Prospective middle grade mathematics teachers’ knowledge of algebra for teaching

Journal of Mathematical Behavior 31 (2012) 417–430

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Prospective middle grade mathematics teachers’ knowledge of algebra for teaching Rongjin Huang a,∗ , Gerald Kulm b a b

Middle Tennessee State University, USA Texas A & M University, USA

a r t i c l e

i n f o

Keywords: Knowledge of algebra for teaching Prospective mathematics teachers Mistakes The concept of function

a b s t r a c t This study examined prospective middle grade mathematics teachers’ knowledge of algebra for teaching with a focus on knowledge for teaching the concept of function. 115 prospective teachers from an interdisciplinary program for mathematics and science middle teacher preparation at a large public university in the USA participated in a survey. It was found that the participants had relatively limited knowledge of algebra for teaching. They also revealed weakness in selecting appropriate perspectives of the concept of function and flexibly using representations of quadratic functions. They made numerous mistakes in solving quadratic or irrational equations and in algebraic manipulation and reasoning. The participants’ weakness in connecting algebraic and graphic representations resulted in their failure to solve quadratic inequalities and to judge the number of roots of quadratic functions. Follow-up interview further revealed the participants’ lack of knowledge in solving problems by integrating algebraic and graphic representations. The implications of these findings for mathematics teacher preparation are discussed. © 2012 Elsevier Inc. All rights reserved.

1. Background Equipping teachers with appropriate knowledge needed for teaching is the key to high-quality teaching that aims at achieving high-quality student learning (Conference Board of Mathematics Science [CBMS], 2001; National Mathematics Advisory Panel [NMAP], 2008). Some studies showed that U.S. elementary teachers revealed their weakness in understanding fundamental mathematics knowledge (Ma, 1999) and U.S. pre-service middle school teachers did not receive an adequate preparation in content knowledge and pedagogical content knowledge (Schmidt et al., 2007) when compared with their counterparts in East Asia. It is a consensus that the U.S. teachers need to be equipped with much more knowledge for teaching (Kilpatrick, Swafford, & Findell, 2001; NMAP, 2008; RAND Mathematics Study Panel, 2003). In particular, some specific requirements for teacher preparation programs have been recommended. For example, the Conference Board of Mathematical Science’s (CBMS, 2001) recommendations call for the teaching of mathematics in middle school (grades 5–8) to be conducted by mathematics specialists. These teachers should have at least twenty-one semester hours in mathematics, including at least twelve semester hours of fundamental ideas of mathematics appropriate for middle school teachers. Yet, it is not clear whether prospective teachers who meet these requirements have appropriate knowledge for teaching. Thus, the purpose of this study is to examine the strengths and weakness of a group of teachers who meet the CBMS recommendations through a deep analysis of their knowledge of algebra for teaching with a particular attention to their errors and gaps in understanding.

∗ Corresponding author. Tel.: +1 615 494 7881; fax: +1 615 898 5422. E-mail address: [email protected] (R. Huang). 0732-3123/$ – see front matter © 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmathb.2012.06.001

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2. Theoretical framework In this section, we describe the conception of teachers’ knowledge of algebra for teaching. We then discuss a model of understanding the concept of function, which is the core component of school algebra. Finally, we summarize research findings on mistakes and misconceptions in algebra learning and teaching. 2.1. Teachers’ knowledge of algebra for teaching In the past decades, researchers have focused on conceptualizing and measuring particular mathematical knowledge needed for teaching (MKT) (e.g., Ball, Hill, & Bass, 2005; Hill, 2010; Kilpatrick et al., 2001; Kulm, 2008). Recently, growing attention has been given to defining and measuring teachers’ knowledge needed for teaching specific areas (e.g., Chinnappan & Lawson, 2005; Even, 1993, 1998; Floden & McCrory, 2007). Since algebra is a core component of school mathematics (National Council of Teachers of Mathematics [NCTM], 2000; NMAP, 2008), researchers have proposed different models defining teachers’ knowledge of algebra for teaching (e.g., Doerr, 2004; Even, 1990, 1993; Floden & McCrory, 2007). For example, Even (1990) described seven dimensions of content knowledge including essential features, different representations, alternative ways of approaching, the strength of the concept, basic repertoire, knowledge and understanding of a concept, and knowledge about mathematics. Doerr (2004) specified the pedagogical content knowledge (PCK) including knowledge about students’ errors and misconceptions in algebra. Different from the categorization of content knowledge and PCK, Floden and McCrory (2007) proposed a model defining knowledge of algebra for teaching (denoted by KAT hereafter). According to this model, the knowledge of algebra for teaching includes school mathematics knowledge, advanced mathematics knowledge, and teaching mathematics knowledge. School mathematics knowledge (denoted by SM) refers to the algebra covered in the curriculum from K-12. Advanced mathematics knowledge (denoted by AM) includes calculus and abstract algebra, which is related to school algebra. Teaching mathematics knowledge (denoted by TM) refers to typical errors, canonical uses of school mathematics, and curriculum trajectories and so on. The contents mainly covered two major themes: expressions, equations and inequalities; and functions (linear and nonlinear) and their properties. Although there are different approaches to dealing with algebra (Bendarz, Kieran, & Lee, 1996), these two brands of content are core components in algebra (NCTM, 2000; Drijvers, Goddijn, & Kindt, 2011). Moreover, Floden and McCrory (2007) also developed relevant instruments for measuring KAT. Based on our research purposes, we then developed our own test through extending their instrument. 2.2. Understanding of the concept of function Process-object duality is a generally accepted model of mathematics concept development (Briedenbach, Dubinsky, Hawks, & Nichols, 1992; Schwartz & Yerushalmy, 1992; Sfard, 1991). With regard to the development of the concept of function, according to the process perspective, a function is perceived as linking x and y values: for each value of x, the function has only one corresponding y value. On the other hand, the object perspective regards functions or relations and any of their representation as entities (Moschkovich, Schoenfeld, & Arcavi, 1993). The term “representation refers both to process and product - to the act of capturing a mathematical concept or relationship in some form and to the form itself” (NCTM, 2000, p. 67). Hence, representation is an essential part of the mathematical activity and a vehicle for capturing mathematical concepts (e.g., Goldin & Shteingold, 2001). Different representations play different roles in helping students understand the concept of function (e.g., Lesh, Post, & Beher, 1987; Schwartz & Yerushalmy, 1992). For example, while the algebraic representations can help students understand a function as a process; graphical representations help students understand a function as an object. Thus, it is important to have a sound understanding of these two perspectives (Clement, 1989; Even, 1998) and to use appropriate representations with regard to different contexts (Cuoco, 2001; NCTM, 2000, 2009). 2.3. Difficulties in algebra learning and teaching Students have faced great challenges in learning algebra (Blume & Heckman, 2000; RAND Mathematics Study Panel, 2003). Various mistakes and misconceptions have been identified in learning algebra and functions (Booth, 1984; Drijvers, 2003; Herscovics & Linchevski, 1994; Matz, 1982). With regard to symbols and expressions, students have difficulties in understanding: multiple meanings of symbols (Booth, 1984); differentiations of parameters, variables and unknowns (Furinghetti & Paola, 1994); multiple meanings of equal sign (Alibali, Knuth, Hattikudur, Mcneil, & Stephens, 2007; Booth, 1984; Kieran, 1981, 1989, 1992); transformation of algebra expressions; and solving equations and transforming equations (Demby, 1997; Kirshner, 1987; Matz, 1982). Regarding functions and their properties, students have problems in three areas (Leinhardt, Zaslavsky, & Stein, 1990): a desire for regularity; a pointwise focus, and difficulty with the abstractions of the graphical world. For example, students preferred one-to-one correspondences and had a strong tendency to default to properties of linearity. For another example, students may not comprehend the significance of the slope of a graph as a measure of rate, or may view a graph as a picture (Janvier, 1978; Stein & Leinhardt, 1989). Furthermore, Drijvers et al. (2011) argued that students’ learning difficulties in algebra are due to its abstraction, generalization and overgeneralization, procedural fluency and symbol sense, and duality of process and object. Pre-service teachers have difficulties in: (1) understanding the concept of function as univalent correspondence between two sets (sometimes, it is not presented by a formula, or it is not

