Maurer, Stephen B., , \u2767

McCallum, W.

English

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Swarthmore College Works Mathematics & Statistics Faculty Works Mathematics & Statistics 10-1-2006 Advising A Precollege Curriculum Project Stephen B. Maurer , '67 Swarthmore College, smaurer1@swarthmore.edu W. McCallum This work is brought to you for free and open access by . It has been accepted for inclusion in Mathematics & Statistics Faculty Works by an authorized administrator of Works. For more information, please contact myworks@swarthmore.edu. Follow this and additional works at: https://works.swarthmore.edu/fac-math-stat  Part of the Mathematics Commons Let us know how access to these works benefits you  Recommended Citation Stephen B. Maurer , '67 and W. McCallum. (2006). "Advising A Precollege Curriculum Project". Notices Of The American Mathematical Society. Volume 53, Issue 9. 1018-1020. https://works.swarthmore.edu/fac-math-stat/5 Advising a PrecollegeCurriculum ProjectStephen B. Maurer and William McCallum1018 NOTICES OF THE AMS VOLUME 53, NUMBER 9We two are members of the content advisory group for the Core-PlusMathematics Project (CPMP), one ofseveral projects funded by the Na-tional Science Foundation (NSF) todevelop a whole secondary school mathematicscurriculum. Serving in this advisory group has beena fascinating experience. We hope we have helpedthe project; we know we have learned a lot in theprocess. We write this article to explain what is in-volved, to suggest to readers that this is a veryworthwhile sort of activity for professional math-ematicians, and to encourage readers to seek outsimilar opportunities.What Is Core-Plus Mathematics?Core-Plus consists of a three-year core program in-tended for a wide range of high school students,plus a fourth-year course continuing the prepara-tion of students for college mathematics. It is published by Glencoe/McGraw-Hill under the titleContemporary Mathematics in Context: A UnifiedApproach. Key features of the curriculum include:teaching algebra and geometry every year alongwith important new topics from statistics and dis-crete mathematics, emphasizing mathematicalmodeling and applications, and teaching studentsto solve more challenging problems.We are part of the content advisory group forthe second edition. Other members are DorisSchattschneider of Moravian College, an expert ongeometry, Richard Scheaffer of the University ofFlorida, a Fellow of the American Statistical Asso-ciation, and Deborah Hughes Hallett of Harvard University and the University of Arizona, founderof the Harvard Calculus Consortium.What We DoCore-Plus invited us to contribute as content spe-cialists and critics of the actual writing. We havealso tried to give advice on the overall goals andcoverage of the program. We discuss some of thethings we have been critical about later on. Thegroup meets with the CPMP development teamonce a year, in late May or early June. In advanceof the meeting we read the current drafts of unitsthat were worked on during the past year and com-ment about them, both in the large and in thesmall. After the meeting, there may be further cor-respondence between individual consultants andindividual writers, for units under revision aboutwhich there were special concerns.How We Got InvolvedBoth of us have been involved in curricular devel-opment at the college level. The first of us hasbeen especially interested in discrete mathematics,so he had his ears perked up for precollege developments. He got further involved when hischildren’s school district adopted the InteractiveMathematics Project (IMP) curriculum (another oneof the NSF-funded projects); initially he had verymixed feelings about IMP but eventually becamemostly positive. He first saw copies of the Core-Plusbooks on a visit to DIMACS and was immediatelyintrigued by how much discrete mathematics wasin them. He had met the discrete mathematicswriter for Core-Plus at various meetings and ex-pressed interest in being part of the project. Even-tually this resulted in an invitation to the contentadvisory group.Stephen B. Maurer is professor of mathematics at Swarthmore College and chair of the Department of Math-ematics and Statistics. His email address is smaurer1@swarthmore.edu.William McCallum is professor of mathematics at the Uni-versity of Arizona and chair of the AMS Committee on Education. His email address is wmc@math.arizona.edu.OCTOBER 2006 NOTICES OF THE AMS 1019The second of us has worked in calculus reformsince the late 1980s, and has recently been work-ing on a college algebra text. This has led to an interest in how algebra is taught in high school. Hemet the Core-Plus authors at a National Acade-mies meeting.Why It’s FunIt’s always fun to be with a lively thoughtful groupof new colleagues, both fellow consultants and thedevelopers of the materials, who are doing valuablework and who bring experiences and insights to thetable that are different from