I’m not a math phobe even if I did hate geometry. I just didn’t think it was helpful in the least. Loved physics and aced calculus by the way. I think it was because I thought proofs were a stupid waste of time–some things just seemed so obvious to me that why would I need to prove it? Perhaps now I might appreciate the process of actually being able to prove something obvious (to me) or disprove it. Maybe that particular lesson got lost in the shuffle.
But teaching critical thinking is not limited to the teaching of geometry.
I maintain that learning simple arithmetic (add, subtract, multiplication, division) and being able to quickly access those facts (so rote memorization after learning how it works) allows a student to more easily manipulate numbers and allows game play with numbers. It just doesn’t happen when tied to a calculator–the brain works faster than that–learning math facts by rote enables ease with numbers later.
The author’s examples of devising a time system, the combinations of coins in the pocket etcsi. are math puzzles which are interesting and can grow more complex but only if you have a good grasp of fundamentals. I didn’t see where he said don’t teach arithmetic–he said make math more interesting.
One of my HS physics projects was to create a perpetual time machine which of course doesn’t exist but it didn’t keep us from trying. And discussing why our designs didn’t work. And how to get it to work etc. It was a fun project and covered a lot of physics. I think that is what the author was getting at–creating interest in the subject rather than just teach formulas.
Well in a way, calculus is “where higher math starts.” So I do see why it’s the focus of highly competitive high schools. Statistics is strongly based in calculus on anything but a very very introductory level (“this is a mean, this is a normal distribution, and ooh look, graphs! Don’t ask where any of this comes from”). Discrete mathematics is important largely because it leads into real analysis, which perhaps should be taught concurrently with calculus because it would make calculus make a lot more sense in the long run.
Instead, the prevailing view seems to promote dumbing down the curriculum to a bare minimum mathematical understanding, which in the process makes a lot of its meaning be lost in translation. It took me until the end of my UG math degree to understand why certain parts of high school math were even taught because they seemed like a pointless waste of space. There was absolutely no reason they needed to be that way - the value of all the tedious stuff had a logical and reasonable explanation that worked better than the “formula memorization” approach that is most prevalent in US high school math.
Caltech and Harvey Mudd do this (though they expect students to have had a regular calculus course previously); some other schools may offer honors calculus or calculus with theory courses as options.
However, most college frosh and advanced high school students would probably find a heavy dose of real analysis in their calculus courses to be too difficult. The way calculus is currently taught is aimed at the majority of students who are not math majors (or others needing a heavy dose of real analysis, like statistics and some economics majors) and who just need a basic understanding of what calculus means in order to be able to apply it to various problems in their subjects. A similar example would be teaching primary and middle school students some of the properties of integers, rational numbers, and real numbers without going deeply into the theory of sets, groups, rings, and fields.
On the other hand, it can be a valid argument that math is often taught too superficially, with too little coverage of the underlying ideas for the majority of students who just need to apply math to their subjects to really understand it well enough to properly apply math to their subjects.
You don’t need to study proofs to understand concepts such as if…then or to learn how to think critically. It’s great that some people find proofs interesting, but most people don’t. The way we teach math is one of the reasons so many people are driven away from it.
I think “learn how to think critically” is a cliche that has lost any real meaning, and that “If X then Y” is a relatively simple concept. But neither of those are what proofs are really for. Doing at least elementary proofs, which is perhaps what geometry tries but effectively fails to do, is an important exercise in structured logic - in starting from definitions and then making connections between those defined structures (lemmas then theorems). That’s always an important skill to have.
Real analysis in full is definitely beyond the level of what most HS/frosh students can do, especially if they have never taken a discrete math course. Though I think the knowledge creep will eventually reach the point where analysis will be at least introduced in high school, that’s going to take some serious restructuring of the entire education system to do well (the first few years of elementary school will have to move along a lot more quickly).
