This method gives an estimate, not the exact decimal equivalent for the fraction. It only works for the denominators 2, 3, 4, 5, 6, 8, 9, 10, and 12, and It can't be used in real life, unless students are to carry around the paper strip "models."
What's going on? Reform math educators believe that standard
paper-and-pencil arithmetic is now obsolete, due to the power of calculators.
In particular, they believe that long division is obsolete. Perhaps
they're right about everyday life. But if a student is to eventually
learn (genuine, not reform) algebra, and move on to learn more advanced
mathematics, then mastery of standard paper-and-pencil arithmetic is the
essential pre-knowledge that sets the stage for migrating up the math learning
curve. Unfortunately, reform math educators are only
concerned about everyday life.
In this section you will find MSU math Professor Tom Parker's opinion of that endorsement. For the source, click on Assessment of CMP Math [URL = http://www.nscl.msu.edu/~tsang/CMP/aaas_p.html].
"Mariam Stricklen, an Okemos parent, wrote an opinion column that appeared in Lansing State Journal on June 18, 1999. The column praised the Connected Mathematics Projects, in large measure because it was rated very well in a recent report by the American Academy for the Advancement of Science (AAAS). Below is a response by Prof. Thomas Parker in the form of a letter to Ms. Stricklen."
Professor Parker's Letter
I would like to make a few comments--- helpful ones I hope--- on your LSJ article. I am a MSU Math professor. While I am a research mathematician, I have been teaching courses in Math Education, and have been closely following current developments in Math education. Evaluating Math programs is a tricky business in the current environment. There are major battles going on, and reports from even trusted sources are usually tinged with the politics of these battles. The AAAS report is a case in point. It was not written by scientists. Rather, the AAAS has lent its imprimatur, under the name 'Project 2061' , to a group of EDUCATORS who are not trained in mathematics or science. These educators have a specific political agenda. They would like to see all education done in group settings with the 'discovery method' and with no instruction from the teacher. They would like to see mathematics classes with long writing assignments, no right and wrong answers, no practice problems, complete reliance on calculators, and a minimization of algebra.
Accordingly, they designed a set of criteria focusing not on WHAT and HOW mathematics is covered by the program, but rather on the extent to which it conforms to the above agenda. Take a close look at the Project 2061 website: "http://project2061.aaas.org/newsinfo/press/attach_a.pdf" Your chart is Attachment A. Note that it is titled 'quality of instruction'. It is a ranking of the INSTRUCTIONAL PRACTICES used by the math program, not a ranking of the overall quality of the program. Attachment B spells out the criteria. The 'Benchmarks' (made up by the 2061 people) describe a dumbed-down middle school program. In other programs and in other countries most of these topics are covered in grades 4-5. Some ('shapes can match exactly or have the same shape in different sizes') are kindergarten level, and not even correct english! For each of these 'benchmarks' the 2061 group applied the 'criteria of quality of instructional guidance' listed at the bottom of Attachment B. Here you see the political agenda I described above. Many of these phrases are educational jargon that require experience to decode. For example, the three listed under Category VI all mean 'no tests, quizzes or worksheets'' and the two in Category III mean 'group work with manipulatives.' The last one, 'supporting all students' means 'dumbed down'.
None of these criteria assess what specific mathematics is covered. Thus the Connected Mathematics Project (CMP) can be the highest rated, even though the topics it covers in grade 8 are covered in grade 7 in competing programs, and covered in grade 6 in Japan, Korean, China, Singapore and Russia. All that one should conclude from the Project 2061 report is that their top-rated programs are politically correct. I have looked over the CMP program carefully. It definitely is politically correct.
What criteria should be used? A good math program has several stages. Grades K-4 should thoroughly cover the basics: addition, subtraction, multiplication, division, fractions, and a little geometry. Grades 5--7 should be preparation for the real math that starts with a solid grade 8 algebra course focusing on quadratic equations and polynomials, and a grade 9 course in Euclidean geometry, with proofs. (That is the plan followed by most countries, and the one recommended by the U.S. Department of Education). Virtually all students find these two courses very hard; many find them TOO hard and drop out of math at that point. But these courses are not intrinsically difficult. In other countries ALL students take them as a matter of routine and, as the TIMSS study shows, master the material. The problem lies in the US grade 5--7 programs. There our children spend their time going over more arithmetic, instead of taking the first steps toward real algebra and geometry with proofs.
