An Overview of IMP Years 1 and 2
by
Kim Mackey
Introduction: Why be concerned about
IMP?
Section 1: Brief Synopsis of Year 1 and Year 2 of the
IMP curriculum
Section 2: A more in-depth look at Year 2 IMP unit
"Solve it!"
Section 3: A more in-depth look at Year 2 IMP unit
"Do Bees do it Best?"
Section 4: Strengths and Weaknesses of IMP
My personal experience with IMP extends back to the late-winter or early spring of 1996 when I discovered references to it on the Internet. I have since that time corresponded with teachers, parents, and students involved with IMP. I have read reports by, and corresponded with, a number of critics of IMP.I have read several studies done by the Wisconsin Center for Educational Research as well as several internal IMP annual reports. I have also purchased years 1 and 2 of the curriculum, the teacher's resource books for two units of year 2, the teaching handbook, and the separate independent unit "Baker's Choice" and its teaching guide, from the the publisher, Key Curriculum Press. I have written two major and numerous minor posts on IMP for the math-teach newsgroup which can be found in their archives at the Math Forum at the Swarthmore University website. The two major posts (one with additions) also appear on the Mathematically Correct website [IMP: A Student's View, with Comments and IMP: A Teacher's Review for Parents]. I will use internal documentation with sources in a bibliography at the end.
Let me be forthright about my bias: I am a skeptic about the replacement of the traditional sequence in high school. In my opinion reform programs (particularly NSF-funded experimental programs) must prove their worth under careful scrutiny before they are allowed to expand. This does not necessarily mean that I am a complete "drill and kill" hidebound traditionalist. My own teaching includes "reformist" practices (alternative writing assessments such as POWs and portfolios, use of calculators, cooperative learning groups, and process credit on problems) as well as more "traditional" practices (emphasis on quizzes and tests for grading purposes and individual accountability, formula memorization for mental math, "sage on the stage" interactive lectures, frequent comprehensive review, etc.).
Why IMP?
Why should we talk about, think about, and discuss IMP? The Interactive Mathematics Program is one of the oldest, best-established reform-oriented programs funded by the National Science Foundation. It has been selected as a Promising Practice by the CPB/Annenberg Project. The NCTM task force on mathematically promising students (what used to be called gifted) includes it in its list of programs worth investigating for promising students. Currently IMP is embarked on a major expansion through several regional centers around the country. Three examples follow.
Example 1
NSF award #963381 in the amount of 1.1 million dollars, was begun 9/96 to run for 4 years with John Bradley as program manager and Carla Oblas of Northeastern University as Principal Investigator. It is designed to do the following according to the abstract of the award:
"Northeastern University is working with twenty school districts in New England to change their curriculum and pedagogy to meet the NCTM Standards. These school districts are joining with the New England Regional Center for the Interactive Mathematics Program (IMP), the Center for Enhancement in Science and Mathematics Education, the Center for Innovation in Urban Education and the Mathematics Department at Northeastern University to implement the IMP curriculum in their schools. Teachers of mathematics at the participating schools receive intensive professional development workshops during the summer and periodic workshops during the school year." [NSF1]
Example 2
Another example is NSF award #9634034 in the amount of 1.07 million dollars also begun in 9/96 to run for 4 years with Judd Freeman as program manager and Linda Jaslow, Nora Ramirez, and Marilyn Carlson as Principal Investigators. According to this abstract:
"Maricopa County Community College District will initiate a 60-month Local Systemic Change project for 187 teachers in grades 9-12 in 5 Arizona school districts and schools in the UCAN RSI. The project's partners, the unified school districts of Chandler, Sunnyside, Nogales, Mingus, and Sedona-Oak Creek, Maricopa Community College, Arizona State University and the Intel Corporation, will implement the Interactive Mathematics Program (IMP) in those districts through the use of seven integrated components: 1) inservice on the mathematics and pedagogy of IMP, 2) follow-up support activities, 3) internal capacity for professional development, 4) school teams for support of teachers, 5) teachers from schools considering IMP, 6) articulation with feeder schools, and 7) impact on preservice teacher training. In addition to the high school teachers, support teams consisting of school administrators, counselors and other teachers, teams of middle school teachers that feed into the participating high schools, teams from schools considering the implementation of new curriculum materials, and personnel from each of the three state universities will take part in the professional development activities. Additional school districts will join the the project in years three and four. There is cost sharing equivalent to 244 percent of the NSF funding." [NSF2]
Example 3
Finally, a third example from Oregon (the reference is the web page http://www.col-ed.org/or/hillsboro.html) is the recently opened Century High School in Hillsboro. This example also includes some of the rationale for moving to IMP.
"Project Context
In the fall of 1997, Hillsboro School District 1J will open its third high school. The new school will be called Century High School and will be equipped with the latest in instructional technology. A planning team of 16 staff members and a principal have been working together since March of 1996 to prepare for the school's opening.
