What’s for Dessert?An Enrichment Course forProspective Middle School

Mathematics Teachers

Jerrold W. GrossmanOakland University

Rochester, Michigangrossman@oakland.edu

MAA/AMS Annual MeetingJanuary 5, 2007

New Orleans, Louisiana

Oakland UniversitySuburban Detroit

One of 15 state universities

About 18,000 students, some parttime

Somewhat selective admissions

Mostly commuter students

Comprehensive (through PhD)

300 Elementary Education majors/yr

(+ a few 6–12 Secondary Ed majors)

Few get K–8 math certification

Content courses taught in Math Dept

Methods courses taught in Ed School

Math Requirements for ALL(all courses are 4 sem. hrs.)

Proficiency through Interm. Algebra

STA 225: Introduction to StatisticalConcepts and Reasoning(General Education course, includesprobability, uses text by Moore)

MTE 210: Numerical Structures (firstsix chapters of Billstein)

EED 302: Teaching Mathematics atthe Elementary–Middle Levels

Math Requirements for MINORS,which gives K–8 certification(all courses are 4 sem. hrs.)

MTH 141: Precalculus (collegealgebra and trigonometry)

MTE 211: Structures of Geometry(last four chapters of Billstein)

MTE 410: Elementary SchoolMathematics and the Computer(programming in BASIC or LOGO)

Math Requirements for MAJORS

10 more math content credits.

Recommended:

MTH 122: Calculus for the SocialSciences (baby calculus, mainly forbusiness majors, no trig, 4 credits)

MTH 118: Mathematical Sciences inthe Modern World (GeneralEducation course, using textbook suchas Excursions in Modern Mathematicsby Tannenbaum, 4 credits)

MTE 405: Special Topics (2 credits)Meets once a week for 2 hours,content up to the instructor, intendedas “capstone”

Do these satisfy the CBMSrecommendations?

Prospective middle grades teachers ofmathematics should be required to takeat least 21 semester-hours ofmathematics, that includes at least 12semester-hours on fundamental ideasof school mathematics appropriate formiddle grades teachers.All courses designed for prospectiveteachers should develop carefulreasoning and mathematical “commonsense” in analyzing conceptualrelationships and in solving problems.Along with building mathematicalknowledge, mathematics courses forprospective teachers should develop thehabits of mind of a mathematicalthinker and demonstrate flexible,interactive styles of teaching.

MTE 405 Content1999–2004

Burger and Starbird topics (infinity,art gallery thm, error-correctingcodes, …): not used as textbook

Problem solving (Car Talk puzzles)Mathematics PentathlonMapleGeometer’s SketchpadProbability (Buffon needle, birthday,

Monte Hall, hatcheck, Bayes’theorem, Simpson’s paradox)

Fair divisionStrategic games (nim, chomp)Computing square rootsWeb surf, book reports

MTE 405 Content2005

Navigating through {Algebra Measurement Probability}in Grades 6–8 (NCTM)

FunctionsChangeLinear relationshipsSymbols of algebra

Accuracy and precisionEstimationPerimeter–area–volumeProportionalityDerived units

Probability distributionsLaw of large numbersPrediction…

MTE 405 Content2006

“Middle School Contest Math”

MATHCOUNTShundreds of problems,free materialscomplete solutions

AMC 8

Written homework (individual andjoint), student presentations atblackboard, discussions, instructormini-lectures and sermons, three shortmidterms (new problems), final exam(old problems)

Special Feature

Guest lecture by Chairman of the

Board of MATHCOUNTS (Oakland

University Adjunct Professor Gary

McDonald), including discussion of

the program, encouragement to have

their schools participate, video of

MATHCOUNTS finals (broadcast on

ESPN), and discussion of Glenn

Commission report.

Student Comments

This class has been an interesting onethroughout the duration of the semester.It was nothing that I expected and Ireally did not think that I would take somuch away from a two-credit course. Ihave learned several different strategiesin problem-solving. … Methods that Ihave become particularly fond of includetrying examples, considering simplercases of the same or easier problem,going away and then coming back to theproblem, making a table or chart, anddrawing a picture.

Exploring problem solving in this classwas a rough journey. I began thesemester feeling frustrated and confused.I was overwhelmed at the difficulty ofmany of the problems, often asking“How was I supposed to think of that?”I was intimidated by the fact that middleschool students were able to do the sameproblems that I struggled with, and yet ina few short months I would be eligible toteach those same students. As thesemester went on, I improved on myability to solve many of the problems …and I gained confidence. … Now at theend of the semester I can see theimportant impact this class has had onme as a math student and future mathteacher.

I felt frustrated several times throughoutthe semester and wondered what waswrong with my brain that it didn’tproblem solve well. Then reflecting onthe experience, I realized that in myschool years as a student, no one hadbothered to show me problem-solvingmethods and strategies. No one everoffered the option of drawing a picture ormaking a table or graph. In fact I neverreally learned from teachers how to takeinformation from a problem and apply itto the solution. … However, toward theend of the course, my confidence grew alittle bit more. … I hope that myconfidence in problem-solving wears offon my students. … The course wasdefinitely an eye-opener to what I don’tknow.

