University of Wisconsin-Milwaukee mathematics focus courses: mathematics content for elementary and middle grades teachers

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University of Wisconsin-Milwaukeemathematics focus courses:mathematics content for elementaryand middle grades teachersKevin McLeod a & Deann Huinker ba University of Wisconsin-Milwaukee , Department ofMathematical Sciences , PO Box 413, Milwaukee, Wisconsin, USAb University of Wisconsin-Milwaukee , School of Education, PO Box413, Milwaukee, Wisconsin, USAPublished online: 26 Sep 2007.

To cite this article: Kevin McLeod & Deann Huinker (2007) University of Wisconsin-Milwaukeemathematics focus courses: mathematics content for elementary and middle grades teachers,International Journal of Mathematical Education in Science and Technology, 38:7, 949-962, DOI:10.1080/00207390701579498

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International Journal of Mathematical Education inScience and Technology, Vol. 38, No. 7, 15 October 2007, 949–962

University of Wisconsin-Milwaukee mathematics focus courses:

mathematics content for elementary and middle grades teachers

KEVIN MCLEOD*y and DEANN HUINKERz

yUniversity of Wisconsin-Milwaukee, Department of Mathematical Sciences,PO Box 413, Milwaukee, Wisconsin, USA

zUniversity of Wisconsin-Milwaukee School of Education,PO Box 413, Milwaukee, Wisconsin, USA

(Received 7 May 2007)

There has been much debate in recent years as to the amount and type ofmathematical knowledge that teachers need to acquire. One set of recommenda-tions is provided by the Mathematical Education of Teachers (MET) report,produced jointly by the American Mathematical Society and the MathematicalAssociation of America. This paper reports on efforts at the University ofWisconsin-Milwaukee to implement the MET report recommendations forpre-service elementary and middle grades teachers, in the contexts of teachereducation programmes and the teacher licensing structure of the state ofWisconsin. Data is also presented on the results of teaching the resulting coursesto in-service teachers.

1. The Mathematical Education of Teachers report and the University

of Wisconsin-Milwaukee

As the twentieth century drew to a close, a series of reports pointed to a crisis inAmerican education and led to a spirited national debate on the extent of theproblem and on possible solutions. The debate has often focused on whether teacherssimply need to take more advanced mathematics courses, whether they need morecourses on the teaching of mathematics, or whether there is something else – courseson mathematics content that is specialized to the profession of teaching. Out of thisdebate has come a recognition of the vital role played by the classroom teacher, anda clearer understanding, especially in mathematics, of the amount and type ofknowledge that an effective teacher needs [1–3]. Teachers need knowledge that anyeducated adult might be expected to know, such as algorithms for multi-digitcomputation with whole numbers; but they also need knowledge specialized toteaching, such as a deep understanding of why the standard algorithms work,knowledge of alternative algorithms, and familiarity with multiple representations ofmultiplication. It is this specialized knowledge that enables teachers to quicklyevaluate students’ work, and to distinguish misconceptions from non-standard but

*Corresponding author. Email: [email protected]

International Journal of Mathematical Education in Science and TechnologyISSN 0020–739X print/ISSN 1464–5211 online � 2007 Taylor & Francis

http://www.tandf.co.uk/journalsDOI: 10.1080/00207390701579498

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correct solutions. Liping Ma [1], comparing groups of US and Chinese teachers,

talks about a ‘profound understanding of fundamental mathematics’; Deborah Ball

and her collaborators at the University of Michigan [2, 3] speak of mathematical

knowledge for teaching (MKT), and this is the terminology we will use.A goal of our work is to better define the specialized mathematical knowledge

needed for teaching and incorporate this into university mathematics courses for

prospective teachers. Teachers need to develop not only a deep and profound

understanding of the mathematics that they will teach, but they need to learn that

knowledge in ways that make it useful and accessible while teaching. Ball [4]

identified several qualities of MKT which we have incorporated into our work. First,

teaching mathematics entails a respect for the integrity of the discipline, and caring

whether a method or an idea is generalizable. Second, teaching mathematics requires

being able to unpack ideas and make them accessible as they are first encountered by

the learner, not only in their finished form. Third, a teacher of mathematics needs to

know how claims are justified and how to investigate and reason about mathematical

propositions themselves. Fourth, teaching mathematics requires knowing where

students have been mathematically and where they are heading as they grow in their

mathematical maturity.At the turn of the new century, in 2001, the American Mathematical Society

and the Mathematical Association of America jointly developed a report on the

Mathematical Education of Teachers, known as the MET report [5], summarizing

current thinking and putting forward a set of recommendations for the

preparation of prospective teachers. While emphasizing that ‘the quality of

mathematical preparation is more important than the quantity’, the report makes

recommendations for the courses pre-service teachers should take at three different

levels: elementary, middle grades, and high school. At the elementary and middle

gradesy, the levels which concern us here, the MET report recommends (Chapter 2):