Century High School will join a select group of local high schools, including West View High School in Beaverton and Grant High School in Portland in implementing the Interactive math Program under the guidance of Portland State University's Mathematics Department.
To be selected by Portland State University to participate in the implementation of IMP, Century High School staff were required to commit to a structured multi-year staff development program. The school district was also required to support this endeavor with resources and staff development." [OR1]
In a separate section labeled "Purpose of the Project" we find this paragraph:
"The transition from a traditional, skill-based curriculum to a problem-based and concept-based curriculum will not be an easy one. Teachers will be trying to learn new mathematical content and to adopt new instructional strategies. Teachers will need to learn to exchange their traditional approaches to teaching mathematics with these new skills. The school district believes that teachers must be given full support in making this transition, through in-service workshops, adequate preparation time, team teaching, and other opportunities to share their experiences with each other." [OR1]
Finally, in a section labeled "Goals" we find:
"The over riding goal of the IMP staff development support project is to increase achievement in higher levels of mathematics for students who typically exit mathematics having met minimum graduation requirements.
The immediate goal of the project is to train mathematics teachers at Century High School to achieve the objectives of Interactive Mathematics Program which are:
1. to improve student achievement in mathematics by learning the
integration of mathematical concepts and problem solving skills through
project-based curriculum;
2. making core mathematical knowledge and skills more accesible and attainable
to women and minorities who traditionally tend to set lower standards for
themselves in mathematics;
3. to more actively engage the learner through a constructivist approach
to teaching and learning;
4. to teach students the use of technology for problem solving and communication
in mathematics; and
5. to help students use and apply their mathematical knowledge for doing
real world problems." [OR1]
IMP is currently being used in about 180 high schools across the United States, with the expansion underway possibly doubling this number (or more) in the next 4 years. IMP seems to have the firm backing of the NSF and the NCTM and is considered by many to be an exemplary example of adherence to the 1989 NCTM Curriculum and Evaluation Standards. Thus, I think it is important for all high school math teachers and others concerned with high school mathematics education to take a careful look at the IMP curriculum. For those of you who are students and parents directly involved with this program or who are being encouraged to join this program, the information that follows will hopefully help you decide whether this program is appropriate for your needs.
The IMP curriculum in years one and two is organized into 5 units each, most of which are said to revolve around a central problem introduced at the beginning of each unit. Each unit generally runs 5-6 weeks of 25-30 days. The units themselves are broken down into sections of 4-7 days resulting in 4-6 sections per unit. The type of work on each page is listed at the top and will fall into one of 5 categories: POW (problem of the week), classwork, homework, supplementary problems, or reference. There are 19 POWs in the first year (at 3 or 4 POWs per unit) and 14 POWs the second year (at 2-4 POWs per unit). In general there is a homework assignment for every day of a unit. Classwork assignments will run fewer than days, thus indicating that the work either extends to more than one day, or that homework is reviewed in class, or that supplementary problems can or should be used. A portfolio of student work is required at the end of each unit. The Supplementary problems will dovetail with the work in each unit and the number varies from 12-23 per unit in the first year and 8-20 per unit in the second year.
Calculators are assumed to be available to all students at any time in the classroom, and some assignments will specifically require them. The time allowed for classwork appears to be the standard 50 minute class period, so adjusting to a block schedule might require considerable tinkering. Computers are not assumed to be available for the assignments and this was a conscious decision on the part of the developers vis a vis calculators.
Cautionary note: When I list the topics of a particular unit, this will not necessarily give you an idea of the extent of coverage for that topic. Also realize that IMP is a very discovery-oriented curriculum and the majority of in-class work is done in cooperative groups (randomized groups is recommended in the IMP teaching handbook). In the lists below the items are generally in order from the beginning to the end of the unit, the caveat being that the central problem is brought back in at the end. Most of the vocabulary comes from the IMP texts themselves and words in quotes come directly from the text.
Section 2: A more in-depth look at Year 2 IMP unit "Solve it!"
In this section I will be comparing the IMP unit "Solve it!" with my own practice and a highly regarded 3 year textbook series I will label XYZ. XYZ actually represents the secondary Japanese mathematics textbook series translated by UCSMP and are used in our equivalent of 7th, 8th, and 9th grades.
Like IMP, XYZ contains a mixture of topics unlike the traditional sequence. This mixture of topics includes algebra, geometry, probability, and statistics.
Why choose "Solve it!"? Because it contains what I regard as one of the most critical topics for student success in higher mathematics classes in high school: solving equations.