yours. Part of the funhas been the interesting debates we have had (seebelow). Part of it has been the experiences andgood humor of the group. In particular, it has beenheartening to meet the teachers who have beenworking with these materials and to hear their sto-ries of how these materials work in the classroom,which lies somewhere between the rosy vision of the developers and the dark vision of their detractors. The dedication and common sense ofthese teachers is remarkable.What We’ve TaughtTo explain what we’ve taught, it is necessary to saya bit about the position of this project in the mathwars. (We do this with some reluctance, because itis not the purpose of this report to take sides inthat war; our purpose is to report on what we havedone and to suggest that the role we have playedwould be of interest to mathematicians of any per-suasion.) The NSF-funded secondary curriculumprojects referred to earlier have received a lot ofcriticism from some mathematicians and from or-ganized Web-based groups. The views of the advi-sory group members vary, but it would probablybe accurate to say that each of us agrees with someof the criticisms and disagrees with others.One thing that all sides in the math wars agreeon is: more students should understand what theyare doing. The disagreement comes in how this isto be achieved. Core-Plus initially took an approachto algebra that, in Courses 1 and 2, emphasized tables, graphs, real-world examples, and verbal descriptions, at the expense of some traditional topics and skills. Then the algebra ratcheted up substantially in Course 3. One of the things wehave helped the writers do is even out the intro-duction of algebraic methods.Other things we have done are: identify mathe-matical statements that are sufficiently vague asto be misleading or wrong, object to unnecessaryneologisms, note terminology whose use is contraryto its use in more advanced mathematics, and askfor more attention to definitions—both includingthem and helping students to understand theirimportant role. We have suggested adding or delet-ing various subjects and problems.What We’ve LearnedThere is more to math education than knowingthe math. Yes, everybody knows that, or says theydo, but working with a curriculum project gives thisstatement a richer meaning than one can otherwiseimagine.First, there are all sorts of constraints on theground. At each yearly meeting, several schoolteachers are in attendance, and they report on thepast year and comment on suggestions we make.Who would have thought that an important issuein deciding the order of units is the density ofclass interruptions at the corresponding time ofyear—vacations, assemblies, special state tests,weather, etc.? Then there are problems with absenteeism, and attitudes of other teachers, administrators, parents. How districts buy text-books also has an effect. In some districts you canbank this year’s money for buying texts; in someyou can’t. In some places this means that theywon’t be able to consider switching to the secondedition until several years after all four new booksare out.Second, introducing a curriculum is an immenseproject. Each of us has found writing the text fora single course a huge project, but Core-Plus isfour years of integrated courses. Plus teacher edi-tions, assessment resources, and software. Plussystematic evaluation at all stages. Plus teacherworkshops. Plus dealing with publishers. And thepublic. And so on. We tip our hats to people whocan pull this off. It’s easy for us to criticize hereand there, but these authors have to put all thepieces together.Finally, what we as mathematicians might thinkis most interesting, or most challenging, doesn’t always jibe with what teachers and students think.We thought that the early discrete mathematicsunits might be disliked, by both teachers and stu-dents, because the material is unfamiliar anddoesn’t meet traditional expectations about whatis mathematics. In fact, students tend to like it asrelatively easy and a nice break from other stuff.But then, if it isn’t on the state test (and it usuallyisn’t), and the class is behind schedule, the discretemathematics material is the first to go.Other things we have learned are more mathe-matical. Core-Plus has a wealth of fascinating examples. We are not statisticians, so we havelearned many things from the statistics examples—fascinating data and even new tests. For instance,suppose you want to test whether two samplesare from the same distribution, but you know nothing yet about normal curves, t-tests, and soforth. There are ways to get at this issue by takingmany random redivisions of the data.The first-named author is an expert in discretemathematics, and yet he has found a fair amount ofthe information in the CPMP discrete mathematics1020 NOTICES OF THE AMS VOLUME 53, NUMBER 9units to be new to him (examples, applications and vignettes). The Core-Plus authors have looked far and wide for useful information.We think