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© Michigan State University 2004 DRAFT: Do not cite or quote 1
Mathematics Textbooks forProspective Elementary and
Middle School TeachersRaven McCroryMichigan State University
Helen SiedelUniversity of Michigan
Andreas StylianidesUniversity of Michigan
© Michigan State University 2004 DRAFT: Do not cite or quote 2
Context of the Study• Agreement on the overall inadequacy of
elementary and middle school teachers’mathematical knowledge
• Need a better understanding of the mathematicselementary teachers need to know and how theycan learn it
• Current research, national policy documents, andprofessional standards offer new views ofmathematical knowledge for teaching
© Michigan State University 2004 DRAFT: Do not cite or quote 3
Questions
What mathematics are we teachingprospective elementary and middleschool teachers in undergraduatecourses?
What mathematics are they learning?
Who teaches this mathematics:Education, Mathematics, or both?
© Michigan State University 2004 DRAFT: Do not cite or quote 4
Question
What do prospective elementary and middleschool teachers have an opportunity to learnin their undergraduate mathematicseducation?
© Michigan State University 2004 DRAFT: Do not cite or quote 5
Plan for the Presentation
• Overview of the study– Overall scope– Textbook analysis, methods and examples
• Analysis: Multiplication of Integers• Analysis: Reasoning and Proof• Conclusions
– Textbook analysis results– Mathematical knowledge for teaching
© Michigan State University 2004 DRAFT: Do not cite or quote 6
Parts of the Study• Identify and analyze mathematics textbooks• Interview textbook authors• Analyze state requirements, state and national
policies, professional standards• Survey instructors• Interview instructors• Review international textbooks and requirements• Investigate the history of such textbooks• Identify and catalog methods textbooks
© Michigan State University 2004 DRAFT: Do not cite or quote 7
Analysis of Textbooks
• Undergraduate mathematics textbooks• For prospective elementary and middle
school teachers
• For courses usually taught in mathematicsdepartments
• For one, two, three, or more semesters,depending on state requirements
© Michigan State University 2004 DRAFT: Do not cite or quote 8
• 18 Books identified– 2 self-published– 2 preliminary editions
• These are all such books currently in print
Analysis of Textbooks: Texts
We are currentlypolling publishers to
be sure we have foundthem all.
One we found late --some slides refer to 17
books
© Michigan State University 2004 DRAFT: Do not cite or quote 9
Bassarear, Tom. 2005. Mathematics for elementary school teachers. 3rd ed:Houghton Mifflin.
Beckmann, Sybilla. 2003. Mathematics for elementary teachers. 2 vols.Bennett, Albert, and Leonard T. Nelson. 2003. Math for Elementary teachers: A
conceptual approach, Sixth Edition
Billstein, Rick, Shlomo Libeskind, and Johnny W. Lott. 2001. A problem solvingapproach to mathematics for elementary school teachers. 7th ed.
Center for Research in Mathematics and Science Education. 2000. Mathematics201. 2 vols.
Darken, Betsy. 2003. Fundamental Mathematics for Elementary and Middle SchoolTeachers: Kendall/Hunt.
Devine, Donald F., Judith Olson, and Melfried Olson. 1991. Elementarymathematics for teachers.
Jensen, Gary R. 2003. Fundamentals of Arithmetic
Jones, Patricia, Kathleen D. Lopez, and Lee Ellen Price. 1998. A mathematicalfoundation for elementary teachers.
http://www.msu.edu/~ravenmw/
© Michigan State University 2004 DRAFT: Do not cite or quote 10
Krause, Eugene F. 1991. Mathematics for elementary teachers: A balancedapproach.
Long, Calvin T., and Duane W. DeTemple. 2003. Mathematical reasoningfor elementary teachers. 3 ed
Masingila, Joanna O., Frank K. Lester, and Anne M. Raymond. 2002.Mathematics for elementary teachers via problem solving
Musser, Gary L., William F. Burger, and Blake E. Peterson. 2002.Mathematics for elementary school teachers: A contemporary approach.6th ed.
O'Daffer, Phares G., Charles, Cooney, Dossey, & Schielack. 2002.Mathematics for elementary school teachers.
Parker, Thomas H., and Scott J. Baldridge. 2003. Elementary mathematicsfor teachers (Volume 1).
