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Fall 2013
The Multifaceted Nature of Mathematics
Knowledge for Teaching: Understanding the Use of
Teachers' Specialized Content Knowledge and the
Role of Teachers' Beliefs from a Practice-based
Perspective
Lauren E. Provost
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Recommended Citation
Provost, Lauren E., "The Multifaceted Nature of Mathematics Knowledge for Teaching: Understanding the Use of Teachers' Specialized Content Knowledge and the Role of Teachers' Beliefs from a Practice-based Perspective" (2013). Doctoral Dissertations. 741.
T H E M U L T IF A C E T E D N A T U R E O F M A T H E M A T IC S K N O W L E D G E F O R T E A C H IN G :
Understanding the Use of Teachers’ Specialised Content Knowledge and
the Role of Teachers’ Beliefsfrom a Practice-based Perspective
by
L A U R E N E. P R O V O S T
B.S. C om puter Science, University o f Texas at A ustin, 2002 M.S.T., M athem atics Teaching, U niversity o f N ew H am pshire, 2008
D IS S E R T A T IO N
Subm itted to the University o f N ew H am pshire in Partial Fulfillm ent o f
the Requirem ents for the D egree o f
D o c to r o f Philosophy in
E ducation
UMI Number: 3575981
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D r. T hom as H . Schram , A ssociate P ro fesso r o f E ducation
D r. Suzanne E. G raham , A ssociate P ro fesso r o f E ducation
Dr. S haron N od ie O ja, P rofessor o f E d ucation
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D r. K aren J. G raham , D irector, T h e Leitzel C enter and Professor, D ep artm en t o f M athem atics and Statistics
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D r. Philip U ri T reism an, D irector, C harles A. D an a C enter and Professor, D ep artm en t o f M athem atics
Oe*. I zo
i't-D ateACKNOW LEDGEMENTS
This w ork w ould n o t have been possible w itho ut the help o f so m any people in so many ways. I w ould like to express my deepest gratitude to my advisor, D r. T o m Schram for his patience through challenging times and truly m odeling the deepest care for students as learners. I could never thank you enough. T o D r. Treism an, for years o f su pp ort th a t I
cannot begin to describe. T o D r. Suzanne G raham — the b est m athem atics teacher I have had and for helping m e build such a vast array o f quantitative knowledge. T han k you, Suzanne, for always being there to help, give advice and cheer m e o n w hen my ow n life was a bit too challenging. I will never forget the offers to walk m e to m y car at night after class, at a time w here I m ight no t have adm itted needing the offer. T hank s for creating a lo t o f fun and laughter throughout the process. I thank my entire com m ittee and the E ducation D ep artm en t for all the support and excellence this dep artm en t has to offer. T o my family - you have all been there at difficult times, especially m y daughters Alexia and K arianna. I love you m ore than anything!
TABLE O F C O N T E N T S A C K N O W L E D G E M E N T S ... iii LIST O F TA BLES... ix LIST O F F IG U R E S ...xi A B S T R A C T ... xii C H A P T E R P A G E I. C H A P T E R 1: IN T R O D U C T IO N ... 1 II. C H A P T E R 2: C O N C E P T U A L F R A M E W O R K ... 9 In tro d u ctio n ... 9
I. U nderstanding SCK and the O rganization o f M K T ... 11
T h e Learning M athematics for T eaching Research G ro u p an d the M K T M odel 15 T h e Em ergence o f Specialized C o n te n t K now ledge (SCK)...18
T h e D evelopm ent o f the Middle School Level M K T Survey: M easuring S C K ... 19
W hat is T here Still to Learn A b o u t the C on ten t o f SCK in M iddle School Level?...24
II: U nderstanding SC K and the Role o f T eachers’ Beliefs in P ractice...25
Pedagogical C ontent Beliefs in the C o n te n t A rea o f M athem atics...28
A n Illustration... 32
W hat is T here Still to Learn A b o u t the M anifestation o f Beliefs w ithin SCK Use?...33
C onclusion... 35
III. C H A P T E R 3: M E T H O D O L O G Y T h e M K T Survey and D ataset... 37
D a ta ... 38
P articipants...39
D istributing the Middle School M K T Survey and G ath erin g R esults...40
Missing D a ta ... 41
A H istory o f the D evelopm ent o f the M easures o f C C K and S C K ... 44
E xpanding M K T W ork to Assess M iddle School M K T ... 44
Reliability and Validity... 45
Setting the Stage for Analysis... 47
Sum m ary... ...55
P art II: U nderstanding H ow T eacher Beliefs are M anifested in U se o f S C K ... 56
In tro d u ctio n ... 57
A Pragm atic A pproach to the E xtensio n o f a Previous S tudy... 58
C hoosing a Participant P o o l— •... 59
Section A: M K T Survey...60
Section B: Beliefs Survey...61
Section C: T he Structured Interview ... 67
W hy M K T Tasks and the T hink-A loud A pproaches W ere C h o se n ...69
Section D: T he Focused Interview ... 70
Section E: O bservations... 71
Reliability and Validity... 74
Sum m ary... 75
IV: C H A P T E R 4: D A TA ANA LY SIS A N D F I N D I N G S ...77
D escriptive Statistics ...79
Testing the Fit M odel 1...84
Testing the Fit for M odel 2 ...95
Testing the Fit for M odel 3 ...101
C ross-validation...105
Sum m ary...107
U nderstanding H ow T eachers’ Beliefs are M anifested in SC K U se ... 113
M ethods O verview ... 115
Study Participants ... 119
Preliminary Analyses: Five D ata S ources... 120
Analysis o f M K T Survey D a ta ... 120
Analyzing the Use o f Specialized C o n te n t K now ledge...132
A dditional Analyses... 134
Results o f A nalysis... 137
Inconsistency in the Beliefs-to-Practice C onnection: Surface Beliefs...138
B ackground and Setting...140
Beliefs A b o u t the N ature o f M athem atics... 143
Beliefs A b o u t M athem atics L earning... 144
Beliefs A b o u t M athem atics T eaching... 145
R ob ert’s U se o f Specialized C o n ten t K now ledge in P ractice...146
Julie’s Use o f Specialized C o n ten t K now ledge in P ractice...150
Sum m ary... 157
Consistency in the Beliefs-to-Practice C onnection: Lim ited SC K U se... 158
B ackground and Setting...159
Beliefs A b o u t the N ature o f M athem atics... 163
Beliefs A b o u t M athem atics L earning...165
Beliefs A bo ut M athem atics T eaching... 145
D ebra’s Use o f Specialized C o n te n t K now ledge in P ractice... 166
N ancy’s Use o f Specialized C o n te n t K now ledge in P ractice... 171
Sum m ary... 176
Consistency in the Beliefs-to-Practice C onnection: O p portunities for SCK U se 176 Background and Setting...177
Beliefs A b ou t the N ature o f M athem atics... 181
Beliefs A b ou t M athem atics L earning...181
Beliefs A b ou t M athem atics T eaching... 182
Sarah’s U se o f Specialized C o n te n t K now ledge in P ractice... 183
Sum m ary... 188
Consistency in the Beliefs-to-Practice C onnection: M ixed Beliefs and an Illustration o f Teaching Quality 189 B ackground and Setting...189
Beliefs A b ou t the N ature o f M athem atics... 195
Beliefs A b ou t M athem atics Teaching and L earning...196
Jake’s Use o f Specialized C o n ten t K now ledge in P ractice... 198
