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2014 ASCD Conference on Educational Leadership

2014 ASCD Conference on Educational Leadership

October 31–November 2, 2014, Orlando, Fla.

Learn the secrets to great leadership practices, and get immediate and practical solutions that address your needs.

 

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Books in Translation

November 2007 | Volume 65 | Number 3
Making Math Count Pages 66-71

From Arithmetic to Algebra

Leanne R. Ketterlin-Geller, Kathleen Jungjohann, David J. Chard and Scott Baker

Teachers can help students make the transition by developing their algebraic thinking early on.

Increasingly, algebra is the focus of mathematics discussions in schools and districts across the United States. Policymakers, professional organizations, and researchers emphasize the importance of developing algebraic reasoning at increasingly earlier ages. The National Mathematics Advisory Panel (2007) has issued initial reports stating that students need to develop understanding of concepts, problem-solving skills, and computational skills related to algebra in grades preK–8. In 2006, the National Council of Teachers of Mathematics published the Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics, which emphasizes connections to algebra as early as kindergarten and promotes the development of algebraic reasoning across the elementary and middle school grades. Finally, mathematicians and mathematics educators are speaking up about the need to increase teachers' awareness and abilities for teaching algebra across the grades (Wu, 1999).

Multiple factors are driving the increased emphasis on algebra proficiency. For many educators, the primary concern is the poor performance of U.S. students on national and international assessments of mathematics ability. On the 2005 National Assessment of Educational Progress (NAEP), only 6.9 percent of 17-year-olds scored at or above proficiency on multistep problem solving and algebra (National Center for Education Statistics, 2005). On the algebra subtest of the 2003 Trends in International Mathematics and Science Survey (TIMSS), U.S. 8th graders scored below many economic competitors, such as Japan, the Russian Federation, Korea, Singapore, and China. These results suggest that a majority of U.S. students are not proficient in algebra by the time they exit middle school or high school.

Although the academic performance of U.S. youth as a whole is important, No Child Left Behind (NCLB) emphasizes the need to monitor the progress of subgroup populations that have traditionally performed below expectations. On the 2005 NAEP, 59 percent of black students, 50 percent of Hispanic students, and 45 percent of American Indian students did not meet proficiency at the 8th grade level. Similarly, 69 percent of students with disabilities and 71 percent of English language learners did not reach this benchmark (National Center for Education Statistics, 2005). These results highlight the crucial need to develop algebraic thinking across the grades and focus on providing the best instructional practices for all students.

Why Algebra?

Employers often expect their employees to translate work-related problems into general mathematical models, from calculating discounts for merchandise to operating technology-based equipment and machinery. Many careers in the fields of science and technology demand high levels of mathematics competence to solve complex problems, such as chemical equations involved in the study of drug interactions. Algebra is also helpful in daily life, from applying formulas for calculating miles per gallon of gasoline to using functions to determine the profit of a business venture.

Research suggests that students who pass Algebra II in high school are 4.15 times more likely to graduate from college than other students are (Adelman, 1999). This has led many state education agencies to raise graduation requirements to include courses in Algebra II. Currently, 13 states require students to take Algebra II to graduate from high school, up from just two states in 2005 (Achieve, 2007). Many states and school districts are considering implementing higher mathematics standards to promote college readiness and future success for their graduates.

What Is Algebra?

When we think about algebra in the curriculum, we often think of a separate area of mathematics concerned with symbols and equations, such as 3x + 7y - 2 = 30. Mathematics curriculums often reinforce the notion of separateness by identifying algebra as a distinct strand with such subtopics as patterning, data analysis, simple functions, and coordinate systems. However, arithmetic and algebra are not mutually exclusive areas of mathematical study.

Basic algebra, as opposed to modern or abstract algebra, extends learners' understanding of arithmetic and enables them to express arithmetical understandings as generalizations using variable notation. Much of the difficulty that students encounter in the transition from arithmetic to algebra stems from their early learning and understanding of arithmetic. Too often, students learn about the whole-number system and the operations that govern that system as a set of procedures to solve addition, subtraction, multiplication, and division problems. Teachers may introduce number properties as “truths” or axioms without developing students' deep conceptual understanding or providing multiple experiences applying these properties.