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a continuous graph); (2) shifting different representations flexibly; and (3) relating formal function notions to contextual situations which produce the function (Even, 1993, 1998; Hitt, 1998). For example, based on 152 prospective secondary teachers’ completion of an open-ended questionnaire concerning their knowledge about function, Even (1993) found that many prospective secondary teachers did not hold a modern conception of a function as univalent correspondence between two sets. These teachers tended to believe that functions are always represented by equations and their graphs are well behaved. 2.4. Research questions In this study, we examined how prospective teachers from a program that meets the CBMS (2001) requirements responded to questions on a survey that was designed to measure knowledge of algebra for teaching. Focusing on the concept of functions, the study examined the research questions: What are the errors and gaps in teachers’ knowledge and understanding of the algebra found in (a) school mathematics, (b) advanced mathematics, and (c) teaching mathematics? 3. Methodology The data used for this study were taken from a comparative study (Huang, 2010) consisting of a survey of 115 U.S. prospective middle mathematics teachers on knowledge of algebra for teaching and follow-up interviews with five participants. The participants were from an interdisciplinary program for preparing mathematics and science middle school teachers at a large public research I university in the southwestern U.S. 3.1. Instrument The study used an existing test of knowledge of algebra for teaching (KAT) (Floden & McCrory, 2007), supplemented by five open-ended items with a focus on knowledge for teaching the concept of function. Thus, the instrument included 17 multiple-choice items and eight open-ended items (3 items from the original KAT test and 5 additional items). The test included three types of knowledge: school algebra knowledge (SM, 7 items), advanced algebra knowledge (AM, 8 items), and teaching algebra knowledge (TM, 10 items). The following are two example items: Multiple-choice item (School algebra). Which of the following situations can be modeled using an exponential function? i. The height h of a ball t seconds after it is thrown into the air. ii. The population P of a community after t years with an increase of n people annually. iii. The value V of a car after t years if it depreciates d % per year. A. i only; B. ii only; C. iii only; D. i and ii only; E. ii and iii only. Open-ended item (Teaching algebra). On a test a student marked both of the following as non-functions (i) f: R→ R, f(x) = 4, where R is the set of all the real numbers. (ii) g(x) = x if x is a rational number, and g(x) = 0 if x is an irrational number. (a) For each of (i) and (ii) above, decide whether the relation is a function; (b) if you think the student was wrong to mark (i) or (ii) as a non-function, decide what he or she might have been thinking that could cause the mistake(s). Write your answer in the Answer Booklet. 3.2. Supplementary open-ended items Based on an extensive literature review on teachers’ knowledge for teaching the concept of function, we focused on two aspects of the concept of function: selection of function perspectives and use of appropriate representations. Including the open-ended problems from the original test, we have three items used to measure flexibility in selecting the perspectives of the concept functions (18, 24, and 25) and four items (19, 21, 22, and 23) used to measure flexibility in using different representations. For example, item 21(teaching mathematics) is: If you substitute 1 for x in expression ax2 + bx + c (a, b and c are real numbers), you get a positive number, while substituting 6 gives a negative number. How many real solutions does the equation ax2 + bx + c = 0 have? One student gives the following answer: According to the given conditions, we can obtain the following inequalities: a + b + c > 0, and 36a + 6b + c < 0. Since it is impossible to find fixed values of a, b and c based on the previous inequalities, the original question is not solvable. What do you think about the reason for the student’s answers? What are your suggestions to the student?

