What is Being Taught in Secondary Mathematics in the USA? a Look at State Standards

By Latterell, Carmen M

Most college mathematics professors in the United States have had the occasion to use some piece of mathematics in class only to be surprised at students’ reactions that they have never seen such material before. While it is possible that students have not seen said material, it is also possible that they have and either do not remember it or want the professor to review it. Regardless, college mathematics professors do not tend to know what is taught in the secondary schools [6]. This article aims to provide some information on just what students are taught in US high school mathematics. Of course, it is not possible to examine what is being taught in every secondary school in the USA. It is possible to take a sample, but another way to evaluate what is being taught is to examine curricula or standards. The USA does not have a national mathematics curriculum. However, national mathematics standards, particularly the National Council of Teachers of Mathematics’ (NCTM) Principles and Standards for School Mathematics, can provide some common ground. The National Council of Teachers of Mathematics lists five process standards: problem solving, representation, communication, connection, and reasoning; and five content standards: number and operations, algebra, geometry, measurement, and data analysis and probability [9].

The NCTM’s first set of standards, Curriculum and Evaluation Standards for School Mathematics [8], definitely influenced states to make significant changes in their standards [2, 3, 11]. Further, No Child Left Behind [10] has standards and assessment requirements that have caused states to look carefully at academic standards, including mathematics standards. Other organizations have also joined the bandwagon.

The American Diploma Project [1] has defined benchmark standards in mathematics that students must have to succeed in college. Their standards include four of the five standards set by NCTM, but exclude measurement as a standard. To accomplish the standards, students should take Algebra I, Algebra II, Geometry, and Data Analysis and Statistics. Understanding University Success [13] has defined standards needed for success in college. Their mathematics standards are computation, algebra, trigonometry, geometry, and statistics. The College Board Standards for College Success [5] require students to take Algebra I, Algebra II, Geometry, and Precalculus.

Standards are clearly growing in presence, but, it is another question as to how closely individual schools align their mathematics curriculum with state standards. Yet, it does seem clear that local school districts pay attention to state standards [12]. If for no other reason, school districts need to be aware of what will be on state mandated tests [4]. State and federal monies depend on local school districts providing evidence that they are meeting the state standards. States must report student scores on mandated tests in terms of their state standards. However, because the requirement of state standards is relatively new, there is a time- delay before it makes sense to analyze student scores. For now, we can only say that it behooves local school districts to implement the state standards.

METHODS

This study aims to partially answer the question What is being taught in high school mathematics courses? No claim is made that the state standards translate to what students actually know when they enter college. Since it is certain that students will not benefit from state standards unless students actually take mathematics courses, this study will also give a measure of how many courses students take. One method for determining this is to examine high school graduation requirements to see how many years of high school mathematics are required for graduation.

During the summer months of 2006, the researcher used the state department of education web sites for all fifty states to find copies of the state mathematics standards and to find high school graduation requirements. These standards and requirements were reviewed, and the data was recorded in an Excel spreadsheet to track three pieces of information for all states: 1) which (if any) of the five content standards given by NCTM are included in the state requirements; 2) what state standards (if any) that are not part of the five content standards given by NCTM; and, 3) how many years of high school mathematics were required for graduation.

Several data gathering decisions were made to make the process uniform across all states. For example, because the researcher was predominately interested in what is being taught in high school, the state standards at the high school level were used as the first step in examining each state’s comprehensive set of requirements. However, several states had an absence of certain standards (e.g., measurement) at the high school level, but had the standard at another level (often the eighth-grade level). The decision was made to record that the state held the standard as a high school standard. In other words, high school standards were viewed as including all previous standards.

A second decision regarded interpretation. It was decided not to infer beyond the written state requirements. For example, in a couple of cases, it appeared to the researcher that the state met the spirit of a specific NCTM standard, but the state did not use the standard’s name or even some variation of the standard’s name. It was decided that in those cases it would be recorded that the state did not hold that particular standard. Although this is a judgment call, states are aware of the NCTM standards and their language. If a state chooses not to use them, that could mean something.

Third, some states list process standards (e.g., problem solving, reasoning), while other states do not. The decision was made to concentrate only on content standards. Although it might seem helpful to college mathematics professors to know if their incoming students know how to solve problems, problem solving is an ill- defined term. In fact, some research suggests that when mathematicians say problem solving, they give it a different meaning from the meaning given in NCTM standards [7]. Until “problem solving” can become well-defined with uniform agreement on the definition, it does little good to say whether incoming students have problem solving ability or not. Further, and perhaps a less fixable problem, process standards are often context specific. A student might be able to solve problems in certain geometric contexts but not in statistical ones. This is also a judgment call made by the researcher; therefore this study concentrates on content standards, only.

RESULTS

The vast majority of states (N = 41 ) include NCTM’s five content strands in their standards. Among the nine that do not cover all five content strands, five states list four of the five content strands as part of their standards. Out of these five, three do not include measurement, one does not include number sense, and one does not include algebra.