(i) Prospective elementary grade teachers should be required to take at least 9

semester-hours on fundamental ideas of elementary school mathematics.(ii) Prospective middle grades teachers of mathematics should be required to take at

least 21 semester-hours of mathematics, that includes at least 12 semester-hours on

fundamental ideas of school mathematics appropriate for middle grades teachers.

(One semester-hour equates to one hour of class meeting time per week over the

course of a semester, or 14–15 hours in all. A typical US college course will meet for

three hours per week, and will count for three semester-hours.) In a later chapter,

the report discusses specific recommendations for coursework in Number and

Operations; Algebra and Functions; Measurement and Geometry; and Data

Analysis, Statistics and Probability. There is also a suggestion that ‘One semester

of calculus could be part of this . . . group of courses if there is (or could be designed)

a calculus course that focuses on concepts and applications’ (Chapter 4).A strict implementation of these recommendations is problematic in Wisconsin

because the state does not award a specific middle grades licence. Instead,

prospective teachers may be licensed to teach in one of three grade bands:

Early Childhood (grades PreK–3), Middle Childhood through Early Adolescence

y‘Elementary’ here refers to US grades Kindergarten through grade 5 (ages 5–11); ‘middlegrades’ to grades 6–8 (ages 11–14).

950 K. McLeod and D. Huinker

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(grades 1–8), or Early Adolescence through Adolescence (grades 6–12). This lastlicence is content-specific: teachers are licensed to teach (for example) mathematics ingrades 6–12. At the lower grades, the licence is a general one, and teachers areexpected to teach any content area. Although the Early Adolescence throughAdolescence grade range does include middle school, the great majority of teacherswith a specialist mathematics licence choose to teach in high school, with the resultthat most middle school mathematics is taught by holders of the MCEA (MiddleChildhood through Early Adolescence) licence; i.e. by generalists. Recognizing thisdifficulty, the state recently revised its requirements for the MCEA licence to includea content area minor, or focus area.

The UWM School of Education offers undergraduate degree programmesdesigned to lead to each of the three state licences. Students in the Early Adolescencethrough Adolescence programme essentially complete a mathematics major inaddition to their education courses, and will not concern us here. Students in theEarly Childhood (EC) and Middle Childhood through Early Adolescence (MCEA)programmes are required by the School of Education to take a two-semesterfoundational sequence, Mathematical Explorations for Elementary Teachers, I & II,prior to admission to the programme. Together with required mathematics methodscourses taught in the School of Education after admission, this sequence can beconsidered to meet the MET report recommendations for elementary teachers, andhence for Early Childhood.

Ideally, all MCEA graduates would be required to meet the MET report middlegrades recommendations, but the general nature of their degree makes it hard,or impossible, to require this much mathematics coursework of all students in theprogramme. What is possible is to require those MCEA students who choosea mathematics focus area to meet the MET report recommendations, but hereanother problem arose: approximately 90% of MCEA students were choosing eitherEnglish and language arts or social science as their focus area. To overcome thisdifficulty, the School of Education introduced the requirement that all MCEAgraduates must have two focus areas, one of which must be either mathematics orscience. (The other must be either English and language arts or social science.) Thishas resulted in one-third of the approximately 150 MCEA graduates each yearchoosing the mathematics minor, providing a sizeable new pool of students who mayreasonably be expected to take the amount and type of mathematics courseworkrecommended by the MET report.

2. The University of Wisconsin-Milwaukee mathematics focus courses

Apart from the two-semester foundational sequence mentioned above, however, theUniversity of Wisconsin-Milwaukee (UWM) Department of Mathematical Scienceshad no courses specifically aimed at prospective elementary or middle gradesteachers. We searched for courses or curricular materials from other institutions thatwould meet our requirements, but failed to find any that were suitable. We did findone other project that was developing courses aligned to the MET report, theConnecting Middle School and College Mathematics project at the University ofMissouri [6], but decided after reviewing their materials that they did not fit ourprogramme. (One reason for the poor fit is that, unlike Wisconsin, Missouri doeshave a specialised middle-grades licence.) There are also graduate-level programmes

University of Wisconsin-Milwaukee mathematics focus courses 951

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in middle-school mathematics, such as that at the Universities of Maryland andDelaware [7, 8], but these also are unsuitable for our undergraduate educationmajors.