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3.3. Data collection Three instructors who taught the mathematics education courses for junior and senior students from an interdisciplinary program of middle grade mathematics and science administered the test within a normal class as part of their work (around 45 min). We collected 115 copies of questionnaires. Among the participants, 87% were junior and senior students; the rest were sophomores. Five of the participants voluntarily agreed to an interview. The first author conducted individual interviews during the week after completing the survey. Each interview lasted about 20 min and was audio recorded. The five interviewees (Larry, Jenny, Kerri, Alisa, and Stacy are pseudonym names of the interviewees) had completed five mathematics courses of Algebra I and II, Geometry, Pre-calculus, and Calculus in high school, except Kerri who only took four of the five courses. They also had completed an average of 27 credit hours of college mathematics and mathematics education courses including Structure of mathematics I and II, Basic concept of geometry, Introduction to abstract mathematics, Integration of mathematics and technology, Problem solving in mathematics, Integrated mathematics, Mathematics methods in middle, Student teaching, Freshman mathematics laboratory, Analytic geometry and calculus, Calculus I, II and III, Foundation of discrete mathematics, Multiple variable calculus, Linear algebra I, and Differential equations. 3.4. Data analysis The data analysis consisted of three phases: (1) quantifying the data; (2) analyzing the items and subscales of the KAT test; and (3) analyzing typical errors in adopting appropriate perspectives of the concept of function, and translations between representations. For each multiple-choice item (items 1–17), the correct choice was scored as 1, while the wrong choice was scored as 0. For each open-ended item, we developed a five-point rubric for scoring the answers: 0 point refers to blank or providing useless statements; 1 point means providing several useful statements without a chain of reasons for the correct answers; 2 points refers to giving a correct answer but the explanations or procedures with major conceptual mistakes; 3 points means giving a correct answer and appropriate explanations or procedures, with some minor mistakes; and 4 points presents giving a correct answer and appropriate explanations and procedures. A secondary mathematics teacher coded all items. The first author double-checked the codes. The agreement was higher than 95%, and the first author made relevant corrections. First, we analyzed overall performance at item and subscale levels. Then, we analyzed and categorized the typical mistakes in terms of the nature of the mistakes. The main mistakes include: (1) inappropriate transformation of functions and equations; (2) inappropriate use of graphic representations, (3) inappropriately logical reasoning in algebra; and (4) inflexibility in selection appropriate function perspectives and in translation between different representations. 4. Results 4.1. Characteristics of knowledge of algebra for teaching There are 7 items (1, 3, 6, 14, 17, 19, and 23) in school mathematics, 8 items (4, 8, 9, 12, 13, 16, 20, and 24) in advanced mathematics, and 10 items (2, 4, 7, 10, 11, 15, 18, 21, 22, and 25) in teaching mathematics. The reliability of the instrument was 0.70 (N = 115). The means and standard deviation (SD) of the multiple-choice items with a brief description of the contents are listed in Table 1. The table shows that the participants performed best on items 1, 3, 10, and 11 (higher than 60%), while they performed poorly on items 4, 5, 8, 9, 12, 13, 14, 15, and 16 (less than 40%). Examining the content of the items, we found that the high performing items related to using algebraic expressions to present quantitative relationships (item 1, 85%), finding the value of a polynomial expression (item 3, 89%), the condition of two lines perpendicular (item 10, 66%), and multiple representations of the concept of slope (item 11, 78%). The participants performed worst in the following areas: transformation of a logarithmic function (item 4, 15%), solving an exponential equation (item 5, 18%), using a geometric representation of a fraction and algebraic formula (item 6, 30%), judging the position of a car based on a time-speed graph (item 8, 16%), the number of roots of tan x = x2 (item 9, 20%), operation rules in different number systems (item 12, 27%), mathematical induction (item 13, 23%), solution of an irrational equation (item 14, 40%), multiple ways to expand an algebraic expression formula (x + y + z)2 (item15, 22%), and slopes of tangent lines of a curve (item 16, 29%). The means of open-ended problems items 18–25 (4 points of each), and subscales of SM (13 points), AM (14 points) and TM (22 points) are displayed in Table 2. The table shows that the participants performed poorly on all of these items, but slightly better on item 18 (38%) and item 22 (35%) and worst on items 21, 23, and 24 (less than 1%). The participants performed better on School Mathematics (36%) and Teaching Mathematics (34%) than on Advanced Mathematics (15%). In sum, with respect to specific content areas, the participants performed better in using algebraic expressions to present quantitative relationship, evaluating a polynomial expression, and multiple representations of the concept of slope, but they performed worst in solving irrational functions and equations (such as logarithm, exponential equations, and trigonometry), and presenting fraction and algebraic formula using geometrical representation and/or reasoning based

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Table 1 Means and Standard Deviation (SD) of multiple-choice items with theme description. Item

Theme

1

Express quantitative relationships in word problems using algebraic expressions Solve quadratic equations 2x2 = 6x (losing roots) Given a quadratic function f(x), find f(x + a) Transform f(x) = log2 x2 Solve equation: 9x − 3x − 6 = 0 using substitution method (adding roots) Represent fractions, percents, and algebraic expressions such as 3/5, 60%, and a(b + c) = ab + ac using the area of rectangle Given two points, find the functions whose graphs passing these two points Given a graph representing speed vs. time for two cars, judge the position of the two cars Judge the number of root of equation: tan x = x2 Judge perpendicular relationship of two lines by using their slopes Multiple ways to introduce the concept of slope of line Judge the proposition “For all a and b in S, if ab = 0, then either a = 0 or b = 0” in different number systems Meaning of mathematical induction √ √ Roots of irrational equation x − 2 = 1 − x (adding roots) Expand algebra expressions by area relationship Find derivative and slope of a function Find value of a composition function

