The focus of school mathematics is shifting from a dualistic mission—minimal mathematics for the majority, advanced mathematics for a few—to a singular focus on a significant common core of mathematics for all students.—Everybody Counts (NRC 1989)
- students who have been denied access to educational opportunities, as well as those who have not;
- African-American, Hispanic, American Indian, and other minority students, as well as those considered to be part of the majority;
- females as well as males; and
- students who have not been successful, as well as those who have succeeded in school and in mathematics (NCTM 1990).
Accommodating such diversity within a three-year common core curriculum of broadly useful mathematics for all high school students is a primary goal of the Core-Plus Mathematics Project (CPMP).
What Is Core-Plus Mathematics?
The Core-Plus Mathematics Project is one of four projects awarded five-year grants in 1992 by the National Science Foundation to design, evaluate, and disseminate innovative high school curriculums based on the recommendations of the National Council of Teacher of Mathematics (NCTM 1989, 1991). In contrast to tracked programs that limit opportunities for students through work-prep, tech-prep, and college-prep sequences, the CPMP curriculum is a single-core sequence during the first three years. College-bound students continue their preparation for college mathematics in a special CPMP fourth-year course.
Depending on the school, employment-bound students completing the three-year core may choose from options such as a computer-aided design course, a statistical process control course, or other emerging technology education courses. The Core-Plus Mathematics Project curriculum keeps options open for all students and gives them the opportunity to engage in appropriate mathematics-related work during their senior year.
The curriculum, which builds on the theme of mathematics as sense-making, has five mathematical and instructional design features: multiple connected strands, emphasis on mathematical modeling, access for all students, full use of graphics calculators, and active learning experiences.
A Look at the Curriculum
Each year the curriculum features four multiple strands: algebra and functions, geometry and trigonometry, statistics and probability, and discrete mathematics connected by fundamental themes and by habits of mind. It enables students to think mathematically about the informal knowledge of data, shape, change, and chance that they, themselves, bring to situations and problems (Hirsch et al. 1995).
Each course in the three-year core is organized in a series of seven connected units and a thematic capstone experience. The units consist of four or five multi-day lessons in which students develop their understanding of major mathematical ideas by investigating rich applied problems. A typical lesson, which lasts from two to five days, is designed around a four-phase cycle of instructional activities.
- What information might the manager want to know about the kinds of shoes the customers prefer? Make a list.
- It's not enough to just have information. The manager needs to be able to organize and manage the information in order to make good decisions. What are some ways the manager might organize the information?
Phase 2. Once the context is set, the students, working in groups of four or sometimes pairs, explore the mathematical features of the situation through a series of more focused activities. The following activity, taken from the athletic shoe store lesson, illustrates the types of work students engage in. Previously, the students had collected class data on different brands of athletic shoes worn by class members.
Students begin by completing a table like the following one for their class data.
Athletic Shoe Brands
A Common Core of Math for All - Table
Men | Women | |
|---|---|---|
| Converse | ||
| Nike | ||
| Reebok |
- The matrix above has three rows and two columns. How many rows were needed to display the class data? How many columns?
- Could you organize the class data using a matrix with two rows? If so, how many columns would it have? Which display would you prefer?
In a follow-up activity, students assume that they are managers of a local Fleet Feet shoe store. Data on monthly sales of L.A. Gear, Nike, and Reebok shoes are shown in the matrix below. Each entry represents the number of pairs of shoes sold.
Monthly Sales
A Common Core of Math for All - table 2
J | F | M | A | MJ | J | J | A | S | O | N | D | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| L.A. Gear | ||||||||||||
| Nike | ||||||||||||
| Reebok |
The students then answer a number of questions, including:
- For each shoe brand, which month has the highest sales? What could be a reason for the high sales?
- How many pairs of all three brands together were sold in February?
- Which brand has more variability in its monthly sales? Explain how you determined variability. What is another way that you could determine variability?
Students are continually encouraged to think about different ways of representing mathematical ideas. Next, students list at least two types of graphs that could be used to represent the monthly sales data. Then they choose the type of graph they think would be most informative and sketch it.
- The Bass Shoe Outlet sells women's shoes sizes 5 to 11 and men's shoes sizes 6 to 13. The manager would like to have an organized display of the number of pairs of shoes sold in 1994 for each shoe size. Explain how a matrix can be used to organize this data. How many rows does this matrix have? How many columns?