And yet, high school math at present has a very awkward inability to properly balance the issues of mathematical rigor and simplicity. A lot of theoretical concepts are introduced, but without the proper context that makes people see their value and bother to remember. Simple proofs are covered in geometry, but none of the structured proof context that would make people appreciate them (this is why I think some discrete math, and some instruction on writing proofs in prose, would help). The number systems are covered in algebra, but there is no reason to care about them or why they exist until discrete math/analysis. The building blocks of derivatives and integrals (the definition of a derivative, Riemann sums, simple numerical algorithms) are mentioned, but without the kind of context that gives that kind of instruction value. Simple elements of basic discrete math and real analysis - sets and functions, relations, the real numbers and the operations on the real numbers, deriving the product/sum rules from the definition of the derivative, a simple explanation of how Riemann integrals work and what the significance of the Fundamental Theorem of Calculus is - would be quite feasible and would help to see the connections. As with all mathematical instruction at all levels, some things will have to be taken “by faith” until further coursework is done, but I don’t think anyone would complain that no one bothered to construct the real number system anyhow.
The other option would be to say, “Here’s how you take a derivative. Here’s how you take an integral. Be on your merry way.” I don’t think that’s what we should be doing.
I remember seeing this stuff at a basic level in high school math including calculus BC (high school was non-elite with about a third of graduates going to four year colleges at the time). Has that been dropped completely from high school math courses?
The real numbers and their significance are pretty rarely mentioned and not part of the AP calculus curriculum. The definition of the addition/multiplication operations is also not part of the curriculum but would perhaps be a good thing to cover. The definition of the derivative and Riemann sums are covered, but also in such a way that their significance is pretty much nil. Fundamental theorem of calculus is covered, but not in such a way that makes people care about what it is or what it means - most people just think, “it’s just an integral” (it is, but it’s what makes integrals work).
What that all is missing is any notion of proofs or of showing that the various methods work, which may just make the subject a bit more mathematically grounded in a useful way. Though perhaps that would be tricky to properly implement because I don’t think the average calculus student would be prepared for continuity with epsilon-delta proofs.
As I pointed out, I loved math and got A’s all through to college advanced calc. But I see no reason for one size fits all education beyond some point. The question is where to set those limits.
People are all different, we don’t end up in the same careers, have same interests. So to what point in life must we all be treated the same?
I just think we do go a little to far with that.
We all agree not to force little boys to play with dolls, we accept some differences early in life.
The question is what to accept. I personally think by 16 kids should have interests they can spend more time exploring, maybe at the expense of other subjects. Arguable, the age and subjects to cover as mandatory through that age, but the idea needs to be explored.
Personally I really resented being forced to take a foreign language while studying math and comp science in college. Am I better for it, maybe in some small way, but it took away from other interests that I could have developed instead. I resented that. I see no difference for math, not going to be biased because I enjoyed math classes.
When I took calculus it was a theory-based version (Leithold). I have taught both the theory based version and a “calculus lite” non-theory based version. The theory version did have some proofs, but was not as rigorous as real analysis, which I also suffered for 2 semesters. When my students complained on their evals that I had done too much theory, I knew I had done what I was hired to do ;).
Drop geometry? It’s probably the first place where people see math’s beauty. Understanding how Archimedes first estimated Pi or Euclid’s proof of Pythagoras’ theorem are both inspirational. Likewise, it’s useful in remarkably surprising (e.g. simple plagiarism detection uses geometry) places.
A few years ago, I read an essay, “The Mathematician’s Lament,” where the author talked about how poorly we teach math. On of the interesting anecdotes involved a student working for a week and a half to come up with a proof of the Pythagorean theorem. From his perspective, it was clear he thought the insight from this was significantly more important than any resulting from acquiring a collection of methodologies. Trying to avoid a political abyss, it’s my observation that the Common Core people understand this and are trying to fix it by developing math intuition over algorithmic execution.
To the person who doesn’t like to prove things that are obvious, do you think you wouldn’t learn anything elegant from proving that the sum of two even integers is even?
