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222 THE CUNY HSE CURRICULUM FRAMEWORK • MATH

What would it look like if

we designed schools to be

places where teachers learned,

alongside their students?

—Dr. Elham Kazemi

THE CUNY HSE CURRICULUM FRAMEWORK • MATH REFLECTIVE TEACHING 223

Reflective Teaching

A Focus on Student Thinking in Problem-Solving

I

f there is one mantra that has stuck with us when it comes to improving our math instruction, it is “make a small change, reflect, and do it again.” In their article, “Math Tasks as a Framework for Reflection: From Research to Practice,” Mary K. Stein and Margaret Schwan Smith cite the NCTM Professional Standards for Teaching Mathematics which argue that a primary factor in the professional growth of teachers is the opportunity teachers have to “reflect on learning and teaching individually and with colleagues.” They go on to say that whereas all teachers informally think about what happens in their classrooms, “cultivating a habit of systematic and deliberate reflection may hold the key to improving one’s teaching as well as to sustaining lifelong professional development.”

But what should teachers reflect on? There is no right answer to that question, but we’d like to share some work we’ve been doing to support teacher reflection, focusing on student mathematical thinking on nonroutine math problems. Below you will find three sets of questions focusing on three important aspects of your teaching—planning, student work, and reflection/ revision. The goal of these questions is to help you learn from your experience and from the experience of your students. I recently heard an inspiring question from Dr. Elham Kazemi , professor of mathematics education and associate dean for professional learning at the University of Washington— “What would it look like if we designed schools to be places where teachers learned, alongside their students?” We offer the process detailed below as a beginning. Even if you only have time to answer a few questions from each section, or if you only do this formally once a year, we hope the experience will be rewarding.

You are a scientist looking into learning. The planning phase is your problem-posing and hypothesizing. The teaching is the experiment, and the student work is the data collection and observation. The reflection is the conclusion and may lead to a revised hypothesis and a new “teaching experiment.” One final suggestion…Consider doing this with at least one other teacher. You can do a problem together and then work together on the Planning Questions. Then each of you tries the problem with their students. You can meet again and discuss what you learned from your students’ reasoning. These questions can help structure any follow-up conversations.

Dr. Kazemi has developed a great observation technique called “teacher time out.” To learn more, visit http://www. shadowmathcon.com/elham- kazemi/

Make a

small change,

reflect, and

do it again.

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QUESTIONS ON STUDENT THINKING

Adult education teachers always talk about how much we learn from our students. Many teachers say they learn more from their students than their students learn from them. Teachers are usually referring to all the stories and experiences our students share, or the inspiration we derive from their decisions to come to our classes, balancing complicated lives and responsibilities for a regular date with struggle. But there is another really important way we can learn from our students—focus on their reasoning. We must really delve deeply into student thinking, to understand the individuals in our class, and also to better understand adult education students in general and how they learn. Our students are trying to teach us, if we take the time to listen. If possible, consider doing this phase with another teachers. Math teachers coming together to analyze student thinking can be a very rich activity. Remember when choosing work to analyze, don’t focus only on students who got the right answer. You may learn more from student mistakes, or solution methods that are interesting but incomplete.

Answer the following questions for each sample of student work you choose:

1 Explain each student’s method/thinking.

2 Why did you choose this sample of student work?

What did you learn from it?

3 How typical was this student’s approach in your class?

4 Any additional comments?

REFLECTIONS/REVISION QUESTIONS

Whatever happens is an opportunity to learn something about your students and how they learn. If something doesn’t go well, you can learn a lot about how to do it better next time. And if things do go well, why did they go well and how could they go better. This section is about looking back at your predictions and comparing them to what happened—as you observe and analyze student thinking you’ll start to improve your sense of how they will make sense of and productively struggle with future problems. Even if you are not going to be sharing this with other teachers, spend some time with the last question. The teacher you are advising might be you.

Our students

are trying

to teach us

if we take the

time to listen.

226 REFLECTIVE TEACHING THE CUNY HSE CURRICULUM FRAMEWORK • MATH

1 What did you learn from using this problem with your students

(about math, about individual students, about your class, about student thinking in general, etc)?

2 What (if anything) would you do differently if you used this

problem again?

3 Comment on how your class (individual students, or as a whole)

may have benefited from their work on this problem.

4 Did students get what you wanted them to get from the problem?

How do you know?

