Textbook Misalignment and Instructional Issues - CiteSeerX

DRAFT

Fractions and Percents Mathematics and Science Textbooks: Coping with Misalignment Problems

Gerald Kulm Curtis D. Robert Professor of Mathematics Education Texas A&M University

Based on a paper presented at the annual meeting of the American Association for the Advancement of Science, February, 1999.

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Fractions and Percents Mathematics and Science Textbooks: Coping with Misalignment Problems Gerald Kulm, Texas A&M University Scientists use mathematics in nearly everything they do. The most well-known version of this idea is xxxxx’s (19xx) statement that “mathematics is the hand-maiden of science.” Although most mathematicians would say that mathematics is much more than a tool for doing science, there are many reasons for school mathematics and science to pay close attention to each other. Teachers in grades 5-8 are often teachers of both mathematics and science. Work with fractions, ratios, and percents is one of the major stumbling blocks for middle school students. Students’ learning and experiences in science can be used to develop and practice skills and understanding of mathematics. There are opportunities in the science curriculum to teach these mathematics topics in meaningful contexts and in ways that connect students’ mathematical ideas with applications. Teaching and extending mathematics, rather than simply applying it in science, can overcome some of the obstacles to high student performance identified in the Third International Mathematics and Science Study (TIMSS). Students often have difficulties connecting their mathematical knowledge with science, especially as they progress through the grades and the subjects are taught more and more separately. Learning, practicing, and applying mathematical ideas and procedures in science requires careful attention to alignment and coherence between science and mathematics instruction. In order for mathematics learning to take place, not only the sequence of ideas, but the type and sophistication of concepts and skills must be in reasonable agreement. In this article, data and examples from TIMSS and other assessments will be used to illustrate concepts and skills in mathematics that are especially important in middle grades science classrooms. Areas of misalignment and incoherence between mathematics and science textbooks will be shown to illustrate challenges to teaching and learning. Finally, examples from science textbooks and other instructional materials will be presented to illustrate effective approaches to help students develop better mathematics understanding in science classrooms. Student Achievement Students’ difficulties with fraction, ratio, and percent concepts is well-known. These difficulties are often exacerbated if the application is unfamiliar or complex, as it can be in science. Some recent mathematics materials [e. g., Mathematics in Context (1998)] have attempted to use science-related contexts and applications but typically, the applications in mathematics textbooks are unlikely to resemble those found in science books. Therefore, it is essential that these connections and applications to mathematics be made as students learn science.

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The National Assessment of Educational Progress (NAEP) and TIMSS results for eighth graders provide achievement data that can help teachers form estimates of what to expect their students to know and be able to do. Although their own students’ knowledge may be somewhat different, these results are a good starting point for assessing student prior knowledge. The following example test items illustrate student achievement in various contexts and offer background information for science teachers who expect students to use these mathematical skills in their science work. The percentages shown in each example are the percent correct for seventh grade student for TIMSS items and for eighth grade students for NAEP items.

Figure 1. Example Fraction, Ratio, Percent Items from TIMSS and NAEP TIMMS Item (14%)

TIMSS Item (79%) Write a fraction that is larger than

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Answer: ____________

If the price of a can of beans is raised 60 cents to 75 cents, what is the percent increase in the price? A.

15 %

B.

20 %

C.

25 %

H.

30 %

TIMMS Item (32%) A car has a fuel tank that holds 35 L of fuel. The car consumes 7.5 L of fuel for each 100 km driven. A trip of 250 km was started with a full tank of fuel. How much fuel remained in the tank at the end of the trip? D.

16.25 L

E.

17.65 L

F.

18.75L

G.

23.75 L

NAEP Item (38%) Of the following, which is the closest approximation of a 15 percent tip on a restaurant check of $24.99? H.

$2.50

I.

$3.00

J.

$3.75

K.

$4.50

E.

$5.00

These results make it clear that middle school students are unlikely to use fractions, ratios, and percents with confidence except in the simplest situations. Percent problems that apply skills in familiar contexts have very low success rates. Ratio problems that involve familiar situations but include metric measures and decimal answers are very difficult for seventh graders. Estimating percents are difficult, even in a setting that might be familiar to students. In order for students to understand and apply these skills and concepts successfully in science, careful consideration must be given to the science contexts and to linking these contexts with students’ learning in the mathematics classroom.