Sgroi, Richard J., and Laura Shannon Sgroi. 1993. Mathematics forelementary school teachers: Problem-solving investigations
Sonnabend, Thomas. 1997. Mathematics for elementary teachers: Aninteractive approach
Troutman, Andria P., and Betty K. Lichtenberg. 2003. Mathematics: A goodbeginning. 6th ed
© Michigan State University 2004 DRAFT: Do not cite or quote 11
Textbooks
Number of Chapters Average Chapter LengthBassarear (2001) 10 59 (38-98)Beckmann (2003, 2 volumes) 12 66 (16-124)
Bennett & Nelson (2004) 11 72 (40-95)Billstein et al. (2004) 12 69 (53-87)CRMSE (2000-2001) 19 20 (8-81)
Darken (2003) 12 57 (32-98)Devine et al. (1991) 14 57 (26-87)Jensen (2003) 9 40 (14-68)
Jones et al. (2000) 8 38 (30-65)Krause (1991) 16 56 (26-79)
Long & DeTemple (2003) 14 70 (53-87)Masingila et al. (2002) 10 36 (11-71)Musser et al. (2003) 17 52 (31-73)
O’Daffer et al. (2002) 13 54 (42-82)Parker & Baldridge (2003)* 9 25 (16-37)Sgroi & Sgroi (1993) 13 37 (28-64)
Sonnabend (2004) 13 60 (13-95)Troutman & Lichtenberg (2003) 20 25 (2-46)
Average 13.6 49.6 (26.6- 79.8)
*For one semester only
PRELIMINARY Data -- subject to change
© Michigan State University 2004 DRAFT: Do not cite or quote 12
Analysis
• Is the content of these textbooks obvious?• Do they all have the same content?
• How does the content vary?
© Michigan State University 2004 DRAFT: Do not cite or quote 13
TEXT # KEY: # indicates chapter number for topics that correspond to chapters, where the topic is part of the chapter title.T# indicates that the topic is included as a section in chapters (#).A indicates topic is in appendix or supplementary pages.I# means topic appears in the index # times but is not a chapter title or section heading.
Bassarear(2001)
10
59(38-98)
T1
T2
T1
I1
3T 5
5 I2
T5
T5
T5
T5
T5
T6
4 7 T7
8 10
9 T2
T2
T3
T356
6
Beckmann(2003, 2volumes)
12
66(16-124)
1 I2
I1
3 4 5 T2
T2
T2345
T245
T245
T2
T2
T23
6 11
12
79
9 8 10
10T4
T3
I1
T5
Bennett &Nelson(2004)
11
72(40-95)
1 2 2 I1
T3, 5, 6
T3
3 5 5 6 6 I7
T6
4 7 8 9 10
11
2 I12
I7
I3
I9
Billstein etal. (2004)
12
69(53-87)
1 2 T1
T1
T3, 4, 5, 6
3 2 5 4 6 6 6 4 8 7 910
11
12
2 T1
T
356
T
356
T
5
CRMSE(2000-2001, 2volumes)
19
20(8-81)
AT2,3,7,
8
AT3, 7,8
A2
AT4
A6A7
AT8
AT4,6
AT8
A9
B B9
AT5
AT5
Darken(2003)
12
57(32-98)
T2
T1
2 T2
4 5, 6 3 T16
T14
T16
T46
T136
2 7 8 9 10
11T3
12
T4
T4
T15
PRELIMINARY Data -- subject to change
© Michigan State University 2004 DRAFT: Do not cite or quote 14
Analysis of Textbooks: Focus
• Multiplication• Reasoning and Proof
• Fractions (Rational Numbers)
© Michigan State University 2004 DRAFT: Do not cite or quote 15
Multiplication
• A basic topic in elementary mathematics• Cuts across grade levels
• Cuts across number systems– From whole numbers to fractions and decimals
to algebraic expressions
• Conceptually easy in some cases,conceptually difficult in others
Repeated addition forwhole numbers
Multiplication of twonegative integers
© Michigan State University 2004 DRAFT: Do not cite or quote 16
Reasoning and Proof
• A mathematical way of thinking and doing• Emphasized in national standards
• Students and teachers have difficulty withreasoning and proof
© Michigan State University 2004 DRAFT: Do not cite or quote 17
Fractions
• Numbers• Taught later in the elementary curriculum
• A key concept for learning algebra
• Conceptually and procedurally difficult formany teachers and many students
© Michigan State University 2004 DRAFT: Do not cite or quote 18
Analysis of Textbooks: Method
• Review of research on each topic:– How students learn– Particular trouble spots for teaching and learning
• Develop a list of topics, concepts, and proceduresfor each focal area– Content– Topic development– Trouble spots
• “Code” and comment on each book with respectto the list
© Michigan State University 2004 DRAFT: Do not cite or quote 19
Methods: Fractions
• Primary Categories:
– Definition– Sequence– Coverage– Representations and models– Properties– Word problems, examples, & applications– Pedagogy
© Michigan State University 2004 DRAFT: Do not cite or quote 20
Fractions
• Definition (For each book, note whichdefinition is primary, if any)– Number line definition– Set theoretic definition (ordered pair)– Definition only by example -- intuitive
definition– As a number system -- rational numbers– As an operation -- division– Other
© Michigan State University 2004 DRAFT: Do not cite or quote 21
Fractions Example 1Billstein, R., Libeskind, S., & Lott, J. W. (2000). A problem solving
approach to mathematics for elementary school teachers (Seventh Ed.).Addison-Wesley Pub. Co.