Fortunately, UWM had recently been awarded a US National ScienceFoundation Math and Science Partnership grant, the Milwaukee MathematicsPartnership (MMP) [9], and this grant provided resources for the development offour new mathematics focus courses. A vital aspect of our approach is that eachcourse is being developed by a design team comprised of at least three people:a mathematician who is responsible for ensuring that the course contains rigorousand correct mathematics; a mathematics educator who ensures that the course isin alignment with current educational thinking and mathematics curricula; anda Teacher-in-Residence (TIR), who ensures that the course material relates toclassroom practice. (The MMP grant funds four TIR positions; each TIR is a teacherfrom the Milwaukee Public Schools, the local urban school district, who is releasedfrom classroom teaching and based at the UWM campus.) A brief descriptionof each of the courses follows, in the order in which they have been developed.

Problem Solving and Critical Thinking. This is perhaps the most innovative of thefour courses, focussing on process rather than content. There is no set syllabus;instead, instructors have built a file of suitable problems and each instructor adds hisor her favourite problems to the file while teaching the course. Students are usuallyshown each new problem in the last few minutes of a class period, and there is a briefdiscussion to ensure that they are reading the problem correctly. Before the next classperiod, they are expected to have made an effort to solve the problem. The discussionat that next class begins with the students’ first impressions on seeing the problem,then moves to possible solution strategies; the instructor ensures that each of thesediscussions runs its course before moving on to any solutions the students may havefound. The discussion of solutions will often take place in a later class, with studentstaking advantage of the discussion of strategies to try to make further progress ona solution of their own. The class may often be working on two or three problems atthe same time, in different stages of solution. Instructors are careful to chooseproblems that are accessible to the students, but which are also rich enough to admitinteresting extensions: a recurring theme of the course is that solving the initialproblem is often only the first step to real understanding.

The course emphasizes communication of mathematical ideas as well as reflectionon the process of problem solving. To this end, students are required to maintaina journal with entries for each problem discussed in the class. Each entry is expectedto include the stages described above: first impressions; initial attempt at a solution;reflections on the class discussion of strategies; solution attempts following thisdiscussion; and a write-up of a final solution.

Some of the problems used in the course are well-known, though not generallyto the students. For example, the first problem is often a magic square:

Place the whole numbers from one through nine into a three-by-three array, sothat the three numbers in each row, the three numbers in each column, and thethree numbers in each diagonal have the same sum.

As with many of the problems, the students can solve this original problem quitequickly by trial and error. The interest comes when solutions are compared: studentsfirst see that different solutions are possible, then, that at some deeper level all of


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their solutions are really the same. This can lead to a discussion of symmetry, or ofwhen we might want to say that two solutions are the same, and hence to the notionof equivalence. At this point, the students will usually recognise an analogy withequivalent fractions. Returning to the magic square problem, the instructor mightpush the students to produce an argument that the problem really does have only onesolution, or he might ask whether there are numbers other than the integers from oneto nine which will fill a magic square. (What about larger squares?) In the discussionof each problem, and also in the choice of problems as the course proceeds, theinstructor needs to be sensitive to the class, keeping the discussion at an accessiblebut challenging level.

Geometry. This course aims to develop the students’ familiarity and facility withgeometry, both as a practical tool for solving problems involving spatial objects andpatterns, and as a powerful logical system. The course begins with a discussion ofmeasurement, including measurement error, and of geometry as a measuring tool.Other course topics include symmetry and rigid motions; spherical geometry; andthe role of definitions, axioms and logical reasoning in geometric proofs.

Discrete Probability and Statistics. The goal of this course is to enable students todetermine the likelihood of various events occurring, and to make inferences basedon such likelihoods. The emphasis is on discrete situations, where probability can becomputed by counting. (Thus, for example, the hypergeometric distribution isstudied rather than the normal distribution.) At the beginning of the course, thestudents study simple models where the counting can be done with tree diagrams ortables; by the end, they have progressed to situations where sophisticated countingtechniques need to be applied.

