Prerequisite Knowledge for the Learning of Algebra!

Transcription

1 Prerequisite Knowledge for the Learning of Algebra! Rachael M. Welder Montana State University

2 Research Overview! Students need to be algebra proficient to succeed in high school and college Algebra achievement of U.S. students on the National Assessment of Educational Progress (NAEP) is poor 1 1. (Chazan & Yerushalmy, 2003)

3 Research Overview! Algebra I is the key and the barrier to students ability to complete a challenging mathematics curriculum in high school 2 Perhaps students are not being properly prepared for their first algebra course 2. (Bottoms, 2003)

4 What do students need to know prior to entering their first algebra course in order to be successful?

5 Algebra I! Operations with positive and negative numbers Evaluation of expressions Solving of linear equations, linear inequalities and proportions Age, digit, distance, work and mixture word problems Operations with polynomials and powers Factoring of trinomials, monomial factoring, special factors (Usiskin, 1987)

6 Algebra I! Simplification and operations with rational expressions Graphs and properties of graphs of lines Linear systems with two equations in two variables Simplification and operations with square roots Solving quadratic equations (by factoring and completing the square) (Usiskin, 1987)

7 Prerequisite Knowledge for Algebra I! The Southern Regional Education Board (SREB) Panel of classroom teachers and curriculum experts Analyzed curriculum materials (Bottoms, 2003)

8 Prerequisite Knowledge for Algebra I! Identified 17 mathematical concepts (Readiness Indicators) essential for middle school students to be prepared for Algebra I (Bottoms, 2003)

9 Prerequisite Knowledge for Algebra I! 5 are general processing skills for all mathematics: 1. Problem Solving 2. Reading and Communicating 3. Estimating and Verifying Answers and Solutions 4. Logical Reasoning 5. Using Technology 12 are content-specific skills for learning algebra (Bottoms, 2003)

10 Research Questions! Do current research findings correspond with the conclusions of the SREB? Which Readiness Indicators are supported by research? Which are not? Does research show that there are topics missing?

11 Readiness Indicators! 1. Read, write, compare, order and represent in a variety of forms: integers, fractions, decimals, percents, and numbers written in scientific and exponential notation 2. Compute (add, subtract, multiply and divide) fluently with integers, fractions, decimals, percents, and numbers written in scientific notation and exponential form, with and without technology (Bottoms, 2003)

12 Numbers! Students must understand numbers prior to algebra introduction 3 A better understanding of number basics will give students a stronger ability to handle algebraic operations and manipulations 4 3. (Booth, 1984, 1986; Gallardo, 2002; Kieran, 1988; Rotman, 1991; Wu, 2001); 4. (Watson, 1990)

13 Integers! Students must have a solid understanding of integers in order to comprehend algebra 5 Beginning algebra students have difficulty conceptualizing and operating with negative numbers 5 Integer misuse is the cause of computational equation-solving errors 6 5. (Gallardo & Rojano, 1988; Gallardo, 2002); 6. (Kieran, 1988)

14 Fractions! Fraction understanding is vital to the transformation from computing arithmetic calculations to comprehending algebra (Wu, 2001)

15 Fractions! Algebra students need a prerequisite understanding of the structure behind: Solving applications The meaning of symbols used in arithmetic The order of operations Basic properties of numbers (especially fractions) (Rotman, 1991)

16 Readiness Indicators! 3. Determine the greatest common factor, least common multiple and prime factorization of numbers (Bottoms, 2003)

17 Fractions! Operating with fractions requires basic number theory knowledge, including greatest common factor, least common multiple, and prime factorization of numbers

18 Readiness Indicators! 4. Write and use ratios, rates and proportions to describe situations and solve problems (Bottoms, 2003)

19 Proportionality! Proportionality is an important contributor to students development of pre-algebraic understanding Useful in algebraic processes: Problem solving Graphing Translating and using tables Other modes of algebraic representation (Post, Behr, & Lesh, 1988)

