From percentages to algebra

From percentages to algebra

Middle school mathematics: New style of teaching based on O.R. and scenario-based learning.

By Kenneth Chelst

In my previous OR/MS Today articles [1, 2], I wrote that the analytics movement aligns fully with the Mathematical Practices part of the Common Core State Standards for Mathematics, which recommends using mathematics in meaningful contexts. Unfortunately, the leadership of the mathematics community and the vast majority of math educators know little about our field, and few are aware of the term “analytics.” Consequently, there is a paucity of mathematics curricula that is engaging and relevant.

Our early work, funded by NSF, resulted in two courses, algebraic modeling and probabilistic decision-making, designed for high school seniors. INFORMS provides continuing funding for teacher workshops during the annual conferences and in selected markets around the country. This year Ford funded both high school and middle school teacher workshops at its Dearborn, Mich., assembly plant that included a Ford presentation on the use of analytics. In 2017, we expect to deliver one-day workshops to more than 300 high school and middle school math teachers in four locations: Dearborn, Newark, N.J., Chicago and Houston.

In recent years, my colleagues and I shifted our focus to K-8 grade mathematics. (My teammates include two professors of math education and two longtime math teachers who have served as math consultants for more than a decade.) We had been dabbling in developing stand-alone activities appropriate for lower grade levels. I say we were dabbling because we had no coherent plan for developing a systematic approach to any specific math skill. Several of the examples were published as articles in math teacher journals [3, 4, 5]. We selected diverse problem contexts that we thought might be interesting. Our energies were focused on ensuring that these problems addressed the following questions discussed in a previous OR/MS Today article [1].

  1. Would anyone care about the problem context and the answer?
  2. Is there a decision to be made?
  3. Would simple counting not answer the question?
  4. Is there something to discuss in the problem context that goes beyond the basic mathematics?
  5. Does the question only ask about the current situation? Are there opportunities to explore how the situation might change?
  6. Can multiple similar examples be created that vary enough to maintain student interest?
  7. Can students embed themselves in the problem context and produce different answers?

Percentages, a critical middle school math skill, requires a coherent and comprehensive teaching approach. Image © kzenon | 123rf.com Image

A year ago, I decided to develop a coherent and comprehensive approach to teaching percentages, a critical middle school math skill. In reviewing current curriculum, I was shocked at the narrow range of applications. Percentages are limited to a few contexts: price discount, sales tax, percent of boys or girls in a group, gratuity and interest on a bank account or loan. Yet “percentage” is one of the top 2,000 words frequently used as reported in the Corpus of Contemporary American English. Almost every news article that has numeric information includes percentages. Yet teachers who see percentages everywhere have failed to translate this experience into a more meaningful and diverse set of examples for their students. It was this educational gap that motivated our team to develop 15 authentic scenarios and related homework. We recently published these as a textbook titled “From Percentages to Algebra.”

In leading the writing of the text, I took a fresh look at all aspects of percentages beginning with a basic question: Why are percentages needed when rational numbers, decimals and fractions can serve the same function? My answer was that percentages more than rational numbers have a natural range of 0 percent to 100 percent. The top percentage of 100 percent represents perfection as on an exam, and the lower the percentage, the poorer the performance. The worst score possible on an exam is 0 percent. The text includes several examples with a goal of 100 percent, such as average grade in a course and passing military special ops training. The text uses scenarios that note that 100 percent may be an unrealistic goal; the standard to be measured against might be a smaller percentage. For example, the best professional basketball free throw shooters approach 90 percent, and the highest winning team percentage in professional baseball in the last hundred years has never exceeded 73 percent. At the other end of the range, 0 percent is not always a bad result. When tracking high school dropouts or homelessness among veterans, 0 percent is an ideal.

I also scanned textbooks for examples involving more than 100 percent. I was disturbed to see examples such as: “A person ordered 1.5 pizzas. Convert this number into an equivalent percentage.” In the contexts we developed, more than 100 percent arises naturally when making comparisons across time. Our contexts include downloads of an app and revenue or customer growth in a business.

Decisions

A principle that guides our development of the scenarios in all of our texts is that each scenario contains an embedded decision. We avoid typical descriptive application. Who cares to answer the question: “If 60 percent of the class of 30 students are boys, how many boys are in the class?” Why would I need to know? We prefer to tackle the following scenarios:

  • Which coupon is best when comparing a percent discount to a fixed discount?
  • How many volunteers are needed given estimates of the percentages who fail training?
  • How may hoodies of each size and color should be ordered given known population percentages for clothing sizes as well as color preferences?

From Percentages to Algebra

We began with percentages but soon realized we could design a transition from percentages to algebra. Traditionally, students are introduced to algebra as a new topic with little continuity to earlier math skill development. Texts begin by presenting the elements of an algebraic expression and equation. The text may include one simple word problem intended to minimally motivate the relevance of algebra. In developing authentic and meaningful contexts for percentages, we found that algebraic representation flowed naturally. For example, a 15 percent coupon for a restaurant can be applied to meals costing different amounts. The middle school student is presented with an algebraic expression to organize his or her approach to calculating savings for different priced meals: savings = 0.15x.

When inserting different values of x, the student learns that x is a variable. If the restaurant also offers an alternative $5 coupon, an algebraic equation can be set up to determine when to use each coupon. Half of our examples logically and meaningfully transition from percentages to algebraic expression to algebraic equation. The algebraic expression is an efficient way of organizing the calculations for different values of x. When deciding between types of discount, solving an equation for x presents an efficient alternative to repeated trials.