Algebraic Structures. This course seeks to provide students with a higher-levelunderstanding of the basic patterns and rules that govern everything from numbersystems to geometric transformations to the manipulation of polynomial expres-sions. Topics include elementary number theory; set theory; functions; and groups,rings and fields, with examples showing where they occur in the elementary andmiddle grades curriculum.

Although only one of these courses is explicitly devoted to problem solving, theyare all taught in an exploratory, hands-on style, and problem solving plays animportant role in all of them.

The Problem Solving course was taught for the first time in the Spring of 2004;the 2005–2006 academic year was the first year in which all four courses were offered.Currently, one section of each course is offered each year. The Problem Solving andGeometry courses are always fully enrolled, and student reaction to these courses hasbeen extremely positive. The Discrete Probability and Algebraic Structures coursesare more recent, and have not yet attracted students in the same numbers, althoughstudents who do take the courses indicate that they find the course material usefuland relevant. (We conjecture that one reason for the lower enrolment in these coursesis that the Department has several other algebra and statistics courses, whereas thereare no obvious substitutions for the Problem Solving or Geometry courses.)

The four focus courses are now recommended for all MCEA mathematics focusstudents. They have the foundational Mathematical Explorations sequence asa prerequisite; altogether, this constitutes 18 of the 21 semester-hours of mathematicsin the MET report. The final requirement for the math focus area is a ‘calculus


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experience’. In practice, this usually means a four semester-hour course entitled BriefSurvey of Calculus which is less than ideal for future teachers; we hope to be ableto develop a more relevant calculus course in the future. (This course also hasa prerequisite course, Intermediate Algebra; mathematics focus students who do nottest out of this course on admission to UWM will take 25 semester-hours ofmathematics as undergraduates.)

3. Project: mathematics fellowships for middle-grades teachers

In December 2004, the Milwaukee Mathematics Partnership received an additionalMath and Science Partnership award from the Wisconsin Department of PublicInstruction, to increase the mathematics content knowledge of middle gradesteachers. Recognizing that this would be an ideal opportunity for us to analyse theeffectiveness of our math focus courses, we designed and offered a programme inwhich current Milwaukee Public Schools teachers could become ‘Math Fellows’ bytaking all four of these courses. Participants were also given the option of taking thetwo additional courses (Intermediate Algebra and Brief Survey of Calculus) andbecoming ‘Advanced Math Fellows’; upon successful completion of all six courses,they would also receive a UWM transcript designation stating that they hadcompleted the mathematics focus area. It should be noted that although these lasttwo are standard UWM mathematics courses, the Math Fellows were taught insections with restricted enrolment. This allowed the instructor to teach non-standardversions of the courses, aligned to the needs of classroom teachers. In particular, aneffort was made to teach a calculus course, which, as recommended in the METreport, focused on concepts and applications.

We hoped that the Mathematics Fellows programme might also help us with thedesign and institutionalization of the focus courses, since they would be taught forthe first time by instructors other than the course developers.

We had 83 initial applicants for the programme, with 53 teachers enrolling for atleast one course. The prerequisite course Mathematical Explorations for ElementaryTeachers, I was added after we looked at the applicants’ transcripts and realized thatmany of them did not have this course.

Due to funding restrictions, it was necessary to complete the programme by theend of August 2006, so the teachers took courses between Summer 2005 and Summer2006, inclusive. Course enrolments for the entire programme are shown in table 1.

Table 1. Mathematics fellows course enrolment.

Summer2005

Fall2005

Spring2006

Summer2006 Total

Mathematical Explorations 18 18Problem Solving 40 6 46Geometry 11 24 35Discrete Probability & Statistics 17 22 39Algebraic Structures 24 24Intermediate Algebra 18 18Survey of Calculus 12 12Total 69 47 40 36 192


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The number of semester-hours taken by the Fellows are summarized in table 2.Given the demanding nature of the programme, we expected a high attrition rate,so we were pleased that 17 participants completed enough semester-hours to earnthe Fellowship, while a further 10 earned the Advanced Fellowship. (In a surveyconducted after the conclusion of the programme, by far the most common reasongiven for not completing all four courses was the time commitment.) Particularlynoteworthy is that four participants were originally placed into the prerequisitecourse and still completed the Advanced Fellowship: a total of 22 semester-hours.The 27 Fellows and Advanced Fellows, together with an additional 14 participantswho had completed at least two courses, were presented with certificates at anawards dinner. The Fellows and Advanced Fellows were also recognized at a meetingof the district School Board.