A nna’s Use o f SCK in Practice...204
Sum m ary... 207
V. C H A P T E R 5 D ISC U SSIO N A N D IN T E R P R E T A T IO N S ... 210
Specialized C ontent K now ledge as the L earner’s Building B locks... 212
T h e D evelopm ent and N atu re o f Beliefs...214
T h e Role o f C o n tex t... 216
Em phasis o f C onceptual K now ledge V ersus S kills/P ractice... 217
Socio-consructivist Versus Traditional B eliefs... 218
O ther D ichotom ies... 219
T he Role o f Teacher E d u catio n ...219
Study Lim itations...220
Future Q uestions and C hallenges...221
LIST O F R E F E R E N C E S ... 222
A P P E N D IX A: SURVEYS A N D IN T E R V IE W G U ID E S ...233
A P P E N D IX B: IRB P A P E R W O R K ...234
LIST O F TABLES
1. Beswick’s (2005) Relationships betw een Beliefs...30
2. T h e O rganization o f M K T Survey Tasks for M odel 1... 48
3. Belief Survey Subscales (Staub & Stem , 2002)... 62
4. A n E xpansion o f Beswick’s (2005) Relationships betw een Beliefs...64
5. Raym ond’s (1992) Criteria for the C ategorization for T eachers’ Beliefs about M athem atics T eaching... 64
6. Ball’s (2005) M athem atical Tasks o f T each ing... 71
7. Raym ond’s (1992) Criteria for the Categorization o f T eachers’ M athem atical T eaching P ractice... 71
8. Teacher Characteristics... 79
9. T eacher’s Educational B ackground (% )... 80
10. O rganization o f Tasks M odel 1...86
11. M odel 1: G oodness o f Fit Statistics... 87
12. O rganization o f Tasks M odel 2 ...95
13. M odel 1, 2: G oodness o f Fit Statistics... 97
14. Residuals...99
15. M odel 1, 2, 3: G oodness o f Fit Statistics: Sample...1...102
16. M odel 1, 2, 3: G oodness o f Fit Statistics: Sample 2 ...105
17. Ball’s (2005) M athem atical Tasks o f T eachin g... 117
18. A n E xpansion o f Beswick’s (2005) Relationships betw een B eliefs... 125 19. Raym ond’s (1992) Criteria for the C ategorization for T eachers’ Beliefs abo ut
M athem atics T eaching... 126
20. Selected Indications o f N ancy’s B elief System ...129
21. Summary o f the Beliefs about T eaching M athem atics C luster C ategorization... 130
22. Ball’s (2005) M athematical Tasks o f T each ing...131
23. Sample D escriptions o f SCK U se... 133
24. Raym ond’s (1992) Criteria for the C ategorization o f T eachers’ M athem atical Teaching P ractice... 134
LIST O F FIGURES
1. A Subset o f the P roposed M K T M odel (LM T Research G ro u p , 2013)... 42
2. T he P roposed M odel 1... 49 3. T he P roposed M odel 2 ... 52 4. T he P roposed M odel 3 ...53 5. M odel 1 O verview ... 82 6. M odel 2 ...96 7. M odel 3 ... 101
8. Overview o f the D ata Collection M ethods in P art II o f the S tudy... 113
9. A bove, All Case Study Participants, O rd ered by IR T Scores on M iddle School Patterns, Functions, and Algebra Section o f the M K T Survey, Ranging from -3 to 3 ... 120
10. A bove, All Case Study Participants, O rd ered by IR T Scores on M iddle School N um bers and O perations Section o f the M K T Survey, Ranging from -3 to 3 ... 121
11. A bove, All Case Study Participants, O rd ered by IR T Scores on Elem entary Patterns, Functions, and Algebra Section o f th e M K T Survey, Ranging from -3 to 3 ... 121
12. Overview o f Them es from A nalysis... 136
13. Julie’s Shape D iscussion... 150
14. A nna’s Shape D iscussion... 205
A BSTRA CT T H E M U L T IF A C E T E D N A T U R E O F M A T H E M A T IC S K N O W L E D G E F O R T E A C H IN G : U N D E R S T A N D IN G T H E U SE O F T E A C H E R S ’ S P E C IA L IZ E D C O N T E N T K N O W L E D G E A N D T H E R O L E O F T E A C H E R S ’ B E L IE F S FR O M A P R A C T IC E -B A S E D P E R S P E C T IV E by Lauren E . P rovost
University o f N ew H am pshire, Septem ber, 2013
This w ork investigates middle school teachers’ m athem atics know ledge for teaching (MKT) as defined by Hill (2007). W ithin this tw o-part dissertation, the level o f M K T was considered as well as the role o f teacher beliefs in actual specialized con ten t knowledge (SCK) use, a specific type o f m athem atics know ledge for teaching vital in quality m athem atics instruction. Additionally, the m odel o f M K T know ledge was explored through confirm atory factor analysis o n a large, national dataset o f m iddle school m athem atics teacher survey responses involving m athem atics know ledge for teaching. SCK was found to be vital in quality m athem atics instruction yet n o t sufficient. T eacher beliefs abo ut the delivery o f m athem atics instruction ultim ately acted as a filter, at times limiting SCK use, even if a teacher held high levels o f SCK. T h e m athem atics know ledge th at teachers hold is highly com plex; confirm atory factor analysis results indicated th a t w e have yet to truly capture the essence o f M KT, yet the im portance o f understanding such know ledge is clearly essential. Im plications for preparing future teachers are discussed.
C H A P T E R 1
T H E IM PO R T A N C E O F S P E C IA L IZ E D C O N T E N T K N O W L E D G E
First, w hat we are doing in this country is unethical. W e let people start teaching w ho have no t yet d em onstrated th at they can perform . A nd, further, the students w ho m o st need skillful and highly effective teachers are least likely to get them. Second, we know h ow to change this and m ust do so deliberately and w ithout delay.1
O ver the last twenty years, w hat constitutes quality m athem atics instruction has draw n increased attention from stakeholders in m athem atics education. Policymakers are particularly concerned with the m athem atics know ledge teachers hold. This is due in p art to accum ulating evidence that students learn m ore w hen they are taught by m ore
mathematically knowledgeable teachers (Hill, R ow an & Ball, 2005; Row an, et al., 2001). In fact, quality m athem atics teaching necessitates an array o f m athem atics knowledge typically interw oven within instructional characteristics such as providing classroom environm ents that are rich in accurate m athem atics language, encourage connections betw een topics and concept generalizations, provide student-accessible explanations, an d offer multiple problem solving procedures o r representations (Hill, et al., 2005; N ational C ouncil o f Teachers o f M athem atics (NCTM , 2000). H ow ever, Hill (2007) found th at m athem atically
knowledgeable teachers are n o t the n orm in classroom s across the U nited States. In fact, m athem atics teachers often enter and rem ain in m athem atics teaching w ith out the
m athem atics knowledge and skills for teaching necessary to deliver quality m athem atics
1 E xcerpt from Dr. D eborah Loew enberg Ball’s Sum m ary o f T estim ony to the U.S. H ouse o f Representatives C om m ittee on E ducation and L abo r o n May 4, 2010.
instruction (e.g., Ball, 1990; Battista, 1999; Ma, 1999; N C T M , 1989, 1991, 2000; N R C , 1989). “Allowing teachers to learn at children’s expense is unethical. W e m u st build a system for ensuring that new teachers have the requisite professional skills and k now how to use th e m ” (Ball & Forzani, 2010).