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A Strategy for Teachers National, state, and local benchmarks and standards in mathematics can be used as tools for helping teachers identify opportunities and obstacles to students’ learning mathematical ideas during science instruction. The content alignment component of the Project 2061 curriculum analysis procedure (Kulm, 1998, 1999) offers a tool for analyzing the mathematics content in science curriculum materials. In the procedure, a benchmark or standard is selected, then used to identify activities, problems, exercises, and other material that target the benchmark. The textbook is examined page by page, using these questions:
Does the lesson or activity address the substance of the specific benchmark or only the general "topic"?
Does the content of the lesson or activity reflect the level of sophistication of the benchmark, or is it more appropriate for benchmarks at an earlier or later grade level?
What parts of the benchmark are addressed by the lesson or activity? A record is made of the page number and description of each lesson, activity, problem, or exercise that meets these questions. These items are called “sightings” for the benchmark. This content analysis of textbooks or other instructional materials can be used to compare and study the alignment and coherence of mathematics and science textbooks. Table 1 provides a partial list of sightings from a seventh-grade science textbook (Morrison, et al., 1997) and a seventh grade mathematics textbook (Burton, et al., 1994) for the following grade 6-8 mathematics benchmark: Use, interpret, and compare numbers in several equivalent forms such as integers, fractions, decimals, and percents [Benchmarks for Science Literacy (AAAS, 1993), 12B #2].

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Table 1. Sample Sightings in Science and Mathematics Textbooks for a Selected Mathematics Benchmark Benchmark: Use, interpret, and compare numbers in several equivalent forms such as integers, fractions, decimals, and percents (12B 6-8#2). Publisher Suggested Lesson Schedule Day 4

7th Grade Science Textbook Activity Description (Science Plus, Holt Rinehart Winston) Making a 5% solution with 1 part soda, 19 parts water

7th Grade Math Textbook Activity Description (Mathematics Plus, Harcourt Brace)

Day 8

Finding the average mpg by dividing miles by gallons, estimating, then using a calculator

Day 29

Meaning of equivalent fractions; writing them in simplest form.

Day 31

Comparing fractions by length of fraction bars, and by using common denominators

Day 32

Comparing fractions on the number line. Meaning of Density Property of fractions, looking for fractions between two given fractions.

Day 35

Making a graph for various masses and volumes; interpreting a mass/volume graph by comparing the slopes of the graphs of the substances

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Type of Content Misalignment

Suggestions for Coping with the Misalignment

Sequence: Percents applied in science several weeks before being introduced in math course.

Introduce percents in science class with a 100x100 grid diagram and relate 1/20 to 5/100

Sequence: Constructing graphs several weeks before the skill is developed in math class. Sophistication: Interpreting fractions as slopes; Comparing fractions using steepness of their

Provide a numerical data table for the graph, using a common standard value for volume. Relate the mass/volume graph lines to their fractions. Compare fractions with the steepness of

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slope.

the graphs.

Day 36

Meaning of density: defined as the ratio of mass to volume. Finding density of an object with mass 11.6g and volume 18mL, then comparing it with densities of known types of wood

Sophistication: Use of density as an application of ratio. Terminology: Use of the same word (density) for a science concept and math concept whose meanings differ. Sequence: Using division to find a decimal expression before students have worked with decimals in math class. Procedure: Reliance on paper and pencil computation

Let students use concrete objects and discuss their experiences with the concept of ratio. Point out the difference between the use of the word “density” in science and in math. Write the density as a ratio, 11.6g/18mL. Write the other densities with a denominator of 18mL. Estimate 11.6/18 before dividing. Consider using a calculator for the computation.

Day 55

Finding the efficiency of a two machines with the fraction work output over work input. Writing the efficiency as a percent.

Sophistication: Use of mechanical efficiency as an application of fractions. Sequence: Using division to find a percentage before students have worked with percents in math class.

Discuss with students their experiences with efficiency and intuitive measures of it. Write 3/6 and ¾ with denominators of 100. Compare and discuss which fraction is greater, then write the fractions as percents.

Day 56

Finding the mechanical advantage by comparing the fraction force exerted by the machine divided by the force exerted on the machine.

Sophistication: Use of mechanical advantage as an application of ratios. Terminology: Use of “fourfold” to describe an increase. Procedure: Reliance on paper and pencil computation

Discuss with students their experiences with mechanical advamtage Discuss the meaning of “fourfold” related to how many times as much. Estimate 10N/2.8N before dividing.

Day 73

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Meaning of ratios, writing of ratios, and finding equivalent ratios

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Day 75

Meaning of a proportion, finding proportions using cross products

Day 80

Finding the density of various materials with mass of 30 mL

Day 81

Interpreting a world map for salinity of sea water, using “proportion” of salt per 1000 parts sea water.