4. Results

The Fellows were pre and post-surveyed for their beliefs and efficacy in themathematical knowledge needed for teaching (MKT), using an instrument developedby the Milwaukee Mathematics Partnership (MMP). They were also pre andpost-tested for MKT in specific content areas using measures developed at theUniversities of Michigan and Louisville. Data were analysed using descriptivestatistics and paired t-tests.

4.1. Beliefs and efficacy

The MMP survey was composed of 53 likert-scale items (1¼ strongly disagree,2¼moderately disagree, 3¼ slightly disagree, 4¼ slightly agree, 5¼moderatelyagree, 6¼ strongly agree) with six embedded scales, three efficacy scales andthree beliefs scales. Responses were summed across all items on each scale togenerate a total score: Efficacy for learning MKT (10 items, score range 10–60),Efficacy to use MKT in teaching (9 items, range 9–54), Efficacy to usestandards-based instructional approaches (11 items, score range 11–66), Beliefsabout MKT (2 items, score range 2–12), Beliefs about teaching mathematics(15 items, score range 15–90), Beliefs about components of mathematical proficiency(5 items, score range 5–30).

The pre-survey with the MMP instrument was administered in Summer 2005,at the start of the Problem Solving course (which was the first course taken bythe Fellows); the post-survey was administered at the conclusion of the programme,in Summer 2006. Results for the Fellows who completed both surveys are shownin table 3. Table 4 reports changes for selected items in the beliefs survey, to provide

Table 2. Number of semester-hours taken by Math Fellows participants.

Number ofsemester-hours* Number of participants Certificate awarded

6–9 14 Math Fellows Participant12 17 Mathematics Fellow19 10 Advanced Mathematics Fellow

*Exclusive of Mathematical Explorations, I.


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an illustration of what is measured by each efficacy scale and the change inparticipants’ mean efficacy scores.

In addition to the surveys, the Fellows completed the standard mathematicsdepartment course evaluation form for each course. Comments on these forms,including comments from participants who did not complete the programme,showed that they recognized their own increased proficiency in mathematics:

‘I avoided problem solving and didn’t like it until I took the course you offered.This was helpful to me because prior to taking the course I limited the numberof problems I gave my students because I didn’t want to tackle them myself.Now my students receive problem solving or world problems with nearly everyassignment instead of next to none at all’.

‘I had always been afraid of problems, and taught them (if I couldn’t avoidthem) in a very dull and uninspiring way. This course made me undertake ALLproblems with trust and almost happiness because now I know I can solvethem’.

‘This course has taught me to look at Geometry in a new light. I learned morebackground into what Geometry is really about. I feel that I will be moreconfident in teaching Geometry in the future’.

‘I lost fear and found that probability is something actually enjoyable andteachable’.

‘This [probability] is a subject I’ve always skirted because I did not feelcompetent to teach it. No longer. I have the fundamentals’.

‘I have a deeper understanding of algebraic reasoning and problemsolving . . . it gave me more confidence as a math teacher for the upper gradelevels’.

Table 3. Results of beliefs about Mathematical Knowledge for Teaching (MKT).

Mean (Standard Deviation)

Scale NPre-testscore

Post-testscore t-value df *

Significance(2-tailed)

Efficacy forlearning MKT

18 44.6 (5.03) 50.8 (5.28) 4.219 17 0.001

Beliefs about MKT 18 7.1 (2.64) 7.5 (2.58) 0.518 16 0.612Efficacy to useMKT in teaching

18 39.8 (6.97) 46.7 (5.84) 3.882 17 0.001

Efficacy to usestandards-basedinstructionalapproaches

18 46.8 (8.86) 56.7 (7.59) 4.650 17 0.000

Beliefs aboutteaching mathematics

18 73.1 (8.86) 76.0 (7.76) 2.247 17 0.138

Beliefs about componentsof mathematicalproficiency

18 26.0 (2.62) 26.1 (1.58) 0.169 17 0.068

*Degrees of freedom.


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Table

4.

Selecteditem

sresultsfortheefficacy

scales(1¼Strongly

Disagree;

6¼Strongly

Agree).

Mean(Standard

Deviation)

Item

NPre-survey

Post-survey

Change

Scale:Efficacy

forlearningthemathem

aticalknowledgeneeded

forteaching

Icanidentify

how

particularmathem

aticalideasgrow

andbuilduponeach

other

insophisticationandabstraction.

