The mathematics experienced by students is so much different from the real nature
of math. What a tragedy!
What is the nature of mathematics?
- Mathematics is human. It is part of and fits into human culture. It is NOT an abstract, timeless, tensely, objective reality…
- Mathematical knowledge is fallible. As in science, mathematics can advance by making mistakes and then correcting them…
- There are different versions of proof or rigor. Standards of rigor can vary depending on time, place, and other things. Think of the computer-assisted proof of four color theorem in 1977…
- Empirical evidence, numerical experimentation and probabilistic proof all can help us decide what to believe in mathematics…
- Mathematical objects are a special variety of a social-cultural-historical object …They are shared ideas like Moby Dick in literature and the Immaculate Conception in religion.
The above description of the nature of mathematics is by Reuben Hersh, from his article “Fresh Breezes in the Philosophy of Mathematics published in American Mathematical Monthly Aug-Sept, 1995 issue. He is also the author of the now classic What Is Mathematics, Really?
What is the ‘nature’ of school mathematics?
The following is a 2002 critic of the US k-12 mathematics by Paul Lockhart in A Mathematician’s Lament. It’s also true in my part of the globe.
The Standard of K-12 mathematics according to Lockhart:
LOWER SCHOOL MATH. The indoctrination begins. Students learn that mathematics is not something you do, but something that is done to you. Emphasis is placed on sitting still, filling out worksheets, and following directions. Children are expected to master a complex set of algorithms for manipulating Hindi symbols, unrelated to any real desire or curiosity on their part, and regarded only a few centuries ago as too difficult for the average adult. Multiplication tables are stressed, as are parents, teachers, and the kids themselves.
MIDDLE SCHOOL MATH. Students are taught to view mathematics as a set of procedures, akin to religious rites, which are eternal and set in stone. The holy tablets, or “Math Books,” are handed out, and the students learn to address the church elders as “they” (as in “What do they want here? Do they want me to divide?”) Contrived and artificial “word problems” will be introduced in order to make the mindless drudgery of arithmetic seem enjoyable by comparison.
ALGEBRA I. So as not to waste valuable time thinking about numbers and their patterns, this course instead focuses on symbols and rules for their manipulation…. The insistence that all numbers and expressions be put into various standard forms will provide additional confusion as to the meaning of identity and equality. Students must also memorize the quadratic formula for some reason.
GEOMETRY. Isolated from the rest of the curriculum, this course will raise the hopes of students who wish to engage in meaningful mathematical activity, and then dash them. Clumsy and distracting notation will be introduced, and no pains will be spared to make the simple seem complicated. This goal of this course is to eradicate any last remaining vestiges of natural mathematical intuition, in preparation for Algebra II.
ALGEBRA II. The subject of this course is the unmotivated and inappropriate use of coordinate geometry. Conic sections are introduced in a coordinate framework so as to avoid the aesthetic simplicity of cones and their sections. Students will learn to rewrite quadratic forms in a variety of standard formats for no reason whatsoever. Exponential and logarithmic functions are also introduced in Algebra II, despite not being algebraic objects, simply because they have to be stuck in somewhere, apparently.
TRIGONOMETRY. Two weeks of content are stretched to semester length by masturbatory definitional runarounds. Truly interesting and beautiful phenomena, such as the way the sides of a triangle depend on its angles, will be given the same emphasis as irrelevant abbreviations and obsolete notational conventions, in order to prevent students from forming any clear idea as to what the subject is about. Students will learn such mnemonic devices as “SohCahToa”…
I feel that this is an important discussion. I believe that math is our way of describing the world around us, it is our language nature that transfers to all people.
I feel that school math and human math need to find a middle ground. In order for students to see math in a descriptive nature they must be able to do computational problems fluently. We need to stay away from any requirements of memorization and stay away from texts. Pose quality problems and then encourage any method of finding a solution, but along the way we need to introduce our knowledge of representation, a textbook language for students to express mathematics. Find a way to make mathematics holistic versus a linear pattern of classes. Allow students to progress at the speed of the patterns they are able to see and communicate. Develop the problem solvers of the next generation at the rate they can learn.
Thank you for raising this discussion. I think “mathematics” means different things to do different people, and to some extent, that is okay. However there are developmental opportunities being lost, as well as a lot of excitement and reward, by avoiding the more challenging versions of mathematics. I tend to agree with Lockhart, that there is great beauty in mathematical exploration and discovery, that students should be encouraged to not only create written arguments, but even to critique each other and to strive for clarity and beauty in their writing.
Personally I profoundly disagree with Hersh about math being a human thing, a product of humans; rather, it seems obvious that mathematics is separate, and eternal, and would be the same mathematics on another planet, done by another species, like Martians. The only difference would be, as we see in different cultures in our own history, that different civilizations would discover different parts of mathematics first, or at all. … In any case, mathematics has tremendous value to exercise the mind, in various forms of reasoning — and to practice precise and rigorous communication. And therefore it deserves a place in any child’s education, even if such things as reading and writing, as social studies, have a more widespread and obvious practical value.
Thanks again to everybody for sharing your thoughts.
In my years around the educational system, I have seen that so many kids don’t understand the pattern of number and the graduation of mathematical skills. So many kids are dependent upon resources outside of critical thinking and reasoning. Math is not hard when you begin to think and extract possibilities. American institutions have gotten away from the enforcement and reinforcement of the valuable critical thinking skills needed to progress as a student. Math is an exact science meaning that it cannot be simply stretched like metaphors in grammar. Its not simply a system of memorizing the patterns. We have to get back to instruction in math by conveying patterns and the purpose that mathematical patterns serve.
That said, I would argue that schools are weak to identify real (authentic) mathematical talent. And if such talent is not identified early, the talented young mathematicians learn that they are, in fact, not talented at all. And the “good” students — those who are good at memorization — go on and somehow gain access to the really good stuff: authentic mathematics and mathematical activity in grad school. I think what this means is that young kids need opportunities to engage in authentic mathematical experiences.
Already, by 6th grade, my students are dumbfounded when I say the purpose of math is to see that the world makes sense; that what you do in math is (and must) make sense. And I tell them that “… because my teacher said so” – especially if that teacher is me – is NOT a valid mathematical reason for doing anything. But they are hard pressed to accept this.