T he Core Curriculum - Teaching Secondary and Middle School Mathematics

Suppose that two students each decided to take four units of high school mathe-matics courses and followed the sequence shown in Table 4.2 :

Table 4.2

Possible Curricular Sequences for Secondary Students College Bound Non-College Bound

Grade 9 Algebra I General Mathematics

Grade 10 Geometry Applied General Math

Grade 11 Algebra II Technical/Vocational Math

Grade 12 Pre-Calculus Pre-Algebra

Upon graduation, each student’s transcript would indicate that he or she had taken 4 years of mathematics. But how do you believe the students would compare if given a standard achievement test or an SAT or ACT exam? Are their mathematics backgrounds equivalent? Clearly not. The college-bound students have a distinct advantage in having studied a more diverse and higher-level set of concepts than other students. Yet this sequence and all of its inequities were very common in schools around the country even into the late 1980s; in fact, in some places, they still exist. Interestingly, in the 1970 NCTM Yearbook, the authors, speaking of changes in mathematics between 1920 and 1945, stated that it became an “increas-ingly common requirement that everyone take at least one year of mathematics in grades 9–12” and that “this new general mathematics [course] developed as the most popular alternative to algebra in the ninth grade. Down to the present day this course has been ill-defi ned and often poorly, or at least unwillingly, taught”

( Jones & Coxford, 1970 , p. 53 ). The General Mathematics course, indeed, still exists today—some 75 years later! The NCTM and Common Core State Stan-dards , however, challenged us to reconsider what mathematical topics were being developed with whom and to think of ways to make mathematics accessible to all students as we have discussed. But why is this so important? Why not keep the cur-riculum the way it has always been?

As the world has moved into the information age, physical skills have been replaced by technical and problem-solving skills, and the workforce continues to require updating as industry changes. The U.S. Department of Labor (2007) esti-mates that two-thirds of all jobs being created today require a postsecondary edu-cation, resulting in a higher demand for individuals with critical thinking skills and creativity. The average worker will change jobs at least 4 or 5 times over a 25-year period ( NCTM, 1989 ), and most experts agree that a job stays the same for only about 5 years, so a need for retraining is inevitable ( Meiring, Rubenstein, Schultz, Lange, & Chambers, 1992 ). The National Research Council (1989) reported that 75 percent of all available jobs require at least a basic understanding of algebra and geometry. A basic understanding of statistics is also necessary in many career areas, just as knowledge of how samples are taken and statistics are analyzed is essential for following debates in politics ( Konold & Higgins, 2003 ). And the need for students to be exposed to discrete mathematics topics, such as recursion and graph theory, is rapidly increasing because of the growth of computer technology and the relevance of discrete mathematics in industry ( Dossey, 1991 ). It is

unrealistic to believe that one segment of the population needs algebra “and beyond,” and another segment will never need more than basic arithmetic; this simply does not appear to be the case anymore.

Consequently, the Common Core State Standards mandate that schools estab-lish a core curriculum—a 3-year high school sequence with common objectives or outcomes—for all students. It is important not to confuse the “Common Core”

with the notion of a “core curriculum.” As we have discussed, the Common Core State Standards is a K–12 publication that describes desired outcomes for each grade level of schooling. The core curriculum, on the other hand, is simply a term for a 3-year program of study intended for all high school students. The core cur-riculum is discussed at length in an NCTM resource book published far ahead of its time, entitled A Core Curriculum: Making Mathematics Count for Everyone ( Meiring et al., 1992 ). The authors make the case for developing a curricular model that would ensure that every secondary student has exposure to important mathe-matical topics that extend well beyond arithmetic. They suggest that college-bound and higher-achieving students will visit many concepts in greater depth than some of their peers but that at least the curriculum would ensure equal access to various areas of mathematics for all students. Meiring et al. suggest three possible models for schools to consider in restructuring their curriculum: the crossover, enrich-ment, and differentiated curriculum models.

The crossover model is probably the easiest to adopt if the school is currently on a very traditional tracking system. Two parallel tracks are established, one for college-bound students and one for students not planning to go to college. There is a 3-year sequence of courses with basically identical content objectives in both tracks. However, although the college-bound students may visit the topics at an advanced level with plenty of abstraction, including the exploration of some optional topics (many of which are included in the CCSS document), the student not planning to attend college will explore these concepts at a more concrete level with, perhaps, an increased emphasis on the use of technology. This is the “cross-over” model because if a student decides, after a year or two

of high school, to go to college after all but has not been in the college-bound track, that student can still switch to the other sequence and know that classmates have explored essentially the same objectives. Compare this situation to more traditional models in which the student in a low-aver-age mathematics class visits less material in a watered-down fashion, making it virtually impossible for that student to ever get out of a low track. The minute a teacher, parent, counselor, or administrator labels the student as below average, the door is shut on the student’s chances for expo-sure to signifi cant mathematics. In traditional basic mathe-matics courses, often reserved for low-achieving students, the classes do no more than review elementary and middle school arithmetic. Also, in the crossover model, the student who is not successful on the college track can switch to the other without a loss of continuity. Finally, the model sug-gests a fourth-year advanced course for college-bound stu-dents. Figure 4.2 illustrates this with a simple fl ow chart.