Day 82

Finding the density of a simulated solution having 259g of salt dissolved in 250mL of water, that represents sea water.

Day 85

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Sophistication: Use of density as an application of ratio. Sequence: Using division to find a decimal expression before students have worked with decimals in math class. Procedure: Reliance on paper and pencil computation Terminology: Misuse of the term “proportion.”

Write each density as a ratio with a denominator of 30mL. Estimate which ones are >1 and which are .38). ♦ Middle school students have difficulty comparing fractions, often treating the numerator and denominator separately. ♦ Early adolescents and many adults have difficulty with proportional reasoning, often influenced by the problem format, the numbers in the problem, and the problem situation. ♦ Middle school students can solve problems in proportions that involve simple numbers and simple wordings but trouble arises with difficult numerical values or contexts.

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♦ Different types of ratios vary in difficulty. For example, speed problems are harder than exchange problems and unfamiliarity with the situation causes even more difficulty when it occurs with a difficult type of ratio. Textbook Alignment and Instructional Issues In order to increase textbook and instructional compatibility so that students will see the connection of their mathematical knowledge to science, teachers need to make informed adaptations. The following comments are organized around issues that arise from the analysis shown in Table 1. The discussion takes into account the research findings, and illustrates the key ideas that teachers might keep in mind to help students build their mathematics knowledge and skills as they learn science. ♦ Sequence of development and use of concepts and skills in the mathematics and science books There are clear mis-matches in the sequence of mathematics concepts and skills. For example, an understanding of percents and their application is assumed from the very beginning of the science textbook, but not introduced in mathematics until more than three months into the school year. ♦ Complexity or sophistication in the science book of contexts or situations Students are expected to apply their developing mathematics knowledge either to learn new, complex ideas in science or to solve problems in science that they also are only learning for the first time. For example, students compare densities using a mass/volume graph at about the same time they are learning in mathematics the meaning of equivalent fractions and how to compare them. Applications such as percentage of solution and percent efficiency of a machine are also much more complex than the typical percent discounts on consumer goods that students see in mathematics. ♦ Use of the same term in the science and mathematics books for similar but different ideas (e. g., Density, proportion) At about the same time that students are learning about the density of substances in science, they are learning the density property of fractions. Not only do these properties have the same names but their meanings are similar enough (although very different) to cause great confusion. ♦ Lack of representation in the science book of mathematics concepts or procedures Applications of mathematical skills in science are seldom represented in a step-bystep form or illustrated with diagrams or figures that would be familiar to students. For example, the idea of a 5% solution made up of 1 part soda to 19 parts water is stated verbally, without a diagram or drawing that might help student visualize the ratio 1/20, then change it to a percent. Computations or rewriting of formulas are not shown in detail to demonstrate how answers are found or expressions derived.

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♦ Emphasis in the science book on the direct application of procedures to obtain numerical answers in science. The science text focuses primarily on using mathematics to find numerical answers to problems either in the explanatory text or the exercises. Little attention is given to making sense of the procedures that are used, making estimates, providing information about rounding or significant digits, showing intermediate steps in finding the answer, or discussing whether the answer makes sense in the problem. ♦ Lack of science contexts for problems or exercises in mathematics book The applications of ideas of fraction, ratio, and percent in mathematics books seem mainly to involve consumer situations such as discounts and interest. The applications themselves and their quantitative meanings and labels are familiar to students. Even though the types of mathematical concepts and procedures are the same (e.g., finding a percent, expressing a comparison as a ratio), the applications and quantities in science (e.g., percentage solution, density, mechanical advantage) are the more complex and unfamiliar. Because they are unfamiliar to both teachers and students, these science contexts are not used as settings for learning mathematics. Suggestions for Teacher Education ♦ Do comparative content analyses of university and K-12 science and mathematics textbooks The kind of content alignment suggested here would be useful to compare sequence and approach to the development of mathematics in textbooks used to prepare teachers at the university. Teachers’ understanding and success in mathematics and science is a continuing issue that requires careful work at the college level. The analysis could also be used to compare key content in the mathematics and science courses taken by teachers with the content they are expected to teach. ♦ In methods courses, practice textbook analysis, using specific benchmarks The analysis outlined here can also illuminate important learning issues, perhaps serving as a capstone experience for future teachers, or as an activity to help students connect their work in science and mathematics methods courses. The opportunity to think carefully about instructional material they are likely to use in the classroom can provide practical work that can help develop their own content understanding, along with practice in planning and coordinating teaching. ♦ Spend time on key research on student mathematics and science learning Much of the research on student mathematics learning is overlooked or under utilized by textbooks and teachers. The analysis provides a setting and motivation for understanding and making use of this research. For future teachers, attending to key research provides a point of view about mathematics and science teaching that emphasizes careful evidence and knowledge over experimentation and “teaching the way I was taught.”

