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Journal of Mathematical Behavior 31 (2012) 417– 430

Contents lists available at SciVerse ScienceDirect

The Journal of Mathematical Behavior

journa l h o me pag e: ww w.elsev ier .com/ locate / jmathb

rospective middle grade mathematics teachers’ knowledge of algebraor teaching

ongjin Huanga,∗, Gerald Kulmb

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

r t i c l e i n f o

eywords:nowledge of algebra for teachingrospective mathematics teachersistakes

he concept of function

a b s t r a c t

This study examined prospective middle grade mathematics teachers’ knowledge of algebrafor teaching with a focus on knowledge for teaching the concept of function. 115 prospec-tive teachers from an interdisciplinary program for mathematics and science middle teacherpreparation at a large public university in the USA participated in a survey. It was foundthat the participants had relatively limited knowledge of algebra for teaching. They alsorevealed weakness in selecting appropriate perspectives of the concept of function andflexibly using representations of quadratic functions. They made numerous mistakes insolving quadratic or irrational equations and in algebraic manipulation and reasoning. Theparticipants’ weakness in connecting algebraic and graphic representations resulted in theirfailure to solve quadratic inequalities and to judge the number of roots of quadratic func-tions. Follow-up interview further revealed the participants’ lack of knowledge in solvingproblems by integrating algebraic and graphic representations. The implications of thesefindings for mathematics teacher preparation are discussed.

© 2012 Elsevier Inc. All rights reserved.

. Background

Equipping teachers with appropriate knowledge needed for teaching is the key to high-quality teaching that aims atchieving high-quality student learning (Conference Board of Mathematics Science [CBMS], 2001; National Mathematicsdvisory Panel [NMAP], 2008). Some studies showed that U.S. elementary teachers revealed their weakness in understanding

undamental mathematics knowledge (Ma, 1999) and U.S. pre-service middle school teachers did not receive an adequatereparation in content knowledge and pedagogical content knowledge (Schmidt et al., 2007) when compared with theirounterparts in East Asia. It is a consensus that the U.S. teachers need to be equipped with much more knowledge foreaching (Kilpatrick, Swafford, & Findell, 2001; NMAP, 2008; RAND Mathematics Study Panel, 2003). In particular, somepecific requirements for teacher preparation programs have been recommended. For example, the Conference Board ofathematical Science’s (CBMS, 2001) recommendations call for the teaching of mathematics in middle school (grades 5–8)

o be conducted by mathematics specialists. These teachers should have at least twenty-one semester hours in mathematics,ncluding at least twelve semester hours of fundamental ideas of mathematics appropriate for middle school teachers. Yet, its not clear whether prospective teachers who meet these requirements have appropriate knowledge for teaching. Thus, the

urpose of this study is to examine the strengths and weakness of a group of teachers who meet the CBMS recommendationshrough a deep analysis of their knowledge of algebra for teaching with a particular attention to their errors and gaps innderstanding.

∗ Corresponding author. Tel.: +1 615 494 7881; fax: +1 615 898 5422.E-mail address: hrj318@gmail.com (R. Huang).

732-3123/$ – see front matter © 2012 Elsevier Inc. All rights reserved.ttp://dx.doi.org/10.1016/j.jmathb.2012.06.001

418 R. Huang, G. Kulm / Journal of Mathematical Behavior 31 (2012) 417– 430

2. Theoretical framework

In this section, we describe the conception of teachers’ knowledge of algebra for teaching. We then discuss a modelof understanding the concept of function, which is the core component of school algebra. Finally, we summarize researchfindings 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 knowledgeneeded for teaching (MKT) (e.g., Ball, Hill, & Bass, 2005; Hill, 2010; Kilpatrick et al., 2001; Kulm, 2008). Recently, growingattention 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 modelsdefining teachers’ knowledge of algebra for teaching (e.g., Doerr, 2004; Even, 1990, 1993; Floden & McCrory, 2007). Forexample, Even (1990) described seven dimensions of content knowledge including essential features, different represen-tations, alternative ways of approaching, the strength of the concept, basic repertoire, knowledge and understanding of aconcept, and knowledge about mathematics. Doerr (2004) specified the pedagogical content knowledge (PCK) includingknowledge about students’ errors and misconceptions in algebra. Different from the categorization of content knowledgeand PCK, Floden and McCrory (2007) proposed a model defining knowledge of algebra for teaching (denoted by KAT here-after). According to this model, the knowledge of algebra for teaching includes school mathematics knowledge, advancedmathematics knowledge, and teaching mathematics knowledge. School mathematics knowledge (denoted by SM) refers tothe algebra covered in the curriculum from K-12. Advanced mathematics knowledge (denoted by AM) includes calculus andabstract 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 dif-ferent approaches to dealing with algebra (Bendarz, Kieran, & Lee, 1996), these two brands of content are core componentsin algebra (NCTM, 2000; Drijvers, Goddijn, & Kindt, 2011). Moreover, Floden and McCrory (2007) also developed relevantinstruments for measuring KAT. Based on our research purposes, we then developed our own test through extending theirinstrument.