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Score Mean

SD

.85

.36

.57 .89 .15 .17 .30

.50 .32 .36 .38 .46

.57 .16

.50 .37

.20 .66 .78 .27

.40 .48 .41 .45

.23 .40 .22 .29 .52

.43 .49 .41 .45 .50

on pictorial representation information, operation rules in different number systems, and mathematical induction. The participants revealed weakness in understanding the concept of function and selecting appropriate representations to solve quadratic equation problems. In the following section, we present the typical mistakes in the knowledge of algebra for teaching (1) expressions, equations, and graphs; and (2) functions and their properties. 4.2. Typical mistakes in expressions, equations, and graphs Some typical mistakes were identified as follows: (1) inappropriate transformation of functions and equations; (2) inappropriate use of graphic representations, and (3) inappropriate logical reasoning in algebra. 4.2.1. Inappropriate transformation of functions and equations It is a common error to neglect the changes of domain and range of variables when transforming equations and functions. For example, on item 4, participants were required to find the equivalent expressions of function f(x) = log2 x2 from: (i) y = 2 log2 x; (ii) y = 2 log2 |x|; (iii) y = 2|log2 x|. The percents of selecting A (i only), B (ii only), C (iii only), D (i and ii only), and E (i, ii, and iii), were 37%, 15%, 6%, 21%, and 0% respectively. This indicates that 37% of them did not consider the domain of independent variable x when transforming a function and 21% of them did not differentiate these four expressions. For another example, participants were asked to comment on the following process of solving the equation: 9x − 3x − 6 = 0: Table 2 Means of open-ended items and subscales. Item

Theme

18 19 20

Definition of functions and students’ misconceptions Solve quadratic inequalities using two methods (algebra and graphic methods) Judge and explain if AB = O (A and B are matrixes, O stands for zero matrix), then either A = O or B = O? Judge the number of roots of a quadratic function with certain constraints using graphic methods. Judge the effects of changing parameters of quadratic functions on its graph translations Given three specific points, find the maximum of a quadratic function whose graphs passing these points Given f(x) and g(x) intersect at a point P on the x-axis, prove the graph of their sum function (f + g) (x) must also go through P Given a figure, find a daily situation that corresponds to the figure. School mathematics Advanced mathematics Teaching mathematics Knowledge of algebra for teaching

Score Mean

21 22 23 24 25 SM AM TM KAT

SD

1.51 .76 .79

1.33 .59 1.25

.18

.51

1.40

1.34

.29

.74

0.02

.13

1.43 4.00 2.10 7.50 13.60

1.21 1.57 1.70 3.35 5.15

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Car A Speed

Car B Time

1 Hour

Fig. 1. A graph of speed over time with two cars.

“Peter denoted y = 3x and gets the equation y2 − y − 6 = 0, which has 2 different roots. He concluded that the given equation also has 2 different roots”. 15% of them agreed with Peter’s solution, and 54% of them believed the substitution method is wrong; only 18% of them realized the mistake in neglecting the range of√y when substituting. √ Again, to find the number of real solutions of the following equation x − 2 = 1 − x, it is crucial to realize the change of domain of the unknown when transforming the irrational equation. While 41% of the participants found the correct answer (no solution), 38% of them neglected the domain of x when transforming the equation (one solution). 4.2.2. Inappropriate use of graphic representations The participants were not able to appropriately use geometrical representations to present algorithms, algebra relationships, the relationship between time and velocity, and roots of trigonometric functions. For example, in responding to the following item, only about one-third of the participants realized that all of these relationships could be presented by the area of rectangles. Which of the following can be represented by the area of rectangles? (1) The equivalence of fractions and percents, e.g., (3/5) = 60%; (2) The distributive property of multiplication over addition: for all real numbers a, b, and c, we have a (b + c) = ab + ac; (3) The expansion of the square of a binomial: (a + b)2 = a2 + 2ab + b2 . In particular, the participants did not realize that the distributive rule of multiplication over addition (26%), the equivalence of fractions and percents (22%), and the formula of square of a binomial (16%) could be presented by the area of rectangles. The participants had difficulties making judgments based on geometrical representations. In the following item, the given graph (Fig. 1) represents a relationship between speed and time for two cars. (Item 8: Assume the cars start from the same position and are traveling in the same direction. Use this information and the graph below to answer. What is the relationship between the position of car A and car B at t = 1 h?) The participants were required to choose from the following: (A) the cars are at the same position; (B) car A is ahead of car B; (C) car B is passing car A; (D) car A and car B are colliding; and (E) the cars are at the same position and car B is passing car A. Only 16% of them made a correct choice (B) and 48% of them selected (E). Participants needed to understand the graph and analyze it analytically in order to make a correct choice. About half of them (48%) made a wrong choice simply based on visual information only: the cars are at the same position and car is passing car A (Choice E) or partially based on the visual information: at the same point (Choice A) or car A is ahead of car B (Choice B). This result implied that around a half of the participants used visual judgment rather than logical reasoning. In another example, participants were asked to judge how many solutions exist for the equation tan x = x2 . Only 22% of them made a correct choice (infinite number of solutions). It is necessary to use the graphical representation and the properties of the tangent function (specially, the periodic property) in order to find the correct answer. These results showed that the majority of the participants were not able to appropriately use graphical representations to represent relevant concepts and solve problems. 4.2.3. Inappropriate logical reasoning in algebra We identified some essential mistakes when making logical reasoning in algebra. These mistakes include overgeneralizing a property to different number systems and inappropriately using logical relationships. One question was to judge whether the statement “For all a and b in S, if ab = 0, then either a = 0 or b = 0” is true in different number systems S such as (i) real numbers, (ii) complex numbers, and (iii) a set of 2 × 2 matrices with real number entries. 44% of the participants selected only (i), 22% selected only (i) and (ii) (correct answer), and 30% selected (i), (ii) and (iii), indicating that 44% of the participants were not able to generalize the rule to the complex number system, while 30% of them inappropriately overgeneralized the rule to matrix system. Taking an open-ended item (item 20) related to matrix operation for another example. The item is:

