The National Math Trail

E-Conference Transcript with Kay Toliver
April 27, 2000
Hosted by the Bell Atlantic Technology Education Center
Harlem, New York

Kay Toliver, who has been teaching elementary and junior high school in East Harlem, New York, for the past three decades, uses the Math Trail as a year-end student activity. In her version of the Math Trail, students work in groups to create graphically depicted trails that they turn into books, complete with narratives, photographs, illustrations, maps and, of course, math problems.

Question:
Where did you get the idea of doing the Math Trail?

Kay:
I got the idea when I was a graduate student at Long Island University. In the FAME program (a math leadership program), we had a presenter, Dudley Blaine from Australia. He lectured about the Math Trail that he did in Sydney, Australia with students. It appealed to me because of the whole concept of showing children very vividly that mathematics is not just out of a text book, it's a part of your environment in the real world, and mathematical concepts can be seen everywhere in our neighborhoods and beyond. I want children to see why they need to learn things and how it will impact their lives.

Question:
Why did you start doing Math Trails with your students?

Kay:
When I started using Math Trails, I was teaching junior high school, seventh and eighth grade. It was at a time when our community was looking sort of distressed. I mean there were a lot of buildings that were vacant, and it wasn't a very beautiful place. Besides that, this community is one that is not often highlighted in the news for very positive things. I wanted the students to start viewing their community with more pride. So I figured we should go out into our community, find the math, take a look at what was happening and discuss some of the social issues that were behind it. I felt that this would give them a better appreciation of their school and the community that their school was in. Our trail began at school, because I wanted the children to know something about the history of the school and how the population of the school changed with the community.

Question:
Kay, can you think of three things that would be important for teachers to keep in mind as they are doing a National Math Trail activity?

Kay:
When you are doing any project, National Math Trail or anything else, you have to understand your reason for doing the project. If your reasoning is for assessment and evaluation, then that is why you are doing it. Then you have to make sure that your students have an understanding of why the class is being involved in the project. And then finally, working in a project-orientated atmosphere, I think is key. Bring parents into the loop, because the one thing for the parents to see is that their child is working on an interesting project that involves math. They will want to know how is this really teaching mathematics. When you start any kind of project like this, I think it's important that parents also have an understanding of what is happening.

Question:
Can you tell us what happens basically on the first day? How you run a Math Trail?

Kay:
The first lesson in the Math Trail project is an introduction to the project. First, I explain my philosophy behind doing the project, that it is (1) to assess what they know in mathematics, (2) to get them to be the creators of problems for people to solve and (3) to highlight their community and their school. This project integrates mathematics with writing, with history, with art, and English. For the first time, students start to really see that mathematics is not an isolated subject, that it does relate to other courses. So the first lesson is really just an introduction as to what you will be doing and how we are going to go about doing it.

Question:
Teachers have lots of math concepts and skills that they need to cover every year, and so they might not think that they have enough time for Math Trails. Please talk about how the Math Trail helps students master specific math concepts and skills.

Kay:
First of all, projects are an integral part of what you are doing in school and with activities like the Math Trail, or any project- centered activity, the whole idea is that skills are being reinforced. Skills may also be re-taught during the time that students are going through the creation of this project. At this point students are working together to qualify their understanding of different math topics. The teacher, as a facilitator and a guide, is also making sure that the students are creating, or trying to create, problems on certain concepts in math that they have learned so far this year. If it seems that the class as a whole has had some misunderstanding with material that has been covered, then a teacher can review these concepts with all of the students. This project is a time to see what needs to be re-taught, or re-addressed in another way.

Question:
In a school in a rural area where you have a multi-skill level class, from migrant students, to severely handicapped, to gifted students, how can you do this project with all the students participating?

Kay:
The Math Trail project is a a team project. In every team, you have the top students working with students that fall in the middle and with those students who may have been struggling during the year in mathematics. The reason for that is so that students will learn from one another and they will work together to bring about understanding for every one in the group.

Question:
How do you do a Math Trail if it's hard to take students out into the community, if they're very young, for example?