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 offunction, according to the process perspective, a function is perceived as linking x and y values: for each value of x, thefunction has only one corresponding y value. On the other hand, the object perspective regards functions or relations andany of their representation as entities (Moschkovich, Schoenfeld, & Arcavi, 1993). The term “representation refers both toprocess 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 capturingmathematical concepts (e.g., Goldin & Shteingold, 2001). Different representations play different roles in helping studentsunderstand the concept of function (e.g., Lesh, Post, & Beher, 1987; Schwartz & Yerushalmy, 1992). For example, while thealgebraic representations can help students understand a function as a process; graphical representations help studentsunderstand 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 under-standing: 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 graphicalworld. For example, students preferred one-to-one correspondences and had a strong tendency to default to propertiesof 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) arguedthat students’ learning difficulties in algebra are due to its abstraction, generalization and overgeneralization, proceduralfluency and symbol sense, and duality of process and object. Pre-service teachers have difficulties in: (1) understanding theconcept of function as univalent correspondence between two sets (sometimes, it is not presented by a formula, or it is not

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R. Huang, G. Kulm / Journal of Mathematical Behavior 31 (2012) 417– 430 419

continuous graph); (2) shifting different representations flexibly; and (3) relating formal function notions to contextualituations which produce the function (Even, 1993, 1998; Hitt, 1998). For example, based on 152 prospective secondaryeachers’ completion of an open-ended questionnaire concerning their knowledge about function, Even (1993) found that

any prospective secondary teachers did not hold a modern conception of a function as univalent correspondence betweenwo sets. These teachers tended to believe that functions are always represented by equations and their graphs are wellehaved.

.4. Research questions

In this study, we examined how prospective teachers from a program that meets the CBMS (2001) requirementsesponded to questions on a survey that was designed to measure knowledge of algebra for teaching. Focusing on theoncept of functions, the study examined the research questions: What are the errors and gaps in teachers’ knowledge andnderstanding of the algebra found in (a) school mathematics, (b) advanced mathematics, and (c) teaching mathematics?

. Methodology

The data used for this study were taken from a comparative study (Huang, 2010) consisting of a survey of 115 U.S. prospec-ive middle mathematics teachers on knowledge of algebra for teaching and follow-up interviews with five participants.he participants were from an interdisciplinary program for preparing mathematics and science middle school teachers at

large public research I university in the southwestern U.S.

.1. Instrument

The study used an existing test of knowledge of algebra for teaching (KAT) (Floden & McCrory, 2007), supplementedy five open-ended items with a focus on knowledge for teaching the concept of function. Thus, the instrument included7 multiple-choice items and eight open-ended items (3 items from the original KAT test and 5 additional items). The test

ncluded three types of knowledge: school algebra knowledge (SM, 7 items), advanced algebra knowledge (AM, 8 items),nd 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.ii. 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 marki) or (ii) as a non-function, decide what he or she might have been thinking that could cause the mistake(s). Write yournswer in the Answer Booklet.

.2. Supplementary open-ended items

Based on an extensive literature review on teachers’ knowledge for teaching the concept of function, we focused on twospects of the concept of function: selection of function perspectives and use of appropriate representations. Including thepen-ended problems from the original test, we have three items used to measure flexibility in selecting the perspectivesf the concept functions (18, 24, and 25) and four items (19, 21, 22, and 23) used to measure flexibility in using differentepresentations. 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 gives a negative number. How many real solutions does the equation ax2 + bx + c = 0 have? One student gives the followingnswer:

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?

