principles, art, and craft in curriculum design: the case of connected geometry

E. PAUL GOLDENBERG

PRINCIPLES, ART, AND CRAFT IN CURRICULUM DESIGN:THE CASE OFCONNECTED GEOMETRY?

1. INTRODUCTION

1.1 What’s to be Learned from Curriculum Designers?

When a curriculum finally hits the schools, it shows little trace of how itcame to be – of the years of thought, discussion, decisions about contentand pedagogy, initial crafting, trials, revision, editing, and so on that wentinto its development. If it is done well, it appears somewhat as did Athena,who sprang forth miraculously from Zeus’s head.

But developing a curriculumis a long, complex business that involvesmany different kinds of contributions – mathematicians, writers, classroomteachers, students, editors, mathematics educators, evaluators, publishers,page designers, and illustrators, to name a few – and a wide variety of activ-ities that interact in complicated ways – brainstorming, outlining, writing,editing, revising, field testing, teaching in the classroom, assessing studentprogress, surveying the market, designing format and layout. All alongthe way, a myriad of decisions must be made, all of which collectivelydetermine the final product.

? This paper was supported by NSF grant MDR92-52952, EDC’sConnected Geometrycurriculum development project (Education Development Center, 1998a), and NSF grantRED-9453864, EDC’s “Dynamic Geometry” project, with additional support (and instig-ation) by the Centre for Educational Technology, Ramat Aviv, Israel. I thank MichalYerushalmy for calling especial attention to the need for chronicling our methods, forhelping to arrange part of the support to do so, and for her deep questioning of severalof my initial ideas, stimulating further thought. The ideas owe a great debt to the entireConnected Geometry team – especially Al Cuoco, Pam Frorer, Jack Janssen, June Mark,and Michelle Manes – and to Susan Janssen’s thoughts about curriculum-craft based onthe experiences of EDC’s “Seeing and Thinking Mathematically” project. Thanks alsoto Natasha Shabat, for editorial help in making this paper communicate. I’m particularlygrateful to Wayne Harvey and Peter Braunfeld, who discussed many of the ideas in thispaper with me, and whose strong arguments (and detailed, insightful comments on mydrafts) challenged my thinking and helped me greatly clarify both what I meant and whatI said, and to Brian Harvey who helped me shape the final version.

International Journal of Computers for Mathematical Learning4: 191–224, 1999.© 1999Kluwer Academic Publishers. Printed in the Netherlands.

192 E. PAUL GOLDENBERG

Though curriculum developersdo research and design, what they arepaid to create is curriculum, not research or design reports: seldom is theresupport for self-conscious examination of how developers do what theydo and why they chose, for their curriculum, to do thingsoneway insteadof some other. As a result, developers’ design rationales, the way theserationales translate into words and pictures on a page, and the knowledgegained from extensive applied research all tend to be left implicit in thematerials, and are not readily available to the research community or toother developers. This paper is written in the hope that by making at leastsome of this explicit, for at least one curriculum, the result will prove usefulto current and future developers.

At the outset it is important to say that, with few exceptions (e.g., themathematics must not be in error, the text must be readable), there arefew absolutes: there’s more than one “right way” of doing most things,including the design and construction of curriculum materials. My purposeis not to promote the particular decisions our development team made:others teams will make other decisions and, while there are certainlywrongways to proceed, there must be very many right ones. My purpose is toillustrate whatkinds of decisions must be made,how we came to makethe particular decisions we made, and by what mechanism philosophy,assumptions, and “high-level” decisions can affect curriculum design, insome cases right down to matters of “low-level” craft.

1.2 Basic Principles

Governing all the rest of our principles and methods were these:

1. Nothing is good when it is overdone.2. No principle should be pure.3. The preceding two principles are exceptions.

1.3 A Taxonomy for Thinking About Curriculum Design: Values, Theory,Craft

It is essential, first, to be clear about the ideals and goals that are thepurposeof the curriculum one is designing. Bypurpose, I mean not thelocal goal of having students learn the content, but some end goal thatthe designerscare aboutand believe will be served by having the studentswork with the content. Like all values, such a purpose can be hotly debated,but not under the banner of science: the goodness or badness of a set ofvalues is not “researchable” in an empirical sense. In mathematical terms,it is more like a postulate than like a theorem.

CURRICULUM DESIGN 193

Given a clear statement of purpose, a designer is then in a position toconsider how best to serve that purpose – a statement of theory thatisempirically testable.

Finally, there remain the nuts and bolts of crafting the materials: thepractical considerations of implementing the theory.

In this paper I will focus mostly on the issues of deciding the purpose ofthe curriculum (clarifying one’s values) and ensuring good craft, the twotopics least discussed in the literature, but I will also list some theories thatguided us.

2. VALUES: DECIDING THE PURPOSE OF THECURRICULUM

Courses inevitably give messages in addition to the facts that comprisethem. This is why there is concern about the influences of college mathe-matics courses on the pedagogy of prospective teachers: while the studentsare learning mathematics, they are also absorbing a pedagogy that wasnever intended to be part of the course’s message, and was not given thethought, planning, and careful analysis that the instructor likely gave thecontent matter. Similarly, elementary school students learn their teachers’fears as well as their facts.

Curriculum alone cannot control all the variables. For a simpleexample, curriculum materials cannot, by themselves, erase the pasthistory of a teacher’s feelings about mathematics, or prevent those feelingsfrom being transmitted as part of what the students learn from the teacher.

But clarifying the purpose of the curriculum can controlsomeof thevariables. For example, a curriculum might set out to get students toperform well at some specified set of tasks. If, at some point, it is notedthat students going through the curriculumaredoing well at the designatedtasks, but are poor at “problem solving” or “critical thinking,” it may nothelp to add sections on these latter topics as a remedy. In fact, such add-onsmight not have a positive impact at all. Without thoughtful re-engineering,small alterations to a design that served the original goal of the curriculummight serve that goal less well, while remaining inadequate for serving thenew goal.

We took the following as an axiom:

4. A curriculum should represent a point of view, and not just content.That point of view is the curriculum’spurpose.

By the way, the notion of “hidden curriculum” makes the relatedassumption:


5. A curriculumdoesrepresent a point of view, even if it is not acknowl-edged (or even recognized) by its designer.

A content goal alone – for example, “geometry” or “these facts” –offers little help in making the practical decisions one must face whenconstructing a curriculum, like which facts to include, when or how (orwhether) to use technology, and when or how (or whether) totell studentsinformation rather than having them work it out on their own. “LearningGeometry” might, on the face of it, seem broad enough, but it does notstate why wecare that students learn geometry. Moreover, it’s ill defined:there is a vast quantity of geometry from which to choose, and it cannot allbe learned. Which facts must one acquire, and to what depth or robustness?And, by the way, there’s a vast quantity ofnon-geometry to learn in thisworld, so why choose geometry at all?1 Even the undeniable usefulness ofmathematics is not enough of a purpose to justify math in the curriculum:linguistics, psychology, economics, learning to draw, and learning to fixyour house are all useful, and some may be more useful to more peoplethan most of the math they learn.

Viewed this way, facts and techniques are ameansby which we achievea necessarily differentpurpose. The purpose might be to satisfy students’pure interest, increase their marketability, enable them to succeed at aparticular task, encourage their desire for intellectual challenge, supportthe high-tech future of America, help students survive in that future, helpthem show off at cocktail parties, help them get in to college, or preparethem to go on to something else they like better than these facts andskills. . . . But it is circular to argue that the purpose of learning somethingis to have learned it.2

Unless it is burlesqued into a savagely reductionist exercise, thisperspective doesn’t demolish disciplinary boundaries. Mathematics is thevehicle that tradition, public concern, and government funding virtuallyguarantee will be available for use, and that our team’s own particularexpertise and interest qualify it to drive. Other teams will have othervehicles. The question to ask about the curriculum development vehicleis: Where Do We Want To Go?

2.1 Why We Selected Geometry

There are many reasons one might select geometry as the vehicle for sometrip – helping students understand space, shape, and dimension, teachingaxiomatics, rounding out a mathematics education, among others.

For reasons detailed elsewhere (Goldenberg, 1997; Cuoco, Golden-berg and Mark, 1996), we chose to focus on students’thinking, but morebroadly than suggested above. In debating the alternatives before writing


our proposal to the NSF, we selectedgeometryas the vehicle for ourexpedition for two reasons.

• Geometry’s deep connections to other mathematical topics and tononmathematical interests could help connect a larger and morediverse group of students with rich mathematics. Given our concernabout thinking, we cared to reach not only those whose futures (as faras could be determined) required the particular facts our course wouldcontain, but a broad population of students including both those whowouldgo on to advanced mathematics and those who wouldnot.

• Geometry’s traditional isolation from the rest of the schoolcurriculum made it a strategically easy point to enter, and yet (if wedid our work right) afforded us an opportunity to get a crowbar intothe rest of the curriculum.