T h e drive for im provem ent in teachers’ m athem atics know ledge stem s from multiple sources. O u r nation is significantly underperform ing in m athem atics as com pared to o ther nations leading to valid concerns abo ut ou r future econom ic status as a country (N ational Com m ission on Teaching and A m erica’s Future, 1996; N ational C enter for E ducation Statistics, 2007; U.S. D epartm ent o f E ducation, 2010). W ithin our nation, the achievem ent gap in m athem atics still exists (N CES, 2011). T h e lack o f an equitable distribution o f quality m athem atics teachers is concerning and n o t unique to students predom inantly w ithin high- poverty and high-m inority areas. H ow ever, students w ithin high-poverty and high-m inority areas are even m ore likely to experience less m athem atically know ledgeable teachers (Balfanz & Byrnes, 2006; Hill, 2007). There is w idespread agreem ent am ong stakeholders in
m athem atics education that understanding and im proving teachers’ m athem atics knowledge is im perative in addressing these concerns (N CTM , 2000; U.S. D ep artm en t o f Education, 2008).
Research aim ed at understanding and im proving teachers’ m athem atical know ledge is n o t new. In the past, m athem atics teacher know ledge has been typically defined as a
com posite o f a teachers’ m athem atics courses taken after Calculus, the n um ber o f m ath m ethods courses taken, and a m ajor o r m inor in m athem atics, typical o f process-product research o f the sixties and seventies (Mullens, M em ane &c W illett, 1996; Rowan, Chiang & Miller, 1997). Im precise definitions o f teachers’ m athem atical c o n ten t knowledge such as these resulted in little progress in understanding teachers’ m athem atics knowledge, as well as
a misspecification o f the causal processes linking m athem atics teachers’ knowledge to students’ learning (Rice, 2003).
In 1986, Lee Shulman and his colleagues developed a conceptual fram ew ork that encom pased pedagogical content knowledge; n o t only the know ledge a teacher has accrued (i.e., the w isdom o f practice) b ut how this know ledge is used in classroom s (Shulman, 1986; W ilson et al., 1987). Shulman was n o t the first to investigate pedagogical content knowledge, although his contributions have sparked decades o f research in m athem atics content knowledge use and continue to b e referenced to date (C arpenter, Franke, and Levi, 2003; G rossm an, 1990; Wilson & W ineburg, 1988). A lthough Shulm an’s w ork was
substantial, the m athem atical knowledge used in the practice o f teaching had yet to be fully understood and conceptualized, particularly as it relates to the actual practice o f m athem atics teaching. C ontinuing this endeavor, Ball, Tham es, and Phelps built u p o n Shulm an’s w ork, developing a practice-based theory o f m athem atics know ledge used in teaching.
“T h e m athem atical content teachers m ust know in order to teach has yet to be m apped precisely.. .past m ethods lack the pow er to p rop ose and test hypotheses regarding the organization, com position and characteristics o f content knowledge [core co n ten t and pedagogical] for teaching” (2008, p. 43).
Researchers in m athem atics education agree th a t in order to further the understanding o f teachers’ m athem atical knowledge use, a m ore direct approach is necessary (Blanton & K aput, 2005).
D ebo rah Ball and the Learning M athem atics for T eaching (LMT) research group at the University o f Michigan have taken a direct, practice-based approach in understanding and creating measures o f m athem atics teacher knowledge. In doing so, the LM T research team has created a practice-based m odel representative o f the types o f knowledge and skills
m athem atics teachers hold, w hich they refer to as the M athem atics K now ledge for Teaching (MKT) m odel (Ball, 2008).
A survey was developed by the LM T research group (i.e., th e M K T Survey),
incorporating tasks that reflect the use o f this know ledge as used in the practice of teaching. M K T survey tasks represent tw o main subdom ains o f teacher knowledge: subject m atter
knowledge and pedagogical content knowledge. This w ork has led to significant
understanding o f m athem atics knowledge used in teaching m athem atics as well as a rob ust survey tool used to further understand the role o f M K T in professional developm ent, teacher preparation, student achievem ent and o th er teacher quality issues w ith policy
im plications (Ball, 1990; Ball, G offney & Bass, 2005; Ball & Hill, 2004; Hill, Ball & Schilling, 2008; Hill, Rowan & Ball, 2005).
M ost notably, past research o f the LM T research group have uncovered a special type o f knowledge used by teachers, w hich the LM T research group has nam ed specialised content knowledge (SCK). This type o f m athem atics teacher know ledge, SCK, is knowledge crucial to quality m athem atics instruction. T eachers use this specific type o f decom pressed m athem atical knowledge to look for patterns in stu dent errors, exam ine stu dent solutions to see if the solution will w ork in other similar situations, and provide an explanation as to why a particular student solution works or not. E ach o f these activities represents m athem atical knowledge unique to the profession o f m athem atics teaching and critical in the delivery o f quality m athem atics instruction.
M ost o f the initial M K T survey tasks, how ever, w ere w ritten specifically for teachers’ m athem atical knowledge applicable to grades three through six. M ore w ork is necessary to understand specialized content knowledge at the middle school level, a key period o f time in
a student’s developm ent o f algebraic ideas and a m ajor indicator o f later success in mathematics.
“ Few er efforts have focused o n teachers’ know ledge o f stud en t thinking about algebraic ideas in middle grades - a period th at m arks a significant transition from the concrete arithm etic reasoning o f elem entary school m athem atics to the increasingly com plex, algebraic reasoning required for high school m athem atics and beyond” (Alibali et al., 2007, p. 251).
F urther justifying the need for understanding M K T at the middle school m athem atics level is that students’ m athem atical perform ance sharply declines in m iddle school years, w here for m ost students, Algebra begins (R A N D M athem atics Study Panel, 2003; U.S. D ep artm en t o f E ducation, 2008). Middle school years and A lgebra in particular, serve as gateway
knowledge to later success in high school and eventually college-level m athem atical courses and widen career choices (NCTM , 1989; 2000). H ence, expanding th e understanding o f teachers’ specialized con ten t knowledge at the m iddle school level is critical.
In light o f these concerns, I have chosen to situate my study w ithin middle school m athem atics teaching, typically encom passing Algebraic concepts. T h e research questions guiding this study are:
1. W hat specialized co n ten t know ledge do m iddle school m athem atics teachers hold? In particular, how does this know ledge differ from com m on content knowledge
2. H ow are pedagogical con tent beliefs, an additional co m p o n en t o f teacher knowledge, m anifested in specialized c o n ten t know ledge use In the first p art o f m y study, I will investigate w hat constitutes m iddle school
m athem atics teachers’ specialized co n ten t knowledge and describe such knowledge, focusing on specific m athem atical topics that play a p rom inen t role in the m iddle school curriculum: num bers and operations, patterns, functions, equations and inequalities (Hill, 2007; N ath an & K oedinger, 2000; Sfard, 1995). Specifically, I will use confirm atory factor analysis techniques to understand th e con tent and organization o f b o th co m m o n and specialized
co nten t knowledge through the analysis o f M K T survey responses resulting from a large, nationally representative sample o f m iddle school teachers. T h e dataset resulted from a 2005-2006 adm inistration o f the M K T survey tasks, as rep o rted by H eath er Hill in her 2007 article, entitled Mathematical Knowledge of Middle School Teachers: Implicationsfor th N o Child N ft Behind Policy Inititive.