420 R. Huang, G. Kulm / Journal of Mathematical Behavior 31 (2012) 417– 430

3.3. Data collection

Three instructors who taught the mathematics education courses for junior and senior students from an interdisciplinaryprogram of middle grade mathematics and science administered the test within a normal class as part of their work (around45 min). We collected 115 copies of questionnaires. Among the participants, 87% were junior and senior students; the restwere sophomores.

Five of the participants voluntarily agreed to an interview. The first author conducted individual interviews during theweek 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 AlgebraI and II, Geometry, Pre-calculus, and Calculus in high school, except Kerri who only took four of the five courses. They alsohad completed an average of 27 credit hours of college mathematics and mathematics education courses including Structureof mathematics I and II, Basic concept of geometry, Introduction to abstract mathematics, Integration of mathematics andtechnology, 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 betweenrepresentations.

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 providinguseless statements; 1 point means providing several useful statements without a chain of reasons for the correct answers; 2points refers to giving a correct answer but the explanations or procedures with major conceptual mistakes; 3 points meansgiving a correct answer and appropriate explanations or procedures, with some minor mistakes; and 4 points presents givinga correct answer and appropriate explanations and procedures. A secondary mathematics teacher coded all items. The firstauthor 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 mis-takes in terms of the nature of the mistakes. The main mistakes include: (1) inappropriate transformation of functionsand 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 advancedmathematics, and 10 items (2, 4, 7, 10, 11, 15, 18, 21, 22, and 25) in teaching mathematics. The reliability of the instrument was0.70 (N = 115). The means and standard deviation (SD) of the multiple-choice items with a brief description of the contentsare listed in Table 1.

The table shows that the participants performed best on items 1, 3, 10, and 11 (higher than 60%), while they performedpoorly on items 4, 5, 8, 9, 12, 13, 14, 15, and 16 (less than 40%). Examining the content of the items, we found that thehigh performing items related to using algebraic expressions to present quantitative relationships (item 1, 85%), findingthe value of a polynomial expression (item 3, 89%), the condition of two lines perpendicular (item 10, 66%), and multiplerepresentations of the concept of slope (item 11, 78%).

The participants performed worst in the following areas: transformation of a logarithmic function (item 4, 15%), solvingan 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 irrationalequation (item 14, 40%), multiple ways to expand an algebraic expression formula (x + y + z)2 (item15, 22%), and slopes oftangent 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) andTM (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%) anditem 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 topresent 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, andtrigonometry), and presenting fraction and algebraic formula using geometrical representation and/or reasoning based

R. Huang, G. Kulm / Journal of Mathematical Behavior 31 (2012) 417– 430 421

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

Item Theme Score

Mean SD

1 Express quantitative relationships in word problems using algebraicexpressions

.85 .36

2 Solve quadratic equations 2x2 = 6x (losing roots) .57 .503 Given a quadratic function f(x), find f(x + a) .89 .324 Transform f(x) = log2 x2 .15 .365 Solve equation: 9x − 3x − 6 = 0 using substitution method (adding roots) .17 .386 Represent fractions, percents, and algebraic expressions such as 3/5, 60%, and

a(b + c) = ab + ac using the area of rectangle.30 .46

7 Given two points, find the functions whose graphs passing these two points .57 .508 Given a graph representing speed vs. time for two cars, judge the position of

the two cars.16 .37

9 Judge the number of root of equation: tan x = x2 .20 .4010 Judge perpendicular relationship of two lines by using their slopes .66 .4811 Multiple ways to introduce the concept of slope of line .78 .4112 Judge the proposition “For all a and b in S, if ab = 0, then either a = 0 or b = 0” in

different number systems.27 .45

13 Meaning of mathematical induction .23 .4314 Roots of irrational equation

√x − 2 = √

1 − x (adding roots) .40 .49

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15 Expand algebra expressions by area relationship .22 .4116 Find derivative and slope of a function .29 .4517 Find value of a composition function .52 .50

n pictorial representation information, operation rules in different number systems, and mathematical induction. The par-icipants revealed weakness in understanding the concept of function and selecting appropriate representations to solveuadratic equation problems. In the following section, we present the typical mistakes in the knowledge of algebra foreaching (1) expressions, equations, and graphs; and (2) functions and their properties.