This last sentence exactly describes the problem with CMP and similar programs. The Project 2061 report does not reveal this fundamental deficiency because its criteria do not include real algebra and geometry with proofs. For different ratings, ones based on the actual mathematics covered and taught, see the website "http://www.mathematicallycorrect.com/" Click on http://mathematicallycorrect.com/books.htm, 'Mathematics Program Reviews for Grades 2, 5, and 7'.' ('Mathematically Correct' is an organization of parents and scientists who are working to restore serious mathematics to schools. A study of their website is essential for understanding the battle lines in the current debate over mathematics education).
I have young children in the Okemos schools. I am VERY concerned about the mathematics programs. You should be very concerned also. If your daughter is taking 8th grade CMP (rather than Honors Algebra I) then she is not on track to take calculus in high school. Calculus is the basic entry point to majoring in science, engineering and medicine. In Korea all students, including the children of rice farmers, take calculus as juniors in high school. You should be asking Okemos officials why it is that your daughter, who is apparently a top student in a top school system, is two years behind the children of Korean rice farmers.
Mathematically Correct assigned an overall "F" grade to the CMP, noting that "it is impossible to recommend a book with as little content as this." It was the only "F" grade awarded to a middle school math program. For a comparative summary, including other middle school math programs, click on Mathematically Correct Mathematics Program Reviews [The URL is http://www.mathematicallycorrect.com/books7y.htm
Mathematical DepthThere is very little mathematical content in this book. Students leaving this course will have no background in or facility with analytic or pre-algebra skills.
Quality of PresentationThis book is completely dedicated to a constructivist philosophy of learning, with heavy emphasis on discovery exercises and rejection of whole class teacher directed instruction. The introduction to Part 1 says "Connected Mathematics was developed with the belief that calculators should be available and that students should decide when to use them." In one of the great understatements, the Guide to the Connected Mathematics Curriculum states, "Students may not do as well on standardized tests assessing computational skills as students in classes that spend" time practicing these skills.
Quality of Student WorkStudents are busy, but they are not productively busy. Most of their time is directed away from true understanding and useful skills.
"This letter is to Plano parents of elementary and middle school-aged children. Its purpose is to make you aware of a math program for 6th-8th grades. . . .
This program is called "Connected Mathematics". In a nutshell, this program is "discovery" learning. The students "discover" the concepts in each unit through a variety of activities. The program sounds like a great idea in theory. The goal is to have students make sense of the mathematics they study and to be able to communicate their reasoning clearly. The teacher begins the unit by presenting the 'problem" to the students. The kids then work in pairs or small groups to solve the problem by gathering data, sharing ideas, looking for patterns, making conjectures, or developing other types of problem solving strategies. When most of the students have made sufficient progress toward solving the problem, the teacher helps the class to discuss ways they found to organize the data and look for patterns and related rules in the data.
Below are concerns which every parent should consider.
Connected Mathematics Project offers eight booklets for each middle school grade. The titles for the grade 6 booklets are underlined below. The indented comments are from a parent whose daughter was subjected to CMP. That parent is Betty Tsang, Professor of Physics at Michigan State University.
"Overall, the [CMP] program seems to be very incomplete, and I would judge that it is aimed at underachieving students rather than normal or higher achieving students . . . The philosophy used throughout the program is that the students should entirely construct their own knowledge and that calculators are to always be available for calculation. This means that:
My Special Number
Choose a whole number between 10 and 100 that you especially like. In your journal:
As you work through the investigations in Prime Time, you will learn lots of things about
Record your number Explain why you chose that number List three or four mathematical things about your number List three or four connections you can make between your number and your world.
From both a mathematical and pedagogical point of view this is unfortunate. Mathematically, no integers except perhaps 0, 1, and -1 are more significant than any others. And pedagogically, this reflects a poor point of view towards the development of the number system. If one prefers one whole number over another, think what a big door this opens for hating complicated fractions and even worse, irrational numbers. Basically, such a project appears to me to be totally unjustified except in remedial situations."
