Burton W. Jones Distinguished Teaching Award Lecture

This address opened the Annual Meeting of the Rocky Mountain Section of the Mathematical Association of America, University of Wyoming, Laramie, April 12, 2002

Growing Students

Abstract: What is the best part of doing mathematics? For you personally? Why do you persist in this crazy, difficult subject? Do the students in your classes get a chance to feel that same exhilaration? If we wish to rejuvenate our undergraduate mathematics programs, and entice more students to learn mathematics, we must rethink our curriculum and our roles as teachers.

Thank you Janet for the kind introduction and to Byron for the warm welcome to University of Wyoming. I thank the Rocky Mountain Section and the Burton W. Jones Teaching Award Committee for selecting me for this award last spring. Teaching is the most rewarding and important professional activity that I do. It is very special to be honored for doing what I value the most.

I thank my wife and sweetie, Dee Marcotte and my colleagues in the Department of Mathematical and Computer Sciences at Metro State. I also thank my friends and colleagues in math departments across the state and especially those at CSU and UNC who were part of the RMTEC experience. Specifically, I want to thank my Chair, Dr. Charlotte Murphy, and my colleague, Dr. Lew Romagnano. All of you have helped to create a supportive environment in which I have grown as a person and as a teacher of mathematics.

Today I’ll be telling you about my journey as a teacher over the last ten years, showing you examples of what I do in my classes and sharing with you, how and why my approach to teaching has changed. I hope you get a glimpse of how I conceptualize my role as a teacher and understand my reasons for doing what I do.

As I look back, I can see that my fire as a mathematics teacher gets lit by seeing my students engage with mathematics and grow in their understanding of themselves. We are lucky to be math teachers because, in our society, math is viewed as difficult. That means we have our student’s attention when they are in our classes and we have a chance to impact them deeply. In fact, we will impact them deeply whether we are aware of it or not.

The coolest student feedback that I ever received happened a few years ago in the hallway. Carly had been my student in a Math for Elementary Teachers course several years earlier. As we passed in the hall, I recognized her, greeted her, and asked how she was doing. She said, “Fine.” and added, “You know, I use your class everyday.” Now every one of us in this room loves to hear our former students say things like that. But “everyday” seemed like an exaggeration. So I protested. That’s always the thing to do when someone tries to compliment you. You get to hear it again.

I said, “Everyday?” As she hurried off, she said “Yep. Whenever I get stuck, I know what to do to get unstuck.”

In a nutshell, that’s my goal as a teacher. I want my students to grow in their understanding of themselves and feel competent. I shamelessly use the arena of mathematics instruction to try to accomplish that.

Let’s start by going back to my College Algebra class at Metro circa 1991. Let’s look at my syllabus. It shows the textbook and the grading system. The final grade is determined as percentage out of 500 points. Each day, class began by going over problems from the homework. I’d lecture on the ideas in the next section and I’d have the students work a few examples just before class ended. It was a pretty standard college mathematics classroom.

I was not satisfied with what my students were learning. I noticed this when I taught Calculus Two in the spring semester after teaching Calc One in the fall. Even the students who had had me for Calculus One did not remember important things. I recalled having done a thorough job on those topics in that class. I bet this is familiar observation for many of you. This is not about our students because it happens too frequently. It is important data about the ineffectiveness of our curriculum.

I began my quest to understand what wasn’t happening, hoping to find a way for my students to learn and remember more. I asked them to write short explanations on quiz and exam questions. I added part b) to the standard “Solve” or “Factor these.” I used simple questions like “Explain what you just did.” Or “Describe when you‘d use this technique.” or “Why does this answer make sense?”

I was not surprised by the results. I was shocked! Most students had been doing the problems correctly as far as I could tell, but even when very good students wrote about their thinking, it didn’t make sense. Apparently, they had learned to memorize steps and repeat them without much thought. There was a real problem here and I wanted a way to do a better job.