.2. Typical mistakes in expressions, equations, and graphs

Some typical mistakes were identified as follows: (1) inappropriate transformation of functions and equations; (2)nappropriate use of graphic representations, and (3) inappropriate logical reasoning in algebra.

.2.1. Inappropriate transformation of functions and equationsIt is a common error to neglect the changes of domain and range of variables when transforming equations and functions.

or example, on item 4, participants were required to find the equivalent expressions of function f(x) = log2 x2 from: (i)

= 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

(i, ii, and iii), were 37%, 15%, 6%, 21%, and 0% respectively. This indicates that 37% of them did not consider the domainf independent variable x when transforming a function and 21% of them did not differentiate these four expressions. Fornother example, participants were asked to comment on the following process of solving the equation: 9x − 3x − 6 = 0:

able 2eans of open-ended items and subscales.

Item Theme Score

Mean SD

18 Definition of functions and students’ misconceptions 1.51 1.3319 Solve quadratic inequalities using two methods (algebra and graphic methods) .76 .5920 Judge and explain if A�B = O (A and B are matrixes, O stands for zero matrix),

then either A = O or B = O?.79 1.25

21 Judge the number of roots of a quadratic function with certain constraintsusing graphic methods.

.18 .51

22 Judge the effects of changing parameters of quadratic functions on its graphtranslations

1.40 1.34

23 Given three specific points, find the maximum of a quadratic function whosegraphs passing these points

.29 .74

24 Given f(x) and g(x) intersect at a point P on the x-axis, prove the graph of their sumfunction (f + g) (x) must also go through P

0.02 .13

25 Given a figure, find a daily situation that corresponds to the figure. 1.43 1.21SM School mathematics 4.00 1.57AM Advanced mathematics 2.10 1.70TM Teaching mathematics 7.50 3.35KAT Knowledge of algebra for teaching 13.60 5.15

422 R. Huang, G. Kulm / Journal of Mathematical Behavior 31 (2012) 417– 430

Car A

Car B

Time 1 Hour

Speed

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 equationalso has 2 different roots”. 15% of them agreed with Peter’s solution, and 54% of them believed the substitution method iswrong; 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 representationsThe participants were not able to appropriately use geometrical representations to present algorithms, algebra relation-

ships, the relationship between time and velocity, and roots of trigonometric functions. For example, in responding to thefollowing item, only about one-third of the participants realized that all of these relationships could be presented by thearea 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 equiv-alence of fractions and percents (22%), and the formula of square of a binomial (16%) could be presented by the area ofrectangles.

The participants had difficulties making judgments based on geometrical representations. In the following item, the givengraph (Fig. 1) represents a relationship between speed and time for two cars. (Item 8: Assume the cars start from the sameposition and are traveling in the same direction. Use this information and the graph below to answer. What is the relationshipbetween 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 ofcar 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 passingcar A. Only 16% of them made a correct choice (B) and 48% of them selected (E). Participants needed to understand the graphand analyze it analytically in order to make a correct choice. About half of them (48%) made a wrong choice simply based onvisual information only: the cars are at the same position and car is passing car A (Choice E) or partially based on the visualinformation: at the same point (Choice A) or car A is ahead of car B (Choice B). This result implied that around a half of theparticipants 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 theproperties of the tangent function (specially, the periodic property) in order to find the correct answer. These results showedthat the majority of the participants were not able to appropriately use graphical representations to represent relevantconcepts and solve problems.

4.2.3. Inappropriate logical reasoning in algebraWe 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% ofthem inappropriately overgeneralized the rule to matrix system.