In Part II o f my dissertation, I take a closer look at specialized content knowledge. Specialized content knowledge, although necessary for quality teaching, is only one type o f knowledge necessary for quality m athem atics teaching. A n o th er fo rm o f teachers’
knowledge, teacher beliefs, has been regarded for decades as having a significant influence on m athem atics teaching (A nderson & Piazza, 1996; Battista, 1994; E rnest, 1991; Yadav & K oehler, 2004). F o r example, teachers w ho believe students co n stru ct their ow n knowledge as a result o f instruction generally m ake significantly different choices in their uses o f m athem atical knowledge, representations and o th e r pedagogical tools im pacting student learning in considerable ways (Staub & Stem , 2002). F o r exam ple, P eterson, Fennem a, Carpenter, and L o ef (1989) found that first grade teachers w ith a constructivist perspective m ore often used specicific specialized counting strategies to teach num eracy as an approach to solving w ord problem s, resulting in significantly greater increases in num eracy w ord problem -solving achievem ent than students o f teachers o f a less constructively-based perspective. In another com pelling exam ple, Raym ond (1997) found that instructional practices in the m ath classroom are m ore influenced by beliefs ab o u t specialized co nten t knoweldge than by beliefs about m athem atical pedagogy.
Studying specialized content know ledge along w ith teachers’ pedagogical beliefs specifically from a practice-based, content-specific perspective has b een suggested to provide a m ore com prehensive picture o f how m athem atics c o n ten t know ledge is used in teaching
(Peterson, et al., 1989; Raymond, 1997). T h ere are very few such studies, however. The M K T tasks developed by the LM T team provide ideal practice-based tasks in w hich to investigate the use o f M K T and teacher beliefs (Ball, Tham es & Phelps, 2008). H ence, after gathering a greater understanding o f the c o n ten t and orgnization o f specialized content knowledge teachers hold resulting from P art I o f my study, I then use m iddle school M K T survey tasks classified as specialised content knowledge in m ultiple ways to understand the m anifestation o f beliefs in SCK use. M ultiple sources o f evidence will be gathered from ten middle school m athem atics teachers in m ultiple state, including: (1) participant results o f the middle school M K T survey, (2) survey results o f teachers’ pedagogical co n ten t knowledge beliefs w ritten to coincide with specialized co n ten t know ledge use, (3) a structured interview, w here participants will reason through specialized co n ten t know ledge tasks via a think-aloud interview protocol, (4) a focused interview w here participants view their ow n first interview by video and provide retrospective com m ents, and (5) observational data in the classroom for a m inim um o f three full-class periods, while participants are engaging in middle school m athem atics lessons. Multiple sources o f data w ere strongly suggested due to the
complexity o f how teachers’ beliefs ultimately u nfold in the use o f specialized co nten t knowledge (SCK), as well as the ongoing struggles to uncover and expand up o n the understanding SCK use in practice (Beswick, 2008; H andal, 2003).
“ K nowledge is im portant, b u t alone it is n o t enough to acco un t for the differences betw een m athem atics teachers” (Ernest, 1989, p. 1). T eachers’ beliefs also im pact alm ost every aspect o f instruction, including specialized co n ten t know ledge use (Sowder, et al., 1998). H ence, in conclusion, a prim ary goal o f this study is to provide a rich and cohesive picture o f how specialized content know ledge is used in the classroom , b o th furthering the understanding o f middle school teachers’ SCK and o f the role o f teachers’ beliefs in the use
o f this knowledge. Only by furthering research in b o th areas will w e have a m ore holistic understanding o f how specialized co n ten t know ledge is used, addressing the intial concerns o f preparing quality mathematics teachers in the future.
C H A P T E R 2
C O N C E P T U A L F R A M E W O R K
T he purpose o f this study is to further the understanding o f tw o key types o f knowledge middle school teachers hpld: m iddle school teachers’ specialized co nten t knowledge (SCK) and pedagogical co n ten t know ledge beliefs, as well as the interaction betw een the tw o in practice. T he first p art o f C hapter 2 consists o f a discussion o f how the existing body o f research on m athem atics teachers’ specialized c o n te n t knowledge currently attem pts to explain the content and organization o f this know ledge and the role o f this knowledge in quality m athem atics teaching, fram ing the first research question: What specialised content knowledge do middle school mathematics teachers hold? In particular, how does this knowledge differ from common content knowledge?
I begin with a discussion o f how cu rren t research th at attem pts to explain w hat constitutes SC K as it presents itself in quality m athem atics teaching and its relationship to student learning. Before I proceed, it is necessary to note w hat is m ean t by quality
mathematics teaching as defined formally by Hill and her colleagues at th e University o f Michigan and the H arvard G raduate School o f E ducation (2010), particularly how it is defined in relation to the Mathematical Quality of Instruction (M QI) observational instrum ent. T he M Q I instrum ent, in particular, was w ritten to identify im p o rtan t dim ensions o f quality classroom m athem atics teaching show n to have a significant im p act o n student learning (Ernest, 1989; Charalam bous, 2006; Hill, et. al, 2008; Hill, et. al, 2010; K ersting, 2008; Sowder, et al. 1998; Stigler, 2009; Swafford, et al., 1997) and includes the following characteristics: the richness o f the m athem atics, student participation in m athem atical
reasoning and meaning-making, and the clarity and correctness o f th e m athem atics covered in class, am ong others.
T h e chapter discussion will then m ove to a description o f th e w ork done by the LM T Research group at the University o f M ichigan, in creating a practice-based m odel o f m athem atics knowledge in teaching. T h e discussion will include all the dom ains and topics in the M athem atics K nowledge for T eaching M odel (M K T M odel); how ever, the focus o f this study is on a specific type o f know ledge m athem atics teachers hold, specialised content knowledge. O ne o f the m o st significant findings in research involving the M K T M odel was that specialized content knowledge is a special type o f know ledge specific to the profession o f mathematics teaching, yet different than m athem atics know ledge non-teachers hold. F or example, a teacher w ho can understand and teach procedures involving operations on fractions may n o t have the specialized know ledge to explain the conceptual basis for the operations or the knowledge that allows for com m unicating, such know ledge to third-grade students (Hill, 2007).
In the second part o f C hapter 2 , 1 discuss the conceptual fram ew ork behind my second research question: How are middle school algebra teachers’ pedagogical content beliefs manifested in specialised content knowledge use? I consider this additional critical co m p o n en t o f teachers’ knowledge heavily influencing m athem atics teaching; teachers’ pedagogical content knowledge beliefs. There is a general consensus am ong researchers in m athem atics education that m athem atics teachers’ pedagogical co n ten t know ledge beliefs play a m ajor role in guiding their classroom behavior an d therefore influence stu d en t learning (Clark & Peterson, 1986; Ernest, 1989; T h om p so n , 1992; Fang, 1996; W ilson & Cooney, 2002). I provide an overview o f research in this area, focusing on understanding the relationship betw een m athem atics teachers’ specialized co n ten t know ledge and the role o f teachers’
pedagogical beliefs in the use o f this knowledge. In particular, I p o in t o ut a lack o f research base in this highly complex area.
Lasdy, each o f the tw o parts o f C hap ter 2 conclude w ith som e additional limitations o f current research in each o f the tw o areas and how the cu rren t study addresses these limitations. B oth sections intend to em phasize the need to u n derstan d b o th teachers’ specialized content knowledge and pedagogical co n ten t know ledge beliefs as well as the interplay betw een both, underlining the im portance o f b o th in providing a holistic picture quality mathematics teaching.
