1997 Journals

1997 NCSM JOURNALS

The following NCSM Journals are available in their entirety because one year has passed since their original publication date. Only the contents and President's Message from more recent Journals are available.

• October 1997 Journal of Mathematics Education Leadership
• July 1997 Journal of Mathematics Education Leadership
• April 1997 Journal of Mathematics Education Leadership

OCTOBER 1997 NCSM JOURNAL OF MATHEMATICS EDUCATION LEADERSHIP

Journal Contents

• President's Message: Where Are We and What Are We Doing?
• From the Editor
• The Implications of "Geometry for All"
  By Zalman Usiskin, University of Chicago
• Enabling Students at Risk of School Failure
  By Thomasenia Lott Adams, University of Florida
• A Progress Report on Student Achievement in the Core Plus Mathematics Project Field Test
  By Harold L. Schoen, University of Iowa, and Steven W. Ziebarth, Western Michigan University

President's Message: Where Are We and What Are We Doing?

By Bonnie Hanson Walker

I am learning that it is interesting, challenging and fast-paced to be president of NCSM. On September 20 and 21 we had the fall board meeting. It was a busy, jam-packed, tiring, and invigorating experience. Let me bring you up to date on what we have been doing in response to your input and our commitments to you.

On September 9 and 10 several board members attended training on the TIMSS Resource Kit in Washington, D.C.. At that time work continued regarding possible funding sources for providing the resource kit to all NCSM members and plans for training on the best use of the materials in the kit. Finding this funding is our biggest challenge.

The kit is made up of four modules:

Education Module

This module presents an overview of the TIMSS findings. It is designed for individual and small-group use and features the following publications and videos:

• 1) Introduction to TIMSS: The Third International Mathematics and Science Study. This is a comprehensive overview of TIMSS' purpose, scope, and findings. The booklet also includes overhead transparencies, talking points for speakers, and other materials to facilitate community discussions about TIMSS.
• 2) Pursuing Excellence: A Study of U.S. Eighth-Grade Mathematics and Science Teaching, Learning, Curriculum, and Achievement in International Context.
• 3) Pursuing Excellence: A Study of U.S. Fourth-Grade Mathematics and Science Achievement in International Context.
• 4) A video Presentation of Pursuing Excellence: U.S. Eighth-Grade Findings from TIMSS.
• 5) Discussion Guide for 'A Video Presentation of Pursuing Excellence".

Achievement Module

This module, designed for individual or small-group use, features the following publications and makes the TIMSS findings relevant to local decision makers, educators, and parents:

• 1) Bench marking to International Achievement.
• 2) Mathematics Achievement in the Middle School Years: lEA's Third International Mathematics and Science Study.
• 3) Science Achievement in the Middle School Years: lEA's Third International Mathematics and Science Study.

Teaching Module

Using videotapes of actual eighth-grade mathematics lessons form the United States, Japan, and Germany, this module vividly demonstrates differences and similarities in teaching styles and techniques of educators in these countries. This module is designed for teachers, and those who work with them, and includes the following publications and videotape:

• 1) Eight-Grade Mathematics Lessons: United States, Japan, and Germany.
• 2) Moderator's Guide to Eighth-Grade Mathematics Lessons: United States, Japan, and Germany.
• 3) Fostering Algebraic and Geometric Thinking: Selections from the NCTM Standards.
• 4) Mathematics Program in Japan (Kindergarten to Upper Secondary School).

Curriculum Module

This module features a guidebook to help those involved in curriculum selection evaluate their own offerings. It includes curriculum analysis models, frameworks, and standards:

• 1) Guidebook to Examine School Curricula- A guidebook for use by school and district educators to evaluate and analyze curricula.
• 2) The executive summary of the TIMSS report on mathematics and science curricula
• 3) A Splintered Vision: An Investigation of U.S. Science and Mathematics Education,
• 4) Annotated bibliography.

It is the goal of this board to provide the complete kit for all NCSM members and to accompany the dissemination with opportunities for training on the use of the materials. We are working jointly with NCTM, ASSM (state supervisors), BBA (Benjamin Bannecker Assoc.) and AMTE (Assoc. of Mathematics Teacher Educators) in this effort. I will keep you posted on our progress through updates on our web site, http://ncsmonline.org

Other activities include continuing work, new work and projects still in the idea development stage. They are as follows:

Source Books

• Equity (near completion)
• Integration (near completion)
• Supporting Leaders in Mathematics Education (revision of 1st source book - in progress)

Position Papers/Other

• Leadership (in progress)
• Civilizing the Debate (addressing the backlash - in progress)

Monograph

• Numerical Power (near completion)

Task Forces

• Technology (in progress)
• Supporting Reform with Research (new)
• Closing the Gap (new)
• Professional Development (new)
• Parents and the Community (new)

In the area of collaboration we are pursuing the following opportunities.

• The Education Development Center (EDC) has written a grant proposal for the development, testing, and publishing of two courses for school and district administrators related to mathematics education reform. One course is on implementing new mathematics curricula and one on teacher supervision. NCSM would serve as one of the vehicles for dissemination of the materials developed and I would serve on the advisory board for the project if the proposal if funded.
• A second grant opportunity involves the development of videocase modules for professional development. NCSM's involvement in this project includes piloting and dissemination of the video modules. This grant is proposed by Nanette Seago, Judy Mumme and other notable mathematics education leaders.
• NCSM, as a member of the Conference Board for the Mathematical Sciences(CBMS), has been participating in the organizational efforts to create IMO 2001 USA, Inc.. This organization is now formed and will host a prestigious, world class International Mathematical Olympiad in the year 2001. Bettye Forte, Second Vice President, is our representative in this project.
• The Leadership Academy held at Hamburger University in Oakbrook, IL in July was a major success. This academy hosted approximately 100 participants who went home with knowledge and skills and a plan for addressing local issues. Watch for follow-up to that academy. It cannot help but be great. The presenters were superior and the mighty work of Jerry Cummins and Grace Kelemanik made it all happen.
• Did you realize that 1998 will be our 30th anniversary? Dr. Mary Ann Norton is creating a 30 year commemorative book for us. She is using the 25th Year Commemoration Book so ably done by Betty C. McDaniel as the basis for the document and will build from there. Be prepared for a birthday celebration! Rita Janes, our Canadian regional director is in charge of those plans.
• You should have received a new membership directory earlier this fall. It is very nicely done by the ever capable Mo Nelson. If you have not received one and your membership is up to date contact Mo at monelson@earthlink.net and she will get one to you.

We need your participation. Please contact us if you want to participate in any of the new activities identified in this article. Help us make this organization be what you need for it to be. I am hoping to hear from you.

From the Editor

by Vince O'Connor

Enclosed with this edition of the NCSM Journal of Mathematics Education Leadership (or MEL, as I like to refer to it) is Volume XXVII, Number 1 of the more familiar NCSM Newsletter.

As we start the 27th year of NCSM Newsletters, I am given to pause and reflect on its future. In particular, I am experiencing some mixed emotions about its ability to continue to serve the membership in this fast-paced, increasingly technological society.

One side of me says that we should be looking for ways to shift much or all of the newsletter to the NCSM web site. That way, members could get information when they need it rather than four times a year when the postal carrier calls. That way, the editor wouldn't be scrambling at the last hour to fit items into a format that is bound by pages in multiples of four. That way, we could encourage instant feedback on features and perhaps even assemble some readership statistics electronically.

But, the other side of me says that in our busy world, we might find that members, caught up in the day-to-day work of preparing, teaching, writing, speaking, and struggling to improve student achievement, would often skip the web. Having that quarterly journal/newsletter arrive in the mail, in hard copy, can be a reminder to take time to catch up on the happenings around the country.

What do you think? Check in at the NCSM web page, look it over, suggest ways to improve it, and give us some feedback on the way that the Newsletter and web page can grow together over the next couple of years. Please.

The Implications of "Geometry for All*"

By Zalman Usiskin, University of Chicago

*This paper is a slightly modified version of the script of a talk presented at the 1997 NCSM annual meeting. The author wishes to thank the reviewers for many helpful comments.

In the 1992 National Assessment, only 23% of high school students in the United States reported not taking a course in geometry, an improvement from 29% only two years before. That means that 77% of 17-year-olds in school reported having taken a geometry course, quite a bit higher than the 53% percent who reported taking geometry 14 years earlier, and very close to the percent at that time who reported taking algebra. In fact, this 77% is greater than the percent of students who graduate from high school on time, so it seems that virtually all students who stay in school are taking geometry.

At the same time, not only is geometry in all elementary and middle school textbooks, but it is on most state assessments, and so elementary and middle school teachers are expected to teach some geometry. So, we already have quite a bit of geometry for all students who stay in school.

These substantial increases in exposure to geometry have occurred with little fanfare - in the past 10 years there have been no special conferences on school geometry, no major reports on geometry like we have had on algebra. The situation at the 1997 NCSM Annual Meeting reflects the relative attention given to the two subjects; there are 11 sessions specifically dealing with algebra, often with algebra for all, and only 2 sessions which discuss geometry. The 1998 program will be similar. There is a strand on "algebra for all" but no mention of geometry.

In the absence of some sort of discussion of this major change in the population of students taking geometry, the responses have tended to be eclectic and short-term. The NCTM Curriculum and Evaluation Standards take the "let's do a little bit of everything" approach at the high school level, recommending both synthetic and algebraic approaches, with and without coordinates and transformations. The connection and coordination of these approaches and ideas with each other is a very delicate matter on which little is said.

The same can be said about the Standards view of the level of formality at which geometry might be taught; at K-8 is the-century old idea of informal geometry in the elementary school; at 9-12 is a more sophisticated formal geometry. This is not much of an improvement over the current circumstance.

Furthermore, the articulation question is relatively ignored. In most places, there is no uniform geometry curriculum in the elementary school, and so middle and junior high school teachers tend to start from scratch, and often senior high school teachers ignore this work, too. Consequently, even when much attention is given to geometry at the K-8 level, we do not have that much to show for it.

At the high school level, in practice, the Standards' idea of teaching all approaches to geometry is rarely implemented. Because few people take advantage of the geometry done at the K-8 level, there is no time to cover all the geometry that is recommended at the high school level, except for the best students.

You might think the problem is due to lack of articulation between K-8 and 9-12 schools but that isn't the cause. Even at the high school level few coordinate the geometry done in the geometry course with anything else done in mathematics. Part of the problem, I believe, is a lack of conceptual and mathematical knowledge by enough people to make the change work.

The concepts are not difficult, but most people just have not had enough work with them. I say this, I think, with good authority: I would not know anything close to the amount that I do about geometry had I not become involved with the writing of geometry materials using different approaches. The processes of these other approaches changed my entire conception of geometry.

If you have not studied coordinate geometry in some detail, then you do not realize how placing figures on a coordinate system enables algebra to be used in the service of geometry. If you have not studied analytic geometry -- and most teachers today have not -- then you do not realize the variety of curves, from conics to conchoids, that can be analyzed and studied using geometry.

If you have not studied transformations in some depth, then you may not realize that fish can be congruent, that photographs are geometric objects and really are similar, that all parabolas are similar because they can be mapped onto each other by composites of size changes and isometries, that the graphs of parent functions like y = x2 are geometric objects than can be related to their offspring, that the graphs of y = cos x and y = sin x are congruent, that matrices have powerful geometric applications, and so on.