To serve our goals, the geometry to be taught would have to makestrong connections in two directions. In order to succeed at all, it wouldhave to capture the imagination, curiosity, and interest of a broad groupof students. Even looking only at art (tiling, perspective. . . ), science(astronomy, biology, crystallography, kinematics. . . ), mechanical design(linkages), and play (billiards), it is clear that geometry can connect wellwith diverse interests.

In order to serve the Trojan Horse role and help makemathematics(and not merely an isolated fragment of it) serve good thinking, it wouldhave to reach beyond the walls that isolate it from the rest of the typicalmath curriculum. If the name “geometric series” seems to imply a connec-tion with geometry, the connection is not at all apparent in the standardcurriculum. Similarly, geometry is deeply connected with combinatorics,number theory, group theory, analysis, linear algebra; subtopics withinschool algebra, trigonometry, and precalculus; and such cross-cutting ideasas limits, functions, and mathematical induction.

2.2 Thinking

Yager and Lutz (1994, p. 338) report that “although a focus on. . . criticalthinking [and] problem solving . . . has been a goal of science educa-tion reformers for over fifty years, there is virtually no evidence. . . [ofsuccess].” That goal is at least as old in math education,3 but still on theurgent list in calls for reform. Are such lofty goals doomed to failure?Perhaps, but I don’t think that the experiences with texts to date (at leastin mathematics) provide the relevant data. That is, I do not believe that theexperiment has really been tried. Inserting problem-solving and critical-thinking sections into existing curricular structures doesn’t fundamentally


change the underlying message; and students are, on the whole, absolutemasters at knowing what’s wanted from them.

If the goal is to serve students’ thinking, then that must be the drivingprinciple throughout.All questions about content, organization, lessonstructure, pedagogy, even page layout, must be answered with the goalin mind. In particular, it is important not to take good thinking for granted,as if learning mathematics was automatically good for one’s mind. It isnot automatic. But the reverse direction seems quite inevitable: improvingone’s mindmustbe good for one’s mathematics! Therefore, if this goal isachieved – and especially if it is achievedthroughmathematics – it shouldserve students who want or need to take advanced mathematics later, andalso those who do not.

But what about thefacts that the advanced students need? Can theyshine through the thinking goals? Yes. In fact, the same facts can beused to tell a large variety of stories. The traditional pop-culture storyis rather ladder-like: mastery of addition strictly before subtraction, two-column subtraction before three-column, all arithmetic strictly before anyalgebra, and so on. The same facts might be arranged differently to tellother stories, like historical development or logical interrelation. A currentfavorite organizer is applications: topics do not appear in their own discretesections; instead, techniques and facts from various branches of mathe-matics appear in whatever combination and order best suits the problem towhich they are applied. In each case, the same facts may be encountered.It is the order and the “glue” between them that determines the story.

The storywechose to tell was about how people think in order tofindthe facts. In part, this follows from our picture of mathematics as “notonly . . . a collection of results and conjectures, but also . . . a collection ofmethods, ways of thinking. . . ” (p. 16; Cuoco and Goldenberg,1996).

Our goals for mathematics education certainly include that students become quite familiarwith the edifice of mathematical results that has been built over the last 2500 years, but theyalso include, on an equal footing, that students develop a style of work, a way of lookingat things, that is characteristically mathematical. (ibid., p. 16)

Moreover, we reasoned that this shared emphasis would benefit notonly those whowouldneed the facts but would also serve those for whomthe thinking, more than the facts themselves, would be the greatest lastingbenefit.

It is important to reiterate that this focus on “habits of mind” is achoicethat we made – more like a postulate than like a theorem, as I said earlier.My purpose here is not to defend this choice (that’s done elsewhere!),but to show brieflyhow such choices are made and what implications


– “theorems,” if we push the metaphor – they have for the content andstructure of the curriculum.

Some of these “habits of mind” are more math-specific than others– some are specialized even to subdisciplines within mathematics (e.g.,algebraic habits of mind, or geometric habits of mind). But some, evenones of central importance to mathematics, are easily recognized outsideof mathematics. Here is a brief list of these more general ones:4

A Visualizing: picturing (and drawing) what is inherently visible aswell as that which is not (either because it is an abstract object orrelationship, or because it is a concrete object that has not yet beenbuilt).

B Using precise language (natural and formal) to describe and analyze;making definitions to bring precision or to name new categories andclassifications. Analyzing one’s language, including examining (andmaking) definitions.

C Tinkering with problems, including posing and critiquing problemsas well as solving them and critiquing the solutions.

D Mixing deduction with experimentation. Seeing the interdependenceof interpreting experiments and making theories.

E Reasoning about discrete processes: algorithms, mathematical induc-tion, calculations. This is a connection with the algebraic end ofmathematics.

F Reasoning by continuity, connecting with the analytic end of mathe-matics.

G Approaching one idea from varied perspectives and in varied systems(e.g., coordinate, synthetic). Related idea: Even when problems haveonly one answer they can almost always be solved in more than oneway.

H Thinking in terms of functions. Seeing functions in flexible ways,strange guises, and unexpected places.

I The inclination to look for invariants.J Seeking logical connections, explanations of how things work, and

proof.

The last two, I and J, deserve special emphasis as they are at theabsolute core – the two hearts, so to speak – of mathematics.

For invariant, we might instead have saidpattern, but pattern is oftentaken in too shallow a way. A major theme in mathematics is about deter-mining what doesnot change while other things do. In a visual pattern,what the eye sees does not change even as one’s gaze shifts location. Ina table that relates two numerical variables, the numbers in each columnmight change from entry to entry, but the mathematician’s interest is to


find what doesnotvary: a function, if possible, that describes the invariantrelationship between the two columns.

Likewise, the mathematical mind is seldom satisfied knowing merelythat something is so, without knowingwhat makes it so. Proof servesa dual role in mathematics: itcodifies an understanding of whysomephenomenon must occur, and itprovides certaintythat the phenomenon isa reliable building block for new ideas. In designingConnected Geometry(Education Development Center, 1998a, 1998b), we felt that to helpstudents appreciate the characteristically mathematical concern with proof,it was the first role – the notion of proof as enlightenment – that neededthe greatest emphasis. For one thing, students often associate proof forcertainty with an unwarranted distrust of one’s observations, or simplyobsessive. This may be especially true in an applications-focused culture,in which using the mathematics on the good authority of others ought tosuffice. Proofsdon’t always increase the novice’s conviction, or sense ofinner authority, for that matter: following a proof without making it one’sown may promote reliance on “authority” rather than on understanding,just as surely as accepting statements without proof. By contrast, theessence of understanding is in learning to care, and know, whatmakesthings work as they seem to work.

Incidentally, this is a good place to point out the close interactionbetween facts/skills and what we call “habits of mind.” While skillswithout the opportunity and inclination to use them are just baggage, thebest inclinations (like the inclination to look for invariants or proof) areuseless without the skills to fulfill them effectively. Knowledge and skillremain important.

In basing a pedagogy on mathematical habits of mind, it helps tosee them as specializations of, rather than departures from, the kinds ofthinking that people use all the time and that have evolved in all cultures.Recognizing a pattern – whether it is in numbers, visual forms, or humanbehavior – is finding something that does not change while other factorsdo. Even such a “primitive” cognitive act as the baby’s development ofobject permanence is the mental construction of an invariant underlyingthe baby’s varying visual (and other) perceptions (and occasional lacksthereof). And while it is wrong to cast mathematical proof as if it werejust like legal proof or casual dormitory bull sessions, it would be equallywrong to cast it as if it were developed by alien life forms. Mathematicalproof is highly specialized, but descended from and recognizably relatedto the kinds of thinking we do all the time. The specialization gives mathe-matics its unique power; the relatedness gives mathematical habits of mindgreat value even to those students who do not go on to advanced mathe-


matics. Andtreatingthe mathematical ways of thinking as a specializationor honing of common thought processes may help students and teachersmore readily recognize, relate to, and develop these sharpened ways ofthinking.5

3. DECISIONS ABOUT CONTENT AND ORGANIZATION

3.1 Too Many Problems, Too Little Time

There is vastly too much good material to include in any curriculum, nomatter what its focus. How does one decide what to use?

6. In developing a curricular sequence, always ask, “Why isthis contenthere?”

The answer that it is beautiful, interesting, useful, and accessible isnot generally enough. Even those vital criteria do not winnow the choicesenough to allow what remains to fit into a tome that the students can carry.(But remember, beautymightbe enough in some cases. In fact, it might attimes be theonly point one is trying to make.)

In craftingConnected Geometry, we constantly found ourselves editingback, removing good topics that did not serve the targeted habit of mind,and problems (however wonderful) that were redundant. A few suchproblems could be saved for practice exercises or assessments, but notmany.