This leaves four states that do not include all five NCTM content strands in their state standards. Out of these four, one state lists all of NCTM’s process standards (but none of the content standards) as their state standards; that is, problem solving, reasoning and proof, communication, connections, and representation. Two states list individual courses (e.g., Algebra I, geometry) and long lists of objectives that should be met in the course. For example, under Algebra I at one state there are 18 objectives listed. An example is “The student will justify steps used in simplifying expressions and solving equations and inequalities. Justifications will include the use of concrete objects; pictorial representations; and the properties of real numbers, equality, and inequality.” One might argue that one could find all of NCTM’s content standards if one searched among these long lists of objectives.

Finally, one state (Washington) is quite unique in their standards, including four out of five of NCTM’s process standards (they do not include representation) and a standard that is not at all from NCTM, which is about understanding the procedures of mathematics (although this state does use all of NCTM’s content words under its description of procedures). So, even among the nine states that do not list all five NCTM content strands among their own state standards, all show evidence of being influenced by NCTM standards.

Four states include a separate logic standard (some logic is included under the geometry standard). Three states include a discrete mathematics standard. Five states have a computation standard. Three states have a standard entitled “patterns, relations, and functions.” Five states have standards too numerous to list, as they give detailed standards under each course, like a course outline.

As of summer 2006, the minimum number of years of mathematics required for high school graduation is one (the requirement of one state, Michigan) and the maximum number of years required is four (10 states require four). The median and mode are both 3 years. The mean is 2.84 (with a standard deviation of 0.766). In fact, 23 states require 3 years of high school mathematics and the remaining 16 states require 2 years. The remainder of this paper switches from a quantitative description of the state standards to a qualitative one. Thus, we attempt to paint a picture of each of the five standards: number and operations, algebra, geometry, measurement, and data analysis and probability. We will do this by giving examples and problems, along with descriptions of each of the standards. Obviously, not every state describes these standards exactly the same way. But, the purpose of qualitative analysis is to give a consensus. To be in the descriptions that follow, the spirit of the illustration has to be in the union of the state standards descriptions, even if the exact example cannot be found in all of them. To illustrate, if an example is given requiring solving a system of equations with two unknowns, then solving systems of equations has to be in all the states standards, although perhaps some states also include system of equations with three unknowns and none of the state have the exact example. Exceptional examples, where only one or a few states note such points, will not be included in the paragraphs to follow.

NUMBER AND OPERATIONS

The number and operations standard is easy to describe at the elementary level (basically arithmetic), but harder to describe at the senior high level (number sense). Number sense is having a “feel” for numbers.

At the senior high level, an example might be that if I add two odd numbers and get another odd number, do I realize that I have made a mistake (because the answer must be even)? If so, that is number sense. Or, if I multiply two numbers and my answer is off by several decimal places, do I realize that it is wrong? If so, that is number sense. Here are three examples.

A store sells CDs for $12.99 each. The sales tax is 7%. Marie selects nine CDs. The clerk tells Marie her bill is $157.18. Without actually calculating the total bill, how can Marie explain to the clerk she has been overcharged? (Kansas)

If you have a $4,000 debt on a credit card and the minimum of $30 is paid per month, is it reasonable to pay off the debt in 10 years? (Kansas)

Given the points with coordinates a, b, c, d, e, f, g, and h as shown below, Which point is closet to ab? To \c\? To a/f? To [radical]e ? To [radical]h? Explain your reasoning. (NCTM, 2000, p. 293)

One important point about this standard is that states differ on the degree to which students should be allowed to use calculators. A few states specifically state that students should be able to perform arithmetic procedures without calculators, and a few states specifically state that students should have free use of calculators at all times. Most states say little about calculator use. The NCTM supports the use of calculators at all levels in the curriculum. Thus, professors need to be aware that it is possible that students’ procedural abilities are calculator dependent.

ALGEBRA

The algebra standard includes solving equations for unknowns. However, the standard has so many other things in it that solving equations does not have the major role in algebra that it once did. Professors may notice that traditional algebra skills are weaker. Proponents of reform mathematics will say that perhaps the ability to do meaningless symbol pushing is weaker, but other skills are stronger.

Those other skills involve mathematical modeling. Students are given a real-life situation and asked to create a function (linear, polynomial, exponential, rational, periodic) to model the situation. Students use the function to answer “what-if” questions and make predictions. These functions are messy. Students do not necessarly solve the equations by symbol manipulation, but they make use of technology. They might use the table function of a graphing calculator, or graph the function and narrow in on the intercepts or other points of interest.

Here are illustrations of the type of problems.