Another approach, too, is said to be following the NCTM Standards. To respond to the increasing percentages of students taking geometry, some schools make all geometry informal, to avoid proof and a mathematical system almost completely. Judging from what we read in our journals, if you want to be up to date, you should employ Cabri or the Geometer's Sketchpad or some other drawing software in the inductive process. This takes our mind off the fact that deduction, an essential quality of mathematical thought, has been discarded.

Still another response at the high school level that is considered "Standards-based" has been to eliminate the geometry course and teach some geometry in all or almost all years. It is called integration, but because of the lack of connections made between the geometry one year and the geometry the next, and between the geometry and other topics, it may be more accurate to call it disintegration.

From the remarks, you can tell that I believe many people have accurately recognized that the traditional geometry course is inappropriate for many students but have reacted inappropriately to the problem. In the rest of this paper I would like to discuss the question of geometry for all, and its implications for the curriculum.

Why Should All Students Learn Geometry?

When a course is taken by all students, you need stronger rationales for it than when that course is taken by only half of students (as geometry was a generation ago). You cannot just say, "You need it for college.", because your population includes students who have no plans at all to attend college. In the case of geometry, because it is new that the course is being taken by almost all students, you cannot say, "You need it to survive", because about 1/3 of all geometry students today have parents who did not take geometry and they know that their parents are surviving. You must give compelling reasons that students need geometry now, as well as reasons that relate to college and job-training programs and a higher quality of life.

With a change in rationale must come a change in content. The content must be more immediately relevant to the students, and it needs to be taught over a longer period of time because the population includes students who do not do daily homework, and some who do not do much homework at all.

To consider changing the rationale and the content for the course, let us go back to basic ideas and examine what geometry is.

Geometry is the branch of mathematics that connects mathematics with the real, physical world. Sometimes geometry is characterized as the study of size and shape. Since everything in the physical world has a size and shape, all objects - from molecules to car fenders to houses to planets - are geometric. The most familiar names of geometric figures, such as triangles, rectangles, polygons, circles, cylinders, pyramids, and spheres, are not the only geometric figures; they are merely shapes that commonly appear. Out of them more complicated shapes may be described. There are also other basic shapes, such as spirals, meanders, branches of trees, and various fractals that are part of the language of today's geometry.

It is obvious that geometry is needed by anyone who works with physical objects, such as carpenters, plumbers, and others in the building trades; earth scientists, biologists, chemists, and other scientists; athletes, musicians, and their coaches; sculptors, artists, and designers.

And, since we all live in some sort of physical surroundings, geometry can be useful in our everyday lives. How big a desk will fit in a room? Will that shirt fit? What's the largest box that can be made from that cardboard? How many streamers can we make from that roll of crepe paper? How will light be dispersed from a lamp or through a window? For these questions and many others, one needs to know various kinds of geometric measurements, the most common of which are length, angle measure, perimeter, area, surface area, and volume.

Geometry is the branch of mathematics that studies visual patterns. Geometry is the study of the visualization, drawing, and construction of figures. It studies the effects of looking at objects from different angles; of reflecting, rotating, and translating them; of changing the sizes of objects or of stretching them. Consequently, geometry is used by anyone who needs to make or examine pictures, such as artists, photographers, architects, engineers, and designers. The phrases "look at this from a different angle" or "put a new perspective on it" have their origins in the geometry of angles and perspective. Traditionally, mechanical drawing courses have done more drawing than secondary school geometry, but now more and more with computer drawing programs and throughout the elementary school, we ask students to draw and to visualize.

There are serious reasons for being good at visualization. From two-dimensional pictures, it is often useful to determine the possible shapes of 3-dimensional objects, and vice versa. For instance, doctors and dentists and others in the health profession often need to determine from x-rays or MRI pictures the precise position and shape of an organ or bone or tooth or tumor. Geometry provides the concepts that assist in this work - concepts like cross-section and contour curve.

Geometry is a vehicle for representing phenomena whose origin is not visual or physical. The Chinese proverb, "a picture is worth more than a thousand words", characterizes this aspect of geometry. Graphs of functions and relations have already been mentioned. Bar graphs, circle graphs, coordinate graphs, and pictographs are geometric objects that do not usually have their origin in physical or visual objects. Usually they describe counts or quantities. Journalists and other writers who need to describe numerical information thus use geometry, and all of us who read their writing need to be able to determine whether the geometric picture of a statistic is accurate or distorted.

Networks and other designs and diagrams are geometric objects that are used to describe relationships among people or objects. They are used by psychologists and sociologists and business managers to describe how people's jobs are related to one another, or how they feel towards each other. My favorite application of this type is the use of regular polygons to schedule teams in a league or round-robin tournament.

Whereas there are three dimensions in the visual world, and four dimensions in the physicist's space-time world, when geometry is used to represent quantities, there may be many more dimensions. For instance, a student's grades in five subjects, (90, 80, 78, 82, 85), is a 5-dimensional object. We cannot visualize this geometry, but we can extend ideas from two-dimensional and three-dimensional geometry to study them. So, for instance, we can find the distance between two students' grades and use that distance to determine the extent to which the students are alike or not alike. Statisticians often use these multi-dimensional statistics.

All of these aspects of geometry require and use somewhat the same language, the mathematical language for describing space. The language of geometry has three basic aspects. First, there are the names of the many kinds of figures: points, lines, angles, segments, and other basic figures; triangles, rectangles, hexagons, and other polygons; circles, ellipses, spirals, and other curves; cones, cylinders, spheres, pyramids, boxes, and other three-dimensional figures. Then there are the measures of these figures: distance, length, angle measure, perimeter, area, surface area, and volume. Finally, there are the relationships between figures: parallel, perpendicular, congruent, similar, reflection images, rotation images, and so on.

This language enables geometric situations to be described and problems to be solved. For instance, when a computer programmer wishes to program an object onto a computer, it is necessary to describe the object in some way that captures its shape and size.

Some Geometry Topics and a Few of Their Applications

Let me get more specific by identifying some specific geometry topics and the ideas in them that I believe should be learned by all students.

Points: To young students, points are dots. We may poo-poo this conceptualization, but it is close to a very familiar use of point today, the pixel that is the fundamental element in a TV, calculator, or computer screen. A point can stand for a location either on a line or curve (such as a highway), a plane or surface (such as on Earth), or in space. Archimedes realized that a point can stand for an entire object such as when one uses the center of gravity to determine how an object will behave under a force. This conception of point is basic for physics. Euclid realized that a point can also stand for a number, for instance, when we say that two numbers are "close" to one another. Fermat and Descartes seem to be the first to realize that a point can stand for an ordered pair of numbers, leading the way to graphing in algebra. A point can also represent a node of a network, as it does in telephone networks or bus route networks, a conception first developed by Euler in solving the Königsberg Bridge Problem. And in this century, we have seen points stand for data and arbitrary commodities, the cornerstone of statistics and linear programming.

Lines: Lines also have many manifestations. The shortest paths between two points are along the line segment connecting them, so measurement along lines is important. Lines of sight are used to locate objects. Lines contain the edges of many plane figures, such as desks and windows and chalkboards, and they are the intersections of surfaces, as in the corners of rooms. Lines are straight and so are fundamental in accuracy of drawings. Lines are graphs of - what else? - linear equations. Lines can also picture data and approximate data, as with lines of best fit. Parallel lines "go in the same direction"; perpendicular lines are fundamental to balance.

Polygons and Polyhedra: Polygons and polyhedra are everywhere we can see and in places we can only see with the aid of microscopes. Whenever you read, if you are inside it is likely that the floors and doors and walls and windows are polygons. The page you are reading is often thought of as a rectangle with an interior margin on which we write that is also a rectangle, but the page can also be thought of as a very thin rectangular solid whose surface we write on. The cartons in which items are packed are polyhedra, so anyone interested in storing items, or inventory, must deal with polyhedra. Chemists are particularly interested in polygons and polyhedra because molecules and crystals are physically arranged in polyhedral patterns that satisfy certain properties.

Congruence: Congruence is a basic property of geometric figures, somewhat like equality is with numbers. Copies of pages made on duplicating machines are congruent; that is why they are called "copies". Artists' prints and stamps and many many other items are congruent. One advantage of mass-produced objects over hand-made objects is that mass-produced objects are congruent; if a part fails, a new part can be found to fit in the location of the old part.

Congruent figures are always related either by a reflection, rotation, slide, or glide reflection. Reflection images in mirrors are particularly important: they are used in dentistry and medicine to see parts of the mouth or other parts of the body that cannot be viewed directly. Intersecting mirrors are often found in clothing stores so that you can see yourself rotated. Sound works in the same way, so reflections are used by those studying acoustics, those who make musical instruments, and other designers who have to worry about sound. Balls and particles bounce off of objects in the same way, so athletes who deal with bouncing balls and physicists who are study atoms use these ideas, also.

Similarity: Objects are often bigger than we can place actual size on a desk or a sheet of paper. So we make models or draw them to scale, and we study the models and drawings. We can only do this because the models and drawings are similar to the actual objects, and many properties in similar figures are invariant. So people who build or collect model figures use geometry.

In fact, Euclidean geometry is the study of similar figures. When we ask students to draw a 3-4-5 right triangle, they may not draw congruent figures because they may use different units, but all their figures will be similar and all the results they get will be identical.

Can there be giants, people many times our size but not our shape? Ideas of similarity explain why there cannot be, and also explain why heavier animals need much wider legs and lighter animals (such as insects) can have such thin legs. These ideas also explain why the horns of sheep and the shells of many sea animals are spirals. For this reason, similarity is an important concept in biology.

Length, Area and Volume: Anyone who makes objects needs to know the amount of material in the object. If you buy the object, you also need to know the amount of material. These amounts may be given as lengths, areas, or volumes. The amount of land in a lot is measured by its area, but the fencing needed to surround it is measured by its perimeter. The amount of material needed to make a box is its surface area; the space the box encloses is measured by its volume. Geometry provides formulas for calculating the perimeter, area, surface area, and volume of any common shape. It also provides ways of estimating these quantities for irregularly-shaped objects.

Trigonometry: How far away is a mountain, or a tall building, or a cabin at the other side of a lake, or a lighthouse? Trigonometry has been used for centuries by navigators, scouts, mapmakers, surveyors, and all others interested in accurately locating objects.

The trigonometric functions have a fundamental connection with rotating objects and waves. This makes trigonometry essential in the study of light, sound, radio waves, and electricity. So trigonometry helps people who repair VCRs and other electronic devices and those who use musical and other sounds. Waves are among the simplest kinds of phenomena that occur periodically, and trigonometric functions are consequently used to describe things that occur repeatedly, such as heartbeats and the orbits of planets.

All of these concepts and uses of geometry should he learned by all students.

A Sampling of Geometry Problems

Let us move from concepts to problems. Out of the huge number of situations and questions that can be examined with high school geometry, here are ten. They are not meant to be new, but to be representative of the kinds of applications of geometry questions all students should be able to treat.

1. Outlined here is a miniature golf hole as seen from above. Find a path that will get the ball in one shot from the tee T to the hole H.

2. A farmer has 100 feet of fencing and wishes to make the largest pen for the hogs. What shape pen should be used, and what will be the area of the pen?

3. Examine one of the drawings by the famous artist Mauritz Escher, in which congruent figures fill the picture with no overlap and no spaces. How did he do this?

4. A ladder is considered to be unsafe it is at an angle of more than 700 with the ground. If this is so, how far up on a vertical wall can a 15-foot ladder reach safely?

5. In 1993, Domino's Pizza sold 12"-diameter pizzas with cheese and one topping for $9.99 (plus tax). Suppose that they base their prices on the amount of ingredients used to make the pizza, and that the pizzas have the same thickness. What should they charge for a 14"-diameter pizza?