I began to hear students asking the very reasonable question that they had been asking for a long time: “What is this stuff good for?” I interpret that question in two very different ways. Sometimes students want to know about situations and contexts where the mathematics could be applied. But more often when we sat down, I’d find the question to be a camouflaged plea. They were asking me to help them make sense of “this stuff.” They wanted to be able to understand the material and see how it connected to other ideas. They wanted to learn about things in a context that was meaningful to them.

I figured out a way to deal with the first interpretation: “Where would we ever apply this stuff?” I added a new requirement for the course. It was a five - six page paper in which students were to find and write about applications from their lives or their major. It was worth 50-points, and could not be dropped. It was developed with a colleague from the English Department. She helped me refine the assignment description and the process so that I got the kind of papers that I wanted. Prior to writing, students had to get their topic approved by submitting an abstract of their paper. The current version of this assignment is on my web site. You are free to look at it and borrow it. Modify it and let me know how it works for you.

Initially, I was terrified of grading papers where writing was involved, so I cheated. While the assignment was officially worth 50 points I told students they could get as much as 80 points for a really good effort – extra credit. At first, I gave everyone more than 50 points so that no one would question my grading. After I got more comfortable making judgments about writing, the grades became more reasonable. I still assign the paper in all my classes except those where students write extensively for other assignments in the course.

I am very pleased with the results. The papers contain mathematical details and meet my standard – no B.S. The students like them because they can tailor the topic to their own interests. I have learned how pilots decide the amount of fuel to order before a jet flight. In a Finite Mathematics class, a student analyzed the price/demand structure for the bowling alley where he worked part-time and found that theoretically the owner was only a couple cents off from maximizing his profit. You’ll have to ask me about the how we worked Ballet into a paper on Trigonometry.

Students are very proud of what they write and get energized by seeing the math at work in their world. Additionally, many students now come to my office for a different reason. Instead of haggling for a few points, they want to get help choosing a topic. That conversation helps them connect this class with their own world. It helps them grow personal connections with the mathematics.

Even with these changes, I was still unsatisfied with the outcome of my teaching. The format and layout of our textbooks together with my own belief that I had to show my students how to do things, kept me from getting the results I hoped for. The students still had little chance to make sense of what they were learning. I had heard about the NCTM Standards. I participated in discussions of how mathematics could be taught differently. I actually read parts of the Standards documents, but I still couldn’t see how to get from where I was as a teacher to where I wanted the students to be in this new vision of mathematics teaching and learning.

The next part of my journey is harder to describe. There was lots of experimentation and rethinking how to be a teacher. I pondered how I could run a classroom where things that we studied made sense to the students and still met the syllabus requirements. There was a never-ending struggle about whether I was doing the right thing for students. For me to tell you the next part of my story, join me now in an adventure, a little thought experiment. Allow your usual reservations and judgments to rest quietly for a few minutes while we take a little voyage.

Suppose you are in a mathematics class. The room is set up with tables rather than desks and students are working together in groups on a problem that you have just posed to them. They find it both interesting and difficult. These students are accustomed to discussing their ideas with each other, probing possible ways to solve the problem and fully expect that, with some cooperative work together and further effort on their own, they will make progress on this problem. In addition these students have come to rely on themselves when they wonder if they are reasoning correctly and whether they have the right answer.

Your attention is drawn to one group that is getting particularly animated. They call you over and proudly start to explain their solution. You tell them with a knowing smile that you can’t quite remember how to do this problem and so, as they are explaining, you interrupt with questions about why they did this or that and why their method makes sense.

They are eager for your questions. They have anticipated most of them and are ready to answer. They go through their whole solution with you. The exchange ends without your ever saying they are correct. They finish feeling very satisfied with having struggled on a problem that looked impossible to them just a few minutes before. You ask them if they would present their solution to everyone. They get ready for the presentation by dividing up who will say what.