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Table 3 Score distribution of items related to the adaption of function perspectives. Items

0 point (%)

1 point (%)

2 points (%)

3 points (%)

4 points (%)

18 24 25

32.2 98.3 31.3

17.4 1.7 17.4

27.8 0.0 33.0

12.2 0.0 13.0

10.4 0.0 5.2













p q t u pt qu and B = , Then AB is defined to be . Is it true that if AB = O, then either A = O or B = O r s r v sw v w (where O represents the zero matrix)? Justify your answer and show your work in the Answer Booklet. 57% of the participants got 0 points (giving up) and 28% of them got 1 point (giving a relevant statement); only 11% of them gave a correct answer and explanation. A common misconception was to use incorrect logical reasoning as follows: p → q ⇔ q → p, namely, using the following logic “if A = 0, then AB = 0 or if B = 0, then AB = 0” to prove: “if AB = 0, then A = 0, or B = 0.” More than one-fourth of them made this mistake. In addition, a few of them inappropriately generalized the same proposition from real number system, namely given xy = 0, then, x = 0, y = 0 (x, y are real numbers) to matrix system. The interviews further confirmed participants’ difficulties in providing a correct proof. Two of the five interviewees (Jenny and Stacy) gave the correct answer with an appropriate counterexample. For example, Stacy explained why she tried to disprove the statement as follows: “if someone wants to prove a proposition, s/he has to provide the whole process of proving. However, if someone just wants to disprove a proposition, s/he only provides a counterexample, so, I considered to find a counterexample”. The others gave a wrong judgment by providing examples such as “A = 0, then AB = 0” or “B = 0, then AB = 0” by “guess and check.” Let A =

4.3. Mistakes in the concept of function and quadratic functions The results are organized into two parts. One is related to the selection of perspectives of the concept of function; the other is about the flexible use of representations. 4.3.1. Selecting perspectives of the concept of function Items 18, 24 and 25 were particularly used to measure knowledge of understanding and applying the concept of function from different perspectives (process and object). Item 18 uses a process perspective; item 24 is easily proved if an object perspective is adopted. It is necessary to connect those two perspectives when solving item 25. The score distribution of these three items is shown in Table 3. Inappropriate selection of a perspective of the concept of function. Answering item 18 requires a sound understanding of the definition of function. About 23% of the participants got a correct answer or almost correct answer with minor mistakes (12%). 32% of them provided nothing or meaningless information about the solution, and about 28% of them gave correct answers without any interpretation or gave one correct answer and relevant explanations. It is more appropriate to adopt the process perspective. Yet, the participants preferred using an object perspective (9%), namely, basing their thinking on features of the expression (constant value; two expressions) and graphic features (one line, many holes/un-continuous), or using essentially corresponding relationship between domain and range (i.e., one-to-one; multiple-to-one) (6%). The following is a typical diagram used to visualize the function relationship and then make a judgment of the two relations given as shown below (Fig. 2). In interviews, three of the five (Larry, Alisa, and Stacy) said they used the vertical line test. Since Jenny had difficulty in drawing the graph of the second relation, she believed it was not a function. However, when asked whether she had heard of the vertical line test, she clearly stated that “one x value can only have one corresponding y value; one x value cannot be corresponded to two y-values.” Kerri said she “is a visual learner, and likes using diagrams to represent the relationship between two sets (one-to-one or multiple-to-one, but not one-to-multiple).” Larry not only explained the vertical line test rule, but also showed an example (x = y2 ), which cannot pass the vertical line test. Alisa and Stacy explained the rule

Fig. 2. Judgment of function relationship using diagrams.

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Fig. 3. A graph presenting a daily life situation.

by emphasizing, “each input [value] should have only one [corresponding] value, but that does not mean that different [input] values cannot have the same [corresponding] value.” They also attributed students’ mistakes to students’ superficial understanding of the vertical line test rule (missing multiple x values correspond to one y-value) or the confusion with “many holes,” or the repeating output values. Inflexible use of the two perspectives of the concept of function. On item 25, participants were required to find daily life situations, which correspond to the given graph. It is necessary for participants to have a full understanding of two perspectives of functions. The item is: When introducing the functions and the graphs in a class of middle school (14-15 years-old), tasks that consist of drawing graphs based on a set of pairs of numbers contextualized in a situation or from an equation were used. One day, when starting the class, the following graph (Fig. 3) was drawn on the blackboard and the pupils were asked to find a situation to which it might possibly correspond. One student answered: ‘it may be the path of an excursion during which we had to climb up a hillside, the walk along a flat stretch and then climb down a slope and finally go across another flat stretch before finishing.’ How could you answer this student’s comments? What do you think may be the cause of this comment? Can you give any other explanations of this graph? 18% of participants gave a roughly correct answer and interpretation (3 or 4 scores), 33% of the participants either gave a correct explanation or an appropriate situation, and 31% gave useless information. Sixteen participants (14%) pointed out that the student’s interpretation could be improved by denoting the x-axis as time and the y-axis as height above sea level. About one-third of the participants (31%) gave the situation of speed over time to illustrate the same diagram. The following is an example of participants’ explanation: “The graph could be showing speed vs. time where a car is accelerating at an exponential rate, then goes a steady rate for period of time, and then slows at a constant rate, then stops.” There are other two explanations of the diagram: a graph of height and time (4%), and a graph of distance and time (4%). Three of the five interviewees (Larry, Alisa, and Stacy) realized that the original interpretation should be improved by pointing out the x-axis represents time while y-axis represents position (or height). All of them gave other examples of describing the diagram as a graph of speed over time, or a graph of temperature over time, or a graph of distance over time. Thus, about two-thirds of the participants could explain the given scenario and give another situation corresponding to the graph. 4.3.2. Inflexibility of selecting representations The items 19, 21, 22, and 23 were designed to measure understanding and applying quadratic functions/equations/inequalities through flexibly using multiple representations. It is crucial for participants to flexibly use appropriate presentations and shift between different representations in order to effectively explain and solve these problems. Regarding item 19, participants are expected to have algebraic and graphic representations of equation and inequality. With regard to item 21, it is necessary to shift between algebraic and graphic representations in order to solve the problem. To solve the problem of item 22, it is necessary to have ability in translating graphic representations to algebraic representations. To solve the problem of item 23, it is required to use appropriate forms of algebraic expressions and transformations of different algebraic expressions, and translation between graphic and algebraic representations. The scores distribution of these four items is shown in Table 4.