Kay:
The Math Trail concept is something you can do no matter where you live. You don't even have to go outside, because remember the school is a community unto itself. A trail, a path, can be developed within your school building, or within your classroom. The key thing to remember is that students are highlighting certain areas in their classrooms or their communities, and they are creating mathematical problems around the situations that they are highlighting. Let's say you are in the lunchroom of your school. You might ask students to calculate its area, or to calculate the perimeter of the lunchroom, or find out how many lunches were served during a day, or how many lunches are served during a month or how many different types of combinations of lunches students can have? There are a vast number of ideas that can develop at any location.

Question:
When you have all your students going out and looking for math in their communities, how do you keep control?

Kay:
Well first of all, this project is usually done six to seven months into the school year. By that time the whole tone of the class has been set with a great deal of respect and responsibility. When we go out into the street there are certain ground rules. We stay together, first of all, we don't ever get out into an area where children are wandering off unattended. Parents are asked to join in so that I'm not the only supervisor.

Question:
Why is photography or other illustration important on the Math Trail?

Kay:
Photography is important because it's from the pictures that the children take that they develop and create their mathematical problem. Number one, it helps anyone who sees their Math Trail books to understand from where the problem has evolved. Number two, students like to take pictures. It gets everyone involved. Not only do we have pictures being taken, but sometimes students draw pictures, so that many talents are being applied.

Question:
Why do you have your students make up math problems instead of just going out and finding examples of mathematics?

Kay:
When they are able to write and create problems using the material that they were learning during the school year, then I get a better sense of how well they really understand the concepts. Just looking for the math isn't very much if they can't really apply it to a situation. Remember, they are also solving the problem that they create. So they have to use logic, they have to use reasoning skills, they have to use their ability to communicate their knowledge, they have to come up with problems that make sense, utilizing the language of mathematics to show they understand what they are talking about.

Question:
How can you prepare a National Math Trail submission if you can't take students on an actual field trip into the community? How about getting a fact sheet from the local zoo and having students make up problems from the fact sheet, and then taking digital photographs of the zoo and/or of the students working on the problems. Is that OK?

Kay:
Well that's one thing about the Math Trail, you have to be creative with what's available for you to use. In this case, if you can't make a trip to the zoo but you want to take pictures, bring them in to class and get facts about the various animals at the zoo, then have the children create Math Trail problems about their local zoo without ever visiting it, that's a very good idea. It's often the best thing that can be done. And there's certainly lots of math at the zoo.

Question:
Do you consider the Math Trail to be an instructional activity, or do you consider it to be an assessment activity?

Kay:
I use the Math Trail concept for an assessment activity, because at the time of year when I utilize the project I want to see just how much mathematics the students do understand. With the writing of the problems and the creating of the scenarios that center around the creation of the problem, students have to use the knowledge that they now have. So it's like the creating their own textbook, in a sense, their own problem solving book. You can't create problems if you don't understand the information. Problems must also have solutions.

Question:
What is it that you look for when you are evaluating your student's Math Trail books?

Kay:
Well, the first thing I look for is that the students have actually taken the time to prepare the book, that they have covered a variety of math concepts and that they have taken pride in their book - presentation is so important. I see whether the students have tried to be creative in writing problems that involve multiple steps in order to solve problems that are more than one part. You can have a problem where you have to solve for one thing and that leads to a problem about something else, and then to something else. When the students create this book they can put so much time and effort into it that they are creating a masterpiece, it should represent the best that they can do.

Question:
The students in ninth grade honors Geometry class are very grade focused. Is there a grading rubric for the Math Trail submissions? How many math problems should the students include in solving their Math Trail?

Kay:
There is no minimum or maximum number of problems that can be submitted for the National Math Trail. A teacher can use this project as a means of taking hold of the competition within the classroom. Each student or group of students should try to come up with what would be considered the best problem. The teacher together with the students can develop a rubric for identifying what will be considered a superior problem as opposed to one that is a mediocre problem as opposed to one that is a poor problem. The rubric would vary according to grade level and what is being taught. The students can hold a mock competition and the Math Trails that win are the ones that get submitted.