Set theoreticdefinition (orderedpair)
N
Definition only byexample -- intuitivedefinition
N
As a number system-- rational numbers
Y As a set of numbers that includes the rational numbers:Rational numbers: “[S]et of numbers of the form a/b, where b ! 0 and a and b
are integers. Moreover, numbers of the form a/b are solutions to equations of theform bx = a. This set, denoted by Q, is the set of rational numbers and is defined asfollows:Q = {a/b"a and b are integers and b ! 0}” (p. 246)
Relation between rational numbers and fractions: “Q is a subset ofanother set of numbers called fractions. Fractions are of the form a/b where b ! 0 but
a and b are not necessarily integers. For example, 1/#2 is a fraction but not a rationalnumber. (In this text we restrict ourselves to fractions where a and b are realnumbers, but that restriction is not necessary.)” (p. 246)
As an operation --division
Y “The rational number a/b [with a on top of b] may also be represented as a/b or asa÷b. The word fraction is derived from the Latin word fractus meaning ‘to break.’” (p.
246)
© Michigan State University 2004 DRAFT: Do not cite or quote 22
Fractions Example 2Krause, E. F. (1991). Mathematics for elementary teachers: Abalanced approach. Lexington, MA, D.C. Heath and Company.
Number line definition N
Set theoretic definition(ordered pair)
Y “A (common) fraction is ultimatelyan ordered pair of whole numberswhose second component isnonzero. For reasons that will bemade clear shortly, the symbol 3/4 ismore appropriate than theconventional ordered-pair symbol(3, 4)." (p. 334)
Definition only byexample -- intuitivedefinition
N
As a number system --rational numbers
N
As an operation --division
N
© Michigan State University 2004 DRAFT: Do not cite or quote 23
Plan for the Presentation
• Overview of the study– Overall scope– Textbook analysis, methods and examples
• Analysis: Multiplication of Integers• Analysis: Reasoning and Proof• Conclusions
– Textbook analysis results– Mathematical knowledge for teaching
© Michigan State University 2004 DRAFT: Do not cite or quote 24
Multiplication of Integers inMathematics Textbooks for
Prospective Elementary and MiddleSchool Teachers
Helen SiedelUniversity of Michigan
© Michigan State University 2004 DRAFT: Do not cite or quote 25
Reasons for Investigation of ThisTopic
• Multiplication is a focus topic for our study
• We observed substantial variation in the presentation ofmultiplication of integers
• The use of real-life contexts and concrete models isawkward
• Negative numbers are challenging for teachers to teachand for children to learn
• More teachers may be teaching about integers
© Michigan State University 2004 DRAFT: Do not cite or quote 26
The Challenge
“Even though models for negative numbers may be lessintuitive to children than models for fractions anddecimals … children generally find learning about thesystem of integers to be easier than working with thepositive rational numbers. The notation for negativenumbers is less complex than that for rationalnumbers, ….Furthermore, the rules for operating onintegers are easier to learn and apply than thecorresponding algorithms with fractions. The challengefor teachers is to assist children in understanding whyas well as how these rules work.”
Cathcart, Pothier, Vance, & Bzuk, N. A. (2003), Learning Mathematics in Elementary andMiddle Schools, p. 381
© Michigan State University 2004 DRAFT: Do not cite or quote 27
Reasons for Investigation of ThisTopic
• Multiplication is a focus topic for our study
• We observed substantial variation in the presentation ofmultiplication of integers
• The use of real-life contexts and concrete models isawkward
• Negative numbers are challenging for teachers to teachand for children to learn
• More teachers may be teaching about integers
© Michigan State University 2004 DRAFT: Do not cite or quote 28
Research Question
What are the variables in authors’presentations?