Taking an open-ended item (item 20) related to matrix operation for another example. The item is:

R. Huang, G. Kulm / Journal of Mathematical Behavior 31 (2012) 417– 430 423

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

Items 0 point (%) 1 point (%) 2 points (%) 3 points (%) 4 points (%)

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18 32.2 17.4 27.8 12.2 10.424 98.3 1.7 0.0 0.0 0.025 31.3 17.4 33.0 13.0 5.2

Let A =[

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], Then A�B is defined to be

[pt qurv sw

]. Is it true that if A�B = O, then either A = O or B = O

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

hem gave a correct answer and explanation. A common misconception was to use incorrect logical reasoning as follows: → q ⇔ q → p, namely, using the following logic “if A = 0, then A�B = 0 or if B = 0, then A�B = 0” to prove: “if A�B = 0, then

= 0, or B = 0.” More than one-fourth of them made this mistake. In addition, a few of them inappropriately generalized theame 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 intervieweesJenny and Stacy) gave the correct answer with an appropriate counterexample. For example, Stacy explained why she triedo disprove the statement as follows: “if someone wants to prove a proposition, s/he has to provide the whole process ofroving. However, if someone just wants to disprove a proposition, s/he only provides a counterexample, so, I consideredo find a counterexample”. The others gave a wrong judgment by providing examples such as “A = 0, then A�B = 0” or “B = 0,hen A�B = 0” by “guess and check.”

.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; thether is about the flexible use of representations.

.3.1. Selecting perspectives of the concept of functionItems 18, 24 and 25 were particularly used to measure knowledge of understanding and applying the concept of function

rom different perspectives (process and object). Item 18 uses a process perspective; item 24 is easily proved if an objecterspective is adopted. It is necessary to connect those two perspectives when solving item 25. The score distribution ofhese three items is shown in Table 3.

Inappropriate selection of a perspective of the concept of function. Answering item 18 requires a sound understanding ofhe definition of function. About 23% of the participants got a correct answer or almost correct answer with minor mistakes12%). 32% of them provided nothing or meaningless information about the solution, and about 28% of them gave correctnswers without any interpretation or gave one correct answer and relevant explanations. It is more appropriate to adopthe process perspective. Yet, the participants preferred using an object perspective (9%), namely, basing their thinking oneatures of the expression (constant value; two expressions) and graphic features (one line, many holes/un-continuous),r using essentially corresponding relationship between domain and range (i.e., one-to-one; multiple-to-one) (6%). Theollowing is a typical diagram used to visualize the function relationship and then make a judgment of the two relationsiven 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 inrawing the graph of the second relation, she believed it was not a function. However, when asked whether she had heard

f the vertical line test, she clearly stated that “one x value can only have one corresponding y value; one x value cannote corresponded to two y-values.” Kerri said she “is a visual learner, and likes using diagrams to represent the relationshipetween two sets (one-to-one or multiple-to-one, but not one-to-multiple).” Larry not only explained the vertical lineest 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.

424 R. Huang, G. Kulm / Journal of Mathematical Behavior 31 (2012) 417– 430

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’ superficialunderstanding 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 situ-ations, which correspond to the given graph. It is necessary for participants to have a full understanding of two perspectivesof functions. The item is:

When introducing the functions and the graphs in a class of middle school (14-15 years-old), tasks that consist ofdrawing graphs based on a set of pairs of numbers contextualized in a situation or from an equation were used. Oneday, when starting the class, the following graph (Fig. 3) was drawn on the blackboard and the pupils were asked tofind 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 alonga 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 giveany other explanations of this graph?

18% of participants gave a roughly correct answer and interpretation (3 or 4 scores), 33% of the participants either gavea correct explanation or an appropriate situation, and 31% gave useless information. Sixteen participants (14%) pointedout that the student’s interpretation could be improved by denoting the x-axis as time and the y-axis as height abovesea 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 isaccelerating at an exponential rate, then goes a steady rate for period of time, and then slows at a constant rate, thenstops.” There are other two explanations of the diagram: a graph of height and time (4%), and a graph of distance andtime (4%).

Three of the five interviewees (Larry, Alisa, and Stacy) realized that the original interpretation should be improved bypointing out the x-axis represents time while y-axis represents position (or height). All of them gave other examples ofdescribing 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 tothe graph.

4.3.2. Inflexibility of selecting representationsThe items 19, 21, 22, and 23 were designed to measure understanding and applying quadratic func-

tions/equations/inequalities through flexibly using multiple representations. It is crucial for participants to flexibly useappropriate presentations and shift between different representations in order to effectively explain and solve these prob-lems. 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 representa-

tions. To solve the problem of item 23, it is required to use appropriate forms of algebraic expressions and transformationsof different algebraic expressions, and translation between graphic and algebraic representations. The scores distribution ofthese four items is shown in Table 4.