Question:
Have your students ever come up with anything that really surprised you on a Math Trail?

Kay:
I think that the greatest surprise is to find out how much information students really do know. On paper and pencil tests, some students don't test well and they may appear not to have gained any knowledge or understanding in the math that they have been taught. There is one thing that sticks out in my mind. During the first year I did the Math Trail with my eighth grade class, I had a student who had not been doing well during the school year. We were out on the street one day and we were looking up at a building that was under construction. She looks up and says, 'Miss Toliver I see two similar triangles'. I looked up and I said, 'Wow, they really do look similar.' And then I turned around to the young lady, 'Could you explain to me why you feel that these two triangles are similar?' And when the student went on with her explanation of why the two triangles were similar well, I was blown off my feet. Because on the tests she never showed any understanding.

Question:
What about accountability issues? SAT's and standardized tests concentrate on memorization of math. How can you support real-world teaching when students need to be prepared for these tests?

Kay:
The bottom line is children learn in different ways. There is more than one way to pass on information to students. Memorization does play a part in all of learning. When you are involved in, lets say, a Math Trail project, many times problems that are created come from what a child remembers from a topic that you have covered. It is key to remember that to do well on any examination students must be empowered. Empowerment comes from their ability to understand. You know students understand when they write and speak about what they have learned. A project like the Math Trail allows students to show off what they really have gained an understanding of.

Question:
You look at the world from a mathematical perspective. Would you give us some examples of math that exists in the most common things?

Kay:
Oh my goodness, math IS everywhere. From the trees that grow - you can examine that the branches form angles coming from the trunk. A large branch branches off into smaller parts, forming yet more angles. There is some sort of numerical sequence that is taking place there. When you take a walk in the street you can sort vehicles according to color, according to size, according to brand. You can be walking in a field--I remember my grandfather's field in North Carolina--and you can start talking about the area that's being covered by certain crops or grasses, or in my case, watermelons that he planted. There's math all around. You can find it anywhere!

Question:
How about inside the school itself - is there a lot of math in a school building and in the schoolyard?

Kay:
Math is certainly all around a school. From the classroom, where there's a one-to-one relationship that exists with the number of chairs and students. There are the shapes that exist in the classroom. You can mark up parts of the floor to find areas. The number of windows that are in a classroom, even the number of panels that may be on the classroom door - is it a prime number? Is it a composite number? Is it an even number? Is it an odd number?

Question:
Kay, can you give some examples of community-based math for higher grade levels?

Kay:
Well, almost anywhere you look you are going to be able to find problems that are based in some way on proportional thinking. At the middle school level these can be expressed in terms of numbers, but at higher grade levels you can ask your students to represent them as functions. And when you start looking at problems involving areas, or volumes, those become quadratic and cubic relationships. For example, the number of people that a cube-shaped building can hold is related to the length of one side of the building by a cubic relationship.
Another good area is statistics. The value of storefront real estate in a downtown area, for example, depends partly on the foot traffic. How do you determine how much foot traffic there is? By sampling throughout the day. How many days do you have to sample to have an accurate estimation of the actual traffic and the way it fluctuates? There's a lot of secondary mathematics involved in those problems.
Students might want to go out and create a survey, actually go out in public and ask people to answer questions, then compile the statistics from that.

Question:
OK. Is there anything else that you would like to let everyone know about today?

Kay:
Yes, the Math Trail concept is something that when I started doing it with my upper grade students, my seventh and eighth graders, they developed a real appreciation of mathematics. Prior to doing the project, the students really didn't care much for mathematics. But once they got into the city streets and they saw how much of what we had talked about in class could actually be seen in the design and the flavor of the city, I think that they developed a real appreciation of the importance of the subject of mathematics. Besides remembering formulas and knowing how to do certain things mechanically, it is very important that our students learn to appreciate and value mathematics along with the other curriculum areas.

Question:
Thank you Kay.

Kay:
It was my pleasure.

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