I was able to learn a tremendous amountby watching students show theirsolutions on the board using severaldifferent methods. … This will prove tohave a huge effect on my teaching. Justbecause there may be one answer doesn’tmean that all students will find it usingthe same way. … I can remember inelementary school being marked down ifyou didn’t find the answer exactly ashow you were taught to. If creativity isencourage in other aspects of theclassroom, it should definitely beencouraged in math as well.

It is our job as future teachers to knowdifferent approaches and strategies tosolving problems so we are able to helpall of our students learn. … To be ableto solve a problem proves that you knowhow to do it, but to be able tocommunicate what is happening in aproblem or solution to others is the heartof what we call teaching mathematics.

I viewed [math] as something that was“black and white”. I thought of math asa subject that was very methodical andprocedural. This class has allowed me tosee … that math is not limited toalgorithms and direct answers, but it is aprocess that involves creativity,resourcefulness, and ingenious thinking.

Many of the problem-solving strategiesthat I learned about in MTE 405 I havebeen using for years. What theMATHCOUNTS program has done forme is to clarify when is the best time touse each type of strategy. One techniquewhich I found very useful was simplygoing away or jumping out of the systemso to speak. On numerous occasions Iused this technique on problems thatwere frustrating me. Nearly every time Idid this I found that when I cam back tothe problem I was able to look at it froma totally different perspective. Thisallowed me to get past the mental blockwhich was initially keeping me fromsolving the problem.

One of the main skills that I will takeaway with me from this class is thatguess and check is not the most efficientway to solve problems.

I have gone through this semester with adifferent attitude than I have had for thelast three years. I am taking two collegeclasses for sheer joy and not because Iam required. I have been able to workthese math contest problems form achallenge and enlightenment point ofview. I would often rush home fromschool and start on some of them rightaway. I even attempted the unassignedproblems. What elation I felt when Idiscovered the twist or just simply foundthe answer after a long struggle.

I learned a lot about myself and a lotabout the others in the class. Every timesomeone presented a way of doing aproblem that was different from the wayI did it, I copied it down. … I plan onusing the notes I took in this class to helpme be a quality teacher in the future. …I have, also, learned a lot aboutMATHCOUNTS, and different schoolprograms that promote math. This is soexciting for me because I love math andwant my students to love it too. … I amquite pleased that this course wasoffered. I want to be a middle schoolteacher and this math applies to me. It isobvious that I need to brush up on myskills, but I think I have learned exactlywhat skills I need to possess.

I have learned that middle schoolstudents are capable of solving problemsmuch more complex than I originallyexpected. Knowing this, I will introducemy students to complex problem-solvingand help them develop workablestrategies. As the semester progressed,my knowledge and outlook on thesubject has matured in a unique way. Ihave always considered myself a strongstudent in math; I learn the materialquickly and I have a distinctive talent tomanipulate numbers in my mind. I wantto instill an appreciation for math in mystudents, to help them realize what afascinating and useful subject it is. Thiscourse is a culmination to my[elementary education mathematics]minor; and, ironically, it has challengedeverything I thought I knew aboutmathematics.

I have learned that mathematics, like allsubjects, requires one to see beyond whatis given. To find the solution to aproblem, one does not always need toapply an equation or memorized formula.In this instance, the solution requiresthought or strategy. To implement aprovided equation or formula isstraightforward -- simply plug in thecorrect components to find the answer.Mathematics is much broader than this.It is meant to be understoodconceptually, not procedurally.

I found this class to be particularlychallenging because I was required torecall upon every mathematical concept Ihad learned in my academic career. ...There was such an abundance of materialthat I often found myself overwhelmed.Generally a math class is designed tocover one concept; this class is designedto encompass all of the concepts andintroduce them in a way that is feasiblefor middle school students to understand.... MATHCOUNTS is an excellentprogram because it is designed tochallenge students in a fun andmeaningful way. Students learnstrategies to solve mathematicalproblems, work as a team member, andtake pride in their work.

Seeing MATHCOUNTS competitions inaction, and working on some of theproblems sixth through eighth gradestudents solve, have opened my eyes tosome of the high achieving students Imay be working with in my teachingcareer. A teacher should be able tochallenge all levels of students in his orher classroom. I have learned that I needto continue to work on problems and staycurrent in some of my highermathematical skills in order to bettereducate my high achieving students.

By learning more about these techniquesand how to apply them to countless typesof math problems, I have certainlyimproved my abilities as a math teacher.By understanding these problem-solvingstrategies and how to teach them, I willundoubtedly be a far more efficienteducator of mathematics.

[Student comments used with permission.]

For more on MTE 405, see:

personalwebs.oakland.edu/~grossman/MTE405