When teachers introduce integers and rational numbers in later elementary grades, many of these “truths” about numbers and operations don't generalize to addition and subtraction of positive and negative numbers or multiplication and division of fractions. By the time algebra is introduced in middle school, many students view mathematical principles as subjective and arbitrary and rely on memorization in lieu of conceptual understanding.

The National Council of Teachers of Mathematics has attempted to bridge the gap between arithmetic and algebra by embedding algebraic reasoning standards in elementary school mathematics. From grades 3 to 5, algebra is embedded with number and operations as one of the three main focal points; beginning in grade 6, algebra is the predominant topic. However, it is not always clear how to develop students' algebraic thinking as they learn about numbers, operations, properties of numbers, data display and analysis, and problem solving. Teachers need support in learning how to integrate these topics and provide rich and explicit instruction to their students in early algebraic thinking.

Teaching Algebra for Transfer

Teachers' understanding of mathematics influences the quality of their instruction. Many elementary school teachers have limited experience with mathematics and lack the knowledge and skills to teach mathematics effectively (Ball, Hill, & Bass, 2005). Moreover, most credentialing programs for elementary school teachers require minimal college-level mathematics courses despite calls for considerably more extensive requirements (Conference Board of the Mathematical Sciences, 2001). Aside from developing their content knowledge in mathematics, these teachers can benefit from some general instructional practices that can help them teach arithmetic for transfer to algebra.

Whenever possible, teachers should model precisely what they want students to be able to do, using multiple examples that illustrate the range of problem types that students must solve on their own. Demonstration models should include careful verbal explanations that explicitly detail for students how to perform each step of the problem. As students develop expertise, teachers can make fewer verbal explanations and focus less on each individual step.

Teachers often have difficulty modeling for students how to think about mathematics problems conceptually. Rather than initially using numeric symbols to solve a problem, teachers might use concrete objects or semi-concrete representations (such as pictures) to help represent the underlying concepts behind specific problems. Teachers will find that explaining the concept of 2/3 ÷ 1/3 is more complex than explaining how to use the “invert and multiply” algorithm. To develop deep conceptual understanding, teachers should draw on different types of examples that represent problems.

For example, teachers can use concrete objects to visually represent that the problem 2/3 ÷ 1/3 = □ means the same thing as “how many 1/3s are there in 2/3?” Presenting the problem this way helps students understand what it means to divide any number by a fraction and “see” that in this example, there are “2 1/3s in 2/3.” However, using visual models to help students understand how to solve problems involving division by fractions breaks down quickly when the numerical values in the problem are not artificially constrained, such as in the problem 9/23 ÷ 11/15 = □. Without using the “invert and multiply” algorithm, this problem becomes difficult to solve. After students understand the meaning of division of fractions, instruction should focus on applying the algorithm in a step-by-step fashion. With clear verbal explanations and explicit modeling, students can understand why the algorithm works and what it means to divide by fractions.

In addition to hearing teachers' verbal explanations, students should share their verbal explanations to further develop conceptual understanding. Here again, carefully chosen examples can provide a rich source of discussion as students explain why 2 × 54 = 2 × 50 + 2 × 4 (an application of the distributive property); why 72 - 6 ≠ 72 (an application of the identity property of subtraction); or why 5 + 2 = 2 + 5 (an application of the commutative property of addition). Students should be able to describe the properties of numbers in their own words—such as through telling a story or describing what is happening in a picture that has an obvious numerical focus—as well as in symbolic notation, and they should be able to apply these principles in multiple contexts.

For example, young students might demonstrate the commutative property of addition by using concrete objects, such as groups of marbles. Students might explain the commutative property by showing that reordering the groups of marbles does not change the sum of the marbles when the groups are added together. Once they understand the concept, the teacher might ask the students to provide multiple representations of the commutative property using symbolic notation.

Students also need to demonstrate their own understanding and skills. Teachers can gauge how well students solve problems in relatively straightforward ways. Students can work different types of problems and apply algorithms to solve them. Teachers can set proficiency goals for students and monitor student progress toward these goals.

Algebra-Specific Instructional Strategies

Algebraic reasoning builds on students' deep understanding of numbers and their relationships. Some mathematics researchers have identified areas of arithmetic that provide the foundations for algebra. These include

  • Numbers and number relationships (quantities and magnitudes).
  • Operations (functional relationships between numbers).
  • Field axioms or number properties (commutative, associative, distributive, identity, inverse, and so on).