© Michigan State University 2004 DRAFT: Do not cite or quote 29
Method
• List the yes/no variables• Select those variables that could be considered
content variables• Group the selected variables• Develop a numerical summary of the yes/no
variables• Analyze the numerical summary• Identify more complex variables• Identify topics for further study
© Michigan State University 2004 DRAFT: Do not cite or quote 30
DefinitionTextbook author(s) Definition
Bassarear (2001)
Beckmann (2003)
Bennett & Nelson (2004)
Billstein, Libeskind, Lott (2004)
CRMSE (2000-2001) Y
Darken (2003)
Devine, Olson, Olson (1991)
Jensen (2003) Y
Jones, Lopez, Price (2000)
Krause (1991) Y
Long & DeTemple (2003)
Masingila, Lester, Raymond (2002)
Musser, Burger, Peterson (2003) Y
O’Daffer, Charles, Cooney, Dossey, Schielack (2002)
Parker & Baldridge (2003)
Sgroi & Sgroi (1993)
Sonnabend (2004)
Troutman & Lichtenberg (2003)
PRELIMINARY Data -- subject to change
© Michigan State University 2004 DRAFT: Do not cite or quote 31
Krause Definition
Definition of Multiplication (•) of IntegersFor all whole numbers m and n:
m•n = n•m
–m•n = n• –m = –(m•n)
–m• – n = m•n
Krause, 1991, p. 306
© Michigan State University 2004 DRAFT: Do not cite or quote 32
Long and DeTemple Theorem
Theorem The Rule of Signs
Let m and n be positive integers so that -m and -nare negative integers. Then the following aretrue:
m•(-n) = -mn
(-m)•n = -mn
(-m)•(-n)= mn
Long & DeTemple, 2003, p., 317
© Michigan State University 2004 DRAFT: Do not cite or quote 33
Table: ModelsTextbook author(s) Patterns Number
line-position
Numberline-changingdirection
Repeatedaddition
Sets ofobjects
Chargedfield orsignedchips
Debit/Credit
Temp Time(ago)
Other
Bassarear Y Y Y
Beckmann Y Y
Bennett & Nelson Y Y Y Y Y Y
Billstein, Libeskind, Lott Y Y Y Y Y Y
CRMSE Y
Darken Y Y Y
Devine, Olson, Olson Y Y Y Y
Jensen Y
Jones, Lopez, Price Y
Krause Y Y Y Y Y
Long & DeTemple Y Y Y Y
Masingila, Lester, Raymond Y Y Y
Musser, Burger, Peterson Y Y Y Y
O’Daffer, Charles, Cooney,Dossey, Schielack
Y Y Y Y Y
Parker & Baldridge Y Y Y
Sgroi & Sgroi Y Y Y
Sonnabend Y
Troutman & Lichtenberg Y Y Y Y
Total 9 4 6 13 7 4 6 1 4 4
PRELIMINARY Data -- subject to change
© Michigan State University 2004 DRAFT: Do not cite or quote 34
PatternsFrom Masingila, p. 90
© Michigan State University 2004 DRAFT: Do not cite or quote 35
Number Line ModelFrom O’Daffer, p. 258
© Michigan State University 2004 DRAFT: Do not cite or quote 36
Chips ModelFrom O’Daffer et al, , p. 255
© Michigan State University 2004 DRAFT: Do not cite or quote 37
Models Summary
• Repeated addition: 13• Patterns: 9
• Sets of objects: 7
• Number Line: 6• Debit/Credit: 6
PRELIMINARY Data -- subject to change
© Michigan State University 2004 DRAFT: Do not cite or quote 38
Table: Notation –mTextbook author(s) Raised
negativeRaisedpositive
Discussion
Bassarear Y Y
Beckmann Y
Bennett & Nelson Y
Billstein, Libeskind, Lott Y Y
CRMSE Y
Darken
Devine, Olson, Olson Y Y Y
Jensen
Jones, Lopez, Price
Krause Y Y Y
Long & DeTemple Y
Masingila, Lester, Raymond Y
Musser, Burger, Peterson
O’Daffer, Charles, Cooney,Dossey, Schielack
Parker & Baldridge Y
Sgroi & Sgroi Y
Sonnabend
Troutman & Lichtenberg Y Y
Total 7 3 7
PRELIMINARY Data -- subject to change
© Michigan State University 2004 DRAFT: Do not cite or quote 39
Notation for Negative Integers
Ultimately, however, these texts invariably change to the standard notation…. We feel that it causes less confusion to do this at the outset, stressingthat the context makes it clear when ‘subtract’ is meant as opposed to ‘the
negative of.’