Since I will be comparing IMP not only to series XYZ but also to my own practice, I will briefly explain the particular "niche" around which my teaching has evolved. I have taught in my current position at Valdez High School for 8 years. In that time I have been the only geometry teacher (3 or 4 periods a day) for 7 years. While I use the 1982 edition of the Houghton Mifflin text "Geometry" by Jurgenson, et al, as the basic student text, I also have created and used my own seatwork assignments, quizzes, and tests. Other materials I use include the three Cal Tech videos Similarity, Pi and Pythagorean Theorem. I have also taught Algebra I in 3 non-consecutive years, once with Saxon and twice with an Addison Wesley text, both supplemented with my own materials. Class sizes run 12-25 in a 7-12 Junior-Senior High School of 450 students. Teachers teach 6 48-minute periods in an eight period day (1 lunch, 1 prep) and generally have 3-5 separate classes to prepare for. Acceleration is used as the principle mechanism for challenging gifted math students although several in the past 5 years have done independent study, the John's Hopkins CTY program, or community college courses. Our current Algebra I text, new this year, is D.C. Heath.
The major difference between my teaching and XYZ versus IMP can best be expressed by this quote from the IMP teaching handbook. "IMP is not based on a mastery approach to learning. While the IMP curriculum seeks to attain the same goal of long-term mathematical understanding as mastery approaches, IMP promotes that understanding through a series of spiraled mathematics experiences which result over time in mathematics proficiency."[IMPT, p. 9]
So what does "Solve it!" look like?
The complete 31 day unit occupies the first 105 pages of the year 2 text and contains 3 POWs, 18 classwork assignments, 29 homework assignments, and 11 supplemental problems. The sections of the unit are:
1) Days 1-5: Solving equations and understanding situations
2) Days 6-12: Keeping things balanced
3) Days 13-21: What's the Same?
4) Days 22-26: The linear world
5) Days 27-31: Beyond Linearity.[IMP2]
The teacher resource book for the unit is a 266 page soft-cover book slightly longer and wider than the year 2 text. Every page of the unit appears on its own page in the teacher resource guide. The guide has a synopsis for each day and an outline on what should be covered for that day and in what order. It has asides, suggested questions and sometimes ideas to discuss with teacher colleagues. The days are fairly consistent in terms of what is to be covered: discuss the homework, start (and often finish) a classwork activity, hand out the next homework. Variations on the beginning of class would include the re-randomization of groups (every two weeks), selection of students to present POWs, and the presentations themselves. Sometimes previous day's activities are discussed after the homework. End of unit assessments are at the back of the resource guide.
Some major differences
Two major differences between IMP and my own teaching are immediately apparent:
1) my homework is practice of concepts and work already learned in class and little discussion occurs on it except 1 on 1 with students to discuss errors and misconceptions. In IMP homework is only rarely just practice. Another difference in homework is that mine is immediately collected, graded, and returned while this does not seem to be the case suggested by the IMP resource book.
2) End of unit assessments are also quite different. In my teaching evaluation of student progress through exams is frequent through quizzes and tests which cover all topics to the point of the exam. In IMP the end of unit assessment is both in class and take home. The in-class assessment for "Solve it!" appears below.
1. Solve each of these equations using the idea of equivalent equations. Show and explain each step of your work.
a. 22M + 19 = 13M + 41
b. 7(x - 3) + 29 = 3(x + 2) + 7
c. 4y - (3y - 6) = 2(y + 4)
2. The formula V = x(12-2x)2 gives the volume for a box built by cutting square corners of side x from a 12-inch-by-12-inch sheet of cardboard.
a. Based on this formula, write an equation that you could use to
find out how big the sides of the corners should be if you want the volume
to be 120 cubic inches.
b. Solve your equation and explain your method. Give all solutions to the
equation to the nearest tenth of an inch.
Note: If you use a graphing calculator, be sure to choose settings for
the viewing window that are appropriate to the situation.[SI, p. 253]
So what about textbook series XYZ?
XYZ presents concepts and problems, provides examples, then has short practice sets of 4-8 problems similar to the examples. A section in year 2 of the series on monomials and polynomials, for example, has 34 practice problems and 11 end of section exercises including 2 word problems. This section seems to be covered in 1 or 2 days.
Both XYZ and my own teaching would start equation solving with integer operations and simplifying expressions. XYZ covers various aspects of this in increasing complexity for two chapters. I tend to go more quickly to equation solving and spend more time there before moving on to word problems. XYZ goes to word problems after simpler equation solving.
Comparisons
In IMP the total number of equation problems provided in the "Solve it!" unit is about 50 with another 30-40 problems that include substitution/evaluation, equivalent expressions, and using the distributive property. XYZ and my own teaching would include about 200 not including exams. In IMP year 2 a student might see another 50 equation problems if you include inequalities and simultaneous equations from the cookies unit. This is therefore the total for 2 years (about 150). A student who was in both my algebra I and geometry classes would see somewhere in the close vicinity of 2,000 problems requiring equation solving of one kind or another. My equivalent estimate for XYZ is 1200-1500. For either IMP or my teaching this does not include more conceptual or discovery-oriented problems.
Section 3: A more in-depth look at Year 2 IMP unit "Do Bees do it Best?"
This post will look at the activities and problems involving the Pythagorean theorem in the IMP year 2 unit "Do Bees Build it Best?". As in section 2 I will compare IMP both with my own practice and with the series XYZ.