Our natural approach addresses a well-known gap in understanding. Many students do not understand what it means for x to be a variable. In addition, standard textbooks often fail to differentiate clearly between algebraic expressions and algebraic equations. In an algebraic equation, students see x as simply an unknown to be uncovered and not a variable. Our scenarios’ contexts require students to first work with an algebraic expression before moving on to solve an algebraic equation.

Compound Percentages

Figure 1: Linear growth and compound percent.

I can recall being introduced to compounding with an example of accrued interest on a savings account. In the days of 5 percent annual interest, daily compounding increased this to 5.12 percent, an observable but modest difference. With compounding, a $1,000 account earned $51.20 instead of $50. However, in our day of 1.3 percent interest, compounding increases the earnings from $13 to $13.08. How many children would care about this type of compounding or even small savings accounts? Instead, we envision an app developed by teenagers that was initially downloaded a thousand times. The number of downloads increases by 25 percent each month. With compound percentages, students find that after the third month the overall number of downloads has increased by 95 percent.

In other scenarios, students compare linear and exponential growth. First, they must choose between two marketing campaigns. One campaign generates linear growth, while the other involves compounding percentage growth. A different scenario explores two programs for reducing the number of homeless veterans. One program generates a linear reduction in the number of homeless, while the other program reduces the number of homeless by a fixed percentage each year. In the latter example, compounding percentages has a diminishing impact.

Other Math Skills

Figure 2: Compound growth of downloads of an app.

Scenarios also motivate and reinforce other related math skills. For example, compound percentages are represented by an exponential function. An algebraic expression leads naturally to a table as different values of x are tried and recorded. In our examples, a table is not just another artifact but rather a tool for organizing and perceiving how the expression changes with different values of x. Graphs are especially useful for illustrating how linear and compounded growth develop and for identifying the point where compounded growth surpasses linear growth as in Figure 1. Figure 2 highlights the exponential nature of compound growth. Pie charts are useful when discussing the percentage of people who need different sized hoodies and contrasting men and women sizes as in Figure 3.

The scenario developed around monthly store sales addresses the misconception that a percentage increase followed by an equal percentage decrease results in no net change. The bar chart in Figure 4 demonstrates revenue fluctuations and seasonal trends better than a table.

The military ops scenario involves two types of training delivered in consecutive weeks. Each training module has a different passing rate. The captain and colonel discuss whether changing the order of training would increase the total percent who pass. It does not. One teacher pointed out this was the first time she had seen the commutative law in a meaningful context. We also demonstrate that the economics of training is sensitive to the sequence. It is better to place the training with the lower passing rate in the first week.

Figure 3: Men’s and women’s sizes.

Math textbook examples almost always produce numbers that are integers. Real-world examples are not so neat. We made sure that our scenarios’ calculations would not produce integer results. When deciding on an order of hoodies, students need to translate these calculated non-integer values to an actual order of hoodies. Several other examples also generate non-integer values that need to be converted to integers in the planning context. These examples stress an important aspect of how math is used. Mathematics is a tool for exploring the range of possibilities to make a decision or reach a conclusion. In contrast, math textbooks present math as a tool for finding the one correct answer.

Mental Math and Discussion Opportunities

When discussing our work, colleagues often chime in with comments about how kids today can’t calculate simple percentages in their heads. To address this concern, each scenario is preceded by a section called Number Talks. We recommend teachers spend five to 10 minutes using these examples to do mental math with percentages before introducing the scenario.

One classic concern about math education is succinctly characterized by the following student question: Why is mathematics the only class in which the teacher is uninterested in my opinion, and there is nothing to discuss except for the mathematical procedure? In contrast, our scenarios involve business operations and personal concerns that students understand and relate to. We have designed other scenarios around important topics such as high school dropouts, training for special military ops, homelessness among veterans, congressional redistricting and growth in the use of apps. All of these offer natural opportunities to expand the discussion. We also include at the end of each scenario a suggestion for a student project.

Figure 4: Monthly revenue.

The Future

I believe this work with middle school mathematics has enormous potential. Unlike our high school O.R. curriculum, middle school teachers will not need to learn new math skills nor will they need to justify the curriculum to decision-makers. They readily understand the mathematical concepts and can immediately incorporate our examples into their classrooms. At a recent workshop, 80 percent of the teachers planned to use at least one activity within the next two months. The biggest challenge is a willingness to adopt a new style of teaching associated with scenario-based learning.

Kenneth Chelst (kchelst@wayne.edu) is a professor of operations research in the Industrial and Systems Engineering Department at Wayne State University and co-founder of a collaboration, Applied Mathematics Practices for the 21st Century (http://www.appliedmathpractices.com/).

References

  1. Chelst, K., 2015, “Introducing Analytics to Adolescents,” OR/MS Today, Vol. 42, No. 4.
  2. Chelst, K. and Edwards, T. G., 2012, “Innovative Education: Early O.R. Education: “Operations Research is aligned with the Common Core Revolution in K-12 Mathematics Education,” OR/MS Today, Vol. 39, No. 4.
  3. Ozgün-Koca, S., Edwards, T. G., and Chelst, K. R., 2013, “Exercise Away the Big Mac: Ratios, Rates, and Proportions in Context,” Mathematics Teaching in the Middle School, Vol. 19, No. 3, pp. 184-188.
  4. Chelst, K. R., Özgün-Koca, S. A., and Edwards, T.G., 2014, “Rethinking ratios, rates and percentages,” Mathematics Teaching (GB), issue 240, pp. 26-29.
  5. Özgün-Koca, S. A., Edwards, T. G. and Chelst, K. R., 2015, “Linking Lego and Algebra,” Mathematics Teaching in the Middle School, Vol. 20, No. 7, pp. 400-405.