5 What challenges came up for your students that you didn’t expect?

6 What strategies/solution methods/questions came out that seemed

helpful to students?

7 What advice/message do you have for a teacher who is considering

using this problem with their class?

A Call to Action

To give readers a real sense of how helpful these reflections can be, we are including three sample write-ups, written by Tyler Holzer, a teacher leader at a community-based organization in Brooklyn, NY. If you find Tyler’s write-ups helpful, consider writing one yourself, using these questions to guide you. Share those write-ups with your colleagues. Write them with your colleagues. If you are a program manager, consider protecting some time for your staff to work on these questions together. We believe in teacher-led professional development of practice. Too often, we teach in our little pocket of the egg carton, isolated from other teachers. Let us turn our classrooms into laboratories to learn about learning and share what we discover.

n Multiples of Nine Problem n The Gold Rush Problem n The Movie Theater Problem

228 REFLECTIVE TEACHING THE CUNY HSE CURRICULUM FRAMEWORK • MATH

Also, when the multiples of 9 are organized into a table, an interesting pattern emerges. By looking at the digit sums and the changes to the ones and tens digit, we see some interesting things.

Table of the First Forty Multiples of 9

9 × 1 = 9 9 × 11 = 99 9 × 21 = 189 9 × 31 = 279

9 × 2 = 18 9 × 12 = 108 9 × 22 = 198 9 × 32 = 288

9 × 3 = 27 9 × 13 = 117 9 × 23 = 207 9 × 33 = 297

9 × 4 = 36 9 × 14 = 126 9 × 24 = 216 9 × 34 = 306

9 × 5 = 45 9 × 15 = 135 9 × 25 = 225 9 × 35 = 315

9 × 6 = 54 9 × 16 = 144 9 × 26 = 234 9 × 36 = 324

9 × 7 = 63 9 × 17 = 153 9 × 27 = 243 9 × 37 = 333

9 × 8 = 72 9 × 18 = 162 9 × 28 = 252 9 × 38 = 342

9 × 9 = 81 9 × 19 = 171 9 × 29 = 261 9 × 39 = 351

9 × 10 = 90 9 × 20 = 180 9 × 30 = 270 9 × 40 = 360

Why I Chose This Problem

I recently discovered this problem, and I really like it for a number of reasons. First, it requires a little bit of vocabulary in order to get started. Students will have to know what a multiple is, they will have to know what digits are—and more specifically, how digits can differ from numbers—and they’ll have to understand the difference between even and odd numbers. I also like how nonintimidating it looks at first glance. “How hard could it be to find a multiple of 9 that has only even digits? I shouldn’t have to count up very far.” Because the problem doesn’t look lengthy or challenging, it comes as a surprise when the correct answer is actually the 32nd multiple of nine. I anticipate a lot of students writing out 9, 18, 27, 36, 45, 54, etc, and then getting frustrated or giving up when they don’t get to the answer fairly quickly.

Second, I like that the problem requires students to perform basic calculations and that it requires precision in order to get the answer right without making mistakes. Also, the repetition involved in either adding 9 over and over or multiplying by 9 over and over is helpful for students with lower math abilities, and it still provides good practice for students who are more comfortable working with numbers. Moreover,

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the 9 times tables are interesting because of the pattern that arises in the tens digit and the ones digit. My hope is that as students start listing out the multiples of 9, they will be able to see the pattern and work with it. I like exposing my students to several different ways of thinking about multiplication. My hope is that they find one that works for them.

And finally, I really like this problem because there are good extension questions. If a student finishes early, they can find the next smallest multiple, and then the next one. Once everyone has had plenty of time to work, the class can talk about divisibility tests, and they could work on finding all the three-digit multiples of 9 that have only even digits. And so on.

Why This Is a DOK 3 Problem

This is a DOK 3 problem because it invites multiple approaches, and even though there is a correct answer, students need to be able to explain why the number they chose is the correct one. They also have to analyze each multiple they come up with to make sure that both digits are even. (I anticipate several students saying that something like 36 or 72 is correct because it is an even number.) Depending on how quickly students finish, they might be asked to investigate the pattern that shows up in the multiples of 9 that only have even digits. Students could then be asked to draw a conclusion about that pattern.