R. Huang, G. Kulm / Journal of Mathematical Behavior 31 (2012) 417– 430 425

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

Items 0 point (%) 1 point (%) 2 points (%) 3 points (%) 4 points (%)

19 31.3 62.6 5.2 0.9 0.021 84.3 14.8 0.9 0.0 0.022 34.8 24.3 15.7 16.5 8.7

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The table shows that overall, the participants had limited knowledge and skills in explaining and solving these prob-ems. Further qualitative analysis revealed several typical mistakes. These mistakes include performing algebraic operationsy guessing rather than following operation properties; sticking to one of algebraic and graphic representations withoutexibility 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

id not result in a correct answer, and about one-third left it blank or gave some meaningless statements. Many mistakesnd 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 realizedhat a and b are possibly negative. In addition, none of them noted the logical operations “or” or “and” between two logicropositions (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

ntention 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

tandard form: x2 + x − 12 > 0. 21% of them stopped with the standard form. 7.6% of them were stuck with further algebraicperation: 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;”

In interviews, Larry just simplified the factor form into standard form (x2 + x − 12 > 0), and then moved forwardy “guess and check” such as “x2 > x − 12, and then x >

√x − 12.” although it did not work. By analogizing the prop-

rty 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 theperations of intersection and combination of two sets. They seemed to be satisfied with the “solution”. When asked ifhey can use a graphic method to solve an equation or inequality, they recalled the graphs of quadratic equation. All of thenterviewees explained they did not know how to use a quadratic function graph to solve an inequality, although they knewhe graphing method of solving a system of linear equations. They said that they learned quadratic functions first (probably

arly at high school) and then inequality late (at high school). These topics were taught separately. The participants wereot taught how to use graphic representations to solve algebraic equations and inequalities. They appreciated the methodf integration of algebraic and graphic representations.

able 5isconceptions or mistakes in solving inequality in Item 19.

Mistake Explanation Example Frequency (%)

1 Misconception: if ab > 0, then a > 0, b > 0 (x − 3)(x + 4) > 0 → x − 3 > 0, x + 4 > 0, then x > 3, x > −4. 37.02 Transforming into standard form x2 + x − 12 > 0 or x2 + x > 12. 21.03 Transforming into standard form and getting stuck x2 + x − 12 > 0 or x(x + 1) > 12 or x(x + 1) = 12. 7.64 Working on the standard form with guess and check x2 + x – 12 > 0 → x(x + 1) > 12; →x > 12, x + 1 > 12→x > 12, x > 11. 12.0

x2 + x – 12 > 0, or (x − 3)(x + 4) > 0 →x1 = 3, x2 = −4. 15.05 Drawing a number line Find partial answer: x > 3 or x < −4. 4.06 Using a table x > 3 (x = 1, 2, 3, 4. or 0, −1, −2, −3,. . .). 4.0

426 R. Huang, G. Kulm / Journal of Mathematical Behavior 31 (2012) 417– 430

Fig. 4. Translations of quadratic functions.

Sticking to algebraic representations without awareness of using other representations. On item 21, only one out of the 115participants 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 threevariables [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 oppositeis 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 algebraicmanipulation to find a, b and c, and had no idea about how to help the student get out of their difficulties. 15% of theparticipants even suggested plugging more values of a, b, and c (such as a = −10, b = −9, and c = 20) to see whether certainpatterns can be found. They did not think to use graphic or geometrical representations to find the answers. Some of themhad 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 outfixed 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. Theother two interviewees (Jenny and Stacy) believed that the problem may be solved, but they did not have any concrete ideasabout how to solve it. What they could suggest is that the student is to “try different ways, such as plugging more numbersbetween 1 and 6.” (Jenny) or “explore in different ways such as plugging more numbers to see whether they can find certainpatterns, rather than being stuck.” (Stacy) When asked whether they can try other methods such as graphical methods tosolve, they tried to sketch the graphs and find the possible roots. Four of them succeeded in finding the number of rootsby examining the intersection points of the quadratic function. All of them said they did not know the graphic method insolving quadratic equation or inequality; they did not have such an experience in solving problems, but they appreciatedthe 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 answersbut failed to explain. 15% of the participants gave partially correct answers and explanations. 23% of them gave sporadicinformation about the effect of a, b and c changes. About 35% of them got lost, either leaving it blank or providing somewrong statements.