the most fascinating things we havelearned are about pedagogy. For instance, what isthe role of a textbook?At a university, students buy their books, andadvanced students may even keep their books andrefer to them in later courses and later in life. Thustextbooks also serve as a reference. University stu-dents are also expected to have the intellectualmaturity to understand complex logical exposi-tion. But in high school, students don’t generallykeep their books, they are just beginning to learnthe conventions of mathematicians, and they learnin roundabout ways. Consequently, the Core-Plusbooks are not meant as references—far from it.Once we understood this, we understood many ofthe design decisions made by the authors. In general, there is a tension between precision andclarity—what is clear at a certain stage of concep-tual development may be far from precise, andcurriculum developers have the daunting task oftrying to find clarity for a wide range of students.That doesn’t mean we accepted everything aboutthe textbook design decisions. For instance, maybestudents don’t need a complete index, especiallyone that covers all four volumes, but teachers andthe mathematical public sure do. So we asked formore and better indexes. Also, some attempts tobalance the tension between precision and claritybackfire and make things worse. Many times wehave urged the writers to try again.A Specific Example of What We Learned:EquivalenceIn Core-Plus the word “equivalent” is used twoways. Equations (and other statements) are equiv-alent if, as usual, each implies the other. Expressionsare equivalent if they are two different forms forthe same thing, that is, if they are equal for all values of the variables.The first-named author was taken aback by thesecond use of equivalent. Why say that 2 and 1 + 1are equivalent, when we already have a better word:they are equal. Why say that 2x and x + x are equiv-alent, when we already can say that 2x = x + x is anidentity? Furthermore, he argued that Core-Plus, by defining equivalent for both equations and expressions, would run the danger of exacerbatingthe unfortunate tendency of students to think that equations and expressions are the same thinganyway.Others (both writers and consultants) were takenaback that this author was taken aback! Although“equivalence” was not used for expressions in thealgebra books this author had checked, some others had seen it and had also used it regularly.As the discussion went on (over many days beforeand after the 2004 meeting), it became clear thatwe were all trying to deal with the same issue, butsimply had different ideas about what would workwell. The issue is that many students don’t have agood feel for expressions (or equations!). Rules forrewriting expressions are just formalisms for them,without much purpose. We wanted to make clearthat different expressions for the same thing havedifferent uses. For instance, x2 − 3x + 2 and(x− 1)(x− 2) are equivalent (ahem, setting themequal creates an identity), but the former is goodfor recognizing a quadratic whose graph opens upand the latter is good for identifying the zeros.We thought about banishing the use of the wordequivalent for expressions, but this has downsidestoo. One has to use longer and somewhat fuzzierphrases such as “different forms of the same thing”(but then what does “same thing” mean?) or “ex-pressions which form an identity when set equal”.Or we could just say “expressions that are equal”,but this could cause confusion between equal ex-pressions in an identity such as 2x + 4 = 2(x + 2),and situations where two expressions are set equalto form an equation such as 2x + 4 = 5.What did we learn? First, that what one mathe-matician assumes is standard is not always stan-dard to another. Various members of the advisorygroup had rather different ideas about what wasnormal usage of “equivalent”. Second, that whatwould appear to be a simple issue—just make a de-finition and be done with it—was not simple at allwhen it came to the ramifications for clarity instudent minds. Either direction we took on equiv-alence had pitfalls for possible student confusion.Third, that people who had spent a lot of time inthe precollege classroom had a lot of wisdom aboutwhat works well and what doesn’t.ConclusionWe have very much enjoyed our four years working with the Core-Plus developers. We look forward to the publication starting in 2007 of thesecond edition. We think we will have helped makethe second edition another step forward; we knowwe have learned a lot in the process and havegained immense respect for the people who actu-ally make it happen. We encourage any readerswho are interested in precollege mathematics education to look for opportunities to be involvedin projects that intrigue them.

Advising A Precollege Curriculum Project

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Advising A Precollege Curriculum Project

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