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♦ Practice developing strategies for making science and mathematics instruction compatible The preparation of adaptations in science lessons that account for how students are learning mathematics provides a setting for teachers to think carefully about instruction. Even if the textbook is used as a guide, this activity requires a reflective, informed, and student-based connection to be made about science and mathematics learning. Rather than focusing on the activity or context, this work focuses on building on students’ knowledge. ♦ Work on curriculum frameworks to make science and mathematics sequences agree more closely Many school districts are working on their mathematics and science frameworks to assure they are compatible with state or national standards and benchmarks. The brief analysis shown here suggests that aligning the content sequence and the instructional strategies for middle grades science and mathematics is an important issue in this effort. Since the selection of textbooks, tests, and the professional development of teachers are often based on these frameworks, it is important they are properly aligned and compatible. Concluding Remarks Both science and mathematics curricula and textbooks are packed too full with far more concepts and skills than students can learn and understand. Ten years ago, Science for All Americans (AAAS, 1989) provided guidelines on the goals for science, mathematics, and technology that students could and should be expected to know and understand. One of the catch phrases of that publication was “less is more,” suggesting that more learning can be accomplished by attempting fewer ideas and to learn those ideas in depth. Another idea of Science for All Americans is that even for a more limited number of literacy goals, careful planning is necessary and that planning should include educators from the various disciplines to work together. The brief analysis presented in this paper underlines the importance of joint planning by science and mathematics teachers, not only in the interest of trimming the curriculum to exclude unnecessary material, but so that students have a chance of learning what is presented. This planning is especially important in the middle grades, where the content becomes complex and students begin to move from one teacher to the next. Also, new content and instructional approaches to science and mathematics are beginning to emerge, causing gaps and misalignments between the two subjects as early as the middle grades. Finally, the effort in doing analysis and joint planning is itself educative for teachers, offering a useful and practical context for professional preparation and development.

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References American Association for the Advancement of Science. (1993). Benchmarks for science literacy. New York: Oxford University Press. American Association for the Advancement of Science. (1989). Science for all Americans. New York: Oxford University Press. Burton, G. M, Hopkins, M. H., Johnson, H. C., Kaplan, J. D., Kennedy, L. and Schultz, K. A. (1994). Mathematics plus. Orlando, FL: Harcourt Brace & Company. Kulm, G. (1998). How to do mathematics curriculum materials analysis. Washington, DC: American Association for the Advancement of Science. Mathematics in Context. (1998). Chicago IL: Encylopædia Britannica Educational Corporation. Morrison, E. S., Moore, A., Armour, N., Hammond, A., Haysom, J., Nicoll, E. and Smyth, M. (1997). Science plus. Austin, TX: Holt, Rinehart and Winston. Roseman, J. E., Kesidou, S., & Stern, L. Identifying curriculum materials for science literacy: A Project 2061 evaluation tool. Paper Presented at the Colloquium, National Research Council, Washington, DC, November, 1996.

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Publisher Suggested Lesson Schedule Day 4

7th Grade Science Textbook Activity (Science Plus, Holt Rinehart Winston) Making a 5% solution with 1 part soda, 19 parts water

7th Grade Math Textbook Activity (Mathematics Plus, Harcourt Brace)

Day 8

Finding the average mpg by dividing miles by gallons, estimating, then using a calculator

Day 29

Meaning of equivalent fractions; writing them in simplest form.

Day 31

Comparing fractions by length of fraction bars, and by using common denominators

Day 32

Comparing fractions on the number line. Meaning of Density Property of fractions, looking for fractions between two given fractions.

Day 35 Day 36

Day 55

Day 56

Day 73 Day 75 Day 80 Day 81

Day 82

Making and interpreting a mass/volume graph, comparing slopes Meaning of density: ratio of mass to volume. Finding density and comparing to known types of wood Finding the efficiency of a machine with the fraction work output over work input. Writing as a percent. Finding the mechanical advantage with the fraction force exerted by the machine over force exerted on the machine.

Finding the density of various materials with mass of 30 mL Interpreting a world map for salinity of sea water, using “proportion” of salt per 1000 parts sea water. Finding the density of a simulated solution that represents sea water.

Day 85

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