Other topics linked to algebra include geometry, data analysis, proportional reasoning, and measurement. These topics provide rich opportunities for developing early algebraic reasoning as students learn about functional relationships in these areas (Van de Walle, 2004).

To develop algebraic reasoning, students must understand the following four key components (Milgram, 2005).

Variables and Constants

As students progress through elementary school, they learn about number systems—from counting, to whole numbers, to integers, to rationals, to real numbers. Studying number systems builds students' understanding that each new system is an extension of the previous system and that all number systems are embedded in the real-number system. As such, each system satisfies the basic rules of associativity, commutativity, and distributivity.

As we introduce students to variables, a key insight for students to grasp is that algebraic expressions, in which variables replace real numbers, will also satisfy the properties with which they are familiar. For example, when teachers introduce the distributive property, they can extend instruction from the context of whole numbers and integers to expressions with variables. They can follow a discussion of the problem

6 × (2 + 9) = 6 × 2 + 6 × 9
with a discussion of
6 × (t + 9) = 6 × t + 6 × 9.

Representing and Decomposing Word Problems Algebraically

Key to abstract reasoning and using algebra to solve problems is using algebraic expressions to describe problems. For example, students who think in algebraic terms easily translate the phrase “if you add 3 to a number times itself” into n2 + 3. Students need to apply this conversion of phrases to solve word problems. Teachers can help students master this skill by modeling and using language that identifies the “unknown” in a problem and then translates the process of finding the unknown into mathematical statements and equations.

Consider the following word problem:

Maria needs to find the weight of a box of cereal using a balancing scale. Maria puts 6 identical boxes of cereal on one side of the scale. To balance the scale, Maria puts 2 more identical boxes of the same cereal and 3 4-pound blocks on the other side of the scale. How much does each box of cereal weigh?

Teachers can model how to solve this problem by first identifying the unknown component (the weight of each box of cereal, labeled y) and the known components (the number of boxes of cereal and the weight and number of the blocks). Next, teachers can help students understand how to translate these elements into a mathematical statement to solve for the unknown (6y = 2y + 12). Students can check answers by inserting various numerical values into equations to verify solutions. This last step is about more than just getting the correct answer; it is an important step in problem solving because it encourages students to reflect on the original problem and determine whether the answer is reasonable.

For many students, improving skills at translating or converting problems to algebraic expressions will pose challenges. Students need to learn to break the problem into separate parts and then convert each part to an expression or equation that acknowledges the restrictions that the problem places on it (for example, the phrase “times itself”). Students will also need to recognize when a problem contains irrelevant information.

Symbol Manipulation

Many adults associate symbol manipulation with algebra because their memories of basic algebra are with the struggles of moving abstract symbols about the page “to solve for x.” Although isolating the variable is still the goal for symbol manipulation, students need to understand that manipulating symbols in an equation merely simplifies the equation in a manner that enables us to get the answer we are seeking. Lawful manipulation of the symbols results in an equation that has the same solutions as the original equation.

Related to this topic is a common misconception about the equality rule and the equal sign. Many students in the early grades view a number sentence or mathematical formula as something “to do,” most often with input on the left and output on the right. Consider the number sentence 5 + 3 = □. Students interpret this as adding the quantities 5 and 3 to find the specific answer of 8. Students may not view the following as possible solutions to the same problem:

5 + 3 = 3 + 5
5 + x = 8
8 = 5 + 3
5 + 3 = 2 + 6

Teaching equality and the meaning of the equal sign as a symbol that indicates both sides are balanced (as symbolized, for example, by a balance scale) provides opportunities for students to see equations as more than something to act on or a problem for which they must seek a single solution. Encouraging students to generate multiple solutions to 5 + 3 prepares them for working with variables, understanding and applying the commutative property and the inverse property of addition and subtraction.

Functions

Students should begin to learn elements of functions early in their school careers. Teachers need to strategically teach students to build patterns in which each input has only one output. Milgram (2005) provides an example of how kindergarten teachers can help their students understand simple functions. By sorting and classifying objects on the basis of unique properties, students can understand the association between objects in one set and unique objects (or features of the object) in another set. For example, students can sort objects by color. If each object has a specific color, the object is the input and the color is the output. Sorting the objects by color is an example of a function. As students progress in their understanding, teachers can explicitly model symbolic representations of functions.