6. Develop a schedule for a league of 10 teams so that each team plays each other exactly once.

7. Trace a lake from a map and cover it with a grid of squares. Estimate the area of the lake.

8. Given a view of a table from an angle, sketch top, front, and right side views of the table.

9. In a Mercator projection commonly used as a map of the Earth, how is area distorted? Which continents look bigger than they are relative to other continents, and which look smaller?

10. According to a Sheraton hotel directory, there is a Sheraton in the Houston area that is 40 miles from the Houston Intercontinental Airport and 10 miles from Hobby Airport. With this information, what can you conclude about the distances between the airports?

These are but a small number of the important kinds of problems that all students should encounter, yet there are teachers and professors in mathematics education who think that all of the geometry you need can be learned in six weeks. To my way of thinking, six weeks suffices only if you wish students to remain ignorant of virtually all of geometry. But it exposes a major problem we have in changing what is done in school geometry. Many people - even mathematics majors - have had no geometry that relates to school geometry other than what they took in tenth grade. Their notion of geometry is completely determined by the aspect of geometry that dominated their school experience, the fourth important aspect to geometry.

Geometry is a set of mathematical systems dealing with points and lines with a unique and significant history. This is the aspect of geometry that causes us to use geometry as a vehicle for proof. Historically, what we call Euclidean geometry was the first part of mathematics to be axiomatized, and what we call non-Euclidean geometry was the first example of a branch of mathematics to be axiomatized whose results were not able to be verified simply by appeal to one's visual senses or by arithmetic. Geometry set the first standards for deduction in mathematics, and today's worldwide view of mathematical reasoning and what constitutes a valid argument in all of mathematics is an outgrowth of work in the Greek empire over 2000 years ago.

The differences between geometries are due to the different assumptions we place on points and lines. We may use transformations or coordinates and still be in Euclidean geometry because our conceptions of points and lines are the same. But if we decide that a point is a dot, with some size, or a pixel on a computer screen, our geometry is different - it is not Euclidean because points (and lines) do not satisfy the same properties that Euclidean points do. I do not wish to discuss the semantics of Euclidean and other geometries; that will take us off the point. But I do wish to emphasize that the existence of different aspects that points and lines can take - mentioned earlier in this article - is a fundamental strength both for the applications of geometry and for the axiomatic treatment of the subject. The fact that points and lines are undefined in most logical treatments of geometry is not a weakness of the mathematics but the key to the wide applicability of the subject.

Let me now turn to the age long nemesis associated with geometry when it is associated with the Study of mathematical systems, proof.

Why Do Students Have Difficulty with Proof?

Let me give five reasons which I believe dwarf any others.

1. Most students have had little exposure to the ideas of proof (if-then statements, justifications, etc.) before the geometry course. We have learned that students have difficulty in algebra when they are not exposed to the ideas of algebra earlier than their concentrated exposure to it in a course. We have thought that it was enough to teach some geometry, but proof is not geometry.

2. Students are asked to prove using concepts to which they have just been introduced. Do you remember your first abstract algebra course, in which you were asked to prove some things about rings or fields on the same day that these ideas were defined? When a student learns what a trapezoid is on Tuesday and is asked to prove things about trapezoids on Tuesday evening or Wednesday, the student is in the same bind. A person needs to have intuition and knowledge about an idea before proofs about that idea are understood.

3. Students are asked to prove things that appear to be so obvious that they cannot distinguish by intuition the given from what is to be proved. One of the first difficulties students have is sorting out what they know from what they do not. They are told to forget what they know. For instance, in 6th grade they may have learned that the sum of the measures of the angles of a triangle is 1800, but then in 10th grade they are told they cannot use this. This is what makes early proofs so difficult. In a way, more difficult proofs are easier, because it is clear what is being proved. Here we have a particular problem with the new drawing programs, a problem pointed out to me by the people at the Education Development Corporation who are among the nation's experts at studying proof in a rich drawing environment. Suppose you have some students who are exploring whether the medians of a triangle are concurrent. They move a figure around, and the medians remain concurrent. Some students believe the reason they continue to be concurrent is that everything has been programmed so that the medians intersect - they still cannot sort out the given from the derived.

4. Students are not particularly good at writing any sort of argument in any class. A proof is a communication which we usually require to be written so that it can be examined. Teachers in every subject area speak of the difficulty students have in writing cogent arguments. In mathematics, we have many rituals for proof, ways in which we state the given, and what is to be proved, what steps we allow to be omitted, special names for certain theorems, abbreviations for others. All this has to be learned by the student.

5. Students get proof in one course and do not see it again. It is amazing that something as important as proof is seen only once by most students. Would anyone in science be satisfied today if students only had one year in which students conducted experiments? Surely proof is as basic to mathematics as experiments are to science.

Two other reasons are often given that I believe to be invalid: that students have trouble with proof because they have not reached a suitable cognitive level or because they lack the ability to do proof. While I know that there are some students whose mental capabilities do not allow them to follow a multi-step argument, they are much fewer in number than the number of students who do not succeed at proof.

I wish I could give you a deductive proof that virtually all students can learn proof. I can only give you a proof by analogy with algebra and arithmetic and point out that the argument not only works with these subjects but also seems to work in some other countries.

I take as Given: All students must learn about proof.

I wish To prove: All students can learn to do proof.

Proof (by analogy with algebra and arithmetic):

Introduce students to the ideas of proof well before making proof central in a course.

Do not introduce proof out of context; develop needs for proof in contexts familiar to students.

An argument does not have to be hard to be a proof. Do not avoid calling an argument a proof just because it is easy.

Recognize that a proof is both a concept and a communication; practice is needed in both.

Do not expect the idea of proof to be learned in a single year.

The Beauty of It All

We all know that it is important to read. Even though most students learn to read in elementary school, we teach the reading of literature in high school and through college. The reason for this practice is not just because reading literature is needed for jobs, or because literature may help in solving everyday problems. It is also because many people find reading literature to be enjoyable. Reading is fun!

It is no different with geometry. Not everyone likes geometry because of its usefulness. Some people like geometry because they like the way that it all fits into a structure, somewhat like the way a book or a piece of music of high quality can be analyzed for its structure. Many people like geometry because of its beauty.

The beauty of geometry is not just in designs like the Escher design on my tie. Some of the beauty comes from its wonder. How did Escher draw that? More generally, we wonder how many results of geometry were discovered, and how do we know they are true?

In this regard, geometry is famous. Here is a result with a surprising use. Start with any triangle. We call ours ABC.

On each of the sides of triangle ABC, construct an equilateral triangle. In the figure below, triangles BCD, ACE, and ABF are equilateral.

Now draw AD, BE, and CF.

It looks as if these three lines intersect at the same point. In fact, it can be proved that these lines do intersect at the same point for any triangle. The point is called the Fermat point of the triangle. When the original triangle has no angle as great as 1200, the Fermat point has the property that the sum of its distances to A, B, and C is smaller than that sum for any other point. So, if three towns wish to have an airport and access roads will have to be built from each town to the airport, or if there is to be a cable network connecting the towns and cables need to be laid from the towns, this point may be the cheapest spot for the airport or central switching location.

The result is not only useful, but beautiful. Notice that the six angles formed by the lines at the Fermat point seem to be equal. In fact, each is a 60° angle. It is surprising and wonderful that even when one begins with a triangle that has no symmetry, one winds up with a point in the middle with as much symmetry as one could have. Many people study mathematics for the enjoyment that seeing, discovering, and proving such results brings. No branch of mathematics brings such wonder as geometry.

If Geometry Is So Important, Why Have Many Adults Been Able to Get Along Without It?

It is common for adults today to speak of algebra and geometry and other mathematics beyond arithmetic as if they are important only to a few people. In fact many people avoid geometry even if they studied it in school. There are three reasons for this: first, even today most students are unlikely to have encountered many of the applications of geometry in their courses. When later they see a geometry situation, they are of necessity like people who go to a foreign country but do not know enough of the language to converse with native speakers in that country. If you visit Mexico but do not know Spanish, you can get along, but you will never appreciate the richness of the culture, and you will not be able to learn as much as you could if you knew Spanish. You will be forced to depend on signs that have been translated into English. And perhaps most significantly, you will not always know what you missed.

A second reason why people do not use the geometry they studied it is that the geometry they studied did not contain many of the basic concepts that are useful. In particular, the geometry most students in the United States still study today is a geometry that only applies to polygons and circles. Even when students study transformations such as reflections, rotations, translations, and size changes, it amazes me that the transformations are not used to legalize any set of points as geometric and to obtain general notions of congruence and similarity that apply to all figures. Congruence is everywhere, but congruent triangles are not. Similarity is everywhere, but similar triangles are far rarer.

And a third reason why people do not use the geometry they studied is that their geometry course was associated with proof, with so strong an association that the content was not considered as important as the ritual of writing statements and reasons.

The Solution

This paper began by presenting a problem, the fact that high school geometry a course originally designed for a time when only a small percentage of an age cohort took it -was now being taken by over 3/4 of students, and the percent seemed to be increasing steadily. Of course, some schools have not reacted to this phenomenon; they simply have lots of students doing poorly in geometry and they think that's the way it has to be, or they blame it on the students. I mentioned three ways somewhat in line with the NCTM Standards in which schools have reacted to this phenomenon. Essentially I believe that there are aspects of all three ways that are important for us.

We must use coordinate and synthetic approaches, with and without transformations, with and without proof. But this is too much for one year, so we must teach geometry over many years, as everyone recommends. It has to start in kindergarten and be built upon, not just reiterated year after year, so that by the time a student is in 7th grade the basic vocabulary and measurement ideas are well-ingrained. We can emphasize exploration with geometry drawing tools, but beginning in 6th and 7th grade we have to begin thinking about deduction and using the language of if-then statements, and starting formal chains of reasoning.

We should integrate geometry with other subjects, but if we have done that as we should over six to eight years of school, students should be ready for a concentrated look at the logical nature of the subject in which the mathematical system is exhibited and some proof competence expected. This cannot be done in a single unit or two; it requires a great length of time because students need to see how the system originates, how it develops, and how it grows. And, then, after we have taken the concentrated look at that subject, by 8th grade for the best-prepared students and by 9th grade for virtually all students, we should come back both to geometry and to proof in later years, not drop them. It is the attitude we have with algebra, and an attitude we should spread to geometry.

A Progress Report on Student Achievement in the Core Plus Mathematics Project Field Test

By Harold L. Schoen, University of Iowa, and Steven W. Ziebarth, Western Michigan University

Contact:
Harold L. Schoen
N287 Lindquist Center
University of Iowa
Iowa City, Iowa
319-335-5433
harold-schoen @uiowa.edu

This project is supported, in part, by National Science Foundation Grant No. MDR-9255257. Opinions expressed are those of the authors and not necessarily those of the Foundation. Co-directors of the Core-Plus Mathematics Project are Christian Hirsch (Western Michigan University), Arthur Coxford (University of Michigan), James Fey (University of Maryland), and Harold Schoen (University of Iowa). Other principal curriculum developers are Gail Burrill (University of Wisconsin), Eric Hart (Western Michigan University), and Ann Watkins (California State University-Northridge).