The "Bees" unit of IMP is the third of five units in year two following the statistics unit "Is There a Difference?". Prior to the Pythagorean Theorem problems in the unit, about ten days are spent on the initial central problem, discovery lessons about containers, geoboard lessons, and work on areas of triangles, parallelograms, and trapezoids. There is also a review and reference section on trigonometric relationships that were first studied at the end of year 1. Below I will list the homework and classwork assignments, then look at each day in more detail.
The lessons on the Pythagorean theorem are in section "Day 11-16, A special property of Right Triangles."[IMP2] In this section are six homework assignments:
Homework 11, How Big is it?
Homework 12, Impossible Rugs
Homework 13, Make the Lines count
Homework 14, The Power of Pythagoras
Homework 15, Leslie and Fertile Flowers
Homework 16, Don't Fence me in[IMP2]
There are 4 classwork assignments:
Tri-square rug games on Day 11
Any Two sides work on Day 12
Proof by Rugs on Day 14
Flowers from Different sides on Day 16.[IMP2]
DAY 11
Mathematical Topics covered according to the resource guide:
1) finding formulas for the area of parallelograms and trapezoids
2) Investigating the relationship among the areas of squares made from the sides of triangles[BEES, p. 77]
According to the Day 11 outline in the teacher resource guide, In-class work consists of:
1) forming students into new random groups
2) discussion of homework 10 with the main goal being "review of the use of the trigonometric functions in problem situations. You may want to remind students of any mnemonics they used previously for remembering which function is which."[BEES, p. 77]
3) Tri-square rug games activity where "students compare areas as a step toward developing the Pythagorean theorem. The activity will be discussed on Day 12."[BEES, p. 77]
4) Assignment of homework 11: How big is it? which is designed to "give students a chance to synthesize their recent work with area and to connect it to the unit problem."[BEES, p. 78]
DAY 12
Mathematical Topics:
1) Stating the Pythagorean theorem as the summary of the investigation of area
2) Applying the Pythagorean theorem.[BEES, p. 81]
Outline:
1) discuss Homework 11, "let students share ideas"[BEES, p. 81]
2) discuss tri-square rug games, "get students to give a clear statement of what they see in sorting the tri-square rugs."[BEES, p. 81]
3) the Pythagorean theorem
a) "get students to express the Pythagorean theorem both in
terms of area and as an equation involving the lengths of the sides."
b) Post the Pythagorean theorem.[BEES, p. 81]
In the discussion of the Pythagorean theorem the resource guide has this to say: "Tell students that the most important part of their discovery is the connection between right triangles and fair games. Ask the class to express this discovery about right triangles in terms of areas. Work with them as needed to help them come up with something like this statement: (the following sentence is separated and in bold print) when a triangle has a right angle, the sum of the areas of the squares on the two shorter sides equals the area of the square on the longest side."[BEES, p. 82] This sentence is then posted in the room.
Continuing the discussion, "Tell students that this principle was known to many ancient civilizations. Though it is unclear who first discovered it, there is evidence of its use in China, India, Egypt, and Babylonia."
"Tell them that today this principle is generally called the Pythagorean theorem, after the ancient Greek mathematician and philosopher Pythagoras, and that many mathematicians think this is one of the most important theorems in mathematics."
"Tell students that traditionally we represent the lengths of the legs of a right triangle by the variables a and b and the length of the hypotenuse by the variable c, as shown here"[BEES, p. 83] (diagram of a right triangle with sides labeled).
"Ask students how they might express the Pythagorean theorem in terms of the lengths a, b, and c. As a hint, ask how the areas of the square rugs are related to these length, or how they could express the area of Al's rug and the sum of the areas of Betty's two rugs. They should be able to come up with the equation a2 + b2 = c2 as an abbreviation of the area principle. Add the diagram and this equation to the verbal statement of the Pythagorean theorem. Tell the class that although they discovered the Pythagorean theorem by thinking about areas, the theorem is most commonly used to find lengths of line segments, especially by viewing diagonal lines as hypotenuses of right triangles."[BEES, p. 84]
4) "Any two sides work" classwork activity
a) "students apply the Pythagorean theorem"
b) "the activity will be discussed on day 13."[BEES, p. 81]
5) homework 12: impossible rugs assigned.
DAY 13
Mathematical Topics:
1) reviewing the triangle inequality
2) applying the Pythagorean theorem
3) working with the square-root symbol and irrational numbers[BEES, p. 89]
Outline:
1) discuss homework 12: impossible rugs.
"Ask students to state their conclusions as a principle about triangles, independently of the context of tri-square rugs. They should be able to come up with something like this: (the following sentence is in bold print) the length of the longest side of a triangle must be less than the sum of the lengths of the other two sides. Identify this principle as the triangle inequality and post it."[BEES, p. 90]
2) student presenters present their solutions to the previous day's classwork which consisted of 4 problems.