I: Understanding SCK and the O rganization o f MKT
A lthough there is little disagreem ent am ong stakeholders in education that quality m athem atics teaching relies significantly o n th e m athem atics know ledge teachers hold, research in this area over the past 25 years dem onstrates a continuing struggle to define and conceptualize such knowledge. Interest in teachers’ co n ten t know ledge dates back to the 1960s. Process-product research, also know n as the educational p ro d u ctio n function literature o f the sixties and seventies, was a beginning attem p t at understanding and potentially measuring teachers’ m athem atical knowledge, as well as establish a relationship betw een this knowledge and student learning (Ball, 1990). D uring this time, teachers’ m athem atics knowledge was typically defined and som etim es m easured as a com posite o f the num ber o f m ath courses teachers had taken after Calculus, the n u m b er o f m ath m ethods courses taken, and a m ajor o r m ino r in m athem atics. Begle (1979), w hile w orking as a professor w ithin b oth the m athem atics and education departm ents a t Stanford, conducted a meta-analysis o f studies betw een the sixties to the m id-seventies th a t focused on
investigating the relationship betw een m athem atics teacher know ledge and student learning; a review that encom passed m uch o f the process-product research o f this time. T h e results
w ere inconsistent. Begle’s results concluded th a t the n um b er o f m ath courses taken was positively associated with student achievem ent in only 10% o f the studies reviewed, calling into question the role o f teachers’ pure subject m atter know ledge in student learning.
T he process-product research reviewed by Begle (1979) did n o t m easure m athem atics teachers’ knowledge direcdy, a practice that by and large led to im precise definitions o f teachers’ knowledge and m isspecification o f the causal processes linking teachers’ knowledge to students’ learning (Rice 2003). Rice also review ed a wide array o f studies exam ining m athem atics teacher characteristics and the im pact o n student learning during a substantial period o f time after Begle. As w ith Begle’s 1979 review, Rice’s 2003 review resulted in an array o f literature th at looked at teacher degree level, level o f
coursework, and other proxy measures o f teacher know ledge w ith m ixed results. In Rice’s review, findings indicated th at teacher coursew ork in b o th m athem atics and pedagogy resulted in increases in student perform ance, yet this result was n o t consistent across grade levels. Rice also found that methods courses, i.e., those courses th at taught a com bination o f co nten t and pedagogy, consistendy co ntributed to teacher effectiveness at all grade levels included in her review.
D r. Lee Shulman also began a long quest for understanding the role o f content knowledge and its use in teaching, b u t from a different perspective, beginning in the eighties. Shulm an’s interest in teacher knowledge was related to his deep com m itm ent to teaching as professional work. Shulm an specifically referred to teaching as a profession and as w ith other professions, teaching requires a specialized know ledge base (Shulman, 1987, 2000, 2005). Shulm an’s work in this area has been primarily conceptual an d quite substantial, as he has been cited frequently over the years and his w ork led to the developm ent o f a fram ework for the N ational B oard for Professional T eaching Standards (Shulm an, 1987, 2000, 2005).
Shulm an’s (1987) w ork drew initially from the w ork o f J o h n D ew ey, in the essay The Child and the Curriculum w ritten in 1902, w ho differentiated betw een knowledge o f the scientist (referred to by Dewey as logical understanding) versus know ledge for teaching (psychological understanding, according to D ew ey [2009]). U n d er this assum ption, Shulman introduced the idea o f pedagogical content knowledge (PCK). H e explains th at this type o f
knowledge (PCK) goes beyond core c o n ten t know ledge and can be th o u g h t o f as the “ special amalgam o f co n ten t and pedagogy that is uniquely the province o f teachers, their ow n special form o f professional understanding” (Shulman, 1987). P C K can be thou ght o f as m ore than knowledge o f a discipline and its facets; P C K is know ledge used in the practice of teaching. PC K consists o f the m ost useful form s o f representations, analogies, examples and explanations m o st useful for learning the key topics in m athem atics. Shulm an states, “ ...in a w ord, [PCK represents] the ways o f representing and form ulating th e subject that m ake it com prehensive to others” (Shulman, 1986, p. 9). Shulm an’s definition o f P C K also includes the m isconceptions students are likely to have, ho w different representations are interpreted by students and how backgrounds o f know ledge students bring w ith th em influence new learning experiences (Shulman, 1986).
A lthough Shulm an’s ideas captured the essence o f the type o f know ledge researchers were trying to understand, researchers have used Shulm an’s fram ew ork to define the term P C K in varying ways over the years. F or exam ple, L einhardt and Sm ith (1985) provided a conceptual fram ework for teacher knowledge, focusing o n lesson structure know ledge and subject m atter knowledge. T he categories w ere fu rth er divided depending o n different knowledge types o f the expert and novice teacher. A lso relying on Shulm an’s work, N athan and Petrosino (2003) found that 48 pre-service secondary m athem atics teachers differed in their understandings and judgm ents o f stu dent difficulties depending o n the teachers’ level
o f m athem atical training. T h at is, teachers w ith m ore advanced training in m athem atics had views o f student perform ance o n algebraic reasoning that differed significandy from actual student perform ance patterns. D espite ongoing w o rk in the area o f teacher knowledge, the concept o f teacher knowledge rem ained a roughly defined yet an im p o rtan t area o f research. T he K now ledge o f M athem atics for T eaching A lgebra P ro ject (Ferrinini-M undy, et al, 2003) involves the creation o f a conceptual fram ew ork for teachers’ algebraic knowledge as well as the developm ent o f K A T (K nowledge o f algebra for Teaching), an assessm ent o f teachers’ algebraic knowledge. This w ork has also furthered the u nd erstanding o f the m athem atics knowledge used in teaching in specific areas o f algebra, yet is limited, however, by its focus on core algebra topics w itho ut the necessary key background concepts used in rem ediation during the teaching o f algebra such as n um ber sense (including fractions and other essential Algebra tools). M iddle school teachers, including th o se teachers teaching algebra, spend a significant am ount o f tim e rem ediating in these areas (see Sleeman et al., 1989; Taylor & Francis, 1990); thus it is essential to understand such know ledge and its use.
M athem atics knowledge for teaching has b een investigated in these many different ways over the last several decades. C haralam bous (2006) com pleted an exhaustive review o f such studies, including w ork involving th e m iddle school teacher population. I now state the findings particular to the middle school teacher population:
• Teacher knowledge that is primarily superficial and p ro cedu ral substantially
constrains a teacher from providing a classroom en v iron m en t w ith the dim ensions o f quality m athem atics teaching (Borko et al., 1992; E isenhart, B orko, U nderhill, Jones et al., 1993; Sowder et al., 1998, Stein, B axter and L einhardt, 1990; Swenson, 1998).
• Teachers’ strong m athem atical know ledge can provide necessary and exceptional
support in providing a classroom en viro nm en t w ith the dim ensions o f quality teaching (Ball, 1992; Charalam bous, 2006; Lloyd & W ilson, 1998; Lubinski, 1993).
A lso along these lines, Swenson (1998) assessed four m iddle school teachers’ m athem atical knowledge and classroom teaching thro ug h pre-observation interview s followed by video taped lessons o f middle school co n ten t know ledge items. F or these particular middle school teachers, they lacked explicit and connected know ledge, held traditional views ab ou t teaching and learning o f m athem atics (teaching by telling), lacked the im p o rtan t “big ideas” necessary for the content covered and held litde understanding o f potential m isconceptions students hold, generally lacking the knowledge and skills needed to provide a classroom environm ent consistent w ith quality m athem atics teaching.
T he Learning M athematics for T eaching R esearch Group and the MKT M odel■ ■■■■■■■■...I,... ■ ■ ' i. n... ... ...
In 2008, after reviewing the above inquiries involving m athem atics knowledge for teaching, Hill, Schilling and Ball offered:
“D espite this wealth o f research, w e argue th a t the m athem atical content teachers m ust know in order to teach has yet to be m apped precisely.. .past m ethods lack the pow er to propose and test hypotheses regarding th e organization, com position and characteristics o f co nten t know ledge [core c o n ten t and pedagogical] for teaching”
(p. 18).