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Table 4 Score distribution of items related to flexible use of representations. Items

0 point (%)

1 point (%)

2 points (%)

3 points (%)

4 points (%)

19 21 22 23

31.3 84.3 34.8 80.0

62.6 14.8 24.3 16.5

5.2 0.9 15.7 0.9

0.9 0.0 16.5 0.0

0.0 0.0 8.7 2.6

The table shows that overall, the participants had limited knowledge and skills in explaining and solving these problems. Further qualitative analysis revealed several typical mistakes. These mistakes include performing algebraic operations by guessing rather than following operation properties; sticking to one of algebraic and graphic representations without flexibility in translation; and sticking to the standard formula regardless specific contexts Performing algebraic operations by guessing and checking. The question (item 19) is: Solve the inequality (x–3)(x + 4) > 0 in two essentially different ways. Only one participant gave two essentially different solutions. About 60% of them gave some algebraic statements, which did not result in a correct answer, and about one-third left it blank or gave some meaningless statements. Many mistakes and misconceptions occurred as shown in Table 5. The table shows that 37% of the participant adopted the inference: if ab > 0, then a > 0, b > 0. None of them realized that a and b are possibly negative. In addition, none of them noted the logical operations “or” or “and” between two logic propositions (such as a > 0 and b > 0 or a > 0 or b > 0). They also were satisfied with the solution “x > 3, x > −4” without any intention to further intersect or combine. In order to find another method of solving the inequality, an automatic alternative is to transform the factor form into standard form: x2 + x − 12 > 0. 21% of them stopped with the standard form. 7.6% of them were stuck with further algebraic operation: x(x + 1) > 12 or x(x + 1) = 12. Some of the participants went further with “guess and check strategies”: Mistake 1 (12%): x2 + x − 12 > 0 → x(x + 1) > 12; →x > 12, x + 1 > 12 → x > 12, x > 11. Mistake 2 (15%): x2 + x − 12 > 0, or (x − 3)(x + 4) > 0 →√x1 = 3, x2 = −4. Mistake 3 (2.5%): x2 + x > 12 → x2 + x − 12 → x > x − 12 In addition, there are some other mistakes as follows: “(x − 3)(x + 4) > 0 → x − 3 > 0, x + 4 > 0,√then x > 3 and x > −4:−4 < x < 3;” “x2 + x > 12 → x2 + 12, x > 12 → x > 12;” 2 In interviews, Larry just simplified the factor form into √ standard form (x + x − 12 > 0), and then moved forward 2 by “guess and check” such as “x > x − 12, and then x > x − 12.” although it did not work. By analogizing the property of equation: (x − 3)(x + 4) = 0 → x − 3 = 0, or x + 4 = 0, the remaining four interviewees made an inference as follows: (x − 3)(x + 4) > 0 → x − 3 > 0, (x + 4) > 0 → x − −4. None of them had the intention to work on “x > 3, x > −4” further, such as the logical operations “and” or “or” and the operations of intersection and combination of two sets. They seemed to be satisfied with the “solution”. When asked if they can use a graphic method to solve an equation or inequality, they recalled the graphs of quadratic equation. All of the interviewees explained they did not know how to use a quadratic function graph to solve an inequality, although they knew the graphing method of solving a system of linear equations. They said that they learned quadratic functions first (probably early at high school) and then inequality late (at high school). These topics were taught separately. The participants were not taught how to use graphic representations to solve algebraic equations and inequalities. They appreciated the method of integration of algebraic and graphic representations.

Table 5 Misconceptions or mistakes in solving inequality in Item 19. Mistake

Explanation

Example

Frequency (%)

1 2 3 4

Misconception: if ab > 0, then a > 0, b > 0 Transforming into standard form Transforming into standard form and getting stuck Working on the standard form with guess and check

5 6

Drawing a number line Using a table

(x − 3)(x + 4) > 0 → x − 3 > 0, x + 4 > 0, then x > 3, x > −4. x2 + x − 12 > 0 or x2 + x > 12. x2 + x − 12 > 0 or x(x + 1) > 12 or x(x + 1) = 12. x2 + x – 12 > 0 → x(x + 1) > 12; →x > 12, x + 1 > 12→x > 12, x > 11. x2 + x – 12 > 0, or (x − 3)(x + 4) > 0 →x1 = 3, x2 = −4. Find partial answer: x > 3 or x < −4. x > 3 (x = 1, 2, 3, 4. or 0, −1, −2, −3,. . .).

37.0 21.0 7.6 12.0 15.0 4.0 4.0

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Fig. 4. Translations of quadratic functions.