Here is a specifi c example of how a high school might use the crossover model when teaching all fi rst-year students

Figure 4.2

The Crossover Model

how to multiply polynomials. The students not planning to attend college might begin with a concrete experience involving algebra tiles. An analogy could be expressed as follows: Think of what it means to multiply 12 * 13. Geometrically, this multiplication problem can be expressed as the process of fi nding the area of a rectangle measuring 12 by 13. The tens and units pieces of a set of base ten blocks have been separated for emphasis in Figure 4.3 .

Figure 4.3

Area Model of 12 * 13 Using Base Ten Blocks

Think of the traditional algorithm. Four multiplications take place: 10 * 10, 10

* 2, 3 * 10, and 3 * 2. The diagram shows one fl at that represents 100, 5 longs that represent 50, and 6 units, modeling the product of 156. Using base ten blocks, a child can visualize what it means to multiply. Similarly, the secondary class might consider ( x + 2)( x + 3). This problem is nothing more than a generalization of 12 * 13, where x = 10. So, with algebra tiles, as described in Chapter 3 , the polynomial multiplication would look like the diagram in Figure 4.4 .

The area of the rectangle is made up of one x ² tile, fi ve x tiles, and six unit tiles, so the product would be x ² + 5 x + 6. Students can work through several examples of polynomial multiplication problems like this, involving positive and

Figure 4.4

Area Model of ( x + 3)( x + 2) Using Algebra Tiles

negative numbers, without having to memorize a rule of any kind. Actually, every student should be exposed to this visual model, because many leave high school knowing “how to FOIL” without realizing that FOIL (First, Outside, Inside, Last) has no mathematical meaning beyond serving as a mnemonic for remembering an algorithm that applies only to binomials and without being able to explain why the problem is done that way. Eventually, the non-college-bound students will be able to move on to an iconic level, according to Bruner, and sketch freehand pictures of tiles that represent polynomial multiplication prob-lems. Finally, the student will invent a procedure for multiplying polynomials without using concrete materials or drawing pictures. But the series of lessons that develop the abstract symbol manipulations are appropriate to the students and allow them to generalize their own rules, based on observations, which is at the core of the constructivist model for learning and teaching , as described in Chapter 3 . The students might also try their hand at writing a function that describes the orange grove problem from Chapter 1 to see an application of bino-mial multiplication.

In the college-bound track, the students will explore the same concept but may be ready to move from concrete to abstract levels within a couple of days, thus leaving them additional time to explore extensions of the concept, such as how to develop a rule for multiplying and factoring the difference of squares or perfect cubes. They may also begin solving higher-level applied problems, such as those that involve maximizing volume. Perhaps you have solved the problem in which you are given an 8¹₂ * 11" sheet of paper and asked how big the four squares you cut off from the corners should be in order to maximize the volume of the resulting open-top box formed when the sides are folded up. Figure 4.5 helps you visualize this.

If the original dimensions of the paper were 8¹₂ * 11", and

the squares cut out measure x * x , then the volume of the resulting box could be found by examining the function y = x(11 - 2x)(8.5 - 2x), where x represents the height of the box, and the expressions (11 - 2 x ) and (8.5 - 2 x ) represent the length and the width of the base, respectively. Students can readily view the func-tion’s graph ( Figure 4.6 ) on a graphing calculator.

Figure 4.5

Making a Box from an 8¹₂ * 11" Sheet of Paper

Figure 4.6

Finding the Maximum Volume with a Graphing Calculator

By tracing the curve using the TRACE function or using the CALC (calculate) button, they can locate the local maximum of the function at about x = 1.59 and calculate the volume by taking 1.59(11 - 3.18)(8.5 - 3.18) ⬇ 66.15 cubic inches (or by simply using the y -value). This entire problem represents a rich, real-istic application of polynomial multiplication that would engage the college-bound classes and extend their understanding of the use of variables for generalizing and solving problems. But again, in the crossover model all students still visit the same major core objectives.

A second curricular alternative suggested in A Core Curriculum is called the enrichment model . Under this model, students are arranged into small groups at the beginning of each unit throughout the 3-year common core. If a group of students completes the core content of a particular unit before the rest of the class, they can be assigned an enrichment topic to explore. These enrichment topics may include historical considerations of the content, such as the history of p; further exploration of a concept, such as studying fractals in a unit on area and perimeter; or looking at new situations, such as an application of a mathematical concept to a career area. In this way, all students are assured of a common core of mathematical concepts, but students who are more able have the opportunity to expand their knowledge even further. The enrichment model, similar to the crossover model, provides a common 3-year core. But these models differ in that classes are heterogeneously (mixed) grouped in the enrichment model, with small groups formed within the class, whereas students choose one of two tracks in the crossover model. Figure 4.7 pro-vides a visual description of the enrichment model.