Long and DeTemple, 2003, p. 285
© Michigan State University 2004 DRAFT: Do not cite or quote 40
Properties
• Lists Properties: 14 books• Uses Properties: 14 books
PRELIMINARY Data -- subject to change
© Michigan State University 2004 DRAFT: Do not cite or quote 41
Other ideas
• Opposite: 17 books• Additive Inverse: 11 books
• Absolute Value: 13 books
PRELIMINARY Data -- subject to change
© Michigan State University 2004 DRAFT: Do not cite or quote 42
Findings
• Definition• Vocabulary
• Variable use of properties
© Michigan State University 2004 DRAFT: Do not cite or quote 43
Finding #1:Few authors specify a definition
for the multiplication ofintegers
© Michigan State University 2004 DRAFT: Do not cite or quote 44
Finding #2:Assumptions are made about the
vocabulary of prospective teachers
© Michigan State University 2004 DRAFT: Do not cite or quote 45
Finding #3:These books do not suggest
consensus about how to makesense of multiplication of integers
© Michigan State University 2004 DRAFT: Do not cite or quote 46
Reasoning and Proof in Mathematics Textbooks forProspective Elementary and Middle School Teachers
Andreas StylianidesUniversity of Michigan
© Michigan State University 2004 DRAFT: Do not cite or quote 47
Plan for the Presentation
• Overview of the study– Overall scope– Textbook analysis, methods and examples
• Analysis: Multiplication of Integers• Analysis: Reasoning and Proof• Conclusions
– Textbook analysis results– Mathematics for teaching
© Michigan State University 2004 DRAFT: Do not cite or quote 48
Types of books
• Multiple Editions?• Role of Publishers?• Changing Standards?• Changing Expectations?
© Michigan State University 2004 DRAFT: Do not cite or quote 49
Big Ideas?
• Hard to find the big mathematical ideas insome of the texts
• Hard to find connections across topics
• Some of the texts might allow one to viewmathematics as a bundle of loosely relatedtopics and rules
© Michigan State University 2004 DRAFT: Do not cite or quote 50
Conceptions of knowledge
Mathematics
Mathematicsfor Teaching
Methods
© Michigan State University 2004 DRAFT: Do not cite or quote 51
Conceptions of Knowledge
Algorithms for multidigitmultiplication can be provedusing the distributiveproperty
Mathematics
35
25x
175
700
875
35
25x
125
75
875
35
25x
25
150
875
100
600
A B C
+ +
+
Mathematicsfor Teaching
A common error amongchildren when firstlearning multiplication isthe following:
35x25
1025615
1640
Methods
This example isfrom Ball, 2001
© Michigan State University 2004 DRAFT: Do not cite or quote 52
Conceptions of Knowledge
The axioms for the field ofreal numbers can be used toderive the rules foroperations with negativeintegers, and they imply thatthese rules must be whatthey are.
Mathematics
Each model has particularmathematical characteristics,including both benefits anddrawbacks. Teachers need toknow what each entails andwhat is given up when usingeach model.
Mathematicsfor Teaching
There are several modelsfor multiplication ofnegative integers thatteachers need to know:signed chips, number line,temperature, debit/credit.
Methods
© Michigan State University 2004 DRAFT: Do not cite or quote 53
Conceptions of MathematicalKnowledge for Teaching
MathematicsMethods Mathematicsfor Teaching
What is the knowledge here? What is in this gap?
MathematicsMethods Mathematicsfor Teaching
© Michigan State University 2004 DRAFT: Do not cite or quote 54
Conceptions of Knowledge
MathematicsMethods
Mathematicsfor Teaching
Who teaches this?
© Michigan State University 2004 DRAFT: Do not cite or quote 55
Conclusions
• Next stage of analysis– Approach– Goals
– Assessment– Conceptions of knowledge
• Problem solving• Mathematical thinking• Fewer topics, in depth• Logical development
What kind ofteachers do wewant?
© Michigan State University 2004 DRAFT: Do not cite or quote 56
Conclusions• What mathematics is OFFERED to prospective
elementary and middle school teachers?– Lots of variation across books
– Many possibilities for constructing a course
– Often hard to tell what the textbook authors consider critical
• Across the texts, the line between “method” and“mathematics” is not clearly drawn
• The next stages of the research will be telling– What do authors intend?
– How do instructors use the books?