My son spent 2nd grade doing “problem solving” enrichment because he’d already done 3rd grade math in 2nd grade, but the teacher refused to send him to 4th grade math. He got soooooo sick of figuring out how many ways you can make change, or how many different numbers three arrows in a dart board could make (I yelled at the teacher about that one because it was so stupid.) I ended up teaching my kid to use an Excel spread sheet, which is at least partly how he ended up catching the computer bug.
“To the person who doesn’t like to prove things that are obvious, do you think you wouldn’t learn anything elegant from proving that the sum of two even integers is even?”
I love and feel very fortunate that there are people who care about this and see the beauty in it because if the world was waiting for me to come up with this stuff it would be a very long wait. My head is already down on my desk waiting for class to be over.
“I’m saying: No, we don’t need that many people studying mathematics. We’re shooting ourselves in the foot. One in five people don’t graduate high school — this is one of the worst records of developed countries. And the chief academic reason is that they fail algebra — of course there are other nonacademic reasons, like prison and pregnancies.”
Why is this guy getting any attention? He is not so much the problem as those who give his ideas more than 5 milliseconds of thought.
Education is not statistics and statistics can reflect systemic problems of education, like fundamental problems of K-6 education. Statistics do not define reality and they especially do not define individual potential. Education is about individual opportunities, not statistics.
Is it his idea to stop setting any level of expectations starting in the earliest grades and leave all expectations and ensuring basic skills to us parents and the booming industry of K-8 math tutoring. They are hiding the tracking at home and never asking us parents what we had to do. We aren’t just modeling a love of education and taking our kids to museums. We are ensuring mastery of basic math skills. Incredibly, I’ve even gotten notes from my son’s schools telling us parents to work on “math facts” because they don’t want to do what they’re supposed to do. This increases the academic gap because they think that math is some sort of magical thinking process. They are like Professor Hill in The Music Man. They want to use “The Think Method.” My math brain son had to get help with basic math skills at home. It was not some sort of magical top-down process driven by no-one-right-answer, student-centered, teacher as the potted-plant-on-the-side educational pedagogy. Full inclusion and differentiated instruction (self-learning) is a complete failure and any attempts to justify its use defines educational incompetence. Magically, this educational thinking disappears in most high schools, but the damage has been done and many students can’t recover. They are even led to believe that it’s their own fault or because of IQ.
Many years ago I got to calculus in high school with absolutely no help from my parents. This is now virtually impossible and CCSS has now officially turned K-6 into a no STEM zone. PARCC officially declares it and sets it’s highest level (“distinguished”) goal as 75% successful in “College Algebra” in college. This is educational incompetence - it hides the academic tracking at home and increases the academic gap. It defines a curriculum gap for all. All one has to do is look at El Sistema. It does not require multi-generations, and math is no different than music - both require skills and understanding. In K-6 schools, however, math skills are never ensured and by 7th grade, it’s all over.
The problem I have with this argument is that without at least a basic understanding of algebra, students are shutting off a large number of career opportunities in a time in their lives when they are not known for good decision making and forward-thinking. Nursing, law, any field in science or medicine, all require some advanced math. Many of the concepts taught in trigonometry can also benefit someone in the construction trades. It’s true that students starting in a remedial math class in college are much less likely to get their degree, but I don’t think the answer is to drop the math requirement, I think we need to strengthen math in the K-12 system. Working in school that’s about half independent study/homeschooling, I also see the math angst of parents passed down to their children; eliminating or reducing math requirements would also affect a large number of future teachers, who without an understanding of higher math I feel can’t fully appreciate the math they’re teaching at the elementary level.
Seeing as how our country has fallen academically below some third world countries in math skills, I do not see how the author can make such claims! Sadly, the math that he’s describing as necessary is NOT learned at the elementary school level. Sadly, many elementary, middle school and high school students cannot do the basic