Sticking to algebraic representations without awareness of using other representations. On item 21, only one out of the 115 participants gave a correct explanation and useful suggestions as follows: 1. The student believes that he needs to know the values of a, b, c. He cannot find these values because there are three variables [unknowns] in two equations. 2. The student needs to think graphically. Since at x = 1, the value is positive, then the graph is above the x-axis. The opposite is true when x = 6. Therefore, the graph has to cross the x-axis. Since it has degree two, it must have 2 solutions. 84% of the participants agreed with the student’s explanation (actually, it is wrong), and they got stuck with the algebraic manipulation to find a, b and c, and had no idea about how to help the student get out of their difficulties. 15% of the participants even suggested plugging more values of a, b, and c (such as a = −10, b = −9, and c = 20) to see whether certain patterns can be found. They did not think to use graphic or geometrical representations to find the answers. Some of them had the following misconception: if you plug more numbers, you can find some patterns, and then you may find the solutions (15%). Three of the interviewees fully agreed with the student’s (wrong) statement, namely “Since it is impossible to find out fixed values of a, b and c based on the previously given inequalities, the original question is not solvable.” They tried to find a, b, and c through algebraic transformation but it did not work. They had no ideas how to help the student find a solution. The other two interviewees (Jenny and Stacy) believed that the problem may be solved, but they did not have any concrete ideas about how to solve it. What they could suggest is that the student is to “try different ways, such as plugging more numbers between 1 and 6.” (Jenny) or “explore in different ways such as plugging more numbers to see whether they can find certain patterns, rather than being stuck.” (Stacy) When asked whether they can try other methods such as graphical methods to solve, they tried to sketch the graphs and find the possible roots. Four of them succeeded in finding the number of roots by examining the intersection points of the quadratic function. All of them said they did not know the graphic method in solving quadratic equation or inequality; they did not have such an experience in solving problems, but they appreciated the graphing method in algebra. Inappropriate translations between algebraic and graphic representations. Item 22 is presented as follows: Mr. Seng’s algebra class is studying the graph of y = ax2 + bx + c and how changing the parameters a, b, and c will cause different translations of the original graph (see Fig. 4). Which of the following is an appropriate explanation of the translation of the original graph y = ax2 + bx + c to the translated graph? A. Only the a value changed. B. Only the c value changed. C. Only the b value changed. D. At least two of the parameters changed. E. You cannot generate the translated graph by changing any of the parameters. Explain your answer choice as much detailed as possible.

About 10% of the participants gave correct answers and appropriate explanations and 17% of them gave correct answers but failed to explain. 15% of the participants gave partially correct answers and explanations. 23% of them gave sporadic information about the effect of a, b and c changes. About 35% of them got lost, either leaving it blank or providing some wrong statements. In summary, one-third of the participants had no ideas about solving and explaining this problem while about one-fourth of them gave a roughly correct answer. With regard to the strategies used to explain, it was found that 11% of the participants used the following reason: “The effects of changes of a, b and c on the changes of graphs of quadratic functions,” and 2% of the participants used the other explanation that “because symmetrical line is x = −(b/2a), and a is invariant, then b can be

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changed only.” Typical mistakes are related to the effects of changing a or b. For example, some participants said, “changing c results in changes of left or right,” (4%) while some of them, said, “changing b results in changes of left or right.” (3%) Two interviewees (Larry, and Kerri) correctly and clearly explained the effects of changing a, b, and c on the graphs of quadratic function. The other two were able to explain the effect of changing a and c on the graph of quadratic function, but they were not sure about the effect of changing b. One of them found the correct answer by explaining that changing b to a negative b, would reflect the graph of quadratic function over y-axis although she could not remember the details of the effect of changing a, b and c on the graph. Sticking to the standard formula without awareness of using specific formulas. There are multiple ways to solve the following problem (item 23). Given quadratic function y = ax2 + bx + c intersects x-axis at (−1, 0) and (3, 0), and its y-intercept is 6. Find the maximum of the quadratic function. Show your work as much detailed as possible in the Answer Booklet. Three out of the 115 participants found the quadratic equation by solving a system of linear equations and found the maximum correctly. About 20% of the participants just drew a graph or list equations based on the given three points. Four-fifths of them left the item blank or wrote something unrelated to solving this problem. 3% of the participants used the standard quadratic formula method: y = ax2 + bx + c to find the expression of the quadratic function, and then, transforming into the form y = a(x − h)2 + k to find the maximum. One typical mistake is regarding the y-intercept as maximum (16% of participants). Two of the interviewees knew the process of solving the problem: finding the expression of quadratic equations by plugging given points, and then finding the maximum by taking derivative. Another realized that the maximum should be at x = 1 due to its symmetrical property although they were not able to find the correct expression of quadratic equations. Other two held a misconception that the y-intercept is the maximum. They had difficulties finding the expression by plugging in the given points. They had no ideas about other formula of quadratic function even when the interviewer showed to them. In summary, the interviewees tried to find expressions of quadratic equations by using the standard formula, and then found the maximum by taking derivative and plugging x = 1. However, they had difficulty finding the correct expressions due to the complexity of algebraic manipulation. In addition, nobody was aware of using other appropriate formulas to find the expressions. In fact, three formulas can be used to find the quadratic function: (1) y = ax2 + bx + c; (2) y = a(x − x1 )(x − x2 ); and (3) y = a(x − h)2 + k. And, three methods could be used to find the maximum of the function: (1) transforming into y = a(x − h)2 + k, then finding the maximum; (2) using formula x = −(b/2a), ymaximum = (4ac − b2 )/4a; and (3) taking derivative: when (dy/dx) = 0, then x = 1, ymaximum = f(1). 5. Discussion and conclusion 5.1. Discussion This study showed that the participants performed poorly in all three areas of knowledge of algebra teaching: school mathematics, teaching mathematics and advanced mathematics, with less than an overall 40 percent mean. The majority of the items in the category of teaching mathematics rely heavily on a relational or conceptual understanding of relevant mathematics concepts (Skemp, 1976; Hiebert & Lefevre, 1986). Thus, we could conclude that the participants were weak in the mathematics content knowledge they need for teaching. The participants had relatively limited knowledge of algebra for teaching, revealing their weakness in selecting appropriate perspectives of the concept of function and flexibly using of representations of quadratic functions. Meanwhile, they made numerous mistakes in solving quadratic/irrational equations, algebraic manipulation and algebraic reasoning. Moreover, the isolation of algebraic and graphic representations resulted in their failure to solve quadratic inequalities and judge the number of roots of quadratic functions. Follow-up interviews further disclosed participants’ lack of experience in solving problems by integrating of algebraic and graphic representations. These findings suggest that the participants neither had a relational understanding of algebra content knowledge (with connections and multiple representations), nor had an instrumental fluency of using the algebra knowledge (algebraic manipulation). The weakness in selecting appropriate perspectives of the concept of function and flexibly using representations is in line with other studies (Even, 1993, 1998; Hitt, 1998). Specifically, the mistakes identified are serious and striking. Some of them such as judging car position based on visual image in a speed and time graph (Clement, 1989), and viewing a graph as a picture (Janvier, 1978; Monk, 1992) have been recognized in other studies. Other mistakes draw attention to improving teaching in algebra. For example, the participants seemed to suffer from a lack of fluency in algebraic manipulations and transformation of different algebraic formulae when solving quadratic equations and inequalities or finding quadratic expressions. This may reflect the learning difficulty in algebraic procedural fluency and symbol sense (Drijvers et al., 2011). The participants’ difficulty in making appropriate algebraic reasoning echoes the call for attention to actors-oriented transfers in algebra generalization (Ellis & Grinstead, 2008). Moreover, the participants were not able to flexibly use algebraic and graphical representations to solve quadratic equations, and irrational equations. They also demonstrated weakness in