My Goal for Student Learning

This problem is intended for a class of new students with low math levels, many of whom struggle with multiplication, and it is going to take a while for most of them to finish. My goal is for them to stick with the problem and not get discouraged as the numbers start getting bigger and bigger. I am giving this problem during the first week of class, and my sense is that the students aren’t used to struggling with math problems for long periods of time. Another goal is for students to come up with an organized approach to tackling this problem. That is, I would like to see some students create tables or lists rather than simply start multiplying 9 by randomly chosen numbers. Also, because it’s so early in the cycle, I would like to see my students feel comfortable talking about their work and the work of their peers.

Challenges for Students

The first challenge I anticipate involves the vocabulary and phrasing of the problem. Because I will be working on this problem with a group of new students, they have only recently been introduced to multiples and factors. They will likely need a quick refresher. Similarly, I expect to see

THE CUNY HSE CURRICULUM FRAMEWORK • MATH REFLECTIVE TEACHING 231

Student Work

FI DE L’S APPROACH

F

idel is one of the strongest students in this group. He attends every class session, asks good questions, and works hard on every problem that he encounters. Even this early in the cycle, Fidel’s classmates have come to recognize him as one of the leaders in the class, and they often rely on him to help them out when they are struggling. However, Fidel had a really hard time with this problem. First off, he needed a reminder on the difference between odd and even numbers, and after we talked about it as a group, he wrote them down just to be sure. Then he started working. If you look closely at Fidel’s work, you’ll see that he started out by writing all of the multiples of 9, but then he erased them. When I asked why, he explained that when he got above 100, he noticed that all of the multiples would have a 1 in them and therefore couldn’t be correct. This is where he gave up on the list and decided to try guessing and checking. His guesses look a little disorganized, but there is a method to them. He was trying to locate multiples of 9 that were in the 200s. His first guesses were much too big, but he kept making adjustments. He erased most of these, but he left a few and, after a while he found 9 × 32 = 288.

What was interesting about Fidel’s work is how he noticed some important qualities about the numbers—namely, that the correct answer would have to start with a 2, 4, 6, or 8—but he didn’t come up with a good way of organizing the work that he was doing. Because he guessed and checked, several students finished the problem before him and began working on the extension questions. This was a case where the strongest student in the class struggled the most because the problem- solving strategy he chose may not have been the most appropriate one.

J EAN MAR I E’S APPROACH

O

f all the students in the class, Jean Marie probably has most difficulty with math. She performs all basic calculations on her fingers, and she has very little confidence in her ability to grow as a math student. This was the first extended problem that she had done on her own.

From the outset, Jean Marie was frustrated by this problem because she noticed that it had to do with times tables, and she reminded me several times that she doesn’t know her nines. You’ll even see at the top of the page that she was drawing circles for the first couple multiples of 9. While everyone else was working on their own, I spent a lot of time sitting with Jean Marie and talking her through the problem. She

232 REFLECTIVE TEACHING THE CUNY HSE CURRICULUM FRAMEWORK • MATH

started with 9 × 1 = 9 but then couldn’t remember 9 × 2. So we talked about how she would figure it out. She seemed a little embarrassed when telling me that she would count on her fingers. But when I told her that her method was fine, she went back to work. She counted up to 18, and then counted up another 9 to 27, and so on. From here, she was able to work on her own, but she tried to give up about every five minutes. It took a lot of encouragement to get Jean Marie through this problem, and she made a lot of mistakes. I made the decision to help her identify her mistakes so that she wouldn’t get more frustrated as she got further and realized she had been working with incorrect numbers. In the end, with a lot of support, Jean Marie did arrive at the correct answer. I liked how well-organized her method was, and I really appreciated her ability to stick with a problem that was so challenging and frustrating to her. In the end, Jean Marie finished before Fidel did! And it was a really important moment for her. She wrestled with a problem that she thought she could never do, and she was successful.

