To reform school mathematics in ways consistent with the vision of the National Council of Teachers of Mathematics (NCTM) is an ambitious task. Realizing this vision will require simultaneous changes in both curriculum and instruction.
NCTM asserts that knowing mathematics is doing mathematics and what students learn depends to a great degree on how they learn it. Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) describes what students should learn and provides a framework for developing curriculums that encourage contextualized problem solving and mathematical discourse. How mathematics should be taught is described in the Professional Standards for Teaching Mathematics (NCTM 1991). The emphasis in both documents is on solving nonroutine problems in meaningful contexts.
A new middle school curriculum designed to support the vision of the Standards is currently in development. The approximately 40 units to be included in Mathematics in Context will explore problem situations based on interesting real-world contexts and emphasize the interconnectedness of mathematical concepts. Within these contexts, all students can explore multiple strategies and engage in sense-making through problem solving and mathematical discourse.
To show what teachers and students are beginning to do with this curriculum, we provide an example from a Midwestern school we refer to here as Andrew Jackson Middle School, where the curriculum was eventually piloted. Observing classes and speaking with teachers gave us a window into the process of change in a school where real differences in mathematics instruction are occurring.
Getting Change Started
To promote the growth of mathematical power, teachers must select appropriate mathematical tasks and orchestrate classroom discourse. One of the first steps involved in changing school mathematics is for teachers to accept that aspects of their textbook-based instruction might be problematic (Cobb et al. 1990).
Linda Howard is one of two learning coordinators at Andrew Jackson Middle School. When she transferred from another middle school, one of the first things Howard did was to interview the teachers and visit their classrooms. Most of the time she found that teachers lectured and then assigned exercises for practice. What the students didn't finish in class, they completed for homework. The next day, the class corrected the homework, the teacher explained the next skill, and students practiced again—with more exercises. Unless students asked questions, there was little interaction: It was pretty much teacher-directed, passive learning. Kids who did well were the ones you'd expect to do well—kids who are pretty good at procedures, who are able to sit still, who are able to learn concepts in a fairly abstract way.
Things began to change when the teachers realized that their instruction was not meeting the needs of all of their students. Howard suggested that the NCTM Standards might be a good place to look. For the next six months, they studied the Standards, met with other teachers, and shared what they had learned at conferences. Teaching according to the Standards requires a high degree of authority and autonomy. Teachers began to see that there were other ways of doing things—and that gave them hope. Howard continued: We also got some time for curriculum planning. I remember very specifically the first year I was here ... we went down to the Board Building and went through all sorts of materials. We were going to redo division, multiplication, decimals....We came back with maybe 20 curriculum activities and ideas. The others looked at me and said, “Why isn't there anything to use?” It seemed to me that there were a lot of things here. But I could tell a lot of it relied on me at that moment. It was kind of a break point. Either a unit was going to be put together that was usable and fun ... or something was going to be put together that was going to be run-of-the-mill, kind of boring—and they would say, “So what, the kids didn't like it.” That's when we put together the hands-on decimals unit. According to Howard, the teachers enjoyed teaching the unit—and they really liked what happened with the kids: They became interested in working on curriculum, and that's when they started working on the fraction unit and trying to bring in the pattern blocks. Part of the real change came the next fall when we began to implement some of that.
Most of the teachers at Jackson also attended a series of district-sponsored workshops designed to increase awareness of the Standards and to provide concrete examples of instructional activities that were consistent with them. Meanwhile, they continued to experiment in their classrooms with many of the ideas from the Standards. They supplemented the existing curriculum with available units and materials, began using more group-work and hands-on activities, and included more problem solving and enriched discourse in their classrooms.
At the last workshop in the series, the teachers eagerly agreed to pilot some units from Mathematics in Context. The new curriculum approach supports the professional development of teachers by providing experiences that help them construct innovative ways to teach mathematics.
View from a Classroom
One of the Mathematics in Context units piloted at Jackson Middle School was about algebra and functions. This unit introduced the concept of generalized formulas and explored arrow language, a new notation for representing relations and changes by showing the individual steps in multi-step problems.
We visited a 6th grade class taught by Mark Valente. Prior to the observations, the students had spent several days working with arrow language and formulas. Valente began with a chapter that explored the exchange rate between U.S. dollars and Dutch guilders. In arrow language, the rate was represented as:
Figure
To make sense of the relationship between the two currencies, the whole class used the inverse of this relationship to convert prices from guilders to dollars. The table they produced using their calculators is shown in Figure 1.
Figure 1. Conversion of Prices from Dollars to Guilders
What the NCTM Standards Look Like in One Classroom - table
U.S. dollars | 0.61 | 1.21 | 1.82 | 2.42 | 3.03 | 3.64 | 4.24 | 4.85 | 5.45 | 6.06 |
|---|---|---|---|---|---|---|---|---|---|---|
| Dutch guilders | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
Valente then shared an idea from the unit about a student named Elise. Because 1 guilder is about $.60, she reasoned, then $6.00 is about 10 guilders. She invented her own strategy for easily estimating conversions from guilders to dollars:
Figure
- Convert the prices in guilders of five items in the unit to prices in dollars by using the general formula. (required)
- Convert the same prices by Elise's rule. (optional)
- Explain any patterns you notice. (optional)
- Propose your own conversion formula. If it works, we'll name it after you. (optional)
The next day, the discussion resumed. Matt and Pablo, who had worked on the assignment together, stood next to the overhead projector and shared the formulas they had devised. Pablo's formula was:
Figure
Valente asked the class, “Does Pablo's formula work for the five items we looked at yesterday?”
