I am a 10 year veteran math teacher who has done extensive research in mathematics education over the past 12 months. Below you will find a posting I did to the NCTM-L newsgroup concerning the IMP or Interactive Mathematics Program. IMP is a standards-based curriculum designed to replace the conventional sequence of Algebra I, Geometry, Algebra II, Precalculus. It has a number of flaws which are discussed within the post and elaborated on in comments at the end. To more fully understand IMP and its philosophical underpinnings, it is extremely important that you get a copy of E.D. Hirsch's 1996 book The Schools We Need...And Why We Don't Have Them. At a minimum, this book can help you decipher some of the jargon that educators (including myself sometimes, I have to admit) throw out with such casual ease. It is likely that if you are here at the Mathematically Correct website, you are upset, frustrated, and feeling hopeless about your situation or that of your children. Buck up. This site and the people who maintain it will give you some of the ammunition and advice you need to fight the battles and win the war.
INTRODUCTION When looking at curricula that purport to be standards-based, it is best to keep in mind two things. One is what Sherman Stein calls, in his book Strength in Numbers, the Action Syndrome.
The Action Syndrome helps a person cope with the stress of action. It reduces several options ultimately to one. It enables the doer of an act, the 'actor', to commit to this one option, to suppress doubts, and to sustain dedication.
Nothing is so persuasive as the force of conviction. Those who cannot convince themselves certainly cannot convince anyone else. The Action Syndrome therefore enables the actor to draw others to the cause.
The other thing to keep in mind is the Hawthorne Effect. In analyzing a math curriculum the Hawthorne Effect might occur when it is not the curriculum that produces improved mathematics achievement, but the enthusiasm teachers have when using the curriculum.
Last week on 1/11/97 I suggested that standards-based curricula can be distinguished from traditional curricula in four ways: 1) integrated curriculum, 2) a heavy emphasis on group learning, 3) extensive writing about mathematics, and 4) a heavy emphasis on "real-world" problems. The curriculum I propose to analyze is the Interactive Mathematics Program or IMP.
FIRST QUESTION: Who wrote the curriculum?
The authors of IMP are two professors at San Francisco State University, Diane Resek and Dan Fendel, and two professional developers, Sherry Fraser and Lynne Alpert. A search through the million volume catalog at Amazon.com reveals that Fendel and Resek co-authored a book entitled Foundations of Higher Mathematics, Exploration and Proof, which is currently used in a 100 level math course at Simon Fraser University (and perhaps others).
An Alta Vista search on Diane Resek produced a number of interesting hits including a short talk she gave at ICME 8 on the role of calculators in the classroom. Another web page was her article Techniques for Cooperative Group Work in the Focus on Calculus documents at the University of Arizona. In the paper she discusses a number of common sense suggestions for using cooperative groups with the Harvard Calculus Consortium text. For example she says, I feel the CCH text is certainly lively, but not lean. Following my department's syllabus, I was able to have students work in small groups for about 20 minutes a week out of a total of 150 minutes.
SECOND QUESTION: Is IMP a true reflection of the NCTM standards?
"The IMP curriculum is designed to make the learning of a core curriculum more accessible, especially to those groups, such as women and minorities, who traditionally have been under-represented in college mathematics classes."
"The IMP is designed to be used with heterogeneous classes. The developers of the program believe that virtually everyone can gain a deep understanding of the curriculum and can make valuable contributions as a member of a learning group."
"...students are organized into small groups (usually four students to a group), and much of the classroom learning is done in the context of these groups."
"...the role of the teacher changes from that of "imparter of knowledge" to that of observer and facilitator."
"Some of you may recall that IMP was a part of the first-year PROMPT workshops as an example of a reformist, standards-based high school core curriculum."
IMP is a four-year mathematics curriculum for secondary students that focuses on open-ended explorations of complex problems. The program was designed in 1989 to fulfill the national mathematics standards and those of the California Mathematics Framework.
Teachers must learn new ways to manage a classroom, since IMP does not use the traditional model of a teacher lecturing to students who then complete a number of paper and pencil exercises.