They know that in a couple days they will be assigned a problem that is related to this one. Individually they will create an extensive write-up in which they show all the work and explain why every step was taken. They’ll write summaries of the big mathematical ideas in the problem and even pose extensions and generalizations of the problem.

They like this class even though it is such hard work. Explaining things carefully is not easy and their homework assignments are quite lengthy. But they like it because the problems are meaningful, they understand the ideas and why they make sense; and because they can see themselves growing as problem solvers and explainers of math. In short, they know they are getting stronger and smarter and they like that feeling.

For every one of us in this room, one of the most exciting things is to be around students who engage in the hard work of learning mathematics. Think back to the last time your heart jumped when your students were in the process of cracking a really tough problem.

Even better, think back to when you were stuck on a hard problem, felt the satisfaction of working on it and getting it. At first you were frustrated, doubted whether you could figure it out. But with more work, you got it. Isn’t that a large part of why you like mathematics? Didn’t you get that indescribable feeling of being stronger than you thought you were?

If your answers to these questions are the same as mine, your next question is:

How can I design and run a classroom that offers these types of experiences to students and still meets the instructional objectives for the class?

There are three main components to this kind of class: the problem-oriented curriculum; teacher goals and behavior in the classroom; and the students and their willingness to buy into this process.

The problem-oriented curriculum

It is absolutely critical that you begin with a small set of solvable problems that “cover” the curriculum. Each problem must be carefully chosen so that the solver will encounter the crucial mathematical ideas of the course. The problems must be engaging or puzzling to the students. And the students have to be able to solve them.

After they are chosen, they have to be woven together so each one prepares the ground for those that follow. This is a big job! Usually these problems turn out to be hard for students. So hard, in fact, that students quickly prefer to work in groups rather than on their own. And finally after the students work them all, they must be able to pass the test for the course.

Finding and assembling these problems is, by far, the most difficult part of creating a class like this. It requires a thorough knowledge of content of the course combined with an ability to think outside of the box. In our traditional curriculum, most ideas are lined up as if they were trees planted in rows – orderly and organized by idea. The problems in this kind of course take students on a very non-linear path through the same forest. As a result, they become agile and resourceful thinkers as they learn the material.

Being able to trust that these problems “cover” the syllabus enables the teacher to focus on other things and let problem solving and the class process teach the course.

Those of you who are familiar with the Moore method will see similarities between these methods and the techniques that he used with his graduate students. However there is a big distinction to be made because his approach created a very competitive atmosphere. Whereas in our setting, cooperation in the problem-solving phase is combined with individual accountability in writing-up portion of each problem.

So the first job is to set out interesting, carefully chosen problems for everyone to work on. In our Math for Elementary Teachers course, my colleague, Lew Romagnano, did a great job of collecting and creating such a set of problems. Over the years working together, Lew, Dr. Don Gilmore and I have modified them and polished the sequencing. This afternoon Don Gilmore will be presenting more information about our class in his session.

On the handout that you received, there are two examples from the first week of our Math for Elementary Teachers course. They are to be solved with no prior instruction on any of the ideas they contain. The first is the famous open-top Box Problem. I first saw it in Calculus One. These days students might even see it in middle school solving it with graphing calculators. In our class, the students generally use a chart of values to solve it. They must explain the formula for volume as part of their solution.

The Box Problem (from Integrated Mathematics I, MTH 1610)

You work for a company that sells candy. You and your working group have been asked by your supervisor to prepare a report for management that describes how to use a warehouse full of cardboard sheets to make boxes for the candy. These are to be open-top boxes, produced by cutting and folding the 20 cm by 26 cm cardboard sheets according to the diagram below. Squares are to be cut from each corner, and the sides folded up to make the box. Your goal is to produce boxes that hold the most possible.

Here is the AIDS Problem. It can be thought of as an application of Bayes’ Formula. We use it to introduce the ideas of probability.