In addition to the focus on habits of mind, another effective winnowingcriterion arose from the importance of aconnectedgeometry.

7. Each activity must be grounded in geometry.8. Each activity mustalso show clear connections to ideas, content, or

methods drawn from other branches of mathematics.

3.2 Some Consequences of Having Chosen Habits of Mind as a Principle

Once we have established that ourpurposeis to strengthen and extendstudents’ ways of thinking, several consequences seem inevitable.

The role of problems and explanations in the Student BookIf the goal is for students to learn how peoplefind the facts and methods,then:

9. We cannotpresentsuch facts and methods and have students merelyapply or prove them.


Somehow, the students must arrive at these facts themselves. This is nota statement of faith that “discovery learning” is “better.” If the goal were tolearn the Pythagorean Theorem, students might equally well “discover” itfrom an experiment, a well-written textbook, or a brilliantly crafted lecture.But if the goal is for students to learnto bepeople who can invent suchtheorems for the first time, then they must experience situations in whichthe only authority is their own ingenuity and reasoning, and a context(perhaps historical) in which they are solving a (subjectively) previouslyunsolved problem.

10. Such a curriculum must be problem based rather than explanationbased.

In such a curriculum, the primary role of problems is not to rehearseideas or exercise techniques that have already been explained, but to struc-ture challenges through which studentsdevelopthose ideas and techniques.It becomes largely the student’s job, not the text’s or teacher’s to find andexplain ways to solve the problems.

11. Definitions, theorems, and proofs are theanswers students pro-duce. They appear not in the student book, but in the solutionsmanual!

Remember: “Nothing is good when it is overdone.” Students cannotpossibly reinvent thousands of years of mathematics in a few years of highschool. Some things must betold or explained. Creativity – whether inmathematics or in art – is not the product of loose, undisciplined, free-form novelty. In both fields, the truly creative geniuses often painstakinglywork through the processes and products of earlier geniuses. But learningto be creative, like learning anything else, requires opportunity to try outand exercise the skill. In fact, learning to be creative is apparently hard, andprobably requires a good dealmoreexercise than learning one’s additionfacts.

12. Learning other people’s facts remains important – too many facts,too little time. And learning how others invented their facts can alsobe useful – in all creative fields it helps to study, even copy, theold masters. But to acquire the skills of the discoverer, one mustalso devote a significant part of one’s experience to the creative anddisciplined act of doing the discovering.

We found not only the crafting of the problems, but also their sequenc-ing to be terribly important. InConnected Geometry, in fact, it is in theconnections among the problems (habits E, F, G, and J), far more than in


the novelty of any one of them, that the stories about mathematics and itsinvention really reside.

Other components of the Student BookIn materials dominated by problems, it is possible for students to “miss theforest for the trees.” For example, a cursory look (in no particular order) atselected problems in the Gelfand and Shen (1993)Algebrafails to distin-guish it well from most other algebra books. In fact, even if one worksthrough the problemsin order, if one never poses oneself the additionalproblem of figuring out whytheseproblems were selected and juxtaposedin this particular way, one misses a lot of what makes this book special.Good teaching is essential, but how are teachers to discover the mathema-tical story line? Problems by themselves rarely make it any more explicitto teachers than to the students. Teaching notes can help the teacher, butwe wanted our student materials to help students reflect on the problemsthey solve in ways that helpthemdiscover the point behind them. Some ofthis reflection can be fostered by carefully crafted problems, but we neededother components as well.

In time, we came to see the curriculum in terms of multiple “voices”that brought different perspectives to the subject matter at hand, much asseveral colleagues might if they were co-teaching the course. The riskof these various influences was incoherence, especially in a curriculumin which students’ investigations might leadthem, too, to move inunpredictable directions. If students were able to find, and were eveninvited to pursue, interesting sub-plots and side-stories, they needed a“voice” to narrate the main story and help them (and their teachers) feelthat therewasa story, and that it had coherence and direction, and thatthey were making “progress” through it.

The Narrator RoleOccasionally, the story was explicitly stated in the text but, for the mostpart, this role was filled by the problems. The story was conveyed notso much by the problems themselves, many of which were (in one formor another) common enough in other texts, but in the way they werearranged. Often, the necessary reading between the lines was encouragedby a “reflective” question, often flagged as such, after a set of problems:an explicit request to the student to say how several problems were similaror different. The following example comes fromConnected Geometry’sHabits of Mind: An Introduction to Geometry(Education DevelopmentCenter, 1998b).


Newborns start out at different birth weights. And you now weigh much more than you didas a baby. Nonetheless, it can be said for certain that there was a time when you weighedexactly 30 pounds.

How could such a thing be knownfor certain?Well, there was a time when you weighed less than that. And now you weigh more. So,

no matter how your weight may have increased or decreased in between, there must havebeen at leastonetime, however brief, when you weighed 30 pounds.

Did this person ever weigh 30 lbs?

1 At 6:00 one morning the temperature in Boston was 64◦F. At 2:00 that afternoon thetemperature was 86◦F. Can you be certain that there wassometime that day whenthe temperature was exactly 71.5◦F? Can you tell what that time was? Explain youranswers.

2 In the 1950s, the town of Sudbury had a population of roughly 5,000. By 1990, thepopulation was well over 40,000. Can you be certain that there wassomedate at whichthe population was exactly 10,000? Can you tell when that date was? Explain.

3 Some car companies advertise how quickly their cars go from “0 to 60.” If a car goesfrom 0 to 60, is there a time when it’s traveling exactly 32 mph? Explain.

Write and Reflect4 Problems 1 and 2 look nearly the same except for the numbers, but they are profoundly

different, and have very different answers. Explain why.This problem is, in someways, the most important one!

Take It Further5 Some people find problems 1 and 3 essentially the same. Others object that they’re

quite different. Where do you stand? Why?

In order to answer problems 1 and 2 correctly, the student must, atsome level, be attending to the characteristics of the range, noticingwhether it is discrete or continuous, but only problem 4 asks the student tomake that idea explicit.

The Kibitzer and Critic RolesProblem 5 is the voice of the Kibitzer, interrupting the normal flow andhinting that there might be some issue of interest lurking beneath thesurface. Students who read the solutions manual or who spontaneously


notice that speed cannot be computed in the usual way if the time intervalis zero will find problem 5 to foreshadow ideas about limits.

Creators of mathematics don’t merely solve problems. Theydiscernthem where others noticed nothing; theyposethem where others had nosense that a puzzle was lurking. Even the solving of problems, as Pólyaeagerly taught, often benefited from altering the problem in various waysthat exposed new problems and potentially valuable connections. In asimilar way, seeing a flaw in a line of thought – “critical thinking skills”in the popular literature and “proof analysis” in mathematical circles – cangenerally only be done by stepping outside of that way of thinking.

Connected Geometryused several devices to help build extendedconnections and divergent ideas into the curriculum. Occasionally it was anexplicit part of the story. In the example given above, problems 1 through3 are such variations. The answer to any one of them is of little interest:the student’s understanding of what makes the problems different is whatreally matters.Thatunderstanding does not come from merely working theproblems, but from deliberately viewing them from a distance – steppingoutside, with a “metaproblem,” like problem 4.

“Write and Reflect” sections became a familiar component of thestudent text, generally encouraging the view from a distance.

13. Experiences are not enough. One must have time and encourage-ment to think about those experiences and to find the orderliness andconnections in them.

“Take It Further” sections provided another mechanism for handlingdivergence, connections, and extensions. These were intended as directionsof significance that were off the main track (and thus could not be followedup in the text) but of potential interest to some students.

Design considerations in support of divergenceFor a teacher to diverge from the text is much easier than for a text todiverge from itself. If discrete elements are to be perceived as meaningfullydivergent rather than merely inchoate, they must be embedded within areasonably well established context or path (so the narrator’s voice mustbe strong), and their effect must be relatively local.

In addition to choosing a page design that was deliberatelynotdistracting in such irrelevant ways as the use of multi-colored boxes andshadings, zillions of fonts, and marginally related artwork, we chose awide margin that we could use for comments that might be distractingor interruptive in the main text, but that had local importance. Sometimesthis marginal space was used just for an illustration or a metacomment,and sometimes for brief notes from the “technical consultant voice,” like


a definition or reminder, but often it was the place where divergent ideaswere tossed in. Because language was so important to us (Habit of mindB), some side-notes encouraged students to play with language, or drewattention to connections among words. For example, one sidenote to adiscussion of polygons says,

If penta-, hexa-, hepta-, andocta-mean 5, 6, 7, and 8, andpoly- means ‘many,’ what does-gonmean? And why is a square not a ‘tetragon’ and a triangle not a ‘trigon’? We don’thave the word ‘trigon,’ but we do have the word ‘trigonometry.’ In what sense is trigono-metry about measuring (-metry) ‘trigons’? By the way,dia-gonalhas the -gon in it, too!Thedia- part means ‘across’ or ‘spanning.’