D eborah Loew enberg Ball and her colleagues at the University o f M ichigan and beyond, the Learning M athem atics for Teaching research team , have been investigating the idea o f m athem atics knowledge in teaching (M KT) for over a decade using a m ore direct m easure o f teacher knowledge; knowledge as captured in the practice o f teaching.
T he LM T research team ’s direct approach stem s fro m their concerns th at current theories on m athem atics teachers’ know ledge w ere conceptually based versus practice-based (Ball, 1990). T he LM T research group presents an effective argum en t th at understanding and conceptualizing this knowledge w ould be b est accom plished by first looking at the practice o f quality mathematics instruction (Ball, Schilling & Hill, 2004). In fact, their conceptualization o f m athem atics knowledge ro o ted in quality instruction is a central and
unique aspect o f their w ork, likely an aspect that has resulted in years o f success in furthering the knowledge o f MKT.
Initial interest in the project was driven by a variety o f reasons, including a 1988 study by Ball w ho developed interview questions th a t revealed significant inadequacies o f in- service and pre-service teachers’ know ledge o f im po rtan t m athem atical topics necessary for quality teaching (Ball, 1990). Additionally, Ball (1990, p. 252) fou nd in a large, university- based study o f 252 preservice teachers, th at teachers “b ro u g h t w ith th em to teacher education from their precollege and college m athem atics experiences, understandings that tended to be rule-bound and thin.” O n e m ight argue that this w ould be resolved by teachers taking m ore m athem atics courses. H ow ever, the participants from this study w ere secondary teachers having m ajored in m athem atics w ith substantial coursew ork in m athem atics prior to entering the teacher education program (Ball, 1990). Also, elem entary teachers w ith a greater n um ber o f m ath classes in the same study w ere found to have similar issues in their lack o f conceptual understandings in the same topic areas as the elem entary teachers having taken fewer classes in m athem atics (Ball, 1990).
Highly concerned with these m atters, Ball continued h er interest in this area in the following years w ith her w ork and the w o rk o f h er colleagues culm inating in the
developm ent o f the M K T m odel and the M K T survey tool, discussed in the research article Content Knowledge for Teaching: What M akes it Special? (Ball, T ham es & P helps, 2008). T he M K T m odel and its survey tool are based u p o n a practice-based conceptualization o f m athem atics teachers’ knowledge and includes tw o m ain dom ains, Subject M atter
K now ledge (otherwise know n as co n ten t knowledge) and Pedagogical C o n ten t K nowledge. E ach dom ain includes teacher know ledge vital to quality teaching. C ore com m on knowledge, for example, “is the m athem atical know ledge and skill u sed in settings o ther than
teaching” (Ball, Tham es & Phelps, 2008). T eachers need to be able to do basic
com putations and com parisons w ith num bers. T hese skills are n o t necessarily distinct from oth er educated adults o r even m athem aticians. A lthough a vital co m p o n en t o f teacher knowledge, this type o f knowledge is n o t sufficient for quality teaching. Specialized content knowledge, the focus o f this study to be described in m ore detail below , is the m athem atical knowledge and skills unique to teaching.
It is im po rtan t to note, particularly in relation to the first p art o f this study, that specialized content knowledge is closely tied to, and som etim es undifferentiable, from com m on content knowledge. A n exam ple discussion given by Ball, T ham es and Phelps (2008) involving a teacher choosing a num erical exam ple useful in investigating a student’s understanding o f decimal num bers dem onstrates th e distinction betw een C C K and SCK. Teachers m ust first be able to choose and o rd er the decim al num bers (CCK), then choose the list in such a way that it will bring the key concepts to the fo refro n t necessary for learning (SCK).
Ball includes three other categories o f m athem atical teacher know ledge beyond C C K and SCK, building o n Shulman’s earlier w ork. K now ledge o f c o n ten t and students (KCS) encom passes the intersection betw een teacher’s know ledge o f m athem atical ideas and how students com e to understand these ideas. S tudent co m m o n m isconceptions and
m athem atical thinking w ould be housed here. M isconceptions students typically have, such as confusion with parentheses, require different representations and explanations in teaching. T h e dom ain K now ledge o f C o ntent and T eaching (KCT) encom passes knowledge used in choosing examples, sequencing instruction, choosing instructional form ats, and finding representations m ost likely to be accessible to a certain grade level. I t is im p ortant to stress the m ulti-faceted interrelatedness o f each know ledge category.
T he Em ergence o f Specialized Content K now ledge (SCK)
A round the year 2000, the LM T research group continued th e conceptualization o f M K T and began the developm ent o f survey m easures o f M K T . A distinctive dom ain o f subject m atter knowledge surfaced during the L M T ’s w ork; specialised content knowledge. Specialized content knowledge for teaching (SCK) is described by Ball, Phelps & T ham es (2008, p. 40) as “ the m athem atical knowledge and skill unique to teaching.” T h at is, teachers use a type o f decom pressed m athem atical know ledge (SCK) to look for patterns in student errors, examine student solutions to see if the solution will w ork in o th er similar situations, and provide an explanation as to why a particular stu den t solution w orks or not.
The identification and description o f Specialized C o n te n t K now ledge was a m ajor advance in the understanding o f m athem atics teachers’ know ledge necessary for quality instruction. A particularly notew orthy finding is th at specialized know ledge for teaching m athem atics at the elementary school level exists indepen den dy fro m C om m on C on ten t K nowledge (CCK). Ball et al. (2005) confirm ed this finding empirically; th at is, the M K T survey developed was used in finding th at M K T SC K survey item s w ere statistically separable from M K T C C K item s through the use o f confirm atory factor analysis, am ong other related analyses (Ball e t al., 2005; Hill, 2007).
SCK is clearly vital for quality m athem atics teaching. T eachers need m ore than com m on content knowledge. Teachers m u st also possess m athem atical knowledge that “goes beyond w hat is needed to carry o u t an algorithm reliably” (Ball et al., 2005, p. 22). Teachers spend m ost o f their time “interpreting som eone else’s error, representing ideas in multiple forms, developing alternative explanations and choosing a usable definition” (Ball, 2003, p. 8). “Teaching quality m ight n o t relate so m uch to perform ance o n standardized tests on m athem atics achievem ent as it does to w hether teachers’ know ledge is procedural or
conceptual, w hether it is connected to big ideas o r isolate into small bits, or w hether it is com pressed or conceptually unpacked” as in the form o f SC K (Hill & Ball, 2004, p. 332). M athematics education researchers agree th at SCK, o r lack thereof, strongly affects teaching quality and ultimately student achievem ent (Ball & W ilson, 1990; G raeber, 1999; Hill, 2007; Lee et al., 2003).
In fact, teachers w ith SCK know ledge and skills can develop unpacked knowledge o f m athem atical knowledge (multiple representations o f core ideas, different interpretations o f mathem atical operations), develop detailed know ledge o f classroom m athem atical practices (using m athem atical language, providing rich m athem atical experiences for students), explain student thinking and m ove students’ forw ard in obtaining understanding based up on the students’ ow n thinking, proving, posing questions, explaining representing (Ma, 1999).
T o understand C C K and SCK, and the differentation betw een the tw o, w hich relates to my first research question, I chose the technique o f confirm atory factor analysis and related analyses to attem pt to clarify the distinction betw een C C K an d SCK. In order to proceed to analysis o f P art I o f my study, it was necessary to review M K T w ork done specifically at the middle school level in understanding C CK , SCK and the differentation betw een the two.