1. (diagram with a 4x4 geoboard is on the right) this diagram shows the longest diagonal on your geoboard.
a) estimate the length of this diagonal. (as usual, use the distance between adjacent pegs as the unit of length).
b) find the exact length of this diagonal using the Pythagorean theorem.[IMP2, p. 228]
2. (diagram of a person on a ladder which is leaning against a wall). An 8-foot ladder is leaning against a wall, as shown in this diagram. The bottom of the ladder is 2 feet from the wall. How high up the wall does the ladder reach?[IMP2, p. 228]
Discussion: "As in question 1, you should probably insist that presenters show which triangle they are using and that they write an equation for the missing height, perhaps writing it in the form a2 + b2 = c2. For example, they might write h2 + 22 = 82, and then simplify this to h2= 60." [BEES, p. 92]
In an aside immediately below this part the resource guide says, "You can take this opportunity to comment on the idea of subtracting the same thing from both sides, as a reminder of ideas from 'Solve it!'"[BEES, p. 92]
3. Marlene wants to check that her door frame makes right angles at the corners. the door is 2.5 meters high and 1.5 meters wide. How long should the diagonal of the door be if the corners are right angles?[IMP2, p. 229]
4. (diagram of a 4x4 geoboard with a triangle drawn on it) A student thought that the triangle at the right looked like a right triangle, but wasn't sure. find the length of each side of this triangle, and use your answer to determine with certainty whether or not it is a right triangle. Explain your reasoning.[IMP2, p. 229]
3) homework 13: make the lines count
Day 14
Mathematical Topics:
1) Applying the Pythagorean theorem
2) proving the Pythagorean theorem. [BEES, p. 97]
Outline:
1) select presenters for tomorrow's discussion of POW 8: Just Count the Pegs.[BEES, p. 97]
2) Discuss homework 13: "have students explain some of the lengths they found."[BEES, p. 97]
3) Proof by Rugs classwork: "students use diagrams to develop a proof of the Pythagorean theorem".[BEES, p. 97]
4) discuss Proof by Rugs: "Be sure to get explanations of the fact that Al's unshaded area is a square."[BEES, p. 97]
5) homework 14: the Power of Pythagoras, "the problems in this activity provide students with some more real-world contexts for the Pythagorean theorem." The problems for this assignment are: (all with diagrams)[BEES, p. 101]
1. how far a billiard ball travels with a trick billiard shot.
2. The total distance a zigzagging football player runs.
3. A race between two girls, one of whom takes the diagonal across a rectangular field and one who runs around the edge.
Day 15
Mathematical topics:
1) finding a formula in two variables (POW 8)
2) applying the Pythagorean theorem
3) tesselations.[BEES, p. 105]
Outline:
1) presentation of POW 8.
2) discuss homework 14: "have students present each of the problems".[BEES, p. 105]
3) Introduce POW 9: Tesselation Pictures.[BEES, p. 105]
4) Let students look at tesselation examples.[BEES, p. 105]
5) assign homework 15: Leslie's fertile flowers. In an aside the resource guide says, "This problem is based on the Pythagorean theorem. The numbers were chosen so that the answer would be fairly easy to guess. The focus of the problem is on how to be sure the guess is correct."[BEES, p. 108]
Day 16
Mathematical topics:
1) analyzing triangle problems in terms of equations
2) reviewing algebra for solving equations.[BEES, p. 115]
Outline:
1) discuss homework 15.
2) solve homework 15 algebraically. "Students need to label the diagram and find the altitude using an equation."[BEES, p. 115]
3) Flowers from Different sides classwork activity, "students find the area of Leslie's flower bed using the other altitudes."[BEES, p. 115]
4) discuss flowers from different sides, "emphasize that area is more than just a formula. Bring out that the method would work for any triangle."[BEES, p. 115]
5) homework 16: Don't Fence me in. "Tonight's homework is the first step in the analysis of a complex problem: Given a particular perimeter, what polygon gives the largest possible area?"
[BEES, p. 119]After day 16, the emphasis on the Pythagorean theorem is over. It seems to be used only 2 or three more times on other problems in the "Bees" unit. One explicit instance is homework 25: Pythagoras and the Box. The resource guide states, "In tonight's homework students are asked to develop a formula for finding the length of the diagonal of a box. You might want to use a classroom object (or your classroom itself, if it's box-shaped) to clarify the situation."[BEES, p. 185]
Looking through this unit, the total number of applied Pythagorean theorem problems appears to be about 10. One of the supplemental problems asks for a proof based on similar triangles. So this is the total number of applied Pythagorean theorem problems for the first two years of IMP.
In the textbook series XYZ, the Pythagorean theorem is introduced at the end of the third year prior to a chapter on probability and statistics. Both the theorem itself and its converse are explicitly stated and proved. Application examples and problems presented include:
1) applying the Pythagorean theorem to plane figures
2) the height and area of an equilateral triangle
3) the length of chords and tangents
4) the diagonal of a rectangular parallelepiped
5) the volume and surface areas of cones and pyramids
6) the radius and arc of a circular section of a sphere.