Later students will learn to graph the Cartesian coordinates of the members of the input and output sets (domain and range). Next, they'll develop an understanding of how the domain and range represent a “rule of correspondence” that can be described using function notation, a convention in mathematics. Ultimately, these early insights into functions assist learners in understanding linear algebra and, later, curvilinear and quadratic functions and the role they play in mathematical relationships.

Finally, to help students develop algebraic reasoning in problem solving, students must develop a degree of certainty about the properties of number systems that allow us to manipulate and operate on numbers. Teachers can build this certainty in students by teaching the process of mathematical induction so students understand that their actions must be verifiable mathematically to be lawful and useful (Milgram, 2005). Teachers often teach mathematical induction as a procedure without sufficiently understanding why induction is so crucial for students' cognitive development in mathematics.

Starting Early

Because the goal of teaching algebra is to help students develop abstract reasoning in problem solving, schools should begin to develop these skills in students at the elementary level. By systematically and explicitly incorporating concepts of algebra in elementary school mathematics, schools can help students avoid developing many misconceptions about number and number relationships, operations, and application of number properties. Teaching mathematics in the elementary grades to transfer to algebraic concepts may promote success for all students engaged in mathematical reasoning.


My “Aha!” Moment


Keith Devlin, Professor of Mathematics at Stanford University, Palo Alto, California. “The Math Guy” on National Public Radio.

Mathematics suddenly interested me when I encountered calculus at age 16. Before then, I never saw much point in the subject beyond basic arithmetic, and looking back I now realize why. Other than basic number skills and a bit of trigonometry, no subject generally taught before calculus shows how mathematics makes a difference in the world. Logical thinking is important in earlier math classes, but not mathematical thinking.

The enormous power of mathematics—and its beauty—lies in the vast range of the subject beyond high school mathematics. The mathematics taught in school is what I call abstracted math—and it really amounts to little more than formalized common sense. You can call it math, but it really isn't.

What our modern world depends on—big time—is what I call constructed math. This is the rule-based, abstract reasoning system that forms the basis of all science and engineering, and a lot else besides. It isn't really abstracted from the world; rather, we humans create it to apply to the world. By and large, this kind of mathematics cannot be learned before the upper levels of high school; it requires too much mental sophistication. But there is no reason why we can't teach such mathematics descriptively, where the goal is awareness and understanding, not the ability to do it. I am sure that if I had been taught that way, I would have been interested in math long before I was.


References

Achieve. (2007). Closing the expectations gap 2007. Washington, DC: Achieve.

Adelman, C. (1999). Answers in the tool box: Academic intensity, attendance patterns, and bachelor's degree attainment. Washington, DC: U.S. Department of Education.

Ball, D. L., Hill, H. C., & Bass, H. (2005, Fall). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator, 14–17, 20–22, 43–46.

Conference Board of the Mathematical Sciences. (2001). The mathematical education of teachers (Vol. 2). Washington, DC: American Mathematical Society.

Milgram, R. J. (2005). The mathematics preservice teachers need to know. Stanford, CA: Stanford University.

National Center for Education Statistics. (2005). The nation's report card: Mathematics 2005. U.S. Department of Education. Available: http://nces.ed.gov/pubsearch/pubsinfo.asp?pubid=2006453

National Council of Teachers of Mathematics. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics. Reston, VA: Author. Available: www.nctm.org/standards/content.aspx?id=270

National Mathematics Advisory Panel. (2007). Conceptual Knowledge and Skills Task Group progress report. New Orleans, LA: U.S. Department of Education.

Van de Walle, J. A. (2004). Elementary and middle school mathematics: Teaching developmentally (5th ed). Boston: Allyn and Bacon.

Wu, H. (1999). Basic skills versus conceptual understanding. American Educator, 23(3), 14–19, 50–52.

Leanne R. Ketterlin-Geller (lketterl@uoregon.edu) is Assistant Professor and Kathleen Jungjohann (kjj@uoregon.edu) is Senior Instructor and Research Assistant at the University of Oregon, Eugene. David J. Chard (dchard@smu.edu) is Dean of the School of Education and Human Development at Southern Methodist University, Dallas, Texas. Scott Baker (sbaker@uoregon.edu) is Director of Pacific Institutes for Research, Eugene, Oregon.

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