Such a detour into etymology serves several purposes. It helps maketerminology less arbitrary for the mathematics student. It connects somemathematical ideas. It helps establish the idea that words and meanings(e.g., definitions) are important in mathematics. It invites attention andcontribution from those students who find words interesting. And, in somecases, it is a place in which students whose first language isnot Englishhave special expertise.

What meaning do the wordsquadrant, quadrilateral, quarter, and the Spanish wordcuatrohave in common?

While attention to language was certainly not the only route to diver-gence, it served in that way strikingly often, perhaps because the narrowingand widening of definitions in mathematics is used as a tool for alteringcontext or point of view. Language is an important enough part of mathe-matics that students must not justusealready-perfect definitions, but mustwrestle with and refine bad ones, make definitions (see de Villiers, 1994),and extend good definitions to fit new purposes. Here are two examples,drawn from Habits of Mind: An Introduction to Geometry(EducationDevelopment Center, 1998b).

For Discussion

• As a practical matter, how can youknow for certain that you are traveling straighton any given surface? Just whatis straightness?

• In our world, what is meant by a “straight road?” A road that is straight like a laser-beam won’t stay on the earth, because the earth is round. A road of any significantlength must curve through three-dimensional space in order to stay on this earth-ballof ours. What, then, distinguishes the road we call “straight” from the road we call“curved?”Assume there are no mountains, trees, rivers, or other obstacles.

• Ignoring details like mountains and oceans, picture two straight roads inour world,both starting at right angles to the equator and about a block apart, and extending(straight!) north for six-thousand miles or so. Do they stay a block apart, or does thedistance between them change?Can two (6,000+ mile long) roads on earth remainthe same distance apart and still both be straight?


By tackling this problembeforea formal definition of “straight” on asphere is given, students get a chance to use and perhaps refine the ideasthey developed in the immediately preceding discussion. Presenting a defi-nition first may cut short some useful additional thinking. A later problemasks if a square can be drawn on a sphere. The answer is neither Yes norNo. It all depends on what one means by a square: a regular quadrilateral(four congruent sides and four congruent angles), a quadrilateral with fourcongruent sides and a right angle, or a special kind of parallelogram. Onthe plane, they are the same thing; on the sphere, they are not! Studentsmust decide whether or not to extend the definition of a square and, if so,how. This apparently simple question cannot be answered without reex-amining the definition of a square, and it calls attention to the distinctionbetween thedefiningcharacteristics, and theconsequencesof that defi-nition. Geometry on a sphere is a diversion, not to be followed up, butthe diversion exercises a central theme (from the habits-of-mind point ofview): distinguishingdefiningfeatures from their invariant consequences.

Where mathematical connections could help teachers make decisionsabout when and how to follow up the various directions students’explorations might take them, these “outside” connections were includedin the Teacher Notes.

The CriticThe Critic provided yet another “stepping out of context” voice, whichwe used sometimes to poke fun at a problem. For example, the followingappeared in a context in which all the problems were, in one way oranother, exercising the use of the Pythagorean Theorem. Though other-wisequite like problems found in many standard texts, this problem endsdifferently, with the Critic’s voice.

A plane left Los Angeles, flew 100 miles north, turned due east and flew 600 miles, thenturned north again and flew 350 miles. About how far was the plane from its starting point?(Challenging: Why does the Pythagorean Theoremnot give the precise distance here?)

Why did we choose to present a problem that we would then declareto be faulty? Why not, instead, start with a problem that was a correctapplication of the Pythagorean theorem?

Our focus on thinking caused us often to draw attention to the condi-tions that surround the “truths” that the students were learning. Forexample, the statement that a triangle’s angle bisectors are concurrent isa fact about triangles. The statement that a triangle’s angle measures sumto 180◦ is less about triangles than about planes. The concurrency of atriangle’s angle bisectors is remarkable only because other figures’ anglebisectors are, in generalnot concurrent.


Presenting the problemand critiquing it – or having the Kibitzerpresent a conflicting result – calls attention to the fact that the PythagoreanTheorem is about geometry on a plane, but there’s even more. Though theTheorem does notproperly apply to this problem at all, studentsshouldthink of it in connection with this problem, and feel quite justified using itssimple computations to derive a rough answer, provided that they realizethat that is what they’re doing. It is useful for them to decidehowbad ananswer the Theorem gives in this case. They don’t have the mathematicsto do this precisely, but they can reason that, on the scale of the earth,600 miles is large enough to matter, but not to matter much for roughestimates. It is therefore quite powerful to include the problem, providedthat a critique of the problem is also included.

The Mentor RoleDespite the primacy of problems – getting students todo the thinking– it was also important to have mechanisms for illustrating some of theimportant ways of thinking. Sections titled Ways To Think About It wereoccasionally inserted in the student book, alwaysafter a problem thatwas particularly demanding and novel, and always with more than oneapproach to the problem. These were not intended as direct hints at thesolution, but rather directions to begin thinking about the problem.

The Solutions Manual was also seen as sharing the Mentor Role, andtherefore as an integral part of the curriculum: the fully worked solutionsare written to be readable by students, and answers without explanatorymethods are avoided. Even where “solutions may vary,” the habits-of-mindfocus sometimes obligates us to say what aspect of the problem makesthe solution nonunique. We selected problems whose various solutionstrategies best illustrated the ways of thinking we were emphasizing.

Especially in the beginning of the sequence, when this whole new wayof looking at mathematics was new to the student, we felt a need foranother kind of explaining – not of the mathematics, but of our perspectiveon it. Why not be as open with the student as with the teacher about ouraims?

In the first few days of a course, you figure out what is really wanted of you – whatreally “counts.” The authors believe that what countsmost is your thinking. So, insteadof beginning with review and exercises, this course begins with problems to think about.

The Social Connections RoleSometimes social connections can be made by setting mathematical ideasand discoveries in their original historical context. We did this some inour materials, but one of our goals led us to another device that we’ve notseen in other materials. Namely, if students are to learn how people find


(present tense!) facts and methods, they must see mathematics as a livingsubject, not a finished product created in its entirety by people long dead.Therefore:

14. Our student book must contain personal and varied stories of livingmathematicians.

Such techniques can fall flat. In an attempt (we hope successful) to keepthese essays alive and interesting, we invited the mathematicians to writetheir own pieces. The results were indeed lively, even quirky, filled withpersonality and ringing with the voice of the author/mathematician. Dosuch devices “work”? We do not know, and that is a matter for research.My reason for mentioning the autobiographical sketches is not becausewe know them to succeed at their purpose, but just to point out thatsuch devices mayhavea purpose (other than window-dressing). For us,they followed from a natural, if unverified, theory about implementing ouroriginal principles; they are also consistent with our belief (for supportiveresearch, see Graves and Slater, 1986) that the writing must be colorful andvivid.

4. THE CHOICE AND ROLE OF TECHNOLOGY

To me it is most natural to discuss curriculum development independentof the medium (blackboard, paper, book, or computer) through which itis conveyed, and much less natural to focus on calculators and computers.Most of the issues are no different: while the roles of paper and electronicmedia are ideally not the same, the underlying principles that shape theirroles must be. Each use of computational technology – not just each kindof hardware or software, but each curricular use of it, right down to eachproblemposed of the student – is its own unique case to judge effectiveor not, and appropriate or not. Having to decide at that level of detail isno surprise at all if one thinks of computational technology the way onethinks of pencils. It is theproblemsthat are posed, not the technology withwhich they are attacked, that make all the difference: with computers, aswith pencils, some problems are great and some are a waste.

But high speed, interactivity, and computational power do raise some oftheir own issues. The increasingly rapid infusion of large amounts of ever-newer electronic stuff into schools forces us to rethink what is importantin mathematics education – what new opportunities we have and what oldburdens we can now shed. This is the Trojan Horse effect: get the computerin and it causes change.


4.1 Keeping the Changes Under Our Control

Computers do bring about change (Trojan Horse) but they also stream-line and amplify what already prevails (Troy’s revenge), and can, inboth directions, influence culture without our intent or even our notice.The greatest influence electronics has had on education is not throughanything, like Logo, that is implemented in classrooms, but rather the shifttoward machine-scorable multiple-choice tests, with consequentincreasedsupport for schooling that focuses on the knowledge of facts. Nationallystandardized tests used for college admissions are, in this sense, the intel-lectual parent of “200 things your second grader needs to know.” The rushto the Internet and the push for data analysis in the curriculum, both madepossible only by the new computational tools, focus attention in the sameway – on gathered information, rather than on the systematization of ideasor the fostering and formalization of reasoning.