Total number of problems is about 40. A short history of the Pythagorean theorem is at the end of this section. Several additional proofs of the theorem are included at this point as well as in an appendix.
In my own teaching I cover similar triangles and the geometric mean as well as square roots before dealing with the pythagorean theorem directly. I prove the Pythagorean theorem in class using similar triangles with student assistance at every step. Students are required to memorize the proof for 2 tests and 3 quizzes as well as the semester exam. Immediately after the proof I show the Pythaogrean theorem video from Caltech and it is discussed in class. Students are also required to take notes on the video for their end-of-quarter portfolio. Problems covered include all of those listed for the XYZ series above and more. Total number of problems requiring use of the Pythagorean theorem is about 75-100 in the second quarter, and another 200 or so in the second semester, most concentrated in the 4th quarter except for those involving coordinate geometry (3rd quarter). Note that this total does not include any Pythagorean theorem problems of a simpler nature that might have occurred in Algebra I.
In the previous sections on IMP I have tried to stay away from value judgements. I have tried to describe the program's first two years using IMP materials and delved more deeply into two units of the second year IMP text while comparing IMP with my own practice and the XYZ series. In this section I will look at the strengths and weaknesses of IMP. This will entail making value judgements based on my classroom experience, email conversations with many teachers of mathematics, mathematicians, and lay people, and my personal investigation into constructivism, cognitive psychology, mathematical cognition and comparisons of foreign mathematics curricula with those found in the United States.
I will look first at what I consider to be the strengths of IMP followed by its weaknesses. To aid readers I will first list each so you can skip around to something you find more interesting. A couple of notes: 1) these strengths and weaknesses might easily appear in other programs whether "traditional" or "reform".
1) some interesting problems and approaches
2) professional development
3) teacher's guides
4) presentations
1) index
2) topic coverage and placement
3) homework
4) practice
5) grouping
6) grading
7) dealing with criticism
8) pace
9) evaluation
10) IMP and the NCTM Standards
1) Interesting problems and approaches
I think IMP has a number of interesting problems and approaches that are worthy of being included in any math teacher's repertoire. Some examples include:
the central problem on linear programming in the year 2 unit "cookies"
the pan-balance model for solving equations in the year 2 unit "solve
it!"
the chefs and and cubes model for dealing with integers in year 1.
There are more, including supplemental problems and POWs.
2) Emphasis on professional development.
IMP urges anyone considering the use of IMP to spend at least 10 days in a summer in-service prior to beginning its use. They emphasize the need for administration support and suggest that teachers somehow be given an extra prep period per day. They also encourage active discussion among IMP teachers about teaching particular units or activities. Professional development could be helpful for any program, including traditional ones.
Teachers with weak mathematical backgrounds are especially in need of such training.
The Teacher's guides for each IMP unit contain detailed lesson plans that include the mathematical topics to be covered for each day, an outline of what is covered for that day with notes, discussion items, and asides which contain relevant or important information. The guides also contain end-of-unit assessments, although as noted before these assessments are not comprehensive in nature. The guides are quite readable and while I felt they contained more information than I might personally use, it is certainly better to err on the side of too much rather than too little.
I see some value in the IMP emphasis on presentations, although my preference and experience would be in favor of "high-stakes" presentations that students actually practice and prepare for like the POWs in IMP. In Academic Decathlon we have "high-stakes" speeches that students give formally about 3-4 times in a year. Over a several year span this improves their public speaking skills enormously. I think IMP places too much emphasis on "low-stakes" presentations. This takes away valuable class time that might better be spent on getting feedback from students on problems or difficulties they are having with the material. Another weakness with the presentations in IMP is that the teaching handbook gives no advice whatsoever on how to evaluate and grade the presentations, yet they are suggested to weight as much as 20 percent of the grade.
Last year when I mentioned this several people thought it was of no consequence. Yet not one high school or college mathematics textbook I have looked at comes without an index except for IMP year 1 and year 2. It does make it difficult to find things if one is looking, for example, for the first use of "frequency bar graph", or "trigonometry". Perhaps for a student text this is not essential. I would consider it more important in a teacher resource guide, yet these too are without an index.
2) Topic coverage and placement
Placement
As regards placement, with my teaching background I see no reason why particular units with the particular topics were chosen to be in the order in which they appear. For example, it makes more sense to me to combine the equation solving unit with the unit containing integer operations within the same year rather than have them separated into different years, particularly when one considers the importance of equation solving in mathematics. Also, unless a developer wants to be able to say "yes, we have a probability or statistics unit in every year of our program!" I don't see the mathematical reason why the probability unit of year one can't be tied more closely to the statistics unit in year two. And the same idea for the similar triangle unit in year 1 and the "Bees" unit in year two. Perhaps those with doctorates or masters in mathematics can help explain why the placement of the units makes sense.