Teacher training is essential for the success of this program, as teachers must not only master new instructional strategies, but new mathematical content and assessment methods as well. Success in an IMP class is measured more by how well a mathematical tool or idea can be used in a meaningful context than by the traditional approach of computation tests.
THIRD QUESTION: So what does it look like?
I recently purchased the 1997 student edition of the year 1 IMP text and have been working my way through the problems. Some initial, non-judgemental observations:
1) There is no index.
2) There is classwork for a total of 137 days.
3) There is a lot of writing by students.
4) Homework problem sets are short, often 5-8 problems.
5) Lots of pictures.
6) There is no procedural drill in the text.
7) There are 19 POWs (problems of the week) for an entire year. The POWs typically do not have a single correct answer.
The text, which represents the entire first year of high school mathematics, centers around 5 sections:
1) Patterns -- a 24-day unit of introduction, integers, angles, and in-out tables.
2) The Game of Pig -- a 29-day unit dealing with probability.
3) The Overland Trail -- a 30-day unit dealing with graphs, variables, rate, and lines of best fit.
4) The Pit and the Pendulum -- a 28-day unit dealing with graphing, equations, and statistics.
5) Shadows -- a 26-day unit dealing with basic geometry including similarity and triangles along with some basic trigonometry.
QUESTION FOUR: How is IMP perceived by the students being taught with it and the teachers who are teaching it?
There are a number of IMP web sites on the Internet. One which has access to most of the others is http://www.azstarnet.com/~quesnel/imppage.html This web page is maintained by Jerry Quesnel, a 20 year veteran math teacher at Desert View High School in Arizona whose mean mathematics ITBS scores are in the 27th percentile. Here are some of Jerry's comments:
Do Not allow too many below-level students into the program. This will make it impossible to work.
Pros and Cons
1) more fun for teachers and students.
2) much more time consuming to prepare for.
3) more students can be successful when compared to a traditional approach.
4) standardized test scores have shown no real change yet.
5) students do not get enough computation practice.
It is interesting to note that some of Jerry Quesnel's comments about IMP mirror those of Professor Hung-Hsi Wu's in his review of IMP at Berkeley high school. At other IMP sites there are letters from students extolling IMP. At the Mathematically Correct website there are several anecdotes about problems with IMP including its inability to prepare students for higher level math and science classes and the remediation sometimes necessary for college-bound students.
Some Judgemental comments on the IMP year 1 text.
1) As a high school math teacher who has taught for ten years, there are a number of areas in the IMP text that I have difficulty with. On the minor side, when dealing with 9th and 10th graders, telling them to write "Some ways to..." or "all you can think of..." on an assignment as occurs fairly frequently on IMP classwork, POWs, and homework, is a recipe for low quality work due to teacher subjectivity and student desire to do the minimum amount of work for the maximum grade.
2) There is an incredible amount of writing required of students in the text. To adequately and honestly grade and provide feedback to this writing would eat up a tremendous amount of time.
3) The lack of an index is very irritating.
4) The topics seem disjointed and I often have trouble seeing the connection between one section of a unit and another.
Comments
It is clear that IMP is a discovery-oriented, constructivist curriculum. Students are encouraged to find their own meanings of mathematical subjects while conversing with their peers and being facilitated by the "guide on the side" once known as a teacher. Unfortuntely, as the book Children's Mathematical Development by developmental psychologist David Geary explains, " ... one of the implicit assumptions of the constructivist approach is that mathematics is a biologically primary domain."
Geary explains that this is true of some areas such as number, counting, and some features of arithmetic, but is definitely not the case for more complex mathematical skills. In an area in which IMP is decidedly lacking, drill and practice, Geary has this to say.
"Finally, the argument that drill and practice and the development of basic cognitive skills, such as fact retrieval, are unnecessary and unwanted in mathematics education fails to appreciate the importance of basic skills for mathematical development. As noted earlier, drill and practice provide an environment in which the child can notice regularities in mathematical operations and glean basic concepts from these regularities. Much of mathematics involves being able to use procedures, equations, and so on. Except for basic numerical and arithmetical skills, most children are not likely to be able to develop mathematical procedures solely on the basis of their conceptual knowledge."