It is precisely at this point that CMP becomes strictly remedial. If students are to go on to higher levels of achievement in mathematics, from geometry through calculus, linear algebra and beyond, they must be able to handle precisely defined abstract concepts. Moreover, these abilities are difficult for even the strongest students to master, and they take considerable time to develop."
Rational numbers are the heart of the middle-grades experiences with number concepts. From classroom experience, we know that the concepts of fractions, decimals, and percents can be difficult for students. From research on student learning, we know that part of the reason for students' confusion about rational numbers is a result of the rush to symbol manipulation with fractions and decimals.The discussion above of the mathematics in Bits and Pieces II represents a highly controversial point of view about the subject. This view is agreed with by less than 1% of the professional mathematicians in California, for example. No one would dispute the argument that rote memorization of algorithms alone does not lead to understanding. However, when an algorithm is introduced together with a careful and precise explanation of how and why it works, students are exposed to material that is critical to the continued development of their mathematical skills. Whether students learn these types of things using discovery methods or other methods is
In Bits and Pieces I, the first unit on rational numbers, the investigations asked students to make sense of the meaning of fractions, decimals, and percents in different contexts. In Bits and Pieces II, students will use these new numbers to help make sense of many different situations.
The Mathematics in Bits and Pieces II
This unit does not teach specific algorithms for working with rational numbers. Instead, it helps the teacher create a classroom environment where students consider interesting problems in which ideas of fractions, decimals, and percents are embedded. Students bump into these important ideas as they struggle to make sense of problem situations. As they work individually, in groups, and as a whole class on the problems, they will find ways of thinking about and operating with rational numbers.
The teacher's role is to help students make explicit their growing ideas about the world of rational numbers and, when students are ready, to inject ideas and strategies into the conversation along with the ideas and strategies generated by the students. Simply giving students algorithms for moving symbols for rational numbers around on paper would be a mistake and the temptation to do something such as how to cross multiply to compare two fractions gives the impression of immediate success. Students can do the algorithm by memorizing. However, evidence from student assessments shows that students do not understand algorithms that are given to them in this way and therefore cannot remember or figure out what to do in a given situation.
This unit provides a rich set of experiences that focus on developing meaning for computations with rational numbers. We expect students to finish this unit knowing algorithms for computation that they understand and can use with facility.
"The discovery method and nothing else philosophy of the authors definitely works to the detriment of the students. The numerical skills involved with these topics require practice on the students' part, and discovery methods do not encourage this. Indeed, the discovery approach is carried to extreme levels here. For example note the quote on page 164 of the teachers manual in a sample letter meant to be sent to parents: 'It is important that you do not show your child rules or formulas for working with fractions. This unit helps students to discover these rules for themselves . . . .' "
"At this point, consistent with the point of view towards algorithms described in the introduction to this booklet, the students are asked to work in groups and find their own algorithms for adding and subtracting fractions. Moreover, as verifications of correctness they are given the following instructions:
Test your algorithms on
a few problems.... If necessary, make adjustments to you
algorithms until you think that will work all the time. Write up a final version of each
algorithm. Make sure they are neat and precise so others can follow them."
"The students do not, in fact, have to learn to actually add, subtract, multiply, or above all, divide decimals, since their calculators will do it for them."
"Unfortunately, as has been shown in work on curricular development, many of the cognate skills implicit in things like learning the long division algorithm become important in different contexts many years later. Consequently, students who have not developed these skills often seem to find themselves at a serious disadvantage when attempting to work in technical fields."
Spatial visualization skills are very important in developing mathematical thinking and are critical to reading graphical information, using arrays and networks, and understanding the fundamental ideas of calculus. In the past several decades, research has raised many questions about spatial visualization abilities. Many studies have found that girls do not reason as well about spatial experiences as do boys, especially starting at about adolescence. The explanation offered by some psychologists, that this difference may be innate, is unacceptable to those of us concerned with teaching children.It might also be worth noting that the role of spatial visualization in topics such as calculus is overstated. In fact the vast majority of the topic takes place in two dimensions. When one finally gets to questions in three dimensions involving solid integrals, surface area, and related topics, virtually all students appear to have serious difficulties, and the 'skills' developed in this booklet are not going to address the problem areas which actually occur."
Copyright 2003 William G. Quirk, Ph.D.