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?

When will it end?

[In the United States] Teachers act as if student interest will be generated only by diversions outside of mathematics. (The Teaching Gap, p. 89)

Schema Activation: Journal Jot

Describe the process of converting three-nineteenths to a decimal

Focus: Changing the Script

We have read in The Teaching Gap how teaching mathematics in the United States typically focuses on preparing students to perform prepackaged procedures. What if we tried to change the focus from mathematical procedures to mathematical practices?

In this workshop, try to refocus your efforts away from simply following a mindless process and explore what new knowledge is waiting to be learned. As you work on the following problem, please keep track of how you are (or aren't) engaging in these practices.

Activity: Find the exact decimal representation for three-nineteenths

[The follow represents how learners have engaged with this activity.]

This seems like a fairly straightforward exercise. Why not just plug the numbers into a calculator? The converting-a-fraction-to-a-decimal procedure requires us to divide the numerator by the denominator until the decimal either repeats or terminates. Three divided by nineteen - easy. Except, the quotient displayed on the calculator screen does not provide enough information since it does not show enough places.

Maybe using a different calculation tool would help. What about an Excel spreadsheet?

This seems to suggest that three-nineteenths terminates. And it matches the calculator's answer. A problem solver would not simply accept these and be done - would she?

What other strategies could apply?

• Work the division out by hand
• See if another related fraction might shed some light on the decimal
• 1/19 since 1/19 multiplied by 3 is 3/19
• 16/19 since this would be the complement of 3/19
• Find x such that 19x = 100 and then calculate 3x
• Multiply 0.157894736842105 by 19 to see if the product is 3

[Learners often explore these approaches with various levels of success. Having monitored their own progress, the learners often turn to other strategies or decide to seek a solution to other related problems. They do not want to be stuck trying the same approach over and over again.

WolframAlpha has added a wrinkle to this activity as it provides an exact decimal representation for three-nineteenths. Originally, we thought this would be problematic as it gives a clear answer to the original question. It is interesting, however, that focusing on a problem solving approach generates similar alternative/extension problems whether or not the answer has been found.]

New problems:

• Which fractions repeat and which terminate? Why?
• Can we predict the period of the decimal representation of a given fraction?

Partial table created by learners in order to look for patterns

[Answers to these alternative/extension question rarely are answered in the time available to the workshop. Learners are encouraged to continue exploring the problems if they are interested. Even though they have not come to any firm conclusions, they usually have some interesting answers to the reflection questions.]

Reflection: Mathematician's Chair

• What did you do as it relates to the mathematical practices?
• So what was important about this work?
• Now what might you do the next time you encounter a problem?

Which way is ... ?

A preservice teacher leading a review for a quiz on rational number computation invited me to watch the lesson and work with her to improve on it. The objectives addressed in the review activity were from the Michigan Grade Level Content Expectations (GLCE):

• N. FL. 07.08 (GLCE): Add, subtract, multiply and divide positive and negative rational numbers fluently 
• B. N. FL. 07.09 (GLCE): Estimate results of computations with rational numbers

But the preservice teacher was also interested in developing conceptual understanding - especially around the idea of how multiplying and dividing by numbers between 0 and 1 impact the result. A good activity from NCTM (pdf) was found and modified in an effort to achieve these goals.

Move down or sideways (never up) through the maze from Start to Finish. You may not retrace any steps. Begin with 10 and as you move along a segment do the indicated computation. Record your steps on the scorecard. Your goal is to find the path that results in the largest (or smallest) value when you reach the Finish.

After the lesson, we discussed ways of using the activity more effectively. The first idea was to model what a path looks like. Because this kind of activity was new to the students, it took them a while to understand what was expected of them. For example:

What if we just followed along the left-most edge?

1. 10 x 0.9 = 9; 
2. 9 x 1.75 = 15.75
3. 15.75 + 5 = 20.75

Next, without doing any computation, we would ask the students to predict the path that would result in the greatest result or the least result. Making predictions is a great way to develop interest in a task. Student would share their predicted path and the rationale for their choice. This would provide some insight into the students' number sense related to multiplication and division of rational numbers. 

Then the students would estimate the results of several paths. This would allow them to check out their predictions and refine our list of which paths might represent the largest (or smallest) value. Also, this would address objective B. N. FL. 07.09 from above.

Finally, the students would be asked to compute the path they believed would result in the largest (or smallest) value [N. FL. 07.08]. Calculators are not allowed in this classroom but we decided that we might allow students to use calculators on up to half of the calculations. That way they would be exposed to the idea of using calculators strategically instead of with an "all-or-nothing" mindset.

As an extension, we might ask the students to find the easiest or hardest path to follow without using a calculator and why. We thought this would offer the students an opportunity to be metacognitive. It would also provide us with information on areas where students could improve on their fluency.

The preservice teacher was able to apply some of these subtle shifts to her later class with success. She writes:

...they did much better!  They were excited to do something "more fun than boring problems."  I was really happy with the responses I got...

What are your thoughts? How would you improve on this activity? Why?

When is it okay to use a calculator?

All too often, I run into teachers (both preservice and inservice) lamenting that kids are using calculators to compute something simple, like 6 x 7. These teachers express their frustration by threatening to not let kids use any calculators until the kids prove that they know their facts. And there will be no calculators for any simple computations. I understand this thinking but I am not sure it will achieve the desired result - wise calculator use (i.e. phronesis).

Problem is, if teachers are the ones deciding when their students can or cannot use calculators, then students are not able to practice this critical-thinking skill for themselves. Consequently, whenever a calculator is available in the future it can be used because that was what the students learned in school. As an alternative to controlling calculator use, I suggest a couple of activities that can help students to decide for themselves when it is appropriate to reach for the calculator.

The first activity comes from the article, John Henry - The Steel Driving Man. This article suggests several different experiments involving routines that can be completed by human or mechanical effort (eg: sharpening pencils). Students are asked to predict which method will take longer, gather data, and compare the results using box-plots. 

The experiment I am most interested in involves computing multiplication facts with and without a calculator. I give pairs of students four worksheets like the one shown below.

Each of the four worksheets is different. As one student completes the worksheet, the other one uses a stop watch to time the effort (errors add an extra five seconds to total time). This is repeated until each student completes two sheets - using the calculator for one but not the other. 

Classroom data are gathered and typically show that the students complete the worksheets quicker without a calculator. Discussing the results can provide students with an opportunity to reflect on whether or not it is efficient to use a calculator for basic facts. In the few cases when it is faster to use a calculator, the issue is usually that the student has a lot of wrong answers when computing without a calculator. Teachers must decide an appropriate course of action in these instances.

A second activity that I use to develop students' wise use of calculators involves a worksheet of multi-digit multiplication items. Instead of assigning the entire worksheet, I ask students to pick four items to compute without a calculator, four items to estimate, and four items to use a calculator on. The students are also expected to explain why they selected the approach to use with each item.

Calculator phronesis, the wise use of calculators, requires opportunities for students to experience activities that involve metacognitive aspects. We teachers will not always be there to guide students' choices, but this is not to suggest that we do not have a responsibility to help students to develop this ability. It is my hope that through these classroom experiences, students will be able to ask and answer for themselves the question, "When is it okay to use a calculator?"