The AIDS Problem (from Integrated Mathematics I, MTH 1610)

After hearing emotional testimony from a woman who apparently got AIDS from her dentist, Congress considered a bill that would have required all health care professionals in the United States to be tested for HIV, the virus that causes AIDS. Let's consider whether this would be a good idea. In this activity you are going to explore the following situation. Suppose that, as a result of this bill, you are one of 100,000 health care professionals around the country who are tested for the HIV virus. Your test comes back positive, meaning that the test says you have been infected. What is the likelihood that your positive test result is accurate; that is, what is the chance that you are actually infected with HIV if you have a positive test result?

The procedure used to test for HIV is extremely accurate. We will assume that the test is 99.5 percent accurate*. That means if someone is actually infected there is a 99.5 percent chance of a positive test result, and if someone is actually not infected, there is a 99.5 percent chance of a negative test result. (Note the careful use of words here.) The most recent estimate is that about one in every 250 people in the United States is infected with HIV. For this activity, assume that the 100,000 health care professionals being tested are infected at that same rate.

Later in the course they might see the Border Problem. Their second major turn-in problem is the Toothpick Problem. I’d be happy to e-mail you complete versions of any of these problems.

In my upper level courses, it is harder for me to find great problems. In Foundations of Geometry, I often use David Henderson’s text, Experiencing Geometry. It is laid out in this style. One of the problems I use in that course is this one from Glenn Bruckhart. Find two triangles that are not congruent and yet five parts of one triangle are congruent to five parts of the other triangle. This problem creates a very natural setting in which to investigate the various congruence criteria, ASA, SSS, etc.

The Five Parts Triangle Problem (From Foundations of Geometry MTH 3650)

Find two non-congruent triangles that have 5 parts of one triangle congruent to five parts of the other. (If six parts were congruent, the triangles would be congruent.) Use the transform menu from Sketchpad. Submit the Sketchpad construction and an explanation of why it works.

Another problem that is challenging to my geometry students is the Best Distance to View a Painting problem.

Best Distance to View a Painting (From Foundations of Geometry MTH 3650)

Suppose that you are walking toward a large painting hung above your head on a wall in the Denver Art Museum. When you are far from the painting, you cannot see small details because of the distance. When you are too close to the wall under the painting, it will appear heavily foreshortened because your angle of view is very small.

The distance that gives you the best view of the entire painting is the distance where the angle of view is as large as possible. In the diagram, below, the top of the picture is at T and the bottom is at B. The horizontal ray from A represents the floor, and the segment DE is eye level. Points A, F, D , E, and C all remain fixed. The variable points are T, B and I. Find a construction of the best viewing position that will stay the best when the picture or the eye height is changed. (In this construction, point I moves in response to changes in position of points T or B.) Prove that your construction yields the best viewing position. TURN IN: Disc and hard copy with explanation, together with conjecture and proof.

To summarize this component again: The set of problems is critical. They deliver the course. But only if the teacher knows how to use them.

Teacher goals and behavior in the classroom

My goal is to make room for students to grow mathematically and personally. In order to get the outcomes that I want, I must not tell students how to do things. I can’t tell them if they are right or wrong or even close. Nor is it my job to protect them from frustration. I can support them in their frustration, but I mustn’t protect them from it. All of these actions would undermine my ultimate goal: I want them to become strong, independent, adventuresome thinkers. This teacher behavior is a very counter-intuitive and it differs from most mathematics instruction.

So what is my role in class? What is it that I do and say?

I strive to create a community of investigators. I create a container in which my students can learn to be mathematicians, a place where they conjecture, question, make mistakes and persevere. That means I am neither the resource for ideas nor the arbiter of correctness. It is my main function to model questioning and show what it looks like to understand an idea; to help them distinguish knowing how to do task from understanding why it works. I help everyone work together. I create a social atmosphere in which the students are respectful of each other even as they argue for and against ideas. I must set clear expectations that when we are in here, we are having fun and working hard on math problems.