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The Core-Plus Mathematics Project (CPMP) is a comprehensive curriculum development project funded initially by a five-year grant from the National Science Foundation. It is developing student and teacher materials for a three-year high school mathematics curriculum for all students, plus a fourth-year course continuing the preparation of students for college mathematics. The organization of the curriculum and its design features are based on the authors' interpretation of curriculum, teaching, and assessment recommendations in the NCTM Standards documents. The curriculum builds upon the theme of mathematics as sense-making. Throughout it acknowledges, values, and extends the informal knowledge of data, shape, change, and chance that students bring to situations and problems. The curriculum materials have the following features:

• Multiple connected strands: Each year the curriculum features multiple strands of algebra and functions, geometry and trigonometry, statistics and probability, and discrete mathematics.
• These strands are connected by fundamental themes, by common topics, and by habits of mind.
• Mathematical modeling: The curriculum emphasizes mathematical modeling, especially the modeling concepts of data collection, representation, interpretation, prediction, and simulation.
• Access: Core topics are accessible to all students. Differences in student background, interest, and performance are accommodated by the depth and level of abstraction to which topics are pursued, by the nature and degree of difficulty of applications, and by providing opportunities for student choice of homework tasks and projects.
• Graphics calculators: Numerical, graphics, and programming/link capabilities of graphics calculators are assumed and capitalized on throughout the curriculum. This technology helps to facilitate the emphasis in the curriculum and instruction on multiple representations (numerical, graphic, and symbolic) and on goals in which mathematical thinking is central.
• Active learning: Instructional practices promote mathematical thinking through the use of rich problem situations that involve students, both in collaborative groups and individually, in investigating, conjecturing, verifying, applying, evaluating, and communicating mathematical ideas.
• Multi-dimensional assessment: Comprehensive assessment of student understanding and progress through both curriculum-embedded assessment opportunities and supplementary assessment tasks enables monitoring and evaluation of each student's performance in terms of processes, content, and dispositions (CPMP 1995)

The CPMP curriculum and instructional model are described in more detail elsewhere (Hirsch, Coxford, Fey & Schoen, 1995; Schoen, Bean & Ziebarth, 1996), and the Course 1 textbook, Contemporary Mathematics in Context Course 1, is now available in published form (Coxford, Fey, Hirsch, Schoen, Burrill, Hart & Watkins, 1997). Carefully developed with teacher input over a three-year period, each CPMP course is field tested in 36 high schools in Alaska, California, Colorado, Georgia, Idaho, Iowa, Kentucky, Michigan, Ohio, South Carolina, and Texas. A broad cross-section of students from urban, suburban, and rural communities with ethnic and cultural diversity is represented. CPMP Course 1 was field tested in ninth-grade classrooms in 1994-95, Course 2 was field tested in tenth-grade classrooms in 1995-96, and Course 3 was field tested in eleventh-grade classrooms in 1996-97. A great deal of quantitative and qualitative data were collected during the CPMP field test. The data includes information about various student outcome variables, teacher attitudes and beliefs, level of implementation of the curriculum and instructional model, and specific site characteristics and experiences.

Thirty-three of the thirty-six field test schools were on a regular schedule, and the other three were on a semester-block schedule in which students completed an entire course each semester. This report primarily focuses on the Course 1 and Course 2 pretest and posttest mathematical achievement of the students on a nationally-standardized test and on the project-developed CPMP Posttest aggregated across the 33 schools that were on a regular schedule. Since the pretest and posttest dates in the three semester-block schools differed from those in schools with regular schedules, a separate analysis was required for them. Due to space limitations, all the details of the evaluation could not be included here. A more complete evaluation report can be obtained from the first author.

RESULTS

Standardized Achievement Test

One measure of mathematics achievement used in the CPMP field test is a standardized test called Ability to Do Quantitative Thinking (ATDQT), which is the mathematical subtest of the Iowa Test of Educational Development (ITED) (Feldt, Forsyth, Ansley & Alnot, 1993). The ITED is a nationally standardized battery of high school tests developed by the Iowa Testing Programs, the same group that writes the widely used elementary school level Iowa Tests of Basic Skills (ITBS).

The ATDQT is a 40-item multiple-choice test with the primary objective of measuring students' ability to employ appropriate mathematical reasoning in situations requiring the interpretation of numerical data and charts or graphs that represent information related to business, social and political issues, medicine, and science. The ATDQT correlates highly with other well-known measures of mathematical achievement. According to research conducted by the test's developers, correlation of the ATDQT, when given in grade nine, with the ITBS Mathematics total score in grade eight is .81; with students' final cumulative high school grade point average in mathematics courses is .59; with the ACT Mathematics test is .84; and with the SAT Mathematics test is .82. The ACT and SAT are usually completed in eleventh or twelfth grade.

The ninth-grade level ATDQT was administered to all CPMP Course 1 classes and to some non-CPMP control classes as a pretest in September 1994 and, in equivalent forms, as a posttest of Course 1 in May 1995 and as a posttest of Course 2 in May 1996. Eleven field test schools volunteered to pretest and posttest students in traditional, control classes. The control classes were comprised of 20 algebra 1, five pre-algebra, three general mathematics, and two honors geometry ninth-grade classes. The box plot in Figure 1 shows the pretest to posttest growth of 2944 students in the 33 regular-schedule schools in CPMP Course 1 and of 527 students in the 11 schools who tested in control classes. In each case, these are all students who remained in their mathematics class all year and completed both the ATDQT pretest and the ATDQT posttest. The box and whiskers on each plot indicate the 5th, 25th, 50th, 75th, and 95th percentiles for each group of students. They are plotted against the percentiles of the national norm group.

Notice first that for the CPMP Pretest, these five percentiles are within two to four percentile points of those of the nationally representative norm group at the beginning of ninth grade. This is evidence that CPMP's goal of having a group of field-test students that is representative of all ninth-graders in the country was met, at least with respect to ATDQT scores.

At the 5th and 95th percentiles, the CPMP students' scores did not change from pretest to posttest which means that at these points in the distribution CPMP students improved at the same rate as the national norm group. When percentiles marked on the graph increase from pretest to posttest as in the cases of CPMP's 25th, 50th and 75th percentiles (also referred to as the first, second and third quartiles, respectively), it means that growth over the year at this point in the distribution is greater than the average national growth. For example, the CPMP median grew from the 54th to the 62nd national percentile.

This pattern of growth of CPMP students was consistent at all quartiles in the distribution and across most schools; in fact, mean scores in 32 of the 33 schools showed pretest to posttest growth and the growth was equal to or greater than that of the national norm group in 24 of the schools. Overall, the CPMP students' mean growth was significantly greater (p <0.001) than that of the control group.

The box plot in Figure 2 shows the two-year growth (Course 1 pretest to Course 2 posttest) of all 2270 students in the 33 regular-schedule schools in CPMP Course 1 and all 201 control group students who completed both the Course I pretest and the Course 2 posttest. As in Figure 1, the box and whiskers on each plot indicate the 5th, 25th, 50th, 75th, and 95th percentiles for each group of students. They are plotted against the percentiles of the national norm group.

In this analysis of growth over two years, both the CPMP and the control students began at a higher level on the pretest than those in Course 1, as can be seen by comparing Figure 1 and Figure 2. The different pretest level is mainly due to the fact that the Course 2 analysis excludes students who transferred out of their grade 9 or grade 10 course (CPMP or control) and students who failed CPMP Course I or their traditional ninth-grade mathematics class.

As in Course 1, the CPMP students' growth was well beyond that of the national norm group across the score distribution. In particular, the 5th, 25th, 50th, and 75th percentiles of the CPMP students grew, respectively, from the 7th to the 15th, from the 34th to the 50th, from the 59th to the 73rd, and from the 82nd to the 89th national percentiles. This pattern of growth was consistent across the field test schools. Mean scores in all schools showed pretest to posttest growth, and the growth was equal to or greater than that of the national norm group in 24 of these schools. The posttest mean of the CPMP group when adjusted for differences in the pretest was significantly greater than that of the control group (p <0.05).

Both the Course 1 and the Course 2 results support the following conclusion:

As measured by ATDQT, students in CPMP grew in their ability to think and reason mathematically more than the national norm group for this standardized test and more than a group of control students in some of the same schools. This pattern of growth was consistent across the entire distribution of scores and in over 70% of the schools.

The CPMP Posttests

For each course, the CPMP evaluation team developed an open-ended achievement test, called the CPMP Course 1 Posttest and CPMP Course 2 Posttest which are each in two parts. Part 1 was designed to be a test of content that both the CPMP and the control students would have had an opportunity to learn that year, algebraic content for Course 1 and both algebraic and geometric content for Course 2. Part 2 of each CPMP Posttest included subtests of Data Analysis, Discrete Mathematics, Probability, and (in Course 1) Geometry; that is, content that the control students did not have the opportunity to study. Thus, control students completed only Part 1 of the CPMP Posttest each year, and CPMP students completed both parts. These tests required students to construct their responses and to show and often explain their work. A five-point general scoring rubric was used as the basis for developing highly specific descriptions of what constituted a score of 0 through 4 on each test item. Scores ranged from 4 for a "complete, correct response with clear unambiguous work or explanation" to 0 for "no response or an irrelevant response."

Part 1 of the CPMP Course 1 Posttest was comprised of three subtests, called Algebraic Thinking I, Algebraic Thinking II, and Procedural Algebra. The first two subtests required students to show that they understood algebraic concepts by applying them in realistic settings and interpreting their meaning within that setting. In particular, they were required to translate between and among contextual problem situations and algebraic (linear) representations of the situation including graphs, equations or inequalities, and tables. These subtests also required students to transform algebraic expressions into equivalent forms (that is, simplify expressions and solve equations) that provided insights into a problem context, and to explain how solutions or equivalent forms represented new information in the problem context. The third subtest required students to solve linear equations in one variable and simplify a linear expression with no context, only symbols, given.

Figure 3 shows the mean Posttest results for a random sample of 1,102 CPMP students and for all 743 students in the traditional control classes who completed this test.

Since CPMP and control students had almost identical median ATDQT pretest scores, a comparison of the posttest means is appropriate. (The control group had a slightly higher mean on the ATDQT pretest.) CPMP students' CPMP Posttest Part ~ mean scores in these schools were higher than control students on the Algebraic Thinking I and Algebraic Thinking II subtests. The average effect sizes (difference in means divided by the standard deviation of the control group) were 0.89 and 0.59, respectively. These differences are highly statistically significant (p <0.001). The control group's mean was higher (effect size = .22) on the Procedural Algebra subtest.

Part 1 of the CPMP Course 2 Posttest also contained three subtests, called Coordinate Geometry, Algebraic Thinking, and procedural Algebra. The Coordinate Geometry subtest presented a contextual situation overlaid on a coordinate system, and students were required to apply concepts and methods of coordinate geometry and explain the meaning of the results in the context. Concepts and methods included finding the equation of a line given two points on it, the point of intersection of a vertical line and a second line, the midpoint of a segment, the distance between two points, an estimate of the area of an irregular closed region, and the plot of the reflection image of a given point across a given line. A related contextual problem situation required the use of right triangle trigonometry to solve a triangle for an unknown side.

As in Course 1, the Algebraic Thinking sub-test required students to show that they understood algebraic concepts by applying them in a realistic setting and interpreting their meaning within that setting. In particular, they were required to translate between and among contextual problem situations and algebraic (in this case, quadratic) representations of the situation including graphs, equations or inequalities, and tables. This subtest also required students to transform algebraic expressions into equivalent forms (that is, solve equations and simplify expressions) that provided insights into a problem context, and to explain how solutions or equivalent forms represented new information in the problem context.