In summary, one-third of the participants had no ideas about solving and explaining this problem while about one-fourthof them gave a roughly correct answer. With regard to the strategies used to explain, it was found that 11% of the participantsused the following reason: “The effects of changes of a, b and c on the changes of graphs of quadratic functions,” and 2% ofthe 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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R. Huang, G. Kulm / Journal of Mathematical Behavior 31 (2012) 417– 430 427

hanged only.” Typical mistakes are related to the effects of changing a or b. For example, some participants said, “changing 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 ofuadratic function. The other two were able to explain the effect of changing a and c on the graph of quadratic function, buthey were not sure about the effect of changing b. One of them found the correct answer by explaining that changing b to

negative b, would reflect the graph of quadratic function over y-axis although she could not remember the details of theffect 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 theollowing 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 ofthe 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 theaximum correctly. About 20% of the participants just drew a graph or list equations based on the given three points.

our-fifths of them left the item blank or wrote something unrelated to solving this problem. 3% of the participants used thetandard quadratic formula method: y = ax2 + bx + c to find the expression of the quadratic function, and then, transformingnto the form y = a(x − h)2 + k to find the maximum. One typical mistake is regarding the y-intercept as maximum (16% ofarticipants).

Two of the interviewees knew the process of solving the problem: finding the expression of quadratic equations bylugging given points, and then finding the maximum by taking derivative. Another realized that the maximum should be at

= 1 due to its symmetrical property although they were not able to find the correct expression of quadratic equations. Otherwo held a misconception that the y-intercept is the maximum. They had difficulties finding the expression by plugging inhe given points. They had no ideas about other formula of quadratic function even when the interviewer showed to them. Inummary, the interviewees tried to find expressions of quadratic equations by using the standard formula, and then foundhe maximum by taking derivative and plugging x = 1. However, they had difficulty finding the correct expressions due tohe complexity of algebraic manipulation. In addition, nobody was aware of using other appropriate formulas to find thexpressions. In fact, three formulas can be used to find the quadratic function: (1) y = ax2 + bx + c; (2) y = a(x − x1)(x − x2);nd (3) y = a(x − h)2 + k. And, three methods could be used to find the maximum of the function: (1) transforming into

= a(x − h)2 + k, then finding the maximum; (2) using formula x = −(b/2a), ymaximum = (4ac − b2)/4a; and (3) taking derivative:hen (dy/dx) = 0, then x = 1, ymaximum = f(1).

. Discussion and conclusion

.1. Discussion

This study showed that the participants performed poorly in all three areas of knowledge of algebra teaching: schoolathematics, teaching mathematics and advanced mathematics, with less than an overall 40 percent mean. The majority

f the items in the category of teaching mathematics rely heavily on a relational or conceptual understanding of relevantathematics concepts (Skemp, 1976; Hiebert & Lefevre, 1986). Thus, we could conclude that the participants were weak in

he mathematics content knowledge they need for teaching.The participants had relatively limited knowledge of algebra for teaching, revealing their weakness in selecting appro-

riate perspectives of the concept of function and flexibly using of representations of quadratic functions. Meanwhile, theyade numerous mistakes in solving quadratic/irrational equations, algebraic manipulation and algebraic reasoning. More-

ver, the isolation of algebraic and graphic representations resulted in their failure to solve quadratic inequalities and judgehe number of roots of quadratic functions. Follow-up interviews further disclosed participants’ lack of experience in solv-ng problems by integrating of algebraic and graphic representations. These findings suggest that the participants neitherad a relational understanding of algebra content knowledge (with connections and multiple representations), nor had an

nstrumental 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

ine with other studies (Even, 1993, 1998; Hitt, 1998). Specifically, the mistakes identified are serious and striking. Some ofhem such as judging car position based on visual image in a speed and time graph (Clement, 1989), and viewing a graphs a picture (Janvier, 1978; Monk, 1992) have been recognized in other studies. Other mistakes draw attention to improv-ng teaching in algebra. For example, the participants seemed to suffer from a lack of fluency in algebraic manipulationsnd transformation of different algebraic formulae when solving quadratic equations and inequalities or finding quadratic

xpressions. This may reflect the learning difficulty in algebraic procedural fluency and symbol sense (Drijvers et al., 2011).he participants’ difficulty in making appropriate algebraic reasoning echoes the call for attention to actors-oriented trans-ers in algebra generalization (Ellis & Grinstead, 2008). Moreover, the participants were not able to flexibly use algebraicnd graphical representations to solve quadratic equations, and irrational equations. They also demonstrated weakness in