I use my energy to help reduce the fear and anxiety that students bring to math class. Also, I need to use my energy to fuel positive beliefs in themselves. They need to know that I have no doubts about whether they can do this. They have to see that being frustrated does not mean they are wrong. Hard significant problems won’t be solved in 10 minutes. I have to sell this idea…repeatedly.

I use humor and silliness to communicate to students that this is a place where un-expected things take place. If they suddenly become confused or excited, there is room for that to come out.

I use my questions to model understanding. Through them I set a clear standard that making sense is the bottom line in the class. A thousand times a semester, I ask “Why is that true?” Notice the reversal here. I am asking why and they are answering.

Let’s take a detailed look at the model I have used for my questions. You all have copy. It comes from page 3 of the Professional Standards for Teaching, the purple book published by NCTM in 1991. When I began to teach our Math for Elementary Teachers for the first time, I decided to not do it in my usual way. I decided to take a chance and try something different. I started by using these questions. They give a precise definition to what I mean here about a new role for the teacher in this kind of class.

Helping students work together to make sense of mathematics.

"What do others think about what Janine said?"

"Do you agree? Disagree?"

"Does anyone have the same answer but a different way to explain it?"

"Would you ask the rest of the class that question?"

"Do you understand what they are saying?"

"Can you convince the rest of us that your idea makes sense?"

Helping students rely more on themselves to determine

whether something is mathematically correct.

"Why do you think that?"

"Why is that true?"

"How did you reach that conclusion?"

"Does that make sense?"

"Can you make a model to show that?"

Helping students learn to reason mathematically.

"Does that always work?"

"Is that true for all cases?"

"Can you think of a counterexample?"

"How would you prove that?"

"What assumptions are you making?"

Helping students learn to conjecture, invent, and solve problems.

"What would happen if...? What if not?"

"Do you see a pattern?"

"What are some possibilities here?"

"Can you predict the next one? What about the last one?"

"How did you think about the problem?"

"What decision do you think he should make?"

"Compare your method of solution to hers. What is alike and what is different?"

Helping students to connect mathematics, its ideas, and its applications.

"How does this relate to...?"

"What ideas that we have learned before were useful in solving this problem?"

"Have we ever solved a problem like this one before?"

"What uses of mathematics did you find in the newspaper last night?"

"Can you give me an example of ...?"

I memorized these questions and spoke them as an actor. I used a tape recorder to record the class. Then I listened to find out how many times I used these questions. The class changed right before my eyes. When I saw the students actually become active participants in the class, I loved it. This was the beginning of my serious involvement in revising all of my classes.

What was different? For one thing, I had stepped out of the lime light and made space for something else. That action alone has been a very important discovery for me. I must back away and trust that the students’ curiosity, their new habit of being able to understand why something works and the atmosphere of inquiry will create a place where students’ energy can step in and fill the void.

I get tremendous satisfaction when I watch the students buy into my values of understanding, of being able to explain how and why something makes sense, of being curious about things mathematical. Often after one of these classes, Lew, Don or I have to find one of the others to share how cool it was in class that day and relate the unexpected ways in which the student discussion evolved.

One of the most delicate parts of this kind of teaching is running the whole class discussion that happens after most groups have worked the problem. I must combine my knowledge of the mathematical content of the problem with the goal of having the class encounter the mathematically big idea(s) contained in the problem. This is the where the huge payoff comes. After students have presented their different solutions and no one else has questions, I get to come to the board and say to them, for example, “This kind of formula is called a linear function. Just like you have seen in the problem, here are the clues that tell you it is linear and not quadratic, or exponential…” I get to give the punch line to a group of students who are already intimately connected to the phenomenon that I want to describe and name. Just as in mathematical research, examples come before definitions. These students already know linear functions because they are the kinds of functions they have been struggling with in the Border Problem and the Toothpick Problem.