18

3.83(1.043)

4.89(0.758)

1.06

Ifinditdifficultto

use

representationsorvisualmodels

toexplain

whymathem

atics

works.(reversedscored)

18

4.50(0.985)

5.06(1.162)

0.56

Itisdifficultformeto

analyse

andcompare

theuse

ofmathem

aticallanguageandnotationforaccuracy.(reversedscored)

18

4.22(1.353)

4.61(1.290)

0.39

Scale:Efficacy

touse

mathem

aticalknowledgein

thetasksofteaching

Icandesignmathem

aticallyaccurate

explanations

thatare

clearandusefulforstudents.

18

4.11(1.323)

5.28(0.752)

1.17

Icanexplain

proceduresto

students

inwaysthatare

both

mathem

aticallyvalidandaccessible

tothem

.18

4.33(0.767)

5.22(0.943)

0.89

Icanunderstandandevaluate

students’mathem

aticalstrategies,

solutions,andconjectures.

18

4.39(0.916)

4.94(0.639)

0.55

Scale:Efficacy

touse

standards-basedinstructionalapproaches

Icanleadaclass

discussionin

whichstudents

analyse

andevaluate

each

other’smathem

aticalthinkingandstrategies.

18

3.50(0.985)

4.94(1.056)

1.44

Icanengagestudents

inform

ulatingandjustifyingtheirown

andeach

other’smathem

aticalconjectures.

18

4.00(0.767)

4.78(0.878)

0.78

Icanansw

erstudents’math-relatedquestions.

18

4.83(0.857)

5.44(0.878)

0.61


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4.2. Number, algebra and geometry

The measures developed by Deborah Ball, Hyman Bass and their collaborators atthe University of Michigan in the Learning Mathematics for Teaching (LMT) [10]project were used to assess the participants’ mathematical knowledge for teaching inthe areas of Number and Operations, Algebra, and Geometry. These assessmentsemerged from the theoretical perspective that the mathematical knowledge neededfor teaching is different from the mathematical knowledge required for other careers.These items measure teachers’ facility in using mathematical knowledge in thecontext of classroom teaching and utilize item-response-theory in evaluating items,constructing scales, and reporting results. The Michigan items were normed usinga group of California teachers selected from schools which were engaged ininstructional improvement. Thus, a mean of zero would indicate a level of knowledgeequivalent to that possessed by an average practicing teacher.

These measures were selected because of their availability, their philosophicalalignment to our courses and the Math Fellows project, and their reputation as validand reliable measures of knowledge comparable to other populations. Theinstrument was comprised of 68 multiple-choice items, with 22, 31 and 15 itemsper scale, respectively. The pre-test was given at the end of the problem solvingcourse in Summer 2005, but before taking any of the other content courses. The sameinstrument was used as a post-test, and was administered at the completion of theprogramme in Summer 2006. The results for each strand and for the composite scoreare shown in table 5.

4.3. Probability and statistics

Because measures for statistics and probability were not available from the LMTproject, we contacted the Diagnostic Teacher Assessments in Mathematics(DTAMS) [11] project at the University of Louisville. They provided us withan instrument to assess the Fellows’ knowledge in these areas. The DTAMSinstrument was composed of 30 items: 20 multiple-choice and 10 open response.Each open-response item was scored on a 0–2 point rubric; total possible score forthe instrument was 40 points.

The DTAMS instruments assess four types of mathematical knowledge: Type I:Memorized Knowledge; Type II: Conceptual Understanding; Type III: ProblemSolving/Reasoning; and Type IV: Pedagogical Content Knowledge. The maximumpossible score for each type of knowledge is 10 points. Since this particular

Table 5. Results of the LMT mathematical knowledge for teaching assessments.

Mean (standard deviation)

Scale N Pre-test Post-test Post - Pre t-value df Significance (2-tailed)

Number 21 �0.07 (0.60) 0.15 (0.88) 0.22 1.154 20 0.262Geometry 21 0.23 (0.96) 0.40 (0.85) 0.17 1.235 20 0.231Algebra 21 �0.63 (0.85) �0.28 (0.87) 0.35 2.946 20 0.008Composite 21 �0.27 (0.81) 0.00 (0.91) 0.27 1.908 20 0.071


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instrument was composed of two subcategory domains, total scores were alsoreported for statistics and for probability; each subcategory had a maximum possiblescore of 20.

The instrument was given to one section of the Discrete Probability and Statisticscourse as a pre-test at the beginning of the course, and again as a post-test at thecompletion of the course. The results, presented as raw mean scores, for each typeof knowledge and for the total scores, are shown in table 6.