Thus, ironically, the prevalence of computers, and the prevalent culturalview of what they are for, may well abet the forcesagainstthe stated goalsof curricular reforms that aim at understanding and reject the view of math-ematics as a list of disconnected facts. No onesuggestsusing computersfor the purpose of elevating facts, but the medium pulls for it, and some-times we follow without noticing. One way to counteract this weight is toorganize our use of computers around the development and formalizationof reasoning (habits of mind again!), rather than around data acquisition(through experiment or lookup) and crunching.

15. Programming, per se, is currently out of favor except as an elective,but having a formal language to express and reason about algorithms(e.g., Logo) is as valuable a part of mathematical learning as havinga formal language to express structure and quantitative relationships(e.g., algebraic notation). Systems like dynamic geometry, though not“linguistic,” also help one learn to construct, play with, reason about,and describe mathematical objects that result from an orderly sequenceof steps. One can certainly gather “data” from experiments with thesetools, but a habits of mind curriculum should choose activities that usethese tools to scaffold reasoning.

It is so often claimed that new technologies can help us shed old burdensthat this argument deserves its own analysis. Without calculators, computa-tional techniques like the division algorithm were indispensable for findinganswers. Some of these, in addition to theircomputationalfunction, alsohave explanatory functions, benefits that remained largely unrecognized,taken for granted because they came “free” with skills that were requiredanyway. The long division algorithm – often used as the examplepar excel-


lenceof a foolish post-calculator topic – is a case in point, a useless pieceof drudgery that makes no sense to inflict upon students in an age in whichabsolutely nobody performs such calculations on paper. But knowing howthe algorithm works, and being able to use it with relative ease – even ifone neverdoesuse it – allows one to know and prove (i.e., knowwhy) thedecimal expansion of a rational number must either terminate or repeat.The division algorithm is also a prototypical example of any algorithm ofapproximation and successive refinement. Chucking a technique becausetechnology has rendered itscomputingfunction obsolete risks losing these“side benefits,” resulting in troublesome gaps in students’ mathematicalknowledge and understanding.

16. When technology can reduce the effort needed to learn or apply a skillor idea, we should use it. When it seems to eliminate theneedfor theskill or idea, we must check to see if it has eliminateall the needs. Ifcontent is to be cut, technology (alone) is not the excuse.

Mathematics is complexly interconnected, and so it should be nosurprise thatmostof its ideas – not just the division algorithm – showup in multiple guises and in many places. Think of the division algorithmas a stand-in for any of the many ideas that have multiple functions withinmathematics, only one of which is replaced by the calculator tool.6

4.2 What Leads? Math or Technology?

I’ve argued that the goal is good thinking, with mathematics as a special-ization; any use of technology is merely a technique toward achieving thatgoal, not a part of the goal itself. But, in a limited way, there might be onetiny exception.

With traditional teaching, many students – even smart ones – learnedjust enough to get by; only a few developed what we sometimes call“mathematical understanding.” Technology offers the lure of an alterna-tive, by which students can gain access to important mathematical ideaswithout the protracted skills-acquisition period that used to be the onlyroute and that, by many accounts, failed anyway (or “succeeded” quitewell, if the goal was to filter out many students). But if we propose alter-natives to the old tools (like algebra) that our students’ parents failed tomaster, mustn’t we aim for the new generation to become true mastersof the new tools: their spreadsheets, dynamic geometry, symbolic algebrasystems, or other technologies? If students’ electronic tool skills remainjust barely passable as were the algebraic skills of their parents, we havemerely replaced one set of barriers with another. Of course, perhaps nolevel of electronic tool skills will make up for a lack of, say, algebraic skill


– that question must await empirical answer – but lackingboth skills isclearly not progress!

With a limited setof tools well matched to the curriculum, there is oneway in which we might cautiously treat the technology as a techniqueanda goal just as algebra is both a technique and a goal. With the best ofthese tools, even technical details (or most of them) can be taught in waysthat are not intellectually empty. Approached in that way, the tools them-selves become worthy content. So, for example, the fact that constructinga perpendicular on geometry software requires one to identify two objects,a point and a line, is not merely an interface idiosyncrasy; instead of asoftware-how-to, learning to construct a perpendicular with the softwarecould be a medium for stating a fact about geometry, or recalling the point-slope analytic representation of a line. Another example: Logo’srepeatis often presented as a technique for saving keystrokes, writing code that’seasier to read and debug, organizing one’s thinking, or revealing the struc-ture of an algorithm. These are all virtues – especially the organization andclarity that is a good habit in any intellectual endeavor – but, in the contextof a mathematics course, they seem peripheral, a distraction from themathematical goals. On the other hand, learningrepeat can be part of thecontent: “factoring”fd 50 rt 90 fd 50 rt . . . into repeat 4 [fd 50rt 90] better expresses the four-ness of the square, its membership in theclass of regular polygons, its exclusively 90◦ angles, and so on. Curriculumshould teach the tool skills needed for understanding and manipulating thecontent, but should (and usually can!) do so in ways that are not uselessoverhead, distracting attention from the content.

17. Curricula that use computational tools should find systematic ways todevelop fluency with them. Students were not masters of the old tools(like algebra). The lack of competence hampered them and left themwithout power. It is no favor to give them new tools that they do notmaster.

Ideally, this power shouldbuild from grade to grade and allow studentsby high school to use the tools easily and appropriately to solve non-routine problems that match their intellectual and mathematical develop-ment. Such an approach is strongly at odds with the too-common tendencyto treat Logo as an elementary school toy, and then drop it altogether (or,for the very few who continue to program, drop it only to replace it withsome other language, which is a bit like using a semester of Spanish aspreparation for French). As a toy, any computer language, even Logo, isfar too expensive in time to be worth the investment. As a fluent language,it rounds out students’ repertoire of expressive means: one’s own naturallanguage is best for conveying the semantics of a mathematical idea or


situation (the “everyday” sense of meaning that modern pedagogy seeks tosupport in its heavy emphasis on oral and written presentations); algebraiclanguage is best at expressing and transforming quantitative or struc-tural relationships; and computational language is optimal for describingprocesses and algorithms.

4.3 Expressive Media

Somewhat more generally, the habits-of-mind orientation leads us toselect tools whose design focuses not so much onsolving problems ason rendering mathematical ideas and problems expressible and explor-able, and leading students into new ways of thinking about the problemsand new connections among the ideas. The distinction is not particularlyobvious in geometry, where software designed tosolvegeometric problemsin the classical sense (i.e., prove theorems) is neither currently availablenor generally even proposed as an educational tool, and where software(like CAD) that is designed to solve the geometric problems of an engineeror architect is also rarely considered relevant to the school curriculum.But it is easy enough to see thatif the object of the geometry coursewere to apply geometry to practical mechanical/architectural design prob-lems, then CAD software might well be the right choice, and things likeSupposer, Cabri, and Logo might well be distracting and inefficient tools.

In the case of geometry, this argument may seem academic, but I feelcompelled to bring it up because of what I believe is its real importancein thinking about the algebra curriculum. So, let’s switch domains for amoment and suppose our “content” goal is to get students to find roots oflinear and quadratic equations. If we chose for this new subject matter thesame habits-of-mind orientation that we chose forConnected Geometry,we might well be disinclined to use a graphing calculator as a support tool.It fully meets the content requirements, but makes the path from problemto solutionso quick and clean that it is almost devoid of ideas. Yes, onecould think deeply about how zooming in on a graph gives the desiredanswer. One underlying idea, for example, is the topology of the set ofreal numbers, but this idea, though important, is not particularly “close” tothe current goal of having students think about roots of equations. Morepromising (given our habits-of-mind point of view) might be the approachthat Schwartz and Yerushalmy take in theirFunction Supposer(Schwartzand Yerushalmy, 1993; Yerushalmy and Schwartz, 1993; Yerushalmy andGilead, 1997), which leads students to think about how their equationrelates to other equations that have the same solution set.

It is important to reiterate that my ranking ofThe Function Supposerover a graphing calculator is not an absolute, but a consequence of having


chosen a habits-of-mind orientation. If, alternatively, we chose to focuson applications of mathematics, we might greatly prefer the graphingcalculator, which cuts to the chase and lets the student get on with therest of the problem situation. The choice of technology depends on thegoal of the course. When the student is practicing acting like an engineeror scientist or statistician (something that probablyshould be part of astudent’s experience atsometime or other) then the tool should reflectthat purpose, and should resemble (in a developmentally appropriate way)the tools that those professionals use. But when the goal is to learn howpeople find mathematical methods and facts, or to develop mathema-tical ways of thinking, it may be too difficult for students to take thenecessary reflective/connective steps if the “answer” comes too quicklyor without rich enough associations and connections. For this latter goal,the calculator is probably the wrong tool.

18. One’s most basic goals determine the role of technology: with a habits-of-mind orientation, a primary purpose of technoogy will be to helpstudents formulate, express, and reason about mathematical ideas;with other orientations, technology may more often serve other func-tions, like reducing the knowledge one needs or computational effortone must expend in order to solve certain problems.