Using their calculators, the students began to check Pablo's rule for each of the five items.
“I got the same answer for the stereo as when I used the general formula,” Julie volunteered.
Richie said, “It works for the TV, too.”
“It works for all five of the items,” added Jen.
Again Valente asked the class whether Pablo's formula worked, and the general consensus was that it did.
“We have lots more,” said Matt. “All you have to do is take 1.65 and divide it by any number and whatever you get as an answer, that will give you two numbers for your formula. See!” Matt showed how 1.65 ÷ .5 = 3.3 and 3.3 × .5 = 1.65. Then he wrote this general formula on the overhead:
1.65 ÷ any # = the other # for the formula
After a few moments of trying this formula on her calculator, Jen said, “I don't think that works. I took 1.65 ÷ 7 and got .2357142, and when I multiplied that number back by 7 again I got 1.6499994, not 1.65!”
Looking concerned, Pablo walked to Jen's desk and watched as she entered the numbers into her calculator again and got the same result.
Richie volunteered, “On my calculator, I got .235714285, and when I multiplied back by 7, I did get 1.65.”
Valente, who had taken a seat at a vacant student desk asked, “Why do you think that is?”
Harper answered, “Jen's calculator is different from Richie's. Richie's calculator has more decimal places than Jen's.”
Valente asked, “Did anyone get a different number from those Jen or Richie got?”
Lisa said, “I get .2357143 on my calculator when I divide 1.65 by 7. And I get 1.65 when I multiply that by 7.”
Jason said: “I get .2357142857 and 1.65 when I multiply that by 7.”
To keep track of the various results, Valente asked the four students to put the numbers they got when dividing 1.65 by 7 on the chalkboard. Here is what they wrote:
What the NCTM Standards Look Like in One Classroom - table
Jen | 0.2357142 |
|---|---|
| Richie | 0.235714285 |
| Lisa | 0.2357143 |
| Jason | 0.2357142857 |
Debra noticed that “all of the calculators had the same numbers in the first six decimal places. Only Lisa's had a different number in the seventh place.”
“It looks like the 3 has been rounded up,” Jason observed. “See, the next digit is an 8. Jen's must have just run out of places so the 8, 5, and 7 got cut off.”
“Jason's must be the right answer,” said Julie, “since his has the most places.”
Richie responded, “How can Jason's be the only right one? Three out of the four get us back to the starting number—1.65—when multiplied by 7. How could three of them do that and not the other one?”
Valente questioned, “How can we be sure that Jason's shows us all of the digits?”
“By dividing it out on paper,” Debra answered.
“This looks like it could be a pretty long problem,” announced Valente. “Why don't you get in your groups and see what you come up with?”
Using the long division algorithm, the students came up with: .23571428. (After this many decimal places, the division process repeats the digits 571428.) The discussion ended with the class recognizing that none of the calculator displays gave them enough information to see the solution as a repeating decimal. Instead of bringing the discussion back to the original idea—whether Elise's formula was reasonable for estimating a price in dollars when given the price in guilders—Valente let the discourse focus on the calculator-precision issues.
What Is Different?
- The students did mathematics that was different (see Lampert 1990). They explored functions by looking at relationships and patterns in real situations rather than through the more typical notations and symbols. Students developed formulas informally but conceptually, using arrow language to represent changes. Such a situated, conceptual approach provides access to some concepts of functions at a much earlier grade level than does the traditional symbol-intensive approach.
- The students actively engaged in nonroutine problem solving and mathematical discourse (see Ball 1991). With his questioning, Valente orchestrated a rich, problem-solving discussion about a genuine dilemma that had arisen for his students. Rather than finding correct answers to routine exercises, he emphasized the process of solving the dilemma at hand as well as understanding mathematical concepts. The experiences also provided opportunities to make mathematical connections during future discussions.
- The students used calculators as problem-solving tools as well as for computational assistance. Students also experienced situations where a calculator and a paper-and-pencil algorithm were each appropriate calculation methods, and they were allowed to make their own choices of tools (see NCTM 1989).
- The teacher and the students learned from each other (see Silver et al. 1990, Schifter and Fosnot 1993). Although this result caused some discomfort for both teacher and students, it also empowered them: the teacher did not need to be the sole authority for answers to open-ended problems. Instead, Valente asked the class to determine the validity and workability of the proposed formulas.
- The activities and discourse provided a view of students' thinking different from that provided by traditional assessments. The problem-solving, reasoning, and communication processes were open to view, and students showed their understanding and thinking through classroom discussions (see Stenmark 1991). Because Valente was listening to what his students were saying, opportunities arose to build on the current understandings of individual students.
In Support of Change
The changes seen in Mark Valente's classroom follow the direction encouraged by the NCTM Standards and reflect a fundamental rethinking of what understanding and teaching mathematics means. Mathematics reform at Andrew Jackson Middle School began to move forward when teachers collaborated and received encouragement and guidance from leaders such as Linda Howard. As she put it: It wasn't a lonely pursuit—not one lonely 6th grade teacher or one lonely 7th grade teacher—but all of the group sort of saying, “It's scary, but we're going to try it. If it doesn't work, we back up a step, but we continue going.“ A lot of things came together at sort of a crucial period, but I think it really took the fact that we were all in this together.
The reform of mathematics education recommended by the NCTM is unquestionably a challenging effort. As schools across the nation rethink their curriculum and instruction in ways consistent with these recommendations, support for the continuing professional development of teachers—the key figures in mathematics reform—is crucial. Perhaps this view from one middle school classroom may provide some direction for others beginning to pursue this vision.