- What are some advantages of using matrices to organize and display data? Disadvantages?
- Can the same information be displayed in a matrix in different ways? Explain.
Phase 4. Following this sharing and summarizing activity, students apply their mathematics to problem situations in different contexts, connect their newly developed mathematics to other mathematical ideas in the same or a different strand, reflect on the development of their mathematical thinking, and extend or formalize their mathematics. Each unit's final lesson gives students an opportunity to look back over the unit and consolidate their learning through a set of new modeling tasks.
Benefits for Students
Currently, we are in various stages of evaluating the effectiveness of the Core-Plus Mathematics Project curriculum. Each course goes through a three-year research, development, and evaluation cycle. We are completing the pilot-testing of Course 3 (for juniors) and the national field-testing of Course 2 (for sophomores). Field-testing yields two kinds of information: (1) the perspectives of teachers using the curriculum, and (2) achievement information gathered from the results of standardized and other tests.
Teachers' perspectives. Students come to mathematics classes with different gifts and talents. By integrating the four strands in the curriculum each year, CPMP emphasizes the connected nature of mathematics and provides variety in content to accommodate and capitalize on individual interest, background, skill, and understanding. The following comments from three teachers who field-tested the first year of the curriculum support the attainment of this goal: The students who are more visual tend to enjoy the geometry and discrete math units, the tessellations, transformations, digraphs. Some students become "techies"; they enjoy program-writing. And then some students think more abstractly and are able to write symbolic representations for patterns. Because all of these diverse personalities are in the classroom together, the students work together and share ideas.I see the program as an equalizer for girls in math. It takes away the advantage the boys tend to have in symbolic manipulation skills.... It allows girls to see that there are other avenues for mathematics.My first-hour class had 10 students with special ed notations by their names.... These 10 students interacted appropriately with their classmates, contributed wonderful insight, and, during exams, showed very creative (and correct!) answers.
Putting mathematics instruction and learning in context helps students see that mathematics is part of their world. It also enables them to construct meanings that make sense to them, which, in turn, helps them make sense out of new situations and problems.
The project's emphasis on group work promotes student engagement, mathematical thinking, and better communication. One teacher noted "an enormous amount of growing, both socially and mathematically" in her students. One student said she "learned to work with people I never thought I would even talk to." And another expressed surprise, "I thought only his ideas would be right; mine are too!"
That comment illustrates another aspect of the Core-Plus Mathematics Project curriculum. Students do not all see the same path to a problem solution. Often they do the unexpected, yet they do a fine job of thinking and problem solving.
Achievement data. We field-tested Course 1 nationally during the 1994-95 school year with a broad, diverse cross section of approximately 5,650 students in 36 high schools in urban, rural, and suburban areas. Achievement measures were administered in a pre/post-evaluation design to all project classes and, for purposes of comparison, to 9th grade students enrolled in traditional courses, primarily algebra, in 10 of the schools.
Our initial analysis of the data, controlling statistically for initial differences, indicates that CPMP students were significantly better at the end of the year at reasoning and applying mathematical concepts than both comparison students in the field-test schools and the standardized test's nationally representative norm group of end-of-year 9th graders.
Mathematics for All
The Core-Plus Mathematics Project is creating new windows of opportunity for all students. The instructional model engages all students in important mathematics. The exploratory work is accessible to all students. Group work provides support for struggling students and helps all students to clarify their understanding by discussing mathematical ideas within the group. The individual work accommodates differences in ability, interest, and mathematical knowledge, and challenges students in heterogeneous classes. In addition, the extensive use of the graphics calculator as a tool for learning and doing mathematics helps students whose limited computational abilities previously prevented them from advancing in the study of important mathematics.
Universities that we've asked to evaluate the CPMP core curriculum for admission purposes have classified it as a college-preparatory curriculum. David Smith, a professor of mathematics at Duke University and a leader in the calculus reform movement, commented that if students in the first-semester reformed calculus course at Duke University had completed just the first three years of the Core-Plus Mathematics Project curriculum, he would be able to eliminate at least a third of the course. High praise, indeed, for a curriculum for all students.