Running a class like this is intuitive, subtle, and delicate. Those of you in this room already know how to do this kind of “bob and weave” because you do it all the time during your office hours. I am sure that you can recall a recent moment when one of your students came in to your office, stuck on a problem. You could immediately see that they were “Oh so close” to solving it. You knew that if you chose your responses carefully, the student would be helped by you, but not so much that they felt that you solved their problem. As they left your office they may even have been a little disappointed that you didn’t solve it for them. But they came in the next day having solved the problem and very proud of their accomplishment.

In that moment, when they came in to see you, and you understood where they were in the problem, your goal was not to solve their problem or show them what was correct. It was to allow them the chance to become stronger. I try to create this experience in my classroom everyday, not just in private moments in my office.

It is difficult to explain to some one else how to do this, isn’t it? It is a strange combination of backing away at the same time being very present that characterizes the support that I strive to give them as they work. And in that moment, it requires that I have a very clear understanding of the mathematics involved in the problem.

I have outlined two of the components that I think must be in place for this type of class to work. A great set of problems and a set of teacher beliefs and behaviors. Together they promote a classroom community that values and welcomes students’ ideas and energies. You have seen a brief picture of how this type of class is run and why it is done that way.

One of my recurring struggles involves trusting. I have to trust the problems I am using. Even harder, I have to trust that the students will buy in. To the extent that I decide that I need to take charge and explain or force something, I violate my basic ground rules. I have to trust and believe that students will engage, question, and get involved. I must trust that they will be motivated to ask why something makes sense. I must trust that they will be bothered until they understand it clearly and can explain it. I hope my excitement conveys to you that my trust is almost always rewarded beyond my expectations.

The students and their willingness to buy into this process.

How often does this happen? I am most successful in our Math for Elementary Teachers course. These students are mostly women, math phobic for sure and all are not excited about taking these two classes. Approximately 90% of them buy into the plan. Many of those are significantly changed by their experiences in this class. It is the first time in their lives that they see mathematics as something that makes sense, a subject where they can feel powerful as problem solvers and “explainers” of math. Hard work and long papers in exchange for not being afraid. That seems like a good deal to them. Even students who have had calculus find this course challenging.

Another exciting thing that happens for me in this course is that my students work very hard. I hear and read about colleagues who complain that the students these days just won’t work hard. That is not my experience in these classes. They turn in five or six 5 – 10 page papers each semester in addition to their daily work. Each paper is graded on a Exceeds – Meets – In Progress grading system that requires that they resubmit papers that do not yet MEET the standard.

In my upper level math major classes, I am much less successful in meeting these goals. I don’t fully understand why, but it seems to be true. These students can see right away that explaining carefully and solving hard problems is far more work than they have had to do before. There must be other factors.

When I read the papers from my Math for Elementary Teachers course, I see the process of mathematics taking place: problem solving, conjectures, writing proofs and explanations and making generalizations and extensions of the assigned problems. So it is disappointing when I read the work from my upper level students. Their papers are much more formulaic and lack vigor and personal involvement.

There is an irony here. My upper level students know a lot of mathematics. They can factor, take derivatives, integrals, and solve long algebra problems. But there is a way in which they are handicapped by their success in math classes. They can do all of this and yet their knowledge is often not very connected or accessible to them. We all see this when we ask them to solve problems that appear out of the context of a class.

It is incredibly rewarding when all three components come together and the class buys in. And this happens almost everyday in the Elementary Teachers course. I feel like I am making a real contribution to the growth of my students. I sense that this class deals with their self-concept as problem solvers and helps them feel more confident and competent. In short, it grows them. And seeing my students grow is what it is all about for me. Because the problems we are doing contain significant mathematics and because I grade strictly, they know they are improving measurably - getting stronger. They are better at figuring out what to do even when they don’t know what to do next. They can explain their ideas. Even better, they know when they don’t yet understand it. No wonder they work hard in these classes. They get immediate feedback that they are learning things and that their skills are growing.