4.4 “Idea Editors”

Returning to the more restricted domain ofConnected Geometry,our initial assumptions and goals led us to conclude that neithertutorial/explanatory nor practice components could play a dominant rolein the student text materials: the same reasoning applies to computationaltools. In theory, a software tutorial with a habits-of-mind focus is imagin-able, but our choice to organize around ways of thinking has, in fact, ruledout all tutorial software we’ve seen to date.

Better suited for our purpose are “idea editors” (named by analogyto text-editors): word-processors, computer algebra systems, drawingprograms (from MacDraw to CAD), programming languages, spread-sheets, and geometry tools like Cabri or Geometer’s Sketchpad. Whatthese diverse tools have in common is that they do not convey subjectmatter content, set the agenda, or take initiative in the interaction: theyallow thinkers to express their own ideas with precision (in whateverdomain the tool is tailored for) and then rearrange and tinker with thoseideas until they work. The nature of the task, the ideas that go into it,and the criteria for completion all rest with the thinker. Contrast thiswith tutorials and games, in which the machine sets the agenda, and the


person responds. Contrast it also with database searches or web browsing,in which the agenda remains with the human searcher, but the primaryfunction of the computer is as asource of informationrather than as atool augmenting the information-seeker’s ability to manipulate ideas. The“idea-editor” category is quite fluid – some simulations, for example, blurthe distinction between information source and idea editor, and betweencomputer-initiative and user-initiative – but the notion of “idea-editing” isuseful when selecting technology to support a ways-of-thinking-(habits-of-mind)-centered curriculum.

Having chosen the sub-discipline of geometry as the springboard forour curriculum, and the goal of connecting it with the rest of mathe-matics, it seemed quite natural to select idea-editors that supported theconnections.

Tools like Geometer’s Sketchpad or Cabri present geometric structuresin an environment that emphasizes the continuous nature of Euclideanspace, and thus serve as an excellent bridge between geometry andanalysis. The geometric structures that one explores are essentially func-tions defined with geometric rather than algebraic rules: vary the positionof this object (independent variable) and watch the effect on that one.Change and rate of change are one’s direct experiences, rather than theresults of interpretations made on static pictures or symbolic expressions.One reasons by continuity.

By contrast, a procedural/functional programming environment (weselected Logo) emphasizes algorithms, iteration, and recursive definition,discrete processes and algebraic thinking. Whether in or out of geometry,tools like spreadsheets and Logo highlight algebraic/computational struc-tures, and emphasize algorithmic thinking and reasoning by mathematicalinduction.

Each tool has its own character, even when applied to the same task.One cannot say whether Logo or dynamic geometry or some other toolis “better,” without saying “better for what.” Because our habits of mindincluded both reasoning by continuity and algorithmic thinking – andbecause we were determined to create a geometry connected with themathematical ideas classified broadly as algebraic and analytic – we choseboth kinds of tools.7

4.5 Aligning Technology with Content and Pedagogy

Even if a piece of technology is employed only to add a new mediumof activity to a curriculum that already exists – which, by the way, isprobably always a bad idea – the technology is almost bound to affectthe curriculum, sometimes in unanticipated ways. I recall when two of


us were excitedly showing the then brand new Geometer’s Sketchpad toa colleague who had been using the Geometric Supposer happily andsuccessfully for several years in her teaching. Her cool reception startledus: “If I want my students to study properties of squares, this thing isterrible. I can’t evenget a square without first going through a lot oftrouble.” Our imagination had been so captured by the new activities thatthis technology made possible, that we failed to notice that there wouldalso be existing activities that it might make more difficult. And even ifsabotaging practice were part of our goal – the Trojan Horse effect – theinterim results,until content and pedagogy change may be worse than whatwe’d had before.

19. When staging a surprise attack, don’t let it surpriseyou. Even if tech-nology is carefully selected to support the goals of a course, it cannotjust be dropped into a structure designed for use without it. Pedagogy,content, and technology must make sensetogether.

5. ISSUES OF CRAFT

Having picked a purpose and strategies for achieving it, one must stillcraft the materials well to implement the strategies. This entails its ownset of benchmarks or local goals. Ours were to create materials that weremathematically sound and significant; that were interesting, appealing, andunderstandable to teachersandstudents; that made classroom “sense” (thatis, were manageable and teachable); that were in line with the currentbest thinking about mathematical learning (research and accepted stan-dards); and that were realistically capable of public acceptance and schooladoption.

Principles govern the craft issues of writing, editing, and illustrationsjust as they do the purpose (goals) and the content. And to implement thoseprinciples, the development staff must combine the right skills. Given thecraft goal thatwearticulated, it was necessary that the writing staff itself befully and deeply fluent both with the mathematics – it is too easy to botch amessage if you don’t fully understand it yourself – and classroom teaching.In addition, we had to have serious input of working mathematicians fromplanning through late drafts and, equally, the full involvement of currentteachers at all stages of the project.

In this paper, I discuss only one aspect of the craft – the craft ofproblems.


The Craft of Problems

The nature, number, and contextual setting of problems required somethinking. Should any problems be open-ended and, if so, how many?Should they be “applied”? What about drill?

Too few problems, too little practice?When current methods seem to have a hard time teaching even the “basics,”what sense does it make to complicate the task with such seeminglyabstruse notions as “Habits of Mind”? In pushing “big ideas,” do we notcare about “low level” skills? Do we believe practice is not necessary, orthat students will just “get it” (whatever It is) if the work is interestingenough?

Exercises created solely for the sake of practice are not apparent in theConnected Geometrymaterials, so we were often asked how we viewedpractice and the need for it.

The thing thatmakesan idea, procedure, or skill truly important is thefact that it appears all over the place in (and often out of) mathematics.We do not have tomakethe important ways of thinking occur frequently;they justdo: that’s what makes them important. They “drill” themselvesby recurring no matter what the local details might be, so the curriculumcan move on to new (small-idea) facts and territories, while the big ideascontinue to mature (through use and exercise) and become habits. The“drill” does not occur all at once: it occurs over time, and so it is notnoticeable.

But what if the practiceis needed all in one place, as when some “lowlevel” skill must be learned prerequisite to other activity?

Two thoughts. First, in a rationally designed multi-year curriculum, onemight never need to put the practice all in one place. A good example isWirtz et al.’s (1966) elementary textbook seriesMath Workshop, in whichimportant ideas are foreshadowed early so that when skills are neededthey are already well under development. But life doesn’t always grantus control over the entire curriculum. When all we get to write is oneyear’s worth, and there isn’t time to spare, we can still take a lessonfrom Math Workshop: Practice at one skill can generally be embedded ininvestigations of something else.

A good example of this latter strategy is in our method for givingstudents practice plotting points. Though that skill appears “all over theplace” (making it important), it is what many people reasonably call a“low-level” skill. Clinical experience tells us that studentsneedpracticeto become fluent at plotting points, and that when they lack such fluency,


the attention that they must devote to the plotting itself can distract themfrom the “bigger” ideas that the plotting was intended to elucidate.

To provide such practice we need not create exercises that serve nofunction other than that drill. Contrast, for example, the following twoways of developing fluency in plotting points.

One approach gives practice on a set of points that contains enoughdiversity to exercise all aspects of the skill (all four quadrants, non-integervalues . . . ).

1. Plot the following points:

(3,1) (−5,−1) (6,−1) (−12,−2) (0,5) (4,−1) (−15,5)

(−4,7) (2,9) (−6,−8.5) (1,8) (5,2) (−2,−0.5) (0,−1)

(2.5,−3) (−4,0) (12,3) (−1,0) (7,7) (12,12) (−7,−7)

A second approach provides the same amount of drill, but embeds it inan investigation of transformations applied to the vertices of a triangle.

2. Look at the table below. Plot the three points in column A, and connect them to forma triangle.Use the rule in column B to find the three vertices for that triangle. (One’s done foryou.) Then draw that triangle. Label that picture “(x,y)→ (x+3,y)” and describe howthat transformation of coordinates transformed the triangle.Do the same for triangles A and C, A and D, and so on.Use separate sheets for each pair of triangles.

Vertices of Vertices of Vertices of Vertices of Vertices of Vertices of Verticestriangle A triangle B triangle C triangle D triangle E triangle F triangle G

(x,y) (x+3,y) (−x,y) (x,−y) ( x2 , y2 ) (2y,2x) (−y,x)

(2,1) (1,12 )

(−5,4) (5,4) (−4,−5)(−4,0) (−1,0)

In both problems #1 and #2, the student gets practice plotting 21 points.In problem #1, the points are varied, but otherwise arbitrary. Designinga problem in which the points create a smiley face or reveal a hiddenmessage is equally arbitrary – the pattern is entirely up to the whim of thewriter and not intrinsic to the situation. But in problem #2, the points arenot arbitrary. And they are not all given explicitly. The student is reviewingone skill to determine the coordinates, exercising a second skill to plotthem, and developing skills in observation and description as they solvethe stated problem.