The Procedural Algebra subtest required students to solve linear equations in one variable and simplify a linear expression involving parentheses with no context, only symbols, given. Students also were required to apply the laws of exponents to transform expressions and to decide if given forms were equivalent. Each CPMP Course 2 Posttest item was scored using a specific rubric that was based on the general framework given earlier.

Figure 4 shows the mean Posttest results for a random sample of 584 CPMP students and for all 157 students in the traditional control classes who completed this test. Since CPMP and control students had nearly identical median ATDQT pretest scores, a comparison of the posttest means is appropriate. (As was true in the Course I analysis, the control group had a slightly higher mean on the ATDQT pretest.) CPMP students' CPMP Posttest Part 1 mean scores in these schools were higher than control students on all three subtests. The average effect sizes on the Coordinate Geometry, Algebraic Thinking, and Procedural Algebra subtests were 0.62, 1.27, and 0.06, respectively. The first two of these differences are highly statistically significant (p <0.001).

The following task from the Algebraic Thinking I subtest of the Course 1 Posttest is provided to illustrate the nature of the thinking required on the CPMP Posttests. Means and standard deviations of the CPMP and control groups on each part of this task are also provided.

Task	CPMP Mean (SD)	Control Mean (SD)
The number of gallons (y) of gasoline left in a large motor boat after traveling x miles since filling the tank is given by: y = 18 - 2x
(a) Explain what 18 and -2 in the equation tell about the number of gallons.	2.5 (1.1)	1.9 (1.2)
(b) Graph this equation. Explain the role of 18 and -2 in the graph.	2.0 (1.2)	1.1 (1.0)
(c) After filling the gasoline tank, Helen drove the boat until there were 10 gallons left. How many miles had she driven? Explain how you can tell from the equation and how you can tell from the graph.	2.2 (1.4)	1.3 (1.3)
(d) How many gallons of gasoline were left after Helen had driven the boat 8 miles? Show or explain your work.	2.6 (1.4)	1.8 (1.6)

The meaning of these results for parts (a) and (b) are discussed below. Parts (c) and (d) can be interpreted in a similar way.

In part (a), the intent was for students to indicate that 18 is the number of gallons of gasoline the boat contained before it traveled at all and -2 means that 2 gallons of gasoline are used by the boat for each mile it travels. Such an answer was given a score of 4. The mean of the CPMP students on part (a) is 2.5, midway between 2 and 3. A score of 3 means either (1) that both parts of the question were answered but with some vagueness such as "18 is the starting point" or "-2 is the slope" or (2) one question was answered at a 4-level and the other was vague or incorrect.

The control students, mainly from Algebra I classes, had a mean of slightly less than 2 on part (a). A score of 2 was assigned if (1) one part of the response was vague but relevant, that is, at the 3-level, but the other part was incorrect or (2) one part of the question was answered at the 4-level but the other part was missing.

In part (b), students were to graph the given linear equation on a grid that was provided. An answer at the 4-level would be an accurate graph with an explanation that indicated that 18 was the y-intercept of the graph and -2 was its slope. The CPMP students' mean was 2.0, a score that was assigned if (1) the graph was accurate but no explanation or a totally irrelevant explanation was given or (2) the graph was linear but had mistakes such as a slightly incorrect slope or positioning and a relevant but vague explanation was given.

On average, the control students scored at about the 1-level on this part. This means that the graph was incorrect in a serious way such as composed of segments, bars, or a saw tooth or curved shape, and no relevant explanation was given. In short, the mean of the control students was at a level that suggests virtually no understanding of the content that was measured in part (b).

In summary, the CPMP Posttest Part I results complement those for the standardized ATDQT.

As measured by the CPMP Course 1 and Course 2 Posttests (Part 1), CPMP students were better mathematical thinkers and problem solvers than comparable students in other mathematics courses at the end of the school year. At the end of Grade 9, control students scored higher on a subtest of procedural algebra, but by the end of Grade 10 this difference had disappeared.

FINAL NOTES

In summary, CPMP students in both Courses I and 2 improved more as mathematical problem solvers than a nationally representative sample of students at the same grade level and more than a control group of students in traditional ninth- and tenth-grade classes in some of the field-test schools. The growth in problem solving and reasoning documented above is quite likely the result of CPMP students engaging frequently in class, homework, and assessment activities that are rich in the opportunity to reason about problems presented in interesting contexts, about mathematical models for those problems, and about connections among and patterns in various representations of those models. Class observation and teacher perception data, while still being analyzed, appear to provide further evidence in support of the above explanation for the positive CPMP student achievement outcomes. Furthermore, analysis of CPMP students' perceptions of their experience in Course 1 revealed two themes that were consistently rated very positively: (1) solving realistic and challenging problems is difficult, especially at first, but later students gain from the experience and find it interesting; and (2) problem solving in groups, with the accompanying discussing and writing of mathematical ideas, is an important aid to learning.

Many researchable questions are still under investigation. For example, student performance results in the three semester-block schools are not reported here, since different testing times required these results to be analyzed separately from those from schools on regular academic schedules. The overall pattern of achievement and attitude results in the three semester-block schools appears, at this time, to be similar to that reported in this article for the schools on regular schedules. When the CPMP field test is completed, a careful comparison of outcomes in schools on different academic schedules will be conducted and reported.

Important questions concerning CPMP's impact on students of different gender, minority group, English language proficiency level, and level of mathematical background and preparation are being carefully monitored. Much more detailed information concerning the implementation process including local implementation issues, concerns, and strategies is being gathered. The performance of eleventh- and twelfth-grade CPMP students on college entrance examinations, college mathematics placement tests, college mathematical sciences courses, and on-the-job mathematical tasks is also of great interest and will be carefully monitored.

The continuing data gathering and analysis notwithstanding, it seems important to emphasize that over a two-year period the large, diverse national field test sample of CPMP students showed consistently strong growth in problem solving across the distribution of students' test scores while maintaining an algebraic skill level comparable to students in traditional mathematics courses. These results were obtained on measures of mainly algebraic and geometric content, but the CPMP students also studied topics in probability, statistics and discrete mathematics. The findings reported here are encouraging, preliminary evidence that CPMP is developing a core curriculum aligned with the 9-12 NCTM Curriculum Standards that, when well-implemented, is an effective alternative to the traditional high school mathematics curriculum.

[A more detailed evaluation report is available from the main author at N287 Lindquist Center, University of Iowa, Iowa City, Iowa 52242. E-mail: harold-schoen @ uiowa.edu]

REFERENCES

Coxford, A. F., Fey, J. T., Hirsch, C. R., Schoen, H. L., Burrill, G., Hart, E. W., & Watkins, A. E. (1997). Contemporary Mathematics in Context A Unified Approach, Course 1. Chicago: Everyday Learning Corporation.

Feldt, L. S., Forsyth, R. A., Ansley, T. N., & Alnot, S.D. (1993). Iowa Tests of Educational Development (Forms K & L). Chicago: The Riverside Publishing Company.

Hirsch, C. R., Coxford, A. F., Fey, J. T., & Schoen, H. L. (1995). Teaching sensible mathematics in sense-making ways with the CPMP. The Mathematics Teacher, 88(8), (November), 694-700.

Schoen, H. L., Bean, D. L. & Ziebarth, S. W. (1996). Embedding communication throughout the curriculum. In P. C. Elliott (Ed.), Communication in Mathematics K-12 and Beyond, pp. 170-179.Reston, VA: National Council of Teachers of Mathematics.

End of October 1997 Journal. Return to top of this Journal or top of this page.

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JULY 1997 NCSM JOURNAL OF MATHEMATICS EDUCATION LEADERSHIP

Journal Contents

• President's Message: An Open Letter to Steve Leinwand
• From the Editor: Thanks to the Trailblazers
• Standards-Based Mathematics Curriculum Reform: Impediments and Supportive Structures
  By Barbara J. Reys and Robert E. Reys, University of Missouri
• U.S. Demographic Trends: Challenges for Mathematics Education
  By Thomasenia Lott Adams, University of Florida

President's Message: "Welcome to Our New Journal!"

by Bonnie Walker

Thank you, Steve, for exemplary leadership. Thank you for setting high standards in quality of work, as well as quantity of work. While I find the role of President of NCSM a daunting, one, I am aware much of the the work I will oversee has already been laid out and begun through your efforts. At this time next year, as we assess our accomplishments and progress, we will be aware and appreciative of your efforts.

Thank you also for making my year as President-Elect an insightful and growth experience. You made it possible for me to prepare myself for the presidency by keeping me informed, allowing me to represent NCSM at various meetings, assigning the task of providing a Wednesday night reception for the conference, and by turning part of the board meeting over to me.

You set high standards in everything you do. I admire your knowledge, writing, and speaking talents, energy, and that you appreciate and uplift people.

I will do my best to maintain those high standards.

Thanks to the Trailblazers

By Vince O'Connor

When Steve Leinwand asked me to resume the editor duties of the NCSM Newsletter, he indicated that he wanted to transform it into a journal. That was two years ago.

Now we have published the second edition of the NCSM Journal for Mathematics Education Leadership and have materials in process for the third edition in October.

This transformation has not been without struggles. I want to publicly acknowledge the people who have helped us get through the problems and pitfalls that we encountered along the way.

Probably the greatest risk-takers were the authors who responded to the various calls for manuscripts and to the personal requests from Steve and Bonnie. Their willingness to write for a new journal with rather sketchy guidelines has provided NCSM with the essential elements to make this journal a success. Beyond that, their patience with review procedures that have been "in development," is greatly appreciated by the editorial panel.

Thanks, too, to the members of the Executive Board and other NCSM members who have volunteered to serve as referees for the journal. We are still working on the communications processes needed to ensure timely and fair review of articles and appropriate feedback mechanisms that will allow for final revisions and page layout on time for publication. The unseen work of the reviewers has been extremely important and we appreciate their contributions to the journal.

The work of our two associate editors, Carol Malloy and Frank Gardella, has also been key to this transformation. Their efforts, in the face of communication difficulties and tight deadlines have brought us through some very tough problems.

Lastly, I want to recognize the contributions of Harry Tunis at NCTM for his willingness to help NCSM launch this journal. We will be making use of the long history that NCTM has in publishing journals to fine tune our procedures in soliciting, reviewing, and editing future editions of this journal. Having the guidelines from the NCTM journals to follow will save us from many problems that would have slowed our progress.

Standards-Based Mathematics Curriculum Reform: Impediments and Supportive Structures

By Barbara J. Reys and Robert E. Reys, University of Missouri

Over a two-year period (1995-97) teams of teachers and administrators representing 24 Missouri school districts learned about the new materials and the instructional approach upon which the programs rest via curriculum conferences led by authors from curriculum development teams. After each conference, participants tested a unit of the materials with their students and talked with their colleagues (fellow teachers and administrators) about the ideas. Utilizing collaborative curriculum investigation as a vehicle for teacher enhancement prompted many issues including assessment, teaching practices and content-related questions to emerge. It provided opportunities and support for teachers to make small changes as they tried out various units of the curriculum including continuous dialogue with other teachers implementing similar changes. As participants became familiar, successful, and confident with the new materials, they increased the level of change in their own middle school program, while providing support for other middle school faculty and assuming active mentoring roles.

The project is based on the premise that significant reform in curriculum and instructional models which leads to more and deeper student learning does not just happen. Neither can change be legislated or forced upon unwilling participants. Teachers, with the input and support from administrators and parents, must understand, believe, and make a commitment to standards-based curriculum reform if it is to translate into systemic change at the classroom and district level.