I don’t think that every class should be taught this way, at least for math majors. But some should. And they should occur early in the program. Wouldn’t it be great to have students with these experiences and beliefs about themselves in your classes. I’ll never forget the Faculty-to-Faculty meeting a few years ago when Dr. Steve Leth from UNC commented about how great it was to have students who had been in a RMTEC class prior to taking his class. They were active, questioning and insisted on understanding.

I offer these experiences as an example of what can be done, an existence proof. It requires thinking of the course content in a very different way. It requires finding problems that when solved, introduce the students to all of the important concepts in your syllabus. It requires sequencing problems so that you can step back and help the class sort out the ideas that come up. It requires structuring the class so that your students expect to actively contribute in each moment as a member of this community of investigators. You must let them tell you how things work instead of the other way around. You must know and trust that your students will get the ideas that they need to learn in your course by working the problems and understanding their solutions. And most difficult, it requires that you trust that your students will want to think mathematically.

As long as we as a collective mathematics faculty believe that we must force feed mathematical concepts and tell the answers, we are crushing students’ interest. We miss a major opportunity to engage them and let them see what we love about our subject. As long as we are stealing from them the joy of hard won victory after a protracted struggle with a worthwhile problem, most of our students will write us off as doing some arcane, irrelevant dance to very old tunes.

That’s it, isn’t it? Math class is like a dance. You can’t make anyone dance. But some choose to get up and do it. They must get some reward from dancing. Some enjoyment, something must motivate them to get out there and expend all that energy.

Have you ever thought of your classes as a dance? Try it on for a minute just for fun. What would you have to do to create a successful dance? You’d hire a good band, greet everyone at the door and help them to expect a good time. And then you’d have to wait and wait. No one dances to the first couple numbers. But pretty soon, the band plays a fun one and a few go out there on the dance floor. And then party starts. Quadratic field extensions here we come!

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The K-12 schools need our involvement more than ever before. There are outstanding new school curricula that have been written at every level: Everyday Math at the Elementary level, the Connected Mathematics Program at the middle level and IMP at the secondary level to mention a few. While these curricula may not be the perfect solution, they do contain the kinds of problems that I have described today. They are meant to be taught in a manner like what I have been describing. But many of the teachers out there have never experienced this kind of classroom as students. It is a tremendous stretch for them to teach this way and they desperately need the support of those of us who know mathematics. If you are at all inclined, I urge to help. If you are looking for a place, you need only look as far as the school down the street.

In Colorado, as well as for many of you in Wyoming and South Dakota, we are into high stakes testing. The CSAP test is a good test – based on the Model Content Standards. It is a high level test that asks students to both solve and explain their work. I support that kind of test. I don’t even mind teaching to that kind of test. But the results are being used to beat up the schools and the hard working teachers who are trying to make them work. They need our support badly. It is a hard time to be a K-12 teacher. Glenn Bruckhart and I will be talking about this topic in our session on Saturday at noon. We invite you to begin to think about what your role might be in helping to prepare the K-12 teachers and schools to graduate math students that you’d like to have in your classes.

I feel so blessed at this point in my life. I am very excited to be a mathematics teacher these days. There is room for experimentation like there never has been. And there is a tremendous need for rejuvenation of our curriculum for many reasons. I invite you to engage in a new and more exciting life as a teacher. Let’s bring all of our knowledge of mathematics to the task of reorganizing the curriculum so that our students leave our classes with an integrated and connected understanding of mathematics.

Now let’s invite back into the room the reservations and the judgments you have about the kind of change that I have been discussing. Together, each of us, with all of our differing viewpoints, can and must begin the hard work of making mathematics more accessible to everyone.

Thank you again for the recognition that goes with this award. And thanks for listening to the story of my journey as mathematics teacher.

Now you know what puts gas in my scooter!