both answered eight.

Then we went back to the original. "What does this mean?"

They said, "What number times itself gives one over sixty-four?"

"Yes, What factor squared is one-sixty-fourth? Now that we understand," I said, "we can begin thinking about what to do."

It's troubling to me that so many years after Pólya wrote How To Solve It, that students race past the first phase of the problem solving process - Understanding the Problem.

Why is that? It can't be that we are focusing on the procedures and not the concepts. No, it can't be that, can it?

Why do I make these things so hard?

During a Math Institute put on by the Public Education & Business Coalition (PEBC), we were asked to monitor our thinking while working on the Coffee Break problem from (warning, this link includes the answer) Discovery Education. Essentially, we were creating a Metacognitive Memoir that would make our thinking, not just our work, visible for others to see. Being familiar with this form of math writing, I quickly began working on solving the problem.

I started by reading the problem - the entire problem. In the past, there were times when I began working on a problem without understanding it. This lead to "wasting" time on unproductive approaches. I did not want something like that to happen on this problem.

The very first step, "Beginning with a full cup of coffee, drink one-sixth of it," got me thinking about similar fraction problems I have worked on. The second step, narrowed the list of similar problems to those involving mixtures. Fortunately, I kept reading because if I had started solving this problem with those previous approaches in mind I would have gotten off to a "bad" start. This problem was quite different from those I had worked on before.

As I was reading, mental images of coffee cups began coming to mind. At first these were simply generic cups of coffee. But pretty soon I realized that I was going to need a model that could represent the quantities presented in the problem.

In order to make the steps of my thinking visible, I decided to make a series of images, like a story board, that showed the ways I was picturing the different steps. First there was the whole cup of coffee. Then there was the cup filled with five-sixths coffee and one-sixth milk. Next I drank a third of the mixture ...

While I initially took the top third, I quickly realized that the milk would need to be mixed with the coffee. I decided to show this by splitting the cup up vertically. But what was I going to do to show drinking a half of the mixture?

As I was debating how to represent this step, and writing down my thinking about it (I could make a three-dimensional model or split each of the eighteenths in half), the facilitator asked us to stop working on the problem so we could share our results. RESULTS?!?! I felt like I had barely gotten started.

Sure enough, some of the other participants had answered the questions:

• Have you had more milk or more coffee?
• How much of each have you had?

What happened next got me wondering, "Why do I make these things so hard?"

Anyone want to guess what happened (what one of the participants said) and why I made this problem so hard? If you'll put your guesses in the comments, I will update this post with the answers next week.

Did I get my money's worth?

The following is not intended to be an endorsement of any product.

UPDATE: 2017 - This year the tumbler cost $42.40 (including tax) and a grande is $2.20 (including tax).


In December, I got a tumbler from Starbucks that allowed me to get free coffee every day this January. The tumbler cost $30 plus tax. My last free grande is shown to the right. I still get a $.10 discount when I use the tumbler but from now on I need to pay the regular $1.95 plus tax for brewed coffee. This got me wondering: did I get my money's worth from the purchase of this tumbler?

Will you (or your students) help me answer this question?

Wait. You need more information - such as? Go ahead and share what you need in the comments.

What are your thoughts?

The following is an adaptation of the workshop I planned for the Scholarship of Teaching and Learning (SoTL) Academy. It represents an activity I do with teachers that helps them to make their thinking visible as they are problem solving. The activity is intended to prepare the way for the problem solvers to write metacognitive memoirs (which I wrote about here).

Metacognitive Memoirs: Making Thinking Visible

Schema Activation: Making Your Thinking Visible [10 minutes]

Why are you here? What do you hope to get out of this presentation (blog post)? Please write your thinking down in order to make it public and permanent.

Focus: Metacognition [5 minutes]

Schoenfeld found that one of the major issues novice problem solvers face when they encounter non-routine problems is an inability to monitor, and therefore regulate, their thinking. It’s what I call flat-lining. Einstein called it insanity.

Not surprisingly, the National Resource Council (2005) wrote in How Students Learn that one of the primary principles associated with learning is The Importance of Self Monitoring. They write:

“Meta” is a prefix that can mean after, along with, or beyond. In the psychological literature, “metacognition” is used to refer to people’s knowledge about themselves as information processors. This includes knowledge about what we need to do in order to learn and remember information (e.g., most adults know that they need to rehearse an unfamiliar phone number to keep it active in short-term memory while they walk across the room to dial the phone). And it includes the ability to monitor our current understanding to make sure we understand. Other examples include monitoring the degree to which we have been helpful to a group working on a project. (p. 10)

The focus of this workshop is to help us to be intentional about our thinking so that we can examine it, share it, and improve it.

Actions: Sowing Seeds [30 minutes]

As you work through this problem, try to be aware of your thinking as you make decisions about how to proceed. You will want to write as much of your thinking down as you can so that it will be available later as you work on your memoir. Also, keep track of choices you decided against and why. Were these possible pitfalls you avoided or just different approaches? What would have happened if you followed these paths instead?

This is a lot to keep track of, which means you probably will not arrive at a solution in the time provided. That is to be expected. In fact, if you can complete a task quickly, it was probably not a problem but an exercise. Only the problems found on sitcoms get rapped up in under twenty minutes.

After about ten minutes stop and on a separate piece of paper write a reasoning recount. What steps have you taken so far, and why did you take them? Don’t forget to include things you chose not to do and the rationale behind those decisions.

Reflection: Where did you ...? [15 minutes]

As you review your reasoning recount, try to identify where you are:

• Assessing (gathering data about your thinking and your progress);
• Analyzing (evaluating what was working and what was not);
• Adjusting (changing course because what you were doing was not making sufficient progress toward your goal); and
• Acting (putting your plan into action).

It is typical that this first attempt might result in your own sort of flat-line – a lot of action without much thinking. Only 16% of my students are able to develop a clear, correct, complete, and coherent metacognitive memoir the first time through. However, with practice and feedback they are better able to monitor their thinking and communicate it to others.

Thus far, most of the memoirs have been written as narratives. A few problem solvers layer their thinking on using sticky notes or the comment function of Word. We have also been experimenting with using two columns – putting the thinking in the left-hand side in order to recognize its importance in the process. Recently, some problem solvers have been using technology to record their thinking as they solve problems (a la this post).

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So what are your thoughts about this approach to making thinking visible?