Is too much going on at once for this to be good pedagogy? This isan empirical question, but one that is highly sensitive to the particularcrafting of the problem, the medium (e.g., computer or paper) in which


it is presented, the prior experience of the students (which the teacher andtext influence), and so on.

In the case of the problems shown above, our experience suggests that,with reasonable counsel from the teacher (or good emerging strategiesfrom the student), thepractice purposes are served as well by problem#2 as by problem #1 even for the “poorest” students. If each stage – deter-mining the coordinates, plotting the points, and recording the observations– is taken separately, they do not interfere with each other. The exercisetakes longer than problem #1 might, but (in different ways for the weakerand stronger students) to good effect. If, in fact, too much is going on forsome students in problem #2, it is the hints of transformations that get lost,and they were not the primary goal of the problem.

Such secondary content does not need to be followed up immediately –this is not a mastery learning model, where everything that appears must bemastered before moving on. Rather, it plants seeds for later development,a foreshadowing of ideas to come. Such foreshadowing we believe to bevaluable in giving students time to build a store of experiences that willbe organized, refined, and formalized later. This belief goes beyond theearlier statement that, done properly, foreshadowing does not detract frompractice. Like the earlier statement, it is empirically testable, but probablynot in a categorical way.

Through approaches like these, a small number of big ideas andmethods that are important for all students are encountered repeatedly, butin a variety of mathematical contexts rather than in problem sets devotedsolely to them. The contextual topics that are used to develop the big ideasand methods, and the facts that are encountered along the way, are lesscentral and recurrent in the curriculum, and receive less practice than insome traditional approaches. For the nonmath-intending student, the prior-ities seem perfect: important ways of thinking first, and forgettable ideasin the background. But even for the math-specializing student, who needsmuch or all of the content used as context for the major habits of mind,there are advantages. Mathematical methodsare at least as important forthese students as the facts that mathematics finds. Moreover, our field-test experiences suggest that the facts, themselves, appear to be easier toassimilate when they are connected through the big ideas and methods.This greater retention appears to work for the nonmath-intending studentas well.

The result is a curriculum that provides depthand practice on theessential big ideas and skills, along with breadth of connections andexperience.


Open-ended problemsIn a group of curriculum developers discussing “What we have learned,”8

the subject of open-ended problems became a significant topic.Open-endedness was certainly “in,” and many of its advantages were

quite clear. For one thing, life itself is often pretty open-ended – its prob-lems may require effort to clarify, and the approaches to them are seldomclear-cut or unique – and so such problems had a kind of reality andhonesty about them. Moreover, open-endedness allowed students to pursueproblems to different depths, accommodating different interests and skills.But several developers9 also pointed out difficulties in using open-endedproblems well.

For one thing, while a major virtue of such problems (if they are goodones) is that wewantpeople to be able to dig in deeply, both students andteachers need some way of being able to tell when to stop. For students,the issue is partly a matter of being able to sense their own progress, or thequality of their work. Conscientious students might put in great effort and,limited by reality, feel like they have not done enough. Less conscientiousstudents could spend mere minutes and have no sense that there’s reallymore to do. How does one decide what is a creditable effort, or whenone has “finished”? The same is true for a teacher, who must somehowdecide when to apply the brakes, or when a tiny investment of additionaltime is really likely to be worthwhile. Open-ended questions, when theyare working well, lead to divergent ideas and approaches. How can thedeveloper help the teacher know which ones to pursue and which to drop?

There is also a question of focus. Where the social climate of the classis good and the intellectual climate lively, good open-ended questions canlead to great depth, great discussions, and great opportunities to find math-ematics, precisely because the problems do not readily “close down” witha straightforward answer. Where things are less rosy, even good open-ended questions can lead to shallowness. Without a clear sense of whatreally matters, students might easily put in lots of work, evenfeel like theyhave worked hard, and not have any mathematics to show for it. Moreover,certain kinds of open-ended problems – for example, “planning a party”– make it extremely hard for teachers to know what to evaluate,10 or howto give students relevant feedback. It’s not clear whatmathematical(or,for that matter,other) message to give back to students. These, too, canbe rich and productive in the hands of a teacher who has a clear sense ofmathematical purpose but, lacking that, too often result in positions like“there are no wrong answers” or “whatever the students do is good.”

So are open-ended problems just a bad idea? No. Life is gener-ally unclear and open-ended, and wedon’t know when we’ve done


enough.. . . At least someof a student’s time seems well spent tryingto make mathematical sense out of open-ended situations, sometimeseven fuzzy ones. The question is how much. To help decide, curriculumdesigners should probably ruthlessly scrutinize any open-ended problemthey choose to include: Is it clear what goal is to be served by leaving thatparticular problem open-ended? Is there evidence from the classroom thatthis goalis being served?

What about applications and the real world?There is a kind of applicability of mathematical learning that feels abso-lutely essential: the ideas that students learn must either give thempowerthat they can perceive, over something they care about, or must appeal insome more personal way. Of course, what a person cares about varies withthe person, but a course that fails to help students be more competent insomedomain they care about is unlikely to feel worth their effort. This“power” need not be visible all the time, but the curriculum developerbenefits all the time by asking: “What does this help the student do, thatthe studentwantsor needsto do?”

One way to respond to this issue is with “real-world applications,” but itis not the only way. The enthusiastic embrace of this approach in contem-porary thinking makes it seem especially important to sound the occasionalskeptical voice. Applicability is avalue. Real-worldliness is but onetheoryabout how to find applicability. In fact, I have deep reservations abouta real-world focus, as it seems generally conceived, and especially as arallying cry or bandwagon on which to jump. Some “real world” situationsare pretexts for the mathematics but not genuine applications. Pretextscome in many forms. If they are appealing enough to tickle students’fancy and loony enough not to be mistaken for reality, then they mightbe justified. But there isno excuse for presenting something unreal as if itwere real. Problems like

Alejandra noticed that she had been waiting at the airport 7/12 of an hour. How manyminutes is that?

or

Tom tripled the number of wolf eels in the school’s aquarium. Now there are 30. How manywere there originally?

are simply unacceptable. Nobody consciously notices the fraction first andback-computes to determine the minutes, and nothing about the airport orAlejandra helps the problem. Unless there is more point to the problemthan “7/12 of 60,” that is probably all that should be stated. If thereis aneed to justify the day-to-day usefulness of this kind of computation, then


this pseudo-contextualization fails to serve that need. And the aquariumscene is hard to square with real life. Other pseudo-applications includethings like the graphing of data one doesn’t (in the “real world”) graph,like the (linearly related) board feet versus weight of a particular size oflumber, or the (quadratically related) diameter versus weight per foot ofdowels. Contexts are OK. Pretexts are not!

At the other end of things, many true applications – such as applicationsof mathematics to mathematics – are not “real world” in the clichéd sense.

Besides, what’s “real world” (or applied in the sense of empowering)for one student isn’t for another. For some, the power to vault sociallycreated hurdles like tests is a good enough application. But not all studentsare motivated by those hurdles. Some (including those who perceive them-selves as “weak”) figure that passing tests is unlikely no matter whatthey do. Others (including those whose future opportunities, as they seethem, make no use of academics or are handicapped more by nonacademicbarriers such as discrimination or poverty than by academic ones) figurethat passing the tests will not buy them much.

I think that any focus onjobs loses exactly the same students: if appli-cations are to help bring in more kids, they must evokeinterestrather thanfuture utility. In fact, for those students for whom the “real life” approachis claimed to be most needed, I have seen no evidence of greater moti-vation by “real life applications” than by good puzzles, and some reasonto believe quite the opposite. Where is it written that adolescents, even incollege, are pragmatic in their approach to life?

Finally, too strong a focus on pragmatic utility of mathematical resultsmay actually workagainstthe development of certain mathematical waysof thinking. In particular, if the value of some result is only in its utility,one hardly needs to understandwhy it works or go through the thinkingrequired toprove that it works, as long as a recognized authority hasalready approved the result.

So, aside from being interesting (which is not a very controllablefactor), what are the characteristics of agoodapplication?

In my opinion, the analysis of the spread of measles, found in thefirst chapter of theCalculus in Context(Callahan et al., 1995) has allthe right feel to it. It starts with questions of obvious importance thatvirtually anyone can understand and relate to: “How fast does the infec-tion spread? When does the epidemic peak?” The questions, while feelinghard, perhaps even intractable, don’t divert attention from the mathe-matics: no protracted data gathering, no special knowledge requirementsto understand the context.