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using geometrical representations to present algebraic and arithmetic relations. These results make clear how difficult, if not impossible, it is for these prospective teachers help their future students develop representational flexibility. With regard to using graphic representations to visually judge the number of root of quadratic functions, Even (1998) found that less than one-fifth of U.S. pre-service secondary mathematics teachers successfully solved this problem. In the current study, only one of the participants gave a correct answer. On another item on the impact of changing parameters on the graphs of quadratic functions, Black (2007) found that one-fifth of sampled U.S. high school mathematics teachers gave correct answers and relevant explanations. However, one quarter of the participants in this study gave correct answers and explanations for the item. These results indicate that even though the participants in this study (who were preparing to be middle school mathematics and science teachers) performed poorly in general, their performance in quadratic equations is similar to pre-service and in-service high school mathematics teachers. This finding suggests that the mistakes or difficulties identified in this study are not unique for this group of prospective teachers. We can see that the content of the mathematics items in the study is covered in high school algebra and entry-level calculus. The participants probably took relevant advanced courses (they averaged 7 courses in mathematics and mathematics education) but they did not demonstrate the facility to apply knowledge and skills in the survey. As far as algebra teaching and learning is concerned, the CBMS (2001) recommended developing a deep understanding of variables and functions as follows: (1) relate tabular, symbolic, and graphical representations to functions; (2) relate proportional reasoning to linear functions; (3) recognize change patterns associated with linear, quadratic, and exponential functions and their inverses; and (4) draw and use “qualitative graphs” to explore the meanings of graphs of functions. Meanwhile, students need to demonstrate the following skills: (1) represent physical situations symbolically; (2) graph linear, quadratic, exponential functions and their inverses and understand physical situations; (3) solve linear and quadratic equations and inequalities; and (4) exhibit fluency in working with symbols (pp. 108–109). Does it matter if prospective middle school mathematics teachers do not understand and master these contents? Does it matter if they had the difficulties in understanding students’ learning difficulties in these areas? It may be possible for them to teach pre-algebra, but it may be hard for them to prepare their students ready for continued learning in algebra I & II because they may not able to help their students to lay a proper foundation for learning advanced algebraic topics, and develop relevant algebraic reasoning and thinking appropriately. 5.2. Implications The findings of this study provide several implications for improving mathematics teacher preparation. At the teacher preparation curriculum level, it is important to provide an appropriate foundation for prospective teachers to obtain a sound and well-structured knowledge base needed for teaching. This study shows it is not a simple issue of adding more mathematics and mathematics education courses. The curriculum should provide a core of coherent content areas and present them with the trajectory of knowledge development by integrating content and pedagogy. At the teacher preparation pedagogical level, it is also important to provide prospective teachers with relevant learning experiences in developing interconnected knowledge. In order to implement school mathematics curricula that emphasize developing connections and representational flexibility, prospective teachers should have the opportunity to develop their own knowledge and skills needed for organizing classroom activities that promote this kind of students’ learning (NCTM, 2000, 2009). The finding that the participants lacked the learning experience in integrating graphic and algebraic representations to solve problems related to quadratic functions highlights the necessity of creating this kind of learning opportunity in curriculum and teaching. The numerous deficiencies identified in this study provide practical resources and references for developing teacher preparation curriculum and implementing particular lessons. In particular, it is crucial to develop prospective teachers’ deep understanding of the concept of function. The study confirmed the finding that prospective mathematics teachers do not hold a modern notion of function as univalent corresponding relationship between two sets (Even, 1993). Recently, researchers have suggested the ways to develop a deep understanding of the concept of function through developing quantitative reasoning and covariational reasoning (Carlson, Jacobs, Coe, Larsen, & Hsu, 2002; Thompson, 2011). With regard to developing flexibility in translation of representations, appropriate use of technology may help make a difference (Drijvers, Boon, & Reeuwijk, 2011; Hoyles et al., 2009). 5.3. Conclusion All in all, the findings imply that beyond providing an adequate number of mathematics and mathematics education courses, it is crucial to provide focused and coherent curricula in high school and teacher preparation programs. Meanwhile, prospective mathematics teachers need to have the learning experiences necessary for building a deep understanding of the concept of function and developing representational flexibility. 5.4. Limitations This study has its own limitations. First, the instrument mainly focused on expressions, equations, inequality, and functions, which may not reflect an entire picture of algebra. What we found may be only pertinent for these content areas.

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Second, the participants were from a program in a single university, which at most presents one of three routes of preparing middle grade mathematics teachers in the U.S. (Dossey, Halvorsen, & McCrone, 2008). We should be cautious interpreting the findings due to this limitation. Nevertheless, these detailed and vivid descriptions and explanations will shed insight into development of mathematics teacher preparation programs.

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