4.4. Calculus

We did not pre and post-test the Advanced Fellows for content knowledge incalculus, but they did complete the standard mathematics department courseevaluation form (as the Fellows had done for each course they took during theprogramme). Their comments indicated that they saw value in having taken thecourse:

‘I plan to make sure my students get an understanding on how subject areasconnect. This was helpful to me because I wasn’t sure about this until I tookcalculus.’

‘Finding area under the graph is a topic that can be modified to be easilyintroduced to students at the elementary level.’

‘I teach middle school physics and this deeper level of understanding will helpme teach some of the more abstract concepts with confidence.’

5. Discussion

The Fellows who completed both beliefs surveys (essentially those who completedthe programme) showed statistically significant increases, at the 0.1% confidencelevel, across all three efficacy scales. Thus, they reported greater effectiveness in theirability to learn MKT and to use it in their instructional practices. From our ownobservations throughout the programme, we know that this increase was ongoing

Table 6. Results for DTAMS mathematical knowledge assessments in Probabilityand Statistics.

Mean (standard deviation)

Scale N Pre-test Post-test Post - Pre t-value dfSignificance(2-tailed)

Type I 12 6.58 (1.78) 7.17 (1.19) 0.59 1.000 11 0.339Type II 12 6.67 (2.27) 7.42 (1.39) 0.75 1.326 11 0.212Type III 12 4.50 (1.50) 6.75 (1.91) 2.25 3.467 11 0.004Type IV 12 4.33 (2.67) 5.75 (1.96) 1.42 3.027 11 0.012Total Statistics 12 12.33 (3.65) 13.424 (2.99) 1.09 1.215 11 0.250Total Probability 12 9.75 (3.72) 13.67 (3.08) 3.92 4.619 11 0.001Composite 12 22.08 (6.05) 27.08 (4.62) 5.00 3.362 11 0.006


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and not the result of any one course. Nevertheless, we will argue below that the

Problem Solving course was responsible for a large part of the difference. We have

remained in contact with many of the Fellows through other Milwaukee

Mathematics Partnership activities, and they continue to report that they are using

skills and knowledge that they acquired through the programme in their own

classrooms; we have sometimes observed this ourselves during school visits.The Fellows showed improvement in all three areas tested with the Michigan

measures, with a significant increase occurring in Algebra. A partial explanation for

the Algebra result is that the algebra courses were offered towards the end of the

programme, and therefore closer to the post-test. Another possible explanation for

the smaller increases in the other areas is that the Michigan measures were not

closely aligned to the courses. This was particularly true of the Geometry items, and

some Fellows even commented after the post-test that it did not address the content

of the Geometry course.The Fellows tested showed improvement in all areas assessed by the DTAMS

instrument, and statistically significant gains were made in areas given greatest

emphasis in the Discrete Probability and Statistics course. This included the heavy

focus on ideas of probability and an emphasis on problem solving and application

(Type II knowledge), particularly of probability ideas. Given that the Mathematics

Fellows were classroom teachers, connection to their classroom practice was

a pervasive issue of reflection, if not specific discussion. This focus is reflected in the

statistically significant growth in Type IV pedagogical content knowledge.One question that could be asked with respect to the MET report recommenda-

tions is whether there is a need for a specific problem solving course, or whether it is

sufficient to build problem solving into an appropriate collection of content courses.

In our experience, both with the Fellows and with our undergraduate MCEA majors,

the separate course is invaluable. Since there is no pressure to complete a required

amount of content, the instructor can allow class discussions to take as much time as

is necessary to thoroughly address multiple solution strategies, or to go in whatever

direction student questions take them. (Of course, a large part of the skill involved in

teaching the course consists of gently guiding the discussion in fruitful directions.)

Without specific content there are also very few content prerequisites, which makes

the course accessible to the greatest possible number of students, including those who

enter convinced that ‘I can’t do math’. To many of these students, who believe that

‘math’ is synonymous with algebraic computation, the course is their first experience

of real mathematical reasoning. It was for this reason that we required the

Mathematics Fellows to take the Problem Solving course first, and we recommend

that our undergraduate students take it as early as possible.A second question concerns the MET report recommendation (really more of

a suggestion) that middle grades teachers should have a knowledge of calculus. We

believe the last set of comments quoted above affirms the recommendation, by

pointing to two important gains made by the Advanced Fellows as a result of taking

the calculus course. First, many of them saw for the first time how the solution of

a single problem may use skills from many different strands of the middle grades

curriculum. Pythagoras’ theorem, rational expressions, manipulation of square

roots: all may show up in the same optimization problem, so students who have been

exposed to these units only in isolation will be unprepared for higher mathematics.