Students first create a qualitative model: three populations to count –those who have not yet caught the measles, those who are ill, and thosewho have recovered. After making some first-order assumptions about howpeople move from one population to another, students work toward quanti-fications of the model, and then study the implications. The entire processrequires very little mathematical machinery: a tiny amount of early first-year algebra, the ability to iterate linear functions, some good thinking, andnothing more. The payoff is big. Studying the model revealsmorethan wasobvious from the assumptions: one learns important things about the actualspread of disease even without having verifiable numerical accuracy in themodel. The ease of entry, the fact that it is about mathematizing a situationand not merely performing the arithmetic on someone else’s mathematiza-tion, the context (spread of communicable disease), the fact that it revealsmore than one puts into it, the believability that these are questions thatneed answers, all make this, in my assessment, a great application. TheStar Logo work (e.g., as reported in Wilensky and Resnick, 1999) has thesame characteristics.

To summarize:

20. Look for ways that apply students’ new knowledge to something, sothat they can experience the power that this knowledge gives them, butbe wary about too narrow a view of the real world.

I’m tempted to stressconnection rather thanapplication and I’mtempted to look for “odd” connections/applications for mathematicalideas, like in language, art, and music. In fact, I would urge:

21. TreatCuriosity andLove of Puzzlementas amajor application. Alsoinclude applications toGames: “this line of reasoning (or this kind ofanalysis) helps you win.” And don’t forget mathematics and reasoningas an application.

Therealreal world.We do want problems to feel “real.” In real life, prob-lems do not come just after we have studied a chapter on solving them.Problems are problems because wedo notknow how to solve them. Someproblems are hard. Wecannotsolve them, so we measure our “worth” byprogress we make, not by completion. If we accept this as a reasonabledescription of reality, then a consequence is,

22. Include hard problems that take more than minutes to solve, or that canonly be solved partially at the time they are first encountered. Note thathard and long-term do not imply open-ended.

23. Honor partial solutions, but be sure students and teachers recognizethat they arepartial.


6. THERE’S MORE, OF COURSE

I’ve only touched on one way that thecraft of materials is influencedby initial principles. There are others: helping groups of writers achieveclarity, focus, and brevity, while maintaining a lively tone; turning “over-head” (like instructions about technology) into contributions to the mathe-matical aims of the course instead of distractions from it; helping teacherswithout constraining them; designing illustrations that expand rather thannarrow students’ images; and so on. Much of this is written, but brevity isas important here as in a curriculum (as the journal editors were quick, andright, to point out). One craft realization that followed from our focus onthinking is that there’salwaysmore and, as one of our teacher-reviewerssaid, leaving things out sometimes stimulates more thinking than puttingthem in.

NOTES

1. This issue – why geometry, and what its role may be in a general education – isdiscussed in detail in Goldenberg, Cuoco, and Mark (1998).

2. My thanks to David Purpel for first getting me thinking in this way. It has made lifemuch more complicated.

3. Back in 1938, in the thirteenth yearbook of the NCTM, Fawcett described a studyaimed at increasing “critical thought.” Through the centrality of proof and the analysisof argument, he felt that “mathematics has a unique contribution to make. . . ” (p. 120)but he recognized (in fact, it was well understood at the time) that “transfer” is anythingbut automatic and that it would not be the mathematics itself, but the way in which itis taught that would realize the contribution.

4. Details about the choice and nature of these habits of mind are beyond the scope of thispaper, but appear elsewhere. Goldenberg (1996) focuses on the more general habits ofmind, and the habits-of-mind rationale and “politics”; Cuoco, Goldenberg, and Mark(1996) discuss the rationale and expand on the habits of mind that are specific tomathematical subdisciplines.

5. Again, this is not a new idea. Sixty years ago, Fawcett was saying that there wereimportantgeneralways of thinking that mathematics could serve. That he wasnotsaying simply “mathematics improves your thinking” is clear from the emphasis heplaces on how it is taught: “Up to the present time teachers of mathematics have. . . assumed that [mathematics’ contribution to critical thinking] can best be made inthe tenth year through the study of demonstrative geometry. The practice resultingfrom this assumption has tended to isolate the concept of proof . . . ” (Fawcett,1938,p. 120). Knowing that transfer had to betaught, Fawcett observed that “if the kindof thinking . . . is to be used innonmathematical situations such situations must beconsidered during the learning process” (1938, p. 13). Among these, he listed “analysisof advertisements, magazine articles, editorials, political speeches. . . ” (p. 108) forneeded definitions, hidden assumptions, and faulty or unwarranted conclusions. Evensuch connections, as Fawcett hints elsewhere, may be too parochial to do the job


if taught only within the mathematics courses or, worse yet, restricted entirely to aone-year geometry course in the tenth grade. (Thanks to James Elander for bringingFawcett’s work to my attention at the St. Olaf Geometry Conference, June 25, 1997.)

6. For an elaboration of the issues surrounding the invocation of technology to supportthe jettisoning of pieces of the curriculum that “only appear vestigial” and that mightyet have important residual benefits, see Goldenberg (in preparation). The remainingissues in this section are also expanded upon in that paper.

7. That choice is still reflected in the modular version of our materials (published on CD-ROM [Education Development Center, 1998b]), but the editing that was required inorder to create a single-year textbook ultimately reduced the use of Logo so much thatwe reluctantly decided to drop it altogether.

8. The NSF-sponsored Gateways V conference in Madison, Wisconsin, 1996.9. Thanks especially to Chris Hirsch and Sol Garfunkel, who presented these issues (new

to my thinking) particularly clearly.10. Although this is an issue even when one is not using the evaluations for assigning

grades, it can be especially problematic in grading, as it increases the potential forunrecognized and unintentional bias. Curriculum materials cannot prevent intentionaldiscriminatory acts, but the more complex or ambiguous the evaluation criteria are, theless help the teacher has in avoiding the unintentional ones. Assessment proceduresthat involve significant amounts of student writing entail some of the same risks: morethan the student’s mathematics is made visible, both for better and for worse.

REFERENCES

Callahan, J., Cox, D., Hoffman, K., O’Shea, D., Pollatsek, K. and Senechal, L. (1995).Calculus in Context: The Five College Calculus Project. NY: Freeman.

Cuoco, A. A. and Goldenberg, E. Paul (1996). A role for technology in mathematicseducation,J. Education178(2): 15–32.

Cuoco, A. A., Goldenberg, E. Paul and Mark. J. (1996). Habits of mind: An organizingprinciple for mathematics curriculum,Journal of Mathematical Behavior15(4): 375–402.

de Villiers, M. (1994). The role and function of a hierarchical classification of quadrilat-erals,For the Learning of Mathematics14(1): 11–18.

Education Development Center, Inc. (1998a).Connected Geometry. Chicago: EverydayLearning.

Education Development Center, Inc. (1998b).Connected Geometry, Modular CD-ROMversion. Chicago: Everyday Learning.

Fawcett, H. P. (1938).The Nature of Proof. Thirteenth Yearbook of the National Councilof Teachers of Mathematics. New York: Teachers College, Columbia University.

Gelfand, I. M. and Shen, A. (1993).Algebra. Boston: Birkhäuser.Goldenberg, E. Paul, Cuoco, A. A. and Mark, J. (1998). A role for geometry in general

education, In R. Lehrer and D. Chazan (Eds),Designing Learning Environments forDeveloping Understanding of Geometry and Space. Hillsdale, NJ: Erlbaum.

Goldenberg, E. Paul (1996). ‘Habits of mind’ as an organizer for the curriculum,J.Education178(1): 13–34.


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Graves, M. F. and Slater, W. H. (1986). Could textbooks be better written and would itmake a difference?American Educator(Spring): 36–42.

Schwartz, J. and Yerushalmy, M., designers (1993).The Function Supposer. Pleasantville,NY: Sunburst Communications.

Wilensky, U. and Resnick, M. (1999). Thinking in levels: A dynamic systems perspectiveto making sense of the world,Journal of Science Education and Technology8(1).

Wirtz, R. W., Botel, M., Beberman, M. and Sawyer, W. W. (1966).Math Workshop, RevisedEdition. Chicago: Encyclopaedia Britannica Press.

Yager, R. and Lutz, M. (1994). Integrated science: The importance of “How” versus“What”, School Science and Mathematics94(7): 338–346.

Yerushalmy, M. and Gilead, S. (1997). Solving equation in a technological environment:Seeing and manipulating,Mathematics Teacher90(2): 156–163.

Yerushalmy, M. and Schwartz. J. L. (1993). Seizing the opportunity to make algebramathematically and pedagogically interesting. In T. A. Romberg, E. Fennema andT. Carpenter (Eds),Integrating Research on Graphical Representations of Functions.Hillsdale, NJ: Erlbaum.

Education Development Center, Inc. (EDC)55 Chapel StreetNewton, MA 02158-1060U.S.A.E-mail: [email protected]

principles, art, and craft in curriculum design: the case of connected geometry

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