FE LICIANO’S APPROACH

T

his was the day when I learned that Feliciano is incredibly good with numbers and loves doing math. He did this problem on his second day in class, and since he was the first to finish, I got to talk to him about some of the extension questions that I was hoping to use. Feliciano started out by listing the multiples that he knew off the top of his head, and then he worked additively from there. This approach was largely typical of what most students did. Each time he arrived at a new multiple of 9, he added 9, wrote the next one down, and repeated. By following this pattern, Feliciano got to 288 pretty quickly, so I asked him to find the next multiple of 9. He kept working additively for a while before figuring out that 9 × 52 was equal to 468. Here, Feliciano stopped and looked a little more closely

Student Work

Student Work

234 REFLECTIVE TEACHING THE CUNY HSE CURRICULUM FRAMEWORK • MATH

I might also give out hundreds charts to students who really struggle with their times tables. It could help them to get started, and it could also help them to identify a pattern that will help them remember their nines in the future. And lastly, if I do this problem early in a class cycle again, I might ask students to write a reflection of what it was like working on the problem.

n Unexpected Challenges I gave this problem again in another class—one with a wider range of math levels—and found that it was a little difficult to manage all of the students. Some students finished the problem quickly, while others needed me to sit with them and keep them working, give them feedback on their work, etc. This made it challenging to keep the higher-level students engaged while still supporting the students who needed individual attention.

n Student Takeaways My students liked this problem, and it fit in well with the work on factors and multiples that we were doing in class earlier in the week. They enjoyed trying out and discussing some of the problem- solving strategies that we had been working on as a class. They also got to hear about different solution methods from their peers, and they had the opportunity to share their frustrations with the problem, as well as the sequence of steps they took to break through that frustration. For one student in particular—Jean Marie—this problem was a major breakthrough. For the first time in class, she stuck with something, got angry at it, settled back down, tried again, failed, tried again, and finally succeeded. She hasn’t given up on a problem since. This is a great exercise to do with students who need to learn how to stick with something. It has a very low entry point, but the discussion can go a lot of different ways. My students were also able to see the importance of pattern recognition in math. Recognizing the patterns for multiples of 9 helped several students write out all of the multiples quickly, rather than adding repeatedly. After we finished this activity, “Look for a pattern” was added to our list of problem-solving strategies, and it has since helped students succeed in other difficult problems.

n Advice for Teachers This is a good low-entry problem for students who are new to your class, but it could be used at any point in the cycle as a warmup exercise. Teachers should be prepared for students to get frustrated and give up, but they should also be prepared with extra questions for students who breeze through the exercise. The problem works

THE CUNY HSE CURRICULUM FRAMEWORK • MATH REFLECTIVE TEACHING 235

best if you allow plenty of time for the class as a whole to debrief, especially because students need to see that there is a bigger takeaway from doing the problem than just crunching numbers. And there’s a lot of rich territory on which to have that discussion. Talk about organizing information, talk about patterns, talk about divisibility tests, and emphasize key vocabulary. Help your students understand that their struggle was a productive one.

THE CUNY HSE CURRICULUM FRAMEWORK • MATH REFLECTIVE TEACHING 237

critical point, and that critical point will give me the x -value that will maximize area. To find that point, I need to know the derivative of A with respect to x ; this will be the equation for slope of the tangent line to the graph. The critical point I’m looking for will have a tangent line with slope 0.

The derivative of my area formula is A’ = 50 – 2 x , where A’ represents the slope of the tangent line at a chosen point. Since I’m trying to find the point where the slope is zero, I substitute 0 for A’. Now I have 0 = 50 – 2 x. When I solve this for x , I get the solution x = 25 meters. So this is the optimal length, which means that my optimal width is also 25 meters. The shape that will maximize area is a square that is 25 meters by 25 meters.

To answer the second part of the question, I applied the same rationale to a rope that is now 200 meters in length. If the optimal shape is a square, then it would be 50 meters by 50 meters, and it would have an area of 2500 square meters. This means that each prospector would get 1250 square meters of land, which is twice as much as they would have before. So it does make sense to “join the ropes.”

Other Ways to Solve This Problem

The method I outlined above is impractical for teaching, and I only tried it to challenge myself and to see if I could remember how optimization problems worked. So after solving it algebraically, I wanted to examine the relationship between area and perimeter just so that I could see how much the area changed when I made slight modifications to the dimensions. I drew a few different rectangles and ended up at the square that was my final answer from before.

40 35 30 25

10 (^15 ) 25

A = 400 A = 525 A = 600 A = 625

The pattern I noticed when drawing the rectangles out in this order— from long and skinny to square—showed me that as a shape becomes closer in form to a square, the area increases.

I also wrote it out in table form, just so that I could have an organized chart showing the areas given by different dimensions. I started the table at 40 by 10, as shown below, and worked my way up.