Second, studying calculus provided the Advanced Fellows with a deeper


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understanding of topics in their own curriculum, such as area and volume, orirrational numbers.

Finally, we wish to comment on the effect of the math focus courses oninstruction in the Department of Mathematical Sciences generally. The influence ofthe mathematics educators and the teachers-in-residence on the course design teamshas resulted in a wider array of instructional practices in these courses than is usualin a mathematics course. There is less lecturing and more student interaction;students work in small groups, with manipulatives when appropriate, to solveproblems. As the courses are institutionalized, more mathematics faculty will beexposed to these different ways of teaching. This has already happened with theProblem Solving course, which has been taught by three faculty members beyond theoriginal design team; the Discrete Probability and Statistics course will be taught bya new faculty member for the first time in the Fall of 2007. We hope and expect that,as the focus courses are taught by more mathematics faculty, a greater appreciationfor alternative instructional styles will lead those faculty to reflect on theirinstructional practice in all the Department’s courses, not only those for prospectiveteachers.

6. Conclusions

Based upon our work, we support the premise that undergraduate mathematicscourses for prospective teachers should be specialized to the profession of teaching.Feedback from the course developers and students had previously led us to believethat our design team approach was resulting in an effective suite of courses forprospective teachers, and the data from the Mathematics Fellows programmeconfirms that students in the courses do acquire a more positive and productivedisposition towards mathematics, as well as increased mathematical knowledge forteaching. We endorse the recommendations of the MET report, and add to thema recommendation that prospective teachers take a course dedicated to problemsolving and critical thinking. We will continue our work in developing courses of thisnature and study the impact on prospective teachers and their instructional practices.

Acknowledgements

This material is based upon work supported by the National Science Foundationunder Grant No. 0314898 and by the Wisconsin Department of Public InstructionTitle II, Part B, Grant No. 05-3619-MSP. Any opinions, findings and conclusions orrecommendations expressed in this material are those of the authors and do notnecessarily reflect the views of the National Science Foundation (NSF) or of theWisconsin Department of Public Instruction.

References

[1] Ma, L., 1999, Knowing and Teaching Elementary Mathematics: Teachers Understandingof Fundamental Mathematics in China and the United States (Mahwah, NJ: LawrenceErlbaum Associates).


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[2] Ball, D., Hill, H. and Bass, H., 2005, Knowing mathematics for teaching: Who knowsmathematics well enough to teach third grade, and how can we decide? AmericanEducator, Fall, 2005, 14–22 and 43–46.

[3] Hill, H.C. and Ball, D., 2004, Learning mathematics for teaching: Results fromCalifornia’s Mathematics Professional Development Institutes. Journal of Research inMathematics Education, 35, 330–351.

[4] Ball, D., 2003, What mathematical knowledge is needed for teaching mathematics?(Washington, DC: US Department of Education). Available online at http://www.ed.gov/inits/mathscience/ball.html (accessed 18 June 2007).

[5] Conference Board of Mathematical Sciences, 2001, The Mathematical Educationof Teachers, (Providence, RI: Mathematical Association of America). Available onlineat http://www.cbmsweb.org/MET_Document/index.htm (accessed 18 June 2007).

[6] Connecting Middle School and College Mathematics. Available online at http://www.teachmathmissouri.org (accessed 18 June 2007).

[7] University of Maryland Masters of Education programme option: Teaching and learningin the middle school. Available online at http://www.education.umd.edu/EDCI/info/math_med.html (accessed 18 June 2007).

[8] A graduate level course of study for middle school mathematics teachers. Available onlineat http://www.udel.edu/mserc/courseofstudyMSM.pdf (accessed 18 June 2007).

[9] Milwaukee Mathematics Partnership. Available online at http://mmp.uwm.edu (accessed18 June 2007).

[10] Learning Mathematics for Teaching project. Available online at http://sitemaker.umi-ch.edu/lmt/home (accessed 18 June 2007).

[11] Diagnostic Teacher Assessments in Mathematics project. Available online at http://louisville.edu/edu/crmstd/diag_math_assess_middle_teachers.html (accessed 18 June2007).


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Documents

University of Wisconsin-Milwaukee mathematics focus courses: mathematics content for elementary and middle grades teachers