Coverage
When I listed the topics covered in IMP year 1 and year 2, the array looked fairly impressive. Yet the coverage of the topics is often superficial or less than complete. As an example, exponents are covered at the end of year two, but only with regard to multiplication involving exponents, i.e., Ax * Ay = Ax+y. [IMP2, p. 393]). Now it is quite possible that the other laws involving exponents will appear in later years, but the principal reason for the lack of coverage at this point seems to be a desire to cover only the material absolutely necessary to solve the central unit problem.
Now many teachers would supplement the text with their own material if they came across gaps like this in a "traditional" text. Yet the IMP teaching handbook specifically advises against this. "Until you have taught the four-year IMP sequence and know where topics are going, you should resist the temptation to interrupt the flow of a unit and supplement the curriculum content with your own materials." [IMPT, p. 25]
But as most high school teachers know, the likelihood that you will be able or allowed to teach all four years is remote. Typically, teachers with seniority get the higher grade levels with the more mature and motivated students, and newer non-tenured teachers get the lower grade levels. Thus the reality is, if one follows the dictum of the IMP teaching handbook, that IMP is never likely to be supplemented with outside materials.
As stated in the IMP teaching handbook, homework is "not just practice of what was done in class that day," but "often provide essential preparation for the following day. This makes it vital that students do their homework regularly."[IMPT, p. 38]
Now this is not necessarily a weakness if the teacher stays on top of every student and checks all the homework. But IMP does not advocate detailed checking of every homework assignment. "There is not enough time in the day to thoroughly grade every piece of student homework that comes in. Most experienced IMP teachers grade the bulk of homework according to completion."[IMPT, p. 32]
Each IMP unit's teacher guide has a recommendation for the homework assignments that should be collected for detailed checking. Each recommends collection of about 4 out of 30 homework assignments and 1 or 2 out of 15-18 classwork assignments. Even though discussion is supposed to occur in class on each homework assignment, this is no guarantee of student understanding or that a teacher will pick up on the students who are missing the boat. And I think this is especially true in IMP where each day has a pretty extensive agenda. In my opinion the students most likely to suffer from this are the one's least likely to be able to afford it: the lower two quartiles.
As stated in the IMP teaching handbook, "IMP is not based on a mastery approach to learning. While the IMP curriculum seeks to attain the same goal of long-term mathematical understanding as mastery approaches, IMP promotes that understanding through a series of spiraled mathematics experiences which result over time in mathematical proficiency."[IMPT, p. 9]
The result of this position is that practice is given pretty short shrift in IMP. I am not an advocate of 40 problems a night of "drill and kill".
But I am an advocate of reasonable, comprehensive practice of problems over a long time to promote learning of skills and concepts and movement of knowledge and understanding into long-term memory. In my view there is not enough practice of basic skills or non-discovery problem solving in IMP. And the difficulties will be the most apparent with those students who have the weakest skills to begin with.
Groups are very important in the Interactive Mathematics Program. According to students and teachers in IMP programs, about 75 percent of all work is done in groups. The IMP teaching handbook recommends random grouping. "The Interactive Mathematics Program believes that random grouping helps eliminate the labeling and tracking that can occur in a classroom. If you try to "seed" each group with a high-achieving student or plant a less motivated student in each group, the students will figure out what is going on in about two minutes."[IMPT, p. 13]
I think it is naive to think that the students aren't going to figure out who is smart and who isn't fairly quickly anyway. Random grouping provides the potential for some disastrous behavior problems as well as equity problems between groups. If the goal is getting all students to understand the material, at least "seeding" provides the teacher with helpers who can assist each group in achieving this goal.
But any kind of grouping will have problems ranging from the "free-rider effect" to the "sucker effect" as outlined by Solomon and Globerson. It takes a very astute and well-motivated and knowledgeable teacher to overcome these problems. Not only that, but having to deal with 6-8 separate groups certainly decreases teacher to group interaction time vis-a-vis whole class instruction. Countries with high achieving mathematics students like Singapore, Japan, and Korea tend to use whole class instruction a much greater percentage of the time than is true in the United States.
While teachers are encouraged to establish their own grading system, a rationale is presented for setting up grades in the following fashion:
30 percent -- homework assignments
20 percent -- problems of the week
20 percent -- oral presentations
20 percent -- write-ups of class activities
10 percent -- end-of-unit assessments (half in class, half take-home)
When combined with other weaknesses in IMP, I think this is a recipe not only for grade inflation, but also for copying without need for understanding. Whether we like it or not as math teachers, most students are not in mathematics because they have a burning desire to be a mathematician. Many are in math to satisfy high school graduation requirements or because a counselor told them they need the credits to get into college. Thus, they often look for the easy way out and the greatest pay-back for the least amount of effort. With a grading system that emphasizes exams, at least there is some individual accountability in the sense that students have to know the material to do well on the exam. In exchanging email with teachers across the country (some reform-minded, some not), very few weight exams less than 65-75 percent of the total grade. And most that do have exams attempt to be comprehensive and cover most of what was taught. This is not true of IMP end-of-unit assessments. And with the weighting being so low, why should a student even bother attempting to do well? And as a teacher I will have no benchmark to assure me that the students truly understand what was covered except for vague subjective "feelings" and assignments that could very easily be someone else's work.