20 Readiness Indicators! 5. Draw with appropriate tools and classify different types of geometric figures using their properties 6. Measure length with appropriate tools and find perimeter, area, surface area and volume using appropriate units, techniques, formulas and levels of accuracy 7. Understand and use the Pythagorean relationship to solve problems (Bottoms, 2003)

21 Geometry! Geometric concepts (area and perimeter) appeared in research investigating algebraic understanding 7 To investigate knowledge of letter usage, students were asked to find the area of this rectangle: 8 7. (Kieran, 1992; Küchemann, 1978, 1981; Miller & Smith, 1994); 8. (Booth, 1984)

22 Readiness Indicators! 8. Gather, organize, display and interpret data (Bottoms, 2003)

23 Graphing! Graphing causes many problems for algebra students 9 Students of lower ability find it easier to handle functions that are given in graphical form (Brenner et al., 1995, Chazan & Yerushalmy, 2003; Kieran, 1992; Leinhardt, Zaslavsky, & Stein, 1990); 10. (Markovits, Eylon, & Bruckheimer, 1988)

24 Readiness Indicators! 9. Determine the number of ways an event can occur and the associated probabilities (Bottoms, 2003)

25 Readiness Indicators! 10. Write, simplify and solve algebraic equations using substitution, the order of operations, the properties of operations and the properties of equality Finding missing values in number frames Using formulas Using algebraic symbols to write and solve simple equations in one variable in problemsolving contexts Solving some equations and inequalities in two variables (Bottoms, 2003)

26 Properties of Operations! Essential for algebraic manipulation and equation solving, and the cause of difficulties of elementary algebra students: Inverse operations Associativity Commutativity Distributivity Order of operations (Watson, 1990; Booth, 1984)

27 Order of Operations! Prerequisite arithmetic skill for algebra 11 Prealgebra should be used as a time to expand students conceptions of brackets 12 Common misconception among algebra students of all ages (Rotman, 1991); 12. (Linchevski, 1995); 13. (Kieran, 1979, 1988; Pinchback, 1991)

28 Order of Operations! Junior high school students do not see a need for such a rule Saw no ambiguity in solving open-ended expressions, such as Underlying rules actually contradict students natural way of thinking (Kieran, 1979)

29 Order of Operations! Instead, edit identities: 3 5 = 15 becomes = 15 Students would realize need for bracket usage (and order of operations) to keep the equation balanced (Kieran, 1979)

30 Equality! Students, at all levels of education, misunderstand equality 14 Beginning algebra students see the equal sign as a procedural marking, rather than a symbol of equivalence (Kieran, 1981); 15. (Behr, Erlwanger, & Nichols, 1976, 1980)

31 Equality! Elementary students believe the number to the right of an equal sign needs be the answer to the calculation on the left hand side = + 5 Most filled in the box with 12 or 17 (Carpenter, Levi, & Farnsworth, 2000)

32 Equality! Algebraic reasoning requires: Full understanding of equality Appropriate use of the equal sign for expressing generalizations 16 The equal sign needs to be expanded while working with arithmetic equalities prior to the introduction of algebra (Carpenter, Levi, & Farnsworth, 2000); 17. (Kieran, 1981)

33 Symbolism! Mentioned in substantial research addressing algebraic understanding and misconceptions 18 Needs to be more directly addressed within the Readiness Indicators 18. (Behr et al., 1976, 1980; Booth, 1984, 1986; Kieran, 1992; Küchemann, 1981)

34 Symbolism! In arithmetic, the plus sign is also interpreted as an action to be performed 19 Inconsistent with its meaning in algebra (Behr et al., 1976, 1980); 20. (Kieran 1992; Küchemann, 1981)

35 Symbolism! In arithmetic, the plus sign conjoins two terms: 2 + 1/2 = 2 1/2 Yet, in algebra 21 : 2 + x 2x Beginning and intermediate algebra students misinterpret concatenation of numbers and letters (4a) as addition instead of multiplication (Booth, 1986); 22. (Kieran, 1988)