Our experience in facilitating participants' understanding of the materials and the need for reform at the middle grades level has convinced us of the value of collaborative curriculum investigation as a vehicle for supporting and promoting change. It has also helped clarify impediments and supportive structures to standards-based mathematics curriculum reform.

What have we learned?

It is said that "experience" is recognizing a mistake when it is made again. As we reflect back on the first two years of this project, it is clear that we have gained valuable experience! Some of our experiences are consistent with research (Ball, 1996; Hyde & Hyde, 1994) and confirm what we believed at the beginning of the project. Other experiences suggest that supporting curriculum reform is more complex than anticipated. We summarize here several general observations regarding mobilizing and supporting the critical mass of key personnel needed to initiate district level reform.

• Collaborative curriculum investigation is a powerful force in helping team members move toward the vision espoused by the Standards and promoting teacher enhancement.

Field testing individual curriculum units provides direct experience, new content, and fresh approaches to fuel "small changes" needed for advancing professional development. Selecting and trying new content reflects a significant commitment toward change. It encourages teachers to try something different and challenges them to examine the new mathematical content included as well as the mathematical content being replaced.

While it might be a stretch to say that all participants in the M3 project became more metacognitive with respect to the mathematical content, it is fair to say that they became more reflective and developed more sensitivity to important mathematical topics and issues. Nearly all participants (more than 80%) indicated that the focus of collaborative curriculum review of standards-based materials helped them gain a better awareness and understanding of the vision of the

NCTM Standards documents. They agreed that the comprehensive sets of curriculum materials offered an "existence proof" that the philosophy and content outlined in the standards could be translated to instructional materials they could use.

• A carefully structured team of reviewers and decision makers (include middle school teachers, administrators, parents, and teachers from other levels) is essential to providing leadership.

A critical mass of middle school teachers is necessary and should serve as the core of the curriculum investigation team. These teachers, who may represent a range of philosophical viewpoints, strengthen the resolve and commitment of those experimenting with reform curriculum including a critical support network that a) encourages teachers to implement change, and b) builds the "grit" to continue if/when the going gets tough. Include as many members on the curriculum investigation team as possible because the lack of "personal" or first-person learning experiences (participation in project activities) makes it difficult for participants to transfer their enthusiasm, knowledge, and emerging beliefs to non-participants.

One of the expectations of the M3 project was that participants would return to their schools and share experiences gained from the institutes, regular meeting and regional meetings. They would also share their units with colleagues who were not directly involved as participants. However, no local planning time was scheduled or budgeted to allow this sharing to occur. It has become clear that despite the best efforts of participating teachers, it is challenging and often times impossible for participants to share what they have learned, indeed even what they are doing with their students, with their local school colleagues. It is critical that blocks of time for teachers to work together within as well as across schools (districts) needs to be scheduled. Although we have been able to address this issue for participants (by reserving planning and sharing time within regular meetings) we have not been able to solve it at the local level, and many of the participants have become frustrated at the lack of interest and commitment of their colleagues in joining in this reform effort.

Early and continuous support from administrator(s) (principal or curriculum coordinator) is essential for teachers who otherwise may be less willing to try new materials and/or instructional approaches. Parents and teachers at other levels (elementary and secondary) provide valuable input and establish an important communication link.

• A multi-year-year commitment to curriculum investigation and implementation is essential to develop the knowledge and esprit de corps essential to promote significant professional growth.

It takes time to think reflectively about the reform curricula. Initial curriculum concerns of project teachers tended to be at a 'micro' rather than 'macro' level. They centered around the mechanics associated with their grade level, units viewed as appropriate and consistent with current practice, and specific lessons within those units which they used. Only after sustained field-testing of several different units (about a year) were teachers able to elevate their thinking to a level that allowed them to observe how the vision of change portrayed in the Standards was reflected in the reform curricula.

In addition to promoting greater sensitivity to important mathematics, investigating standards-based curriculum materials over a period of time advanced teacher's understanding of different ways of organizing instruction. Teachers are allowing students to assume greater responsibility in their mathematical learning, as well as helping others. The use of models and manipulatives to support instruction has increased. Group work is more frequent and assessment efforts are more varied, including an increase in attention to culminating projects. For nearly all of the participants, all of these changes were incremental and grew in accordance with the teacher's self confidence.

• It is important to identify, acknowledge and address potential "hot" or "controversial" issues and discuss alternatives for addressing them. Among the "hottest" issues facing our participants were: how to inform and "gain approval" from parents; how to assess student learning when mathematics content is less discrete; what about algebra in the middle school; and how to align elementary and secondary programs with new middle school curricula.

Parents wondered about these reform materials. They wanted to know why they were being used and how their children would be affected. This was best addressed by scheduling "parents' night" sessions focusing on mathematics curricula. The format varied but typically engaged parents in doing the same mathematics their children were learning, followed by teachers providing additional insight into the philosophy and direction of standards-based programs.

Assessment was a multi-faceted issue including: How will the new curriculum impact student performance on standardized tests (state and district mandated)? How do I assess what students are learning when the mathematics is more "big idea" oriented? Project teachers scrutinized the units with a nervous eye toward statewide assessment expectations, always wondering in the back of their minds if the time spent using these units would have an adverse effect on their student's performance. Parents as well as administrators also wondered how students using reform mathematics programs would score on state and national assessments. Research regarding the long range impact of these middle school reform programs on test scores is promising but quite limited. We are fortunate in Missouri that significant changes are being implemented in state assessment efforts and the new mathematics assessment (which includes some performance and open ended responses) is more closely aligned with the reform-based curricula than to traditional mathematics textbooks. Thus, examining sample items from state assessment encouraged teachers to continue using the new materials and strengthened their faith in the broader reform effort.

The dilemma of algebra (for whom, what kind of algebra, and when) continues to linger. Despite the "algebra for all" thrust by NCTM, there remain significant challenges to be met. For example, some districts allow (encourage) the "best" eighth grades to take algebra. They wonder if the eighth grade units from the new curricula would be better for these students than a traditional first year algebra course. In addition to placement, another issue is related to the amount of algebra in the middle grade reform materials. If a student successfully completes 6-8 of a reform based curriculum, will they have experienced the equivalent of a year's study of algebra? Although these topics are being addressed by NCTM (participants reviewed focus issues of the Mathematics Teacher, Mathematics Teaching in the Middle Grades and Teaching Children Mathematics), most feel the need to hear a clearly articulated position from a "higher source" (secondary teachers, expert consultant, etc.)

Program alignment with elementary and secondary is an important issue raised by participants as well as parents. More specifically, the following questions were often raised:

What will happen to our middle school students when they go to high school?"

Will the high school mathematics programs be commensurate with our middle school reform?

Will our students from the elementary schools come to us with sufficient understanding and experience with: cooperative learning, writing about their thinking, facility with "basic" tools of mathematics (basic facts, efficient tools for computing, etc.)

These questions are a vivid reminder that change does not occur in a vacuum. Programmatic changes in middle school create ripples (waves) throughout the K-12 arena.

• An electronic communication network serves as a valuable vehicle for ongoing communication between teachers experimenting with standards-based curriculum.

At the beginning of the project, each district reported having the capability of electronic communication. A portion of early meetings was dedicated to helping participants use e-mail and become familiar with some of its benefits, such as sharing and bulletin boards. The success of electronic communication as a vehicle for networking clearly varies by teacher and district. Lack of 'easy access' to computers with Internet connections is the major barrier. Even participants with access within their districts were required to make a special effort to receive, read, and respond to electronic mail. This ranged from having their librarians send and receive the messages, to gaining access to a computer within a lab that is constantly being used, to having to travel across town to another building to access a networked computer.

Lack of technical expertise on the part of the teachers (which may be in part due to the lack of time and easy access) and lack of accessible technical support within the district contributes to the difficulty. At the district level, problems include a reluctance to assign accounts to teachers, lack of accounts available for teaching staff, lack of equipment, and lack of technical support. In some cases gaining access to electronic mail appeared to be as much a political issue as a logistical or technical issue (e.g., the limited number of accounts were reserved for teachers participating in a particular district-sponsored inservice). Although some of the early electronic communication problems have been resolved, our original goal of regularly communicating electronically with all participants remains a pipe dream, and one which seems to create new challenges almost daily A stronger up-front commitment from the district to implement and support electronic communication for the participants might have avoided many problems we encountered.

Summary

Teachers in the M3 project have taken their place on the "front lines" of considering and implementing standards-based curriculum reform. This effort has provided a better grasp of the vision of the Standards and the challenges which emerge from any sincere effort to implement standards-based reform. Those teachers who are committed to the reform based on what they've done and what their students are learning are "apostles" for change. Many are opening their classroom doors for others to visit and observe, thus providing ongoing support for teachers contemplating programmatic changes. We intend to continue to encourage, facilitate, monitor and support their efforts.

References

Ball, D. (1996). Teacher learning and the mathematics reforms: What we think we know and what we need to learn, Phi Delta Kappan, 77,500-508.

Hyde, A., Ormiston, M., & Hyde, P. (1994). Building professional development into the culture of schools. In D. Aichele and A. Coxford (Eds.). Professional development for teachers of mathematics (pp. 49-54) Reston, VA: National Council of Teachers of Mathematics.

National Council of Teachers of Mathematics (1989). Curriculum and evaluation standards for school mathematics, Reston, VA: The Council.

National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics, Reston, VA: The Council.

Reys, B. I., Reys, R. F., Barnes, D., Beem, I. K., Papick, I. (In press). Collaborative curriculum review as a vehicle for teacher enhancement and mathematics curriculum reform. School Science and Mathematics.

Contact:
Robert Reys
212 Townsend Hall
University of Missouri
Columbia, MO 65211
FAX 573-884-7492
cirr@showme.missouri.edu

The Missouri Middle Mathematics (M3) Project is a three-year teacher enhancement effort supported by the National Science Foundation designed to provide a forum for standards-based mathematics curricula study and implementation (Reys, Reys, Barnes, Beem & Papick, in press).

Four emerging NSF-funded mathematics curricula projects are being investigated via a collaborative, cross-district model (Six Through Eight Mathematics, University of Montana; Connected Mathematics, Michigan State University; Mathematics in Context: A Connected Curriculum for Grades 5-8, University of Wisconsin; and Seeing and Thinking Mathematically: Connections in a New Middle School Mathematics Curriculum, Education Development Center).

U.S. Demographic Trends: Challenges for Mathematics Education

By Thomasenia Lott Adams, University of Florida

Abstract

The purpose of this article is to encourage mathematics educators to acknowledge the changing demographics of our nation as preparations are made to teach the nation's children. All children will need to experience and exercise the power of mathematics in order to be productive citizens in the current and future global, quantitative, and technological society. If mathematics is to enable all children to be potential leaders in the twenty-first century, then we must make every attempt to communicate with, to include, to reach, to value, and to respect all children as we engage them in meaningful mathematical experiences.

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Mathematics educators are constantly searching for and developing ways to help all children become mathematically literate and competent, attain success in school mathematics, and develop into lifelong learners and users of meaningful mathematics. "A11 children" represents our goal in mathematics education to transcend barriers to learning for each child. Unfortunately, in many cases, these barriers are in the form of race, gender, ethnicity, culture, socioeconomic status, and other characteristics which make a person or group of persons different from another person or group of persons. While we focus on preparing for diversity, practicing equity, and presenting multicultural curricula in mathematics education, we often neglect to review the demographic characteristics of the nation which have great impact on the characteristics of the children we teach. Diversity, equity, multiculturalism, and multicultural education are not stagnant phenomena. Indeed, what represents the four phenomena today may not represent them ten, five, or even two years from now.