238 REFLECTIVE TEACHING THE CUNY HSE CURRICULUM FRAMEWORK • MATH

Length Width Area

Length Width Area Consec. Diff.

25 25 625 MAX

The table is interesting because it provides the opportunity to see the consecutive difference in area each time that the dimensions are adjusted by 1 meter. I noticed a pattern, which is added in the updated table below:

240 REFLECTIVE TEACHING THE CUNY HSE CURRICULUM FRAMEWORK • MATH

My Goal for Student Learning

One goal in presenting this problem is that my students will be able to apply the basic concepts of area and perimeter in a setting that is a little different from what they might be used to. I also hope to see that they’re able to think creatively about the problem and make adjustments to the shape of their plots in order to see how the shape of the rectangle has a significant effect on its area. In other words, I want them to be able to create a possible plot but then—without my intervention—try drawing other rectangles as a way of checking to see if their answer is correct. Another goal is for students to verbalize the relationship between the shape of a rectangle and its area. What the students should notice is that, as the rectangles become more square-like, the area increases. I also would like to see an organized approach to solving this problem, although I realize that the way students organize their work will differ greatly.

Challenges for Students

I anticipate a number of students drawing one rectangle and thinking that they’ve answered the question after they’ve successfully calculated its area. Moreover, I anticipate some resistance when I prompt them to try drawing other rectangles so that they can compare the areas of each. I also anticipate some issues with understanding the situation. Even though the prompt specifically mentions rectangular plots, I have a feeling that some students will miss this part. They’ll understand that they’re getting four stakes and a rope, but they won’t really know where to go from there. So I might have to intervene a bit just to clarify exactly what the question is asking them today. I also foresee students jumping to a quick conclusion about the second part of the question. That is, I think that some will gloss over the part about joining two ropes together and just assume that because you’re sharing with another prospector, you would get less land. To support students who are struggling, I will first ask them to tell me what is happening in the problem. I would want to make sure that they understand exactly what they’re getting from Billy and why they are getting those materials. If they are unable to make a rectangle, I might ask them to draw one, and then I would ask what the length of the rectangle could be. They could then try a few things and check their work. For students who try to stop after drawing one rectangle, I’ll ask how they know that the one they drew provides the most land to work with. So after they try one more, I’ll ask that they try another. And so on. I have some students in my class who really struggle to do long multiplication, and so I may allow them to use calculators. The goal of this activity is to encourage reasoning about shapes; it’s not about

THE CUNY HSE CURRICULUM FRAMEWORK • MATH REFLECTIVE TEACHING 241

crunching numbers.

Extension Questions

If some students finish early, I would ask what would happen if three, four, or five people joined their ropes together. How much land would each person get in these cases? And is there a pattern to the increase in land you get by working together with other prospectors? How could you organize the data to see what the pattern might be? Could this be viewed as an input/output table, or a function? If so, what would be the rule of the function? How do you know? How many ropes would you need to join together so that you could get 7500 square meters to work with?

Student Work

E LISA AN D B E LE N’S APPROACH

B

elen is one of the brightest students in the class. Elisa struggles and has missed several classes because of her work schedule and other issues. This group had a hard time getting started, but once they figured out a pattern, they were able to make progress. What I like about their representation is how organized it is. They begin with a rectangle that is 30 meters by 20 meters; it has an area of 600. The next rectangle they drew had dimensions of 28 by 22, with an area of 616. When I talked to B. and E. about this, they said that they were surprised about what happened to the area. They explained that they noticed how, when they decreased the length and increased the width, the area got bigger. So they kept doing this until they arrived at the dimensions 26 by 24, for an area of 624. This was the greatest area possible, they said. When I asked why they didn’t go a step further and try 25 by 25, they reasoned that it wasn’t allowed: The plot had to be a rectangle, and 25 by 25 would be a square. I was interested in this solution because I predicted that students would get hung up on the square/rectangle issue, and these two were adamant that the plot could not be a square. So we talked about this. Also, notice their reasoning at the bottom. It says, “After a while we figure it out that if you increase the width, then you have to decrease the length in order to have the same perimeter, but bigger Area.” I understood what they meant, but we talked about it for a while to get some clarification. Is there a point at which decreasing length and increasing width doesn’t increase area anymore? What is that point? Why does it work this way? This group’s graphical approach was very typical of what other students tried.

Student Work