In February 2000 the cost of sending a letter by first-class mail was 33 cents for the first ounce and an additional 22 cents for each additional ounce or portion thereof through 13 ounces. Using C as the cost in cents of mailing a letter and P as the weight of the letter in ounces, then express P as a Junction of C with a table, a graph, and symbols. [9, pp. 297-298]

Conduct research on some text messaging plans available in your area. Find a mathematical model that represents each plan. Given your textmessaging habits and using the mathematical models, evaluate these plans, and choose the one that is best for you. Explain how you made your choice and why you think it’s the best plan for you. (Iowa Model Core Curriculum)

GEOMETRY

The geometry standard emphasizes conjecturing and justification. Students conjecture often in the environment of technology. Dynamic geometry software, such as Geometer’s Sketchpad, is a common high school tool. By using the software, thousands of examples can be constructed in little time. Students might be asked to investigate situations like the following two examples.

Ask students to draw a triangle in a dynamic geometry environment, and construct a new triangle by joining the midpoints of its three sides, and calculate the ratio of the area of the midpoint triangle to the area of the original triangle. [9, pp. 311- 312]

If the ratio of the sides of two similar triangles is 3 to 5, what is the ratio of their areas? (South Dakota)

While this process falls short of formal proof, concepts such as counterexamples are learned. The presence of two-column proofs has diminished through the years and been replaced with logical arguments.

In addition to Euclidean geometry, transformational geometry and coordinate geometry are common in secondary schools. As in the algebra standard, this standard includes more curriculum than in the past. Thus, many things receive much less emphasis than they once did, but the students may have more ability to reason in a geometric environment than they once did.

MEASUREMENT

The measurement standard, like the number and operation standard, is more easily understood at the elementary level. Elementary students are taught to use instruments such as rulers to measure. In addition, at the elementary level, students are taught to calculate areas, perimeters, and volumes. At the senior high level, there is little emphasis on measurement by the NCTM standards. Many states do not have a measurement standard at the senior high level, but do have one in earlier grades. Nevertheless, there is some emphasis on measurement even in senior high.

Students in science courses certainly use mathematical measurement in conducting experiments and working problems. Units, scientific notation, and significant digits are all mathematical concepts that may be covered in science courses. Experiments involving length, time, weight/mass, and temperature all use measurement. In addition, concepts of accuracy and precision in measurement and measurement tools are also a part of this standard. Here is a multiple-choice question from South Dakota’s standards to illustrate:

In determining the area of a flower garden. an unreasonable answer is:

a) 25.8 square feet

b) 26 square feet

c) 26.5 square feet

d) 26.5278394 square feet.

It is certainly possible that mathematics courses emphasize measurement techniques and/or problems involving measurement (e.g., volumes and surface area of rectangular solids, cylinders, cones, or pyramids). Whether the standard is covered in science or mathematics makes little difference, as long as the student understands the concepts. Here is an example that would be considered measurement and might be a senior high level mathematics problem.

While driving through Canada in the late 1990s, a U.S. tourist put 60 liters of gas in his car. The gas cost Can$0.50 a liter (Can$ stands for Canadian dollars). The exchange rate at that time was Can$1.49 for each US$1.00 (United States dollar). The price for a gallon of gasoline in the United States was US$0.99. The driver wanted to compare prices and decide whether a tank of gas was cheaper in the United States or Canada. [9, pp. 322-323]

DATA ANALYSIS AND PROBABILITY

The data analysis and probability standard is probably the easiest to describe. Students are often required to collect data of some nature (e.g., height and shoe length of each class member) and describe it by calculating such things as mean, median, mode, standard deviation, and range. Students also display data through histograms, dotplots, stem-and-leaf plots, and box plots. In the area of probability, students work simple counting problems with combinations and permutations, as well as calculate simple probabilities. Fair Games is a typical item used in senior high. The standards also emphasize the normal curve and using it to make predications. A series of examples follow.

Determine the probability that a 90% free throw shooter will make exactly one of his/her upcoming two free throws. (Maine)

A set of normally distributed data representing the heights of a population of 18-year-old females has mean 65 inches, with standard deviation 2 inches. In a group of one hundered 18-year-old females from this population, approximate the number who are taller than 67 inches. (South Dakota)

Using a standard deck of 52 cards, find the probability of drawing a king followed by another king, without replacing the first king back into the deck. (South Dakota)

CONCLUSIONS

College mathematics professors could ease the transition difficulty for students as they leave high school mathematics and enter college if they had a better idea what mathematics is taught in high school. Although the United States does not have a national curriculum, the vast majority of the fifty states follow the NCTM’s five content standards. Thus, if we assume that local schools follow their state standards and it certainly behooves them to do so (in order to receive state and federal monies), then we can assume that high schools teach algebra, data analysis and probability, number sense, measurement, and geometry. REFERENCES

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13. Standards for Success, Understanding University Success, Eugene, OR, Center for Educational Policy Research, Online http:// www.s4s.org/cepr.s4s.php (2003).

Carmen M. Latterell

Department of Mathematics and Statistics

University of Minnesota Duluth

140 Solon Campus Center

1117 University Drive

Duluth, Minnesota 55812

clattere@d.umn.edu

Copyright Mathematics and Computer Education Fall 2007

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