36 Symbolism! Students believe that signs (such as + and -) cannot be left in an algebraic answers (3 + a is acceptable) (2 + 5 is unacceptable) Leads to the misuse of concatenation (3a) (Booth, 1988)

37 Letter Usage! Algebra is where students are first introduced to the usage of letters in mathematics Student have major misconceptions of this new notation 23 and this causes difficulties for many students (Booth, 1984, 1986, 1988; Usiskin, 1988; Watson, 1990); 24. (Küchemann, 1978, 1981; Macgregor & Stacey, 1997; Sleeman, 1984)

38 Letter Usage! Commonly interpret letters as standing for objects or words 25 Associate letters with their positions in the alphabet 26 View letters as representing specific unknown values rather than numbers in general 27 (as in 3 + x = 8, versus x + y = y + x) 25. (Macgregor & Stacey, 1997); 26. (Watson, 1990); 27. (Booth, 1986)

39 Letter Usage! Difficulties in beginning algebra result from the inconsistent usage of letters in arithmetic and algebra (Booth, 1984, 1986, 1988)

40 Letter Usage! Teachers read A = l w as area equals length times width 28 Students claim that the y in 5y + 3 could stand for yachts or yams, but must represent something that starts with y (Booth, 1986); 29. (Booth, 1984)

41 Letter Usage! Letters such as m and c are used as labels for meters and cents, not the number of meters or the number of cents, as they would in algebra 30 Conversions (6m = 600cm) are read, 6 meters are equal to 600 centimeters (Booth, 1988); 31. (Booth, 1986)

42 Letter Usage! Students read algebraic equations based on their arithmetic knowledge 6P = S is read 6 professors are equal to 1 student Intuitively implies there are 6 times as many professors as there are students Algebraically represents the exact opposite (Booth, 1986)

43 Equation Writing! Misunderstanding of letters causes incorrect translation of word sentences into algebraic equations. Most common error in symbolizing there are six times as many students as professors, is a reversal 6S = P (Wollman, 1983)

44 Equation Writing! A key component of students performance in algebra is their inability or negligence to check their answers in a meaningful way Answer checking needs to be taught prior to algebra (Wollman, 1983)

45 Readiness Indicators! 10. Write, simplify and solve algebraic equations using substitution, the order of operations, the properties of operations and the properties of equality (Bottoms, 2003)

46 Readiness Indicators! 11. Represent, analyze, extend and generalize a variety of patterns (Bottoms, 2003)

47 Patterning! Variable introduction should be based upon pattern generalization (Watson, 1990)

48 Readiness Indicators! 12. Understand and represent functions algebraically and graphically Graph functions from tables & patterns Create tables & graphs of linear functions Determine output Understand rates of change, functions created by familiar formulas, & non-linear functions. (Bottoms, 2003)

49 Functions! Functional relationship between two variables is a central concept in prealgebra courses Functions are notoriously difficult for many students to understand (Brenner et al., 1995)

50 Functions! Misconceptions among ninth and tenth grade students: Vocabulary terms: preimage, image, domain, range, and image set Constant functions Functions with disconnected graphs Every function is a linear function (Markovits, Eylon, & Bruckheimer, 1988)

51 Conclusion! Prerequisite algebra knowledge: Numbers (especially integers and fractions) Ratios and proportions Order of operations Equality Algebraic symbolism (including letter usage) Algebraic equations and functions Graphing

52 Conclusion! 8 of the 12 Readiness Indicators were at least partially, if not fully supported: #1, #2, #3, #4, #8, #10, #11, and #12

53 Conclusion! 4 of the 12 Readiness Indicators were not clearly identified by the research as prerequisite knowledge or cause of misconception in algebra #5, #6, #7, and #9

54 Further Study! Results will be triangulated with topics tested by Iowa Algebra Aptitude Test Results will be used to: Develop tests of prospective elementary teachers content and pedagogical content knowledge regarding prerequisite algebra concepts