428 R. Huang, G. Kulm / Journal of Mathematical Behavior 31 (2012) 417– 430

using geometrical representations to present algebraic and arithmetic relations. These results make clear how difficult, ifnot 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 thecurrent study, only one of the participants gave a correct answer. On another item on the impact of changing parameters onthe graphs of quadratic functions, Black (2007) found that one-fifth of sampled U.S. high school mathematics teachers gavecorrect answers and relevant explanations. However, one quarter of the participants in this study gave correct answers andexplanations for the item. These results indicate that even though the participants in this study (who were preparing to bemiddle school mathematics and science teachers) performed poorly in general, their performance in quadratic equations issimilar to pre-service and in-service high school mathematics teachers. This finding suggests that the mistakes or difficultiesidentified 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 cal-culus. The participants probably took relevant advanced courses (they averaged 7 courses in mathematics and mathematicseducation) but they did not demonstrate the facility to apply knowledge and skills in the survey. As far as algebra teachingand learning is concerned, the CBMS (2001) recommended developing a deep understanding of variables and functions asfollows: (1) relate tabular, symbolic, and graphical representations to functions; (2) relate proportional reasoning to linearfunctions; (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 todemonstrate the following skills: (1) represent physical situations symbolically; (2) graph linear, quadratic, exponentialfunctions 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 mathematicsteachers do not understand and master these contents? Does it matter if they had the difficulties in understanding stu-dents’ learning difficulties in these areas? It may be possible for them to teach pre-algebra, but it may be hard for themto prepare their students ready for continued learning in algebra I & II because they may not able to help their studentsto lay a proper foundation for learning advanced algebraic topics, and develop relevant algebraic reasoning and thinkingappropriately.

5.2. Implications

The findings of this study provide several implications for improving mathematics teacher preparation. At the teacherpreparation curriculum level, it is important to provide an appropriate foundation for prospective teachers to obtain asound and well-structured knowledge base needed for teaching. This study shows it is not a simple issue of adding moremathematics and mathematics education courses. The curriculum should provide a core of coherent content areas andpresent them with the trajectory of knowledge development by integrating content and pedagogy. At the teacher preparationpedagogical level, it is also important to provide prospective teachers with relevant learning experiences in developinginterconnected knowledge. In order to implement school mathematics curricula that emphasize developing connectionsand representational flexibility, prospective teachers should have the opportunity to develop their own knowledge andskills needed for organizing classroom activities that promote this kind of students’ learning (NCTM, 2000, 2009). Thefinding that the participants lacked the learning experience in integrating graphic and algebraic representations to solveproblems related to quadratic functions highlights the necessity of creating this kind of learning opportunity in curriculumand teaching.

The numerous deficiencies identified in this study provide practical resources and references for developing teacherpreparation curriculum and implementing particular lessons. In particular, it is crucial to develop prospective teachers’ deepunderstanding of the concept of function. The study confirmed the finding that prospective mathematics teachers do not holda modern notion of function as univalent corresponding relationship between two sets (Even, 1993). Recently, researchershave suggested the ways to develop a deep understanding of the concept of function through developing quantitativereasoning and covariational reasoning (Carlson, Jacobs, Coe, Larsen, & Hsu, 2002; Thompson, 2011). With regard to developingflexibility 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 educationcourses, 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 theconcept 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 func-tions, which may not reflect an entire picture of algebra. What we found may be only pertinent for these content areas.

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econd, the participants were from a program in a single university, which at most presents one of three routes of preparingiddle grade mathematics teachers in the U.S. (Dossey, Halvorsen, & McCrone, 2008). We should be cautious interpreting

he findings due to this limitation. Nevertheless, these detailed and vivid descriptions and explanations will shed insightnto development of mathematics teacher preparation programs.

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