In researching the Interactive Mathematics Program I have exchanged email with a number of proponents and critics of the program. In these exchanges it has often been my impression that not only are critics not listened to, they are often personally attacked and their points, good and bad, roundly denounced with no give whatsoever. I consider this a weakness of the program because to an outsider looking in on the debate, proponents act as if there is no room for improvement, as if they have discovered the keys to some fabulous kingdom that only the righteous and anointed will be allowed to enter. When I first discovered IMP on the internet I had lots of questions about the program. What often dismayed me was the lack of coherent replies to substantive questions dealing with structure, teaching, groups, etc.
The pace of the individual classwork assignments in IMP often seems out of synch with what I would expect students to be able to do. For many students the assignments will seem rediculously easy, thus leading quickly to boredom. This is especially true of end of unit assessments where the teaching handbook suggests a complete period for problems that I would expect most students to be able to finish in less than 15 minutes.
In the past 8-9 years, IMP has been funded by the NSF and private corporations to the tune of at least 16 million dollars. They are embarked on a substantial expansion of the program through regional centers. Yet the only substantive evaluation documents released to the public have been flawed transcript studies of three high schools in California done by Dr. Norman Webb at the Wisconsin Center for Educational Research. IMP has used these studies to suggest that IMP students will do just as well, on average, as students from traditional programs. Some flaws in the studies are:
1) student selection for IMP versus the traditional programs was not random
2) The studies were transcript studies, thus no information was available about extra tutoring or classes outside of school that might have affected the results
3) the post-test used was the SAT, a high-stakes exam that does not give a good indication of the extent of the mathematical curriculum understood or covered by a student. Students with weaknesses in their mathematical background are also likely to seek remedial help before taking the SAT.
10) IMP and the NCTM Standards
In 1989 the National Council for Teachers of Mathematics came out with their Curriculum and Evaluation Standards. These Standards have become the basis for many of the math standards adopted by states throughout the United States. Within the NCTM Standards many items are listed for "decreased attention" within the document, including at the 9-12 level such things as "The use of factoring to solve equations and to simplify rational expressions" [NCTM, p. 127] and "Paper-and-pencil manipulative skill work" [NCTM, p. 129]. Many proponents of the NCTM Standards like to emphasize that "decreased attention" does not mean "elimination". But this does not seem to be the case with IMP, one of the flagship NSF-funded reform mathematics programs. I suggested last year on the math-teach newsgroup that IMP was using the term "decreased attention" as a virtual synonym for "elimination" for many items. This suggestion is bolstered by this quote from the 1992 initial project description for the Interactive Mathematics Program submitted to the NSF:
Three years ago, IMP began meeting these needs. In its initial funding proposals, IMP set forth the following goals, in line with the recommendations of the above reports:
..... to expand what mathematics is learned:
--- by including those concepts which are recommended by NCTM's Curriculum and Evaluaton Standards and by other recent curriculum guides, and by omitting those topics which the reports suggest be de-emphasized;" [IMP_PD, p. 2]
I believe that this is a weakness for IMP since they place themselves in a straight-jacket of adhering only to items in the NCTM Standards that are supposed to receive "increased attention" without regard for the mathematical potential of the "decreased attention" items and how they might be used to enhance their curriculum.
[BEES] "Do Bees Build It Best?" Teacher's Guide. Interactive Mathematics Program. Dan Fendel and Diane Resek with Lynne Alper and Sherry Fraser. Key Curriculum Press. 1998.
[IMP1] Interactive Mathematics Program Year 1. Dan Fendel and Diane Resek with Lynne Alper and Sherry Fraser. Key Curriculum Press. 1997.
[IMP2] Interactive Mathematics Program Year 2. Dan Fendel and Diane Resek with Lynne Alper and Sherry Fraser. Key Curriculum Press. 1998.
[IMP_PD] "IMP Project Description", NSF Document 92-55262. March 2, 1992.
[IMPT] Teaching Handbook for the Interactive Mathematics Program. Lori Green. Key Curriculum Press. 1997.
[NCTM] Curriculum and Evaluation Standards For School Mathematics. National Council Teachers of Mathematics. 1989.
[NSF1] On the web at http://www.nsf.gov/awards/awards_1996/awd_1996_33/a9633811.txt
[NSF2] On the web at http://www.nsf.gov/awards/awards_1996/awd_1996_34/a9634034.txt
[OR1] On the web at http://www.col-ed.org/or/hillsboro.html
[SI] "Solve It!" Teacher's guide. Interactive Mathematics Program. Dan Fendel and Diane Resek with Lynne Alper and Sherry Fraser. Key Curriculum Press. 1998.
Kim Mackey
Valdez High School
Box 1996
Valdez, Alaska 99686