When discussing issues in mathematics education related to "children", we must keep in mind that this context is relative to the specific demographics of a particular place. We discuss mathematics education issues more effectively when we are sensitive to issues of and the constant changes in the demography of the United States. If we are to address the mathematics needs of all children, then we must at some point put aside "all children" in a collective sense and attend to "all children" in an individual sense. By looking at those things which account for diversity in the classroom, we will be more able to make decisions which benefit all children, collectively and individually. As we maintain a closer view of the changing demographics in the United States, we can make more informed educational decisions which support the mathematics learning of each child.

In the following section, I present an objective overview of several demographic factors which greatly impact the characteristics, indeed diversity, of the children in our mathematics classrooms. These factors are age, race, ethnicity, socioeconomic status, family characteristics, and education achievement.

Age

The last national census was conducted in 1990. At that time, there were 249 million persons in the United States. Of these persons, approximately 64 million were school-age children (persons age birth to 18). The ratio of male to female in the school-age population during the 1990 census was approximately 95 to 100 (Haupt & Kane, 1991).

Race

The Census Bureau uses five racial classifications to describe the population of the United States: (a) White, (b) Black, (c) Asian and Pacific Islander, (d) American Indian, Eskimo, and Aleut, and (e) other (Bureau of the Census, 1994). For consistency and clear communication, these are the racial classifications I will use throughout the remainder of this article.

In 1990, Whites accounted for 80.3% of the population, Blacks accounted for 12.1%, Asian and Pacific Islanders accounted for 2.9%, American Indians, Eskimos, and Aleuts accounted for .8%, and other accounted for 3.9% (Bureau of the Census, 1994). For the purpose of the census, Hispanics are classified as an ethnic group, not a race, and are identified as Mexican, Puerto Rican, Cuban, or other Spanish origins. Hispanic persons may be of any number of races. Most often, they identify with White or other for the census-race question (Haupt & Kane, 1991). For other demographics discussions, Hispanic (ethnic group) is discussed separate from White (race) to provide for a clearer picture of the U.S. population.

As of the 1990 census, 30% of all school-age children were people of color. By the year 2010, with a continuation of the current population growth, it is possible that 38% of all school-age children will be people of color (Hodgkinson, 1991, 1993). While the population of people of color is disbursed throughout the entire United States, in 33 states the enrollment of school-age children who are people of color is 20% or more (Mathematical Sciences Education Board (MSEB), 1991).

Ethnicity

The ethnicity of a people implies certain customs, cultures, beliefs, languages, standards, religions, and behaviors that are unique to the people. Ethnicity also implies diversity or differences between people. We most often use the term when referring to people of color. While ethnicity may be directly related to race, there are variations of ethnicity within races. That is, there may be many ethnic groups within a race of people. In addition, there may exist many differences within ethnic groups (Banks, 1992). In many instances, the ethnicity of a group of people is obvious within the context of the neighborhood or area in which those persons live. For example, there are many ethnic neighborhoods (e.g., Caribbean, Irish, Hispanic, Jewish) in large and small cities in the United States. Within these neighborhoods, the members of the ethnic group develop personal, social, and professional relationships supported by their commonalities. Cohesiveness, shared history, and common philosophies among the residents of ethnic neighborhoods are what often distinguishes them from neighborhoods in which ethnicity does not serve as a common thread among the residents.

Socioeconomic Status

Poverty is certainly a factor in the education of children. Poverty impacts the nutritional and health status of all who experience poverty. There are 14 million poor children in the United States. Of these children, an estimated 60% are White, 33% are Black, and 7% are Hispanic. Approximately 43% of all Black children, 41% of all Hispanic children, and 16% of all White children in the United States live below the poverty level. (personal communication, Population Reference Bureau, June 24,1997). In fact, since 1987, 25% of children under five have been living in poverty conditions (Hodgkinson, 1991). The trend is not improving: The Carnegie Commission on Science, Technology, and government (1991) estimate that by year 2000, 25% of America's children will be classified as poor.

Family Characteristics

In 1990, there were 94 million households (Hodgkinson, 1991). The average size of each household was 2.6 persons. On average, 30% of households with children were headed by a single parent (divorced, widowed, separated, or never married). Of those families, 90% were headed by women (Haupt & Kane, 1991). Over 50% of the children in families headed by female parents are poor (Howe, 1986).

Education Achievement

Thirty-three percent or more of the nation's children are at risk of school failure. This is due to many issues, such as poverty, neglect, sickness, handicapping conditions, and malnutrition, (Hodgkinson, 1991). These children are in danger of academic failure and failure to complete educational goals with sufficient knowledge and skills (Bryson & Scardamalia, 1991; Slavin & Madden, 1989). This situation is especially documented in the nation's drop-out rate of school-age children. The high school dropout rate in the United States is about 30% (Williams, 1992).

Academic achievement is influenced by many variables. However, when social and economic background are constant, race fades as a predictor of academic achievement. For example, children of wealthy Hispanic families perform better in mathematics than do children of poor Asian families although it may be a person's perception that all Asian children are high achievers in mathematics and science (Hodgkinson, 1991).

We have only taken a glance at several demographic variables which impact our attempts to help all children develop their mathematical abilities. Age is a given factor, because it is important to define what we mean by "all children." Essentially, we are referring to all school-age children in the United States. Race, ethnicity, socioeconomic status, family characteristics, and education achievement are all variables which greatly impact our ability to support the mathematics learning of all children.

Challenges and Implications for Mathematics Education

Each of the previously discussed variables present challenges and implications for mathematics education. I will address four of the components of mathematics education which are directly impacted by demographic trends in the United States: (1) curricula, (2) instruction, (3) assessment, and (4) achievement.

Curricula

As we engage in efforts to reform the mathematics curricula of our schools, we must be mindful of the diversity which exists in our school-age population. When we consider the various needs of children from various races, ethnic groups, socioeconomic backgrounds, family designs, and education achievement levels, we need to provide mathematics curricula which promote relevancy, reflectivity, and responsibility.

Mathematics curricula are more effective when they are relevant to children and their life experiences. We can design these kind of curricula when we know who our children are and when we know something about their lives. When we present situations that are to model the use of mathematics, we should make some effort to understand issues in children's lives which might impact the children's experiences in these situations. The mathematics assignments we create for children should at some time relate to the children's experiences in the classroom, in the home, and in the community. By making attempts to make curricula relevant to children's lives, we are giving children an opportunity to make connections between school mathematics and real-world mathematics. Through these connections, the children may be more able to develop better understandings of mathematics concepts and procedures. Relevancy gives children a platform from which they can construct their own mathematical knowledge. As children interact with the curriculum and with other learners and as children acknowledge the relevancy of the curriculum to their own lives and experiences, then the children are better equipped to assign meaning to school mathematics. When we ignore the need for mathematics curricula to be relevant, we increase the risk for children's failure in mathematics.

The ability to exercise mathematical knowledge, rests upon reflectivity. One of our goals in mathematics education is to encourage children to be life-long learners and users of meaningful mathematics. Thus when we engage children in mathematics tasks, we should aim to provide experiences that will require children to reflect upon their own thinking and actions. All children need opportunities to reflect on their mathematical learning. Reflection gives children opportunity to create, ponder, and extend ideas in mathematics. Classroom discourse is an excellent vehicle for promoting reflectivity in mathematics. Classroom discourse includes both oral and written communication. All children need to communicate with and about mathematics and with the ability to communicate as such, children can reflect upon and assess their own mathematics understanding. Because classroom discourse is an important learning tool for all children, we must alleviate barriers to discourse. One of the most dominant barriers is language. However, with language differences, we also get different meanings and interpretations which can add to the richness of discourse in the mathematics classroom. Diversity affords us an advantage and an avenue for helping children to engage in mathematical tasks and experiences that promote reflectivity. When we create an environment where children have opportunities to reflect on their mathematics learning and share their reflections with others, we can help all children to develop better meanings of the mathematics.

As we try to engage all children in mathematical experiences, we must enable children to be responsible to the curriculum. Part of this responsibility occurs when we aid all children in developing mathematical power. Power generates responsibility and responsibility encourages and enables children to engage in meaningful learning experiences. In addition, for children to be responsible to the curriculum, the curriculum must be accessible to the children. If we want all children to have meaningful mathematical experiences and to be responsible to the curriculum, we must enable children to participate in these experiences. The following are four ideas for enabling children in the mathematics classroom.

1

Recognize that our mathematics curricula originated from the ideas and experiences of many people from a variety of races and ethnic groups. Provide opportunities for children to obtain information about the multiculturalism of mathematics. More sources are now available to assist educators in developing mathematics curricula which reflect the demographic characteristics of the nation's children. Consider the following literature list as a starting point:

Addison-Wesley. (1993). Multiculturalism in mathematics, science, and technology: Readings and activities. Reading, MA: Addison-Wesley.

Cuevas, G., & Driscoll, M. (Eds.). (1993). Reaching all students with mathematics. Reston, VA: National Council of Teachers of Mathematics.

Krause, M. C. (1983). Multicultural mathematics materials. Reston, VA: National Council of Teachers of Mathematics.

Nelson, D., Joseph, G.G., & Williams, J. (1993). Multicultural mathematics: Teaching mathematics from a global perspective. New York, NY: Oxford Press.

Secada, W.G., Fennema, E., & Adajian, L.B. (Eds.). (1995). New directions for equity in mathematics education. New York, NY: Cambridge University.

Zaslavsky, C. (1973). Africa counts: Number and pattern in African culture. Brooklyn, NY: Lawrence Hill Books.

Zaslavsky, C. (1987). Multicultural mathematics: Interdisciplinary cooperative-learning activities. Portland, ME: J. Weston Waich.

2

Make certain that mathematical assignments and projects are not demographically specific in the sense that they present barriers to learning. For example, an educator may focus on mathematics games which derived from China. All children should experience this focus, not just children from China or children with a Chinese heritage or background. Another area of concern focuses on the celebration of holidays in the classroom, as some mathematics lessons may be implemented in conjunction with holidays. "In far too many schools, certain holidays are celebrated that automatically exclude some students. The celebration of these holidays should be conducted in such a way as to make the celebration a learning situation for all students" (Baker, 1994, pp. 121-122). There should not be any mathematics assignments and projects which all children cannot participate in or be enabled to participate in because of artificial demographical barriers (i.e., differences in holiday celebrations as related to culture, ethnicity, etc.).

3

Stay alert for potential learning barriers. When the opportunity presents itself, change a learning barrier into a learning asset by developing appropriate contexts for mathematics learning. For example, in the case of barriers presented by socioeconomic status (e.g., children's parents can afford to purchase supplies for mathematics projects), help children exercise their imaginations and use their creative skills to solve real-life problems. Help children explore resources to alleviate barriers to learning. Ignoring demographic characteristics, such as socioeconomic status, because of their unpleasantness does not serve to benefit the child who must face this issue on a daily basis. Real-life contexts can serve to support mathematics problem solving, and at the same time, perhaps help children and their families to solve everyday, real-life problems.

4

Create a learning environment that is conducive to learning for all children. Enough will never be said about the need for children to be valued and to be given opportunities to participate and contribute to the learning community. Helping children develop a strong sense of self gives them the background they need for taking academic risks in the mathematics classroom. One way of accomplishing this is to implement cooperative learning in the classroom. Coelho (1994) suggests that cooperative learning "...is particularly valuable in multilingual, multi-racial, and multicultural classrooms..."(p.85) As children work together in the learning community, they have an opportunity to learn about and support their own cultures and the cultures of fellow learners.