A variation of the enrichment model that is used in some schools, particularly at the middle school level, is called clustering . Using this model, approximately 6 to 8 gifted students are selected and placed in each class, with the rest of the stu-dents in the class having mixed abilities. Grouping is fl exible, and these stustu-dents work together on some topics but are mixed with the rest of the class at other times. A teacher who is well prepared to work with gifted students in a regular classroom can provide challenges for them. This is often much more realistic when there is a critical mass of such students in a class than if only one or two individuals

Figure 4.7

The Enrichment Model

are in need of additional challenges. A book by Susan Winebrenner and Dina Brulles entitled The Cluster Grouping Handbook (2008) provides details on how to manage cluster groups in a regu-lar classroom. Clustering is viewed by many as a preferred alternative to the more traditional grouping of students into full classes of honors or gifted stu-dents, while other classes serve average or struggling students. Unlike tracking practices, neither gifted nor struggling students are isolated from one another so that the diversity of thinking can enhance learning for all students ( Winebrenner &

Devlin, 2011 ).

The third curricular sequence sug-gested by the NCTM is the differentiated model . As is the case with the enrichment

model, students are heterogeneously mixed into a shared 3-year core of mathematics courses, with a possible common fourth-year experience as well. Within each class, students are organized into small groups at the beginning of each unit. However, instead of all students completing a core of objectives and some moving on to addi-tional topics, all students explore the same topics but at a variety of levels, as was described in the crossover model. So, there might be one group of students working on multiplying polynomials with algebra tiles, while another group is devising a shortcut for fi nding the square of a binomial in a more abstract manner. Both groups would be learning to multiply polynomials, but the content would be at a level appropriate to each particular group. Figure 4.8 illustrates how all students explore the same core of objectives but at a different level of depth in the differentiated model. The differentiated model organizes classes in a heterogeneous manner similar to the enrichment model but structures the work of each

group by depth of coverage as the crossover model does.

Common to all of these models is the notion that every student deserves to explore the same core of objectives regardless of perceived ability level or future plans. Lessons should be rich in the use of hands-on materials (manipula-tives) and technology and are taught with the students’

needs in mind. The idea of a core curriculum for all, how-ever, can be diffi cult to implement, particularly at the high school level, because it requires a radical change from the curriculum of the past—the idea that the “best” students get algebra, geometry, and pre-calculus, and the less-able students take basic math and, perhaps, a pre-algebra or algebra course. In a core curriculum, algebra, geometry, statistics and probability, and some discrete mathematics topics are given equal weight, so the secondary and middle school curriculum should provide experiences in all of the content areas. In order for a wide array of mathematical

Cooperative learning strategies are used to promote interaction among students.

Figure 4.8

The Differentiated Model

SPOTlight on Technology

A popular tool used for teaching mathematics is the computer spreadsheet (or the table function on a graph-ing calculator). Suppose that a teacher posed the fol-lowing problem to a class:

A local video store offers the following plans for renting videos: For Plan 1, customers join the Rental Club for $20 per year and can rent each video for $1.50. For Plan 2, the customer does not have to pay a membership fee but pays $2.95 for each movie rental. Under what conditions is Plan 1 better than Plan 2? Which would you choose and why?

Using a spreadsheet application (such as Excel) on a computer, a student can input the number of video rentals in Column A and write formulas for calculating the total cost of video rentals using Plan 1 in Column B and Plan 2 in Column C, as illustrated in Figure 4.9 .

Using the FILL DOWN option on the spreadsheet, the students can create a table (see Figure 4.10 ) that shows the relative cost of renting from 0 to 15 videos a year. Students can scan the table to look for patterns and to draw conclusions. They should notice, for example, that the total costs for 10 video rentals under Plan 1 and Plan 2 are $35.00 and $29.50, respectively.

More importantly, the values in the spreadsheet allow them to compare the relative costs of the options and fi nd the break-even point. They should notice that Plan 2 is cheaper until about 14 video rentals, at which point the costs are almost the same. At 14 videos or more, Plan 1 becomes the less expensive choice. In fact,

if the table is continued even further, a family renting an average of one video per week will save more than

$50 per year if they choose Plan 1.

The spreadsheet, then, serves as a tool for compar-ing the costs of the options, given a particular number of video rentals. Students gain experience in setting up and reading a table, looking for patterns, and using algebraic symbols to defi ne the spreadsheet formulas.

The last of these activities is extremely important because a signifi cant amount of time is generally spent on the representation of patterns and functions with variables at the secondary and middle school levels. In addition, the spreadsheet saves the students from hav-ing to guess and check their way through 14 different video rental scenarios to fi nd the break-even point.

Once the cost of an option has been calculated, the spreadsheet allows the calculation to be mechanically repeated to display a wide variety of possibilities.

Finally, the spreadsheet option allows the teacher to ask such a question as, “What if the video store raised the membership fee to $30 but dropped the cost of renting tapes to $1.25 for members? What is the new break-even point?” With a quick change in the spread-sheet formula, students discover that members need to rent 18 or more videos to make the new Plan 1 the bet-ter choice. The spreadsheets allow a class to examine a number of scenarios to gather data and draw conclu-sions, with the computer doing each of the individual calculations. The focus of the problem, then, becomes the analysis of data and the formulation of an answer rather than computation. Furthermore, research sup-ports the use of spreadsheets in teaching mathematics,

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