In our previous IMACS blog post, we began our response to Professor Andrew Hacker’s op-ed piece entitled “Is Algebra Necessary?” by taking a critical look at his reasoning in favor of eliminating the requirement for high school algebra. We argued instead that the approach to teaching algebra, and more broadly all of mathematics, should be changed significantly in the US to benefit all students, from those who are struggling to ones who are at maximum achievement under the current limited system. In this week’s post, IMACS discusses key elements that we believe should be part of any effective curriculum in mathematics.
We were pleased to see that Prof. Hacker quoted mathematics professor Peter Braunfeld of the University of Illinois as saying, “Our civilization would collapse without mathematics.” Prof. Braunfeld is not new to the mathematics education debate, having co-authored an article* on the subject with IMACS principal founder, Burt Kaufman, and IMACS curriculum contributor, Professor Vincent Haag, nearly 40 years ago. (Prof. Braunfeld was also a contributor to the IMACS curriculum.). Their article outlined five principles that have and continue to guide IMACS in our curriculum development.
Everything Old is New Again
A 40-year-old article!? How can that be relevant now? Sadly, the circumstances lamented then by the co-authors remain a plague on our US math curriculum to this day. Have you heard anything like the following excerpts lately?
On the mindless drudgery that passes for school mathematics: “A student has simply been shortchanged if after nine to 12 years’ study of mathematics, he leaves school with the notion that mathematics consists of a large collection of routine and boring algorithms that enable him to get ‘correct answers’ to certain, usually contrived, questions.”
It’s no wonder that students find math dull and tedious. The trivialized curriculum forced upon them has been stripped of all the wonder and beauty of mathematics.
On technology as the cure: “Some educators appear to believe that the basic problem lies not in the meager and often irrelevant content of school mathematics but in the inadequacy of the delivery systems. … [I]t is surely putting the cart before the horse to concentrate on improving delivery systems without at the same time making a concerted effort to improve and reorganize the mathematics that these systems are to deliver.”
In just the past year, the articles we’ve read suggesting that video tutorials, massive open online courses, and the iPad are going to “revolutionize” education are too many to count. The drive-thru window may have changed how Mickey D’s was served, but the stuff in the paper bag remained of questionable nutritional value for a long time. (Yet even the Golden Arches eventually overhauled its menu!)
Describing the approach then referred to as “behavioral goals”: “As we understand it … we must first very carefully set down our aims—just exactly what we expect the children to know at each stage in their progress. … Once this is done, materials can be produced that explicitly address themselves to the stated aims. Periodic tests and checks should be administered to determine whether the children have met the prestated behavioral goals, i.e., they can indeed ‘do’ the things that the materials purport to teach.”
Can we say teach to the test? The idea of “industrialized education” came about long before No Child Left Behind. What’s unfortunate is that such an ill-conceived notion wasn’t what was left behind.
Five Guiding Principles of Mathematics Education
So what are the five guiding principles that the co-authors proposed? The excerpts that follow summarize how they believed mathematics should and can be taught to children and how IMACS teaches today:
“1. Mathematics is an important intellectual discipline—not merely a collection of algorithms for performing calculations. One of the primary aims of a good mathematics curriculum should be to exhibit mathematics as a method of inquiry that enables us to answer interesting and important questions. We will never achieve this aim if we set our sights so low that we teach only the trivial—we must not, for example, become obsessed with teaching only algorithms.”
“2. The subject-matter of mathematics is ideas, not notation. … [T]he unfortunate fact is that more often than not mathematics is presented to children as if it were the study of certain kinds of printed marks on paper. A good example is the standard treatment of polynomials in high school algebra: students are told that polynomials are ‘expressions of a certain form’ and are then simply given a number of rules on the ‘proper’ way to ‘manipulate’ such expressions. We submit that if mathematics is presented as a subdiscipline of typography, it cannot play a significant role in the intellectual life of children.”
“3. Mathematics is an organized body of knowledge. … A mathematics curriculum has not done well by a student if it leaves him with the impression that mathematics consists of a myriad of unrelated bits and pieces. … If we are to present mathematics to the student as a coherent whole, we shall first have to become clear on what is fundamental and central to the discipline and what is peripheral. The fundamental ideas should be introduced to the student as early as possible so that they can then serve to unify the entire curriculum.”
“4. Mathematics gives us understanding and power over the ‘real’ world. … [T]he power of mathematics to give us solutions to ‘real’ problems is certainly not well exhibited by the stilted and artificial ‘applications’ we actually see in most curricula. … What we must provide, rather, is a wide variety of situations and problems with genuine life and spirit in them—problems that engage the student’s attention and arouse his curiosity. Surely a problem is ‘practical’ for a child if, and only if, it is one to which he would really like to know the answer.”
“5. Mathematics is a form of artistic expression. … A mathematics program that takes the poetry out of mathematics is a bad program for the simple reason that mathematics, like poetry, music, painting or dancing, deals in aesthetic values. … Nothing can replace the importance of a child’s pleasure in seeing an elegant piece of mathematics or, even better, in creating a piece of mathematics for himself. Learning mathematics and doing mathematics may at times be hard work, but it must never become mere drudgery.”
Exasperated students continue to ask when they are ever going to need high school math in the real world, and who could blame them? Yet you never hear such broad and fervent protest about high school science even though relatively few kids will put that knowledge to work. Why less complaining? Because science curricula, for the most part, still incorporate grand ideas that elicit awe in young minds. Why are parents appalled by cuts in the arts at school? Not because most think their kids will pursue careers in creative fields, but because they understand that the aesthetic experience lifts the human spirit.
Mathematics is brimming with this kind of elemental beauty too. Yet decade after decade, schools use curricula that deprive math students of the good stuff. It’s like feeding the cardboard box instead of the cereal to a kid and saying “See, it says ‘cereal’ right there on the label. How can you say it’s tasteless?” Children have been telling us for too long that we need to change the menu. It’s time we listen.
Editor’s Note: IMACS will soon be rolling out a series of interactive online math courses designed with the five guiding principles discussed above. These courses will allow talented students to complete all of middle and high school mathematics with the exception of calculus before leaving middle school. Check back at www.eimacs.com or like us on Facebook for exciting details to come!