In essence, all children should have equal access to the curriculum regardless of demographic characteristic. Otherwise we create situations where children cannot sufficiently respond to the curriculum and have opportunity to engage in and value meaningful mathematics experiences.

Instruction

The instructional methodologies used to present curricula are most important to facilitate meaningful mathematics experiences for children. Decisions regarding how curricula are to be implemented should be made with consideration of the characteristics of the children in the classroom. The following are a set of questions which should be answered in light of the demographics of the classroom and the goals for children's mathematics learning:

1. What mathematics concepts and procedures are best delivered in a whole class setting? Small group setting? Individual setting?
2. When are heterogeneous groups appropriate? Homogeneous groups?
3. Is the mathematics instruction developed with consideration of curricula and assessment? (Assessment will be addressed in the next section.)
4. Do all children have ample access to calculators, computers, and other technological learning tools?
5. Do all children have ample access to mathematics manipulatives, and concrete objects for developing and making connections between mathematical ideas?
6. Is instruction supported by classroom discourse? Are all children given opportunities to communicate by speaking, writing, and/or acting out ideas?
7. Does instruction promote active learning? Cooperative learning?
8. Does instruction take into account the various learning styles, strategies, and strengths of all children?
9. Is each child given an opportunity to make a contribution as a member of the learning community?
10. Does instruction provide equal access to learning for each child?

As these questions are answered, decisions about classroom instruction can be made with a greater potential for creating classrooms where all children, regardless of demographic characteristics, are given a chance to gain mathematical power. As children respond to the instruction, the classroom teacher can continue to make changes in instruction to meet the children's needs.

Assessment

We are in the midst of reforming the way we support mathematics teaching and learning. As we engage children from many backgrounds in mathematical tasks, it is very important that our assessment techniques provide authentic information about the children's mathematical learning. We must remember that one of the purposes of assessment is to improve the mathematics learning of all children. Because of the diversity which exists among school-age children, we certainly need to have diversity in assessment of children's mathematical learning. When designing assessment models, we need to make certain that the techniques are authentic in nature, that is, that the techniques will provide real and accurate information which leads to the improvement of teaching and learning. This takes us beyond the numerical grade, as a simple score does not always provide sufficient information about children's learning and classroom instruction. There are a variety of authentic assessment techniques which can be implemented for the improvement of children's mathematics learning. Consider these following techniques as beneficial for children:(a) journal writing, (b) oral presentations, (c) performance presentations, (d) cooperative learning assignments, (e) portfolio (collections of children's work), (f) group projects, (g) written test (subjective and objective items), (h) interview, (i) classroom discourse, j) classroom questioning, (k) observations of children at work, (l) children's self-assessment, and (m) peer assessment.

Assessment, of course, is an integral part of curriculum and instruction. This means that as one develops assessment techniques for classroom use, one does so with the curriculum and instruction in mind. Too often, we teach one concept or procedure and assess other concepts or procedures We also teach one way and assess another way. For children to see the value of assessment as a means of improving learning, children need to be able to see the relationship between assessment and curriculum and between assessment and instruction, and children need to be participants in the assessment process and not simply an object of assessment.

Achievement

Many children who are people of color and female are underrepresented in advanced mathematics classes and are also underrepresented in careers which are supported by mathematics (MSEB, 1991). To change this, we must enable all children to attain achievement in mathematics. One of the first steps in improving children's achievement in mathematics is to engage them in mathematics tasks which are rich in meaning and which teases the children's natural curiosity. We cannot afford to underestimate the mathematical power which children of all demographic strands can attain when given the encouragement and opportunity to do so.

There are many reasons why children might experience school failure, as such, the reasons also cause children to experience failure in mathematics. While there is no panacea for this problem, we can help prevent children's failure in mathematics by presenting mathematics curricula, instruction, and assessment which are developmentally appropriate. That is, we are to consider the social, emotional, physical, and intellectual needs of each child when educational plans are made and implemented (National Council of Teachers of Mathematics, 1995). In doing so, we can attend to the academic needs of all children. As children from different backgrounds make their way into classrooms, the demographic characteristics which they bring with them are not to be ignored, especially if those characteristics are factors in the children's achievement potential. We can help all children achieve in mathematics by engaging children in meaningful mathematics experiences and challenges and having high expectations for children's achievement (National Research Council, 1989).

Conclusion

The nation's children are an important resource. While they travel through the schools, we have a chance to engage them in meaningful mathematics learning. As we reflect on the demographic trends of the United States, we will be better able to prepare curricula, instruction, and assessment which will help each child who enters the classroom obtain achievement in mathematics. In addition, as leaders in mathematics education, we are to keep the nation's demographic trends in focus during preparation for professional development of preservice and inservice teachers of mathematics. It is important to disclose to teachers of all levels the necessity of considering the changing demographics of schools and individual classrooms in order to facilitate meaningful mathematics learning for all children.

References

Addison-Wesley. (1993). Multiculturalism in mathematics, science, and technology. Reading, MA: Author.

Baker, G. (1994). Planning and organizing for multicultural instruction. Reading, MA: Addison-Wesley.

Banks, J. A. (1992). The stages of ethnicity. In P. A. Richard-Amato, & M. A. Snow (Eds.), The multicultural classroom (pp. 93-101). Reading, MA: Addison-Wesley.

Bryson, M., & Scardamalia, M. (1991). Teaching writing to students at risk for academic failure. In B. Means, C. Chelemer, & M. S. Knapp (Eds.), Teaching advanced skills to at-risk students (pp. 141-167). San Francisco, CA: Jossey-Bass.

Bureau of the Census. (1994). Celebrating our nation's diversity. Washington, DC: Author.

Carnegie Commission on Science, Technology, and Government. (1991). In the national interest: The federal government in the reform of K- 12 math and science education. New York, NY: Carnegie Commission.

Coelho, E. (1994). Learning together in the multicultural classroom. Markham, Ontario: Pippin Publishing Limited.

Cuevas, G., & Driscoll, M. (Eds.). (1993). Reaching all students with mathematics. Reston, VA: National Council of Teachers of Mathematics.

Haupt, A., & Kane, T. T. (1991). Population handbook. Washington, DC: Population Reference Bureau.

Hodgkinson, H. (1991). Reform versus reality. Phi Delta Kappan, September, 9-16. Hodgkinson, H. (1993). American education: The good, the bad, and the task. Phi Delta Kappan, April, 619-625.

Howe II, H. (1986). The prospect for children in the United States. Phi Delta Kappan, November, 191-196.

Mathematical Sciences Education Board. (1991). Making mathematics work for minorities. Washington, DC: National Academy Press.

National Council of Teachers of Mathematics. (1995). NCTM goals, leaders, and positions. Reston, VA: Author.

National Research Council. (1989). Everybody counts: A report to the nation on the future of mathematics education. Washington, DC: National Academy Press.

Nelson, D., Joseph, G. G., & Williams, J. (1993). Multicultural mathematics. New York, NY: Oxford University Press.

Slavin, R. E., & Madden, N. A. (1989). What works for students at risk: A research synthesis. Educational Leadership, February, 4-13.

Williams, B. F. (1992). Changing demographics: Challenges for educators. Intervention in School and Clinic, January, 157-163.

Correspondence concerning this article should be addressed to Thomasenia Lott Adams, who is now at the Department of Instruction and Curriculum, University of Florida, 2403 Norman Hall, Box 117048, Gainesville, Florida 32611-7048. The phone and fax numbers are (352) 392-0761 x243 and (352) 392-9193, respectively. Electronic mail may be sent via Internet to tla@coe.ufl.edu.

End of July 1997 Journal. Return to top of this Journal or top of this page.

APRIL 1997 NCSM JOURNAL OF MATHEMATICS EDUCATION LEADERSHIP

Journal Contents

• President's Message: "Welcome to Our New Journal!"
• The STEM Experience: Some Things We've Learned
  By Dr. Rick Billstein, University of Montana
• Enabling Students at Risk of School Failure
  By Thomasenia Lott Adams, University of Florida
• Editorial

President's Message: "Welcome to Our New Journal!"

by Steven Leinwand

Welcome to the first issue of the new and exciting Journal for Mathematics Education Leadership!

With this new publication, you have tangible evidence of the growth and transformation of the old and comfortable National Council of Supervisors of Mathematics into today's more activist NCSM: Leadership in Mathematics Education and eventually into the more accurate NCLM: the National Council for Leaders in Mathematics. But changes in name aren't as important as changes in the delivery of services and products that respond directly to the diverse needs and interests of our members. And that is what this new Journal is about.

Daily, we are bombarded with an alphabet soup of data, opportunities, and programs. Our desks and our e-mails are filled with NAEP and TIMSS; there's TAAS or CAPT or IBEST depending on which high-stakes state assessment program runs your life; and there are SSIs and USIs depending on your state or city; and most recently we must choose from the likes of CMP and CPM, ARISE and IMP.

In the midst of this blizzard of reform initiatives, mere information is no longer enough. Newsletters still have a role - and ours is now the Journal's insert - but leaders need commentary and reflections. They need to know about the experiences of colleagues and the successes and failures of reform efforts. And that's why we've launched this new Journal.

It is increasingly clear that the reforms we advocate to truly achieve our vision of powerful mathematics for all require skilled advocates, effective leaders, and resilient change-agents - the NCSM membership - in the form of teacher-leaders, department chairs, resource teachers, supervisors, coordinators and mathematics educators at our institutions of higher education must be prepared to:

• shift the beliefs and mindsets about mathematics and about teaching mathematics among our colleagues and among those in the broader community;
• develop, implement and nurture a wide array of vehicles to increase professional sharing and interaction throughout our profession;
• garner the understanding and support of principals, superintendents and members of Boards of Education' and most importantly,
• craft a coherent, standards-based program of mathematics that significantly raises achievement levels for all.

This new Journal has been designed to help you meet these awesome responsibilities. And that too is why the NCSM Board of Directors supported the creation of this Journal.

On behalf of our entire membership and every mathematics education leader who will benefit from this Journal I would like to recognize and thank Vince O'Connor, our tireless editor, for his commitment to this project and for the quality of his efforts to launch this important new periodical.

Finally, a reminder that this is your Journal. Read it. Use it. Quote from it. Learn from it. Copy it. Share it. Discuss it. Contribute to it. Make it better.

The STEM Experience: Some Things We've Learned and Their Implication for Teacher Preparation and Inservice

by Dr. Rick Billstein
Director, STEM Project University of Montana

The Six Through Eight Mathematics (STEM) Project is one of the five National Science Foundation (NSF) funded projects created to develop comprehensive mathematics curricula for the middle grades. The STEM curriculum reflects the content and teaching methods suggested by the National Council of Teachers of Mathematics (NCTM) Curriculum and Evaluation Standards for School Mathematics (1989) and Professional Standards for Teaching Mathematics (1991).

As pointed out in the Curriculum Standards,

Instruction has emphasized computational facility at the expense of a broad, integrated view of mathematics and has reflected neither the vitality of the subject nor the characteristics of the students.

An ideal 5-8 curriculum would expand students' knowledge of numbers, computation, estimation, measurement, geometry, statistics, probability, patterns and functions, and the fundamental concepts of algebra. The need for this kind of curricula is acute.

STEM is developing Standards-based materials that are consistent with the fundamental principles outlined by the Mathematical Sciences Education Board (MSEB) in Reshaping School Mathematics: A Philosophy and Framework (1990). Thes