To me it is clear that IMP represents a curriculum which is standards-based. Thus its flaws rest either with the standards themselves or with the philosophical orientation of reformers in the mathematics educational community.
As a teacher who has taught Algebra on four different occasions and geometry for 6 years, I have to say that it would take an extraordinary teacher to teach with IMP and adequately prepare students for higher level math classes. Students who enter high school in the United States often have weak arithmetic skills. Building algebra skills requires extensive practice in a number of areas including the following:
In algebra a problem practicing this would be -2/3 x - 5/6x = 12
The ratio of the length of a rectangle to the side of a square is 8 to 5. If the area of the square is 2500 square meters, and the width of the rectangle is 15 meters, what is the perimeter of the rectangle?
Problems like this require students to practice problem solving skills such as: deciphering the problem, drawing appropriate diagrams, setting up correct ratios and equations, remembering or looking up appropriate formulas (my students are required to have them memorized. Less need to stress working memory load), and finding a correct solution path. Note that while there may be more than one correct solution path for the problem, it is not open ended like an IMP problem.
Now there are a number of things that I have not included that it would be nice for algebra students to have in their mental quivers as well. This includes things like scientific notation (important for science classes), quadratic equations, basic coordinate geometry (equation of a line, slope, etc.), pythagorean theorem, square roots, etc. But what I covered in the first five points are what I consider absolutely essential for my kids coming into geometry. And realize that my geometry classes are heavy on practice in these skills because even traditional students tend to be weak in these areas unless they are practiced. In my own classes, however, practice of these skills is embedded in more complex problems, not in isolated drill sheets. Here is what David Geary recommends in his book Children's Mathematical Development.
The goal or end point of problem solving should be explicitly stated when the topic is first introduced.
The stated goals should be immediate, "The goal for this type of problem is to find the answer for X", as well as long term,"This type of problem solving is used in many different types of jobs, including..."
Mastery of mathematical procedures requres extensive (sometimes boring) practice.
Practice should include the following:
1) Small doses (e.g. 20 minutes) over an extended period of time.
2) Practice on a variety of problem types mixed together. Do not practice on only a single problem type -- this leads to procedural bugs.
3) Practice until the procedure is executed automatically, that is, until the child can use it without having to think about it.
4) Once automaticity is reached, include some additional practice of the procedure as part of review segments for more complex material. This facilitates the long-term retention of the procedure.
1) When possible, present the material (e.g. word problems) in contexts that are meaningful to the child.
2) Make one goal to solve the problem in as many different ways as possible rather than simply teaching a problem-solving algorithm. This goal can be achieved as a feature of class discussion.
3) Discuss problem-solving errors. Use errors as a diagnostic for conceptual misunderstandings and an opportunity to clarify the misconception.
As you can see, Geary states ideas that are similar to the way Saxon math is taught, as well as the way that the Japanese teach math. Both procedures and concepts are important and need to be melded together. This is what IMP does not do unless the teachers take it upon themselves to teach outside the text.
So what can you do? See if your child can do simple problems like those outlined above. If not, request that your teacher explain how your child will be able to do higher mathematics without automating basic skills so that working memory load does not interfere with higher problem solving. If your teacher wants an example of a higher math problem that might depend on some automated skills, here are a couple you can give them to see if they can do them, the first from precalculus (although it could easily be done in a good algebra class) and the second from calculus. Both depend on having some good algebraic manipulation skills. (although the first might be solved by guess and check. But if the teacher does this as the only way of solving the problem, it is not a good sign). If they can solve the second one in less than 10 minutes, they are very good.
1) A marketing company in Watertown discovers that 15,000 people will buy raffle tickets for 95 dollars, and that for every 1.56 decrease in price another 450 people will buy a raffle ticket. How many people will buy a raffle ticket if the price is 33.50 dollars?
2) A full 10,000 liter tank contains 40 kilograms of salt in solution. Brine with a concentration of .02 kg/liter enters the tank at a rate of 20 liters per minute. The tank is well mixed and drains at the same rate. How much salt is in the tank after 2 hours?
Kim Mackey