T he D evelopm ent o f the M iddle School L evel MKT Survey: Measuring- SCK
Building u p o n the w ork done by the LM T research team at the elementary school level, Hill (2007) developed the middle school M K T survey. In doing so, Hill relied u p o n the w ork that was done at the elem entary level as a foundation. H ence, I first discuss the relavant w ork that was done in building the elem entary level M K T survey. T he initial M K T item -writing team included m athem atics education researchers, psychom etricians, teachers and other professionals, pulling from their ow n extensive experience in m athem atics
education and m o st im portantly, teaching, to create the M K T survey questions based on essential m athem atical topics (Ball, Hill & Schilling, 2004). T h e goal was to capture the essence o f teacher knowledge in quality teaching, basing m uch o f their w ork on previous research show n to correlate w ith student learning (Abell, 2008; B lanton & K aput, 2005; Carpenter, Fennem a, & Franke, 1996; Rasm ussen & M arrongelle, 2006). As H eather Hill described in an O cto b er 2010 M K T w orkshop ho sted at the H arvard G raduate School o f Education:
“W e w atched hours and hours o f thousands and thousands o f teachers teaching from across the country. T hese w ere videotapes that m ultiple team s o f researchers in California and Michigan have accum ulated over 10 years, teachers w ith a diverse set o f backgrounds in a diverse set o f schools and school populations. These videos captured teachable m om ents in m athem atics classroom s th a t expert teachers wait a lifetime o r m ore to be exposed to ” (Personal com m unication, N o v em b er 3, 2010). T here were initially 138 items developed for the M K T survey w hich were further subdivided into four topic types (num ber concepts, operations, and patterns, functions and Algebra). A lthough it was n o t initially clear how item s w ere categorized, through extensive exploratory and confirm atory factor analysis and a significant o f tim e rewriting, the items factored into the dom ains and topics show n in Figure X on page 17, the hypothesized M K T m odel (Ball, Phelps & Tham es, 2008).
T he M K T surveys were initially piloted in C alifornia’s M athem atics Professional D evelopm ent Institutes including 40 sites serving 23,000 K -12 teachers. T h e sample participants were paid for their w ork over several w eek-long professional developm ent sessions. Three form s o f the test w ere adm inistered: 640 participants too k form A, 535 took form B and 377 took form C. Item s o n all three form s tended to perfo rm consistently across the three form s in the factor analysis w ith m ino r exceptions (Hill, Ball & Schilling,
2004).
T he LM T research group also addressed the issues o f reliability and validity
throughout this time. First, the LM T research group obtained access to the teachers in th e above sample in a form at m ore suitable for understanding the validity o f the M K T co n ten t knowledge; the classroom (Hill, et al., 2008). Additionally, teachers participated in structured think-aloud M K T task sessions to w ork through each o f the survey tasks to verify that survey task knowledge did in fact reflect know ledge teachers use in the classroom . These interviews led to significant insights into teacher thinking and the use o f M K T (Ball, 1990). A fter this was com pleted, th e surveys w ere reevaluated w ith a team o f psychom etricians to rectify any outstanding issues with reliability an d validity. Since 2001, the LM T research group has continued its w ork with psychom etricians, practicing teachers, and m athem atics educators from across the country to expand and revise M K T survey item s, reevaluating the reliability and validity o f the survey measures. M ultiple nationally representative samples o f teachers in multiple contexts have been used to ensure reliability an d validity (Hill, et al., 2008).
T h e best argum ent for the use o f the elem entary level M K T survey as a partial foundation for the middle school M K T survey, how ever, com es from how it has been used to uncover consistent findings involving understandings o f teacher know ledge, teacher education, and student learning. F or example, H eather Hill and her colleagues found that teachers’ M K T did relate to increases in students’ perform ance after controlling for key student and teacher-covariates (including stud ent socioeconom ic status, teacher’s credentials and experience) through the use o f the M K T survey to ol o n a large national representative sample o f bo th students and teachers (Hill et. al, 2005). S tudents taug ht by teachers in the top M K T quartile gained tw o weeks o f instruction com p ared to their counterparts taught by teachers w ith average M K T scores (Flill et. al, 2005). Additionally, the effect o f teacher
knowledge was investigated in relation to students’ socioeconom ic status, finding that “while teachers’ m athem atical knowledge w ould n o t by itself overcom e the existing achievem ent gap, it could prevent the gap from grow ing” (Hill, e t al., 2005, p. 44).
Additionally, Hill and Ball (2004) found gains in M K T for teachers w ho participated in professional developm ent program s focused on m athem atics teaching m ethods.
Specifically, the gains were larger for teachers w ho participated in program s th at focused m ore on proof, analysis and use o f representations than o th er program s available. C urrent studies are under way to replicate these above findings (G eoffrey Phelps, LM T Research Team , personal com m unication, N o v em b er 14, 2010). T h e M K T survey continues to be used to dem onstrate a relationship am ong M K T , quality instruction (M QI) and student learning based at multiple research cites via in depen den t research groups (Hill, et. al, 2008; Hill; Um land, Litke & Kapitula, 2010).
O f the items previously w ritten from the elem entary level M K T survey, tw o concepts were chosen for the construction o f the m iddle school survey: (1) num bers and operations and (2) patterns, functions and algebra. M ath educators, mathem aticians, professional developers, the research project group and cu rren t and form er teachers constructed additional middle school survey item s (Hill, 2007). T h e majority o f the item - writers were the same professionals w ho w rote the initial elem entary level M K T survey items. K eeping in line with the elem entary school w ork, item w riters drew on their extensive experience in and knowledge o f teaching m athem atics, cutting edge research on m athem atics knowledge used in teaching, and observing classroom instruction w hile w riting and
reviewing items over a year’s period o f time (Hill, 2007).
T h e middle school M K T survey item s fell in to tw o categories: num bers and operations, and patterns, functions and algebra. T hese tw o categories w ere chosen for
several reasons. A n estimate from the T h ird International F IX (according to Hill, 2007) shows that 40% o f eighth grade lessons in the U nited States focus o n num bers and
operations (Peak, 1996). C oncepts included in Patterns, F unctions and A lgebra also play a critical role in the middle school years (H ackenburg, 2005; Stephens, 2007; N C TM , 2010). H ence, the 2005 M K T Survey resulted in 92 item s and fell approxim ately evenly betw een the two concepts o f num bers and operations and patterns, functions an d algebra, in specific areas such as: whole, rational and integer num bers and operations, ratio, p ro p o rtio n and percent, radicals, linear, quadratic and exponential functions, A lgebraic expressions and equations, absolute value and inequalities (Hill, 2007).
As discussed earlier, the delineation betw een C o m m o n C o n te n t K now ledge (CCK) and Specialized C on tent K nowledge (SCK) is n o t always clear. W hen Hill (2007)
adm inistered the middle school M K T Survey on a smaller subset o f a dem ographically representative population o f approxim ately 1,000 m iddle school teachers in the U nited States, scales created to represent the SCK and C C K theoretical constructs “w ere correlated at .79, w here .81 w ould be a perfect correlation, accounting for m easurem ent error. T hese strong correlations, along w ith the factor analysis results, suggest a o ne-factor m odel” (p. 73). D espite these initial findings o f a one-factor m odel, there is com pelling research to suggest otherwise (Hill, D ean & G offney, 2006; Hill & Ball, 2004; Hill et al., 2006; Hill, Rowan & Ball, 2005). In my study, I will use techniques similar to H ill’s (2007) in attem pts to understand and clarify the organization o f C C K and SCK. In particular, I will use confirm atory factor analysis and related techniques to differentiate betw een C C K and SCK, if possible, within the current M K T m odel. I will reflect u p o n these results and the overall conceptualization o f the M K T m odel in attem pts to revise the original m odel to clarify the organization o f M KT, specifically the im p o rtan t C C K and SC K distinction.
What Is There Still to Learn about the Content and O rganization SCK at the M iddle School Level?
D espite the extensive w ork o f the LM T rese arch g roup and others researching in the area o f m athem atics knowledge use in teaching, there is further w ork to be done to fully uncover and understand the m ulti-faceted and com plex specialized co n ten t knowledge effective m athem atics teachers hold. In particular, the first notable lim itation is that m o st o f the capturing o f the specialized knowledge has previously focused o n the elementary school level. U nderstanding teachers’ specialized co ntent know ledge at the m iddle school level, and in particular o f Algebra, is a critical. H ence, in the first p art o f my study, I focus specifically o n understanding teachers’ M K T use o f topics that are key in the m iddle school curriculum, including topics central in the study o f Algebra.
A n additional need is to farther verify the organization o f the hypothesized M K T m odel w ith a focus on the subdom ains SC K and CCK . T h e difference has been established
empirically at the elementary school level (Hill, D ean & G offney, 2006; Hill & Ball, 2004; Hill et al., 2006; Hill, Row an & Ball, 2005). T h e delineation o f the tw o constructs SCK and CC K is n o t as clear at the middle school level (Hill, 2007). H ence, in my study I will use datasets gathered by Hill (2007) to readdress the hypothesized delineation betw een SCK and C C K through similar confirm atory factor analysis (and related techniques) used by Hill (2007). I will use these results to inform the cu rren t M K T m odel, w ith respect to understanding C CK , SCK and the differentiation betw een the tw o constructs.
Specialized content knowledge is n o t used in isolation and is m ore often than n o t enm eshed w ith o th er key knowledge that influences teaching. O n e o f the m o st significant forms o f knowledge influencing teaching practice is teacher pedagogical co n ten t beliefs. In order to obtain a greater understanding o f SCK use, I investigate SC K in the practice o f
teaching with the consideration o f this m ajor teacher filter in decision-making: teachers’ pedagogical con tent beliefs. I now turn to fram ing my second research question: How are pedagogical content beliefs, an additional component o f teacher knowledge, manifested in specialised content
knowledge use?
II: Understanding Specialized Content K now ledge and the R ole o f Teachers’ Beliefs in Practice
Specialized content knowledge is n o t used in isolation and is m ore often than n o t enm eshed with o th er key types o f know ledge th at influence teaching, such as teachers’ pedagogical content beliefs. In fact, teachers’ pedogogical co n ten t beliefs have been
regarded for decades as critical to the reform o f m athem atics teaching practices (A nderson & Piazza, 1996; Battista, 1994; Cooney & Shealy, 1997; E rnest, 1989). Specifically, teacher pedagogical con tent beliefs are thought to play a substantial role in how content knowledge is used in the classroom (Beswick, 2008; Buehl & Fives, 2009). H ow ever, few studies have investigated teacher beliefs along w ith the use o f m athem atics c o n ten t knowledge, including SCK, due to the highly com plex nature o f both. H ence, in P art II o f my study, I have chosen to use qualitative m ethods to explore and explain teachers’ beliefs as teachers use specialized co n ten t knowledge in the classroom . I begin by discussing past and current research in the area o f pedagogical co n ten t beliefs.
D espite limited understanding o f how teacher pedagogical c o n ten t beliefs (PCK beliefs) actually influence instruction and varying results in research o n teacher beliefs, there remains a general consensus that P C K beliefs play a m ajor role in guiding their classroom behavior and therefore influence student learning (Clark & P eterson, 1986; E rnest, 1989; Fang, 1996; T hom p son , 1992; W ilson & Cooney, 2002). Past research dem onstrates that we do n o t fully understanding the role o f teachers’ beliefs w ithin the com plex context o f the
classroom. T o begin, there is n o agreed u p o n definition o f beliefs across research in this area (McLeod & M cLeod, 2002). M ost broadly, the term belief is defined in research on teacher education as “a psychologically held understanding, prem ises o r p rop ositio n abo ut the w orld that is felt to be true” (Richardson, 1996).
A lthough som e conceptualizations o f teachers’ beliefs treat beliefs and know ledge as being entirely different, there are several conceptualizations th at p rovide a strong argum ent for teachers’ beliefs as teachers’ know ledge (Ernest, 1989; Leatham , 2006). As Leatham (2006, p. 7) states, having done extensive research in the area o f m athem atics teachers’ beliefs:
O f all the things we believe, there are som e things th a t we “ just believe” and other things we “m ore than believe — w e know .” T h o se things we “m ore than believe” we refer to as know ledge and those things w e “just believe” we refer to as beliefs. T hus beliefs and know ledge can profitably be viewed as subsets o f the things we believe.
T he classification o f teachers’ beliefs as know ledge is one o f the many com plexities in this area o f research. A n additional controversy involves the nature o f the link betw een teacher beliefs and practice, w ith som e authors reporting consistency betw een teacher beliefs and practice (e.g., Stipek, G iw in , Salm on & M acGyvers, 2001; T h o m p so n , 1984) and others finding inconsistent relationships betw een the tw o (Cooney, 1985; Shield, 1999). M ost agree, however, th at the beliefs-to-practice connection can be clarified in a consistent m anner, if contextual factors influencing the en actm ent o f beliefs are taken into
consideration. A ccording to H andal (1995), som e constraints in th e school system such as the school com m unity, school adm inistration, o r classroom environm ent, may be reasons why inconsistencies occur betw een teacher beliefs and their actual teaching practice.
The heart o f the inconsistiencies ultimately fall o n the actual definition o f teacher beliefs and teacher belief systems. In the seventies, G reen (1971) created a conceptual fram ew ork
for teachers’ beliefs, a fram ework that continues to b e used effectively in understanding teachers’ beliefs across disciplines to date (Beswick, 2005; H andal, 1995; M cleod, 1998). A ccording to G reen (1971), teachers h o ld systems o f beliefs th a t are highly complex. This is because teacher beliefs are n o t held in isolation from each o th e r b u t are inter-related in com plicated ways. Further com plicating the understanding o f such systems, teachers may or may no t be conscious o f the beliefs they hold. Also, individual beliefs may take on a variety o f forms; a belief may be fact or opinion or an attitude th a t is m anifested as a belief,
ultimately taking the form o f knowledge in w hen a b elief is p u t into action (Liljedahl, 2008). G reen (1971) identifies three dim ensions o f teacher beliefs systems th at continue to be vital in investigating teacher beliefs:
• T here is a logical relation betw een beliefs (beliefs are prim ary or a derivative o f other beliefs).
• Relations betw een beliefs are influenced by the strength o f beliefs (central beliefs are strongly held, peripheral beliefs are less strongly held in relation to central beliefs).
• Beliefs are organized in clusters (a cluster o f beliefs can be held in isolation from o th er clusters).
G reen ’s first property o f beliefs can be illustrated as follows. C onsider a teacher w ho believes that constructivism is an im p o rtan t philosophy o f teaching to hold. This same teacher m ight also believe that cooperative learning is necessary for the successful application o f a constructivist teaching philosophy. F o r this particular teacher, there is a logical relationship between these beliefs. T h a t is, if a teacher believes in this philosophy, there is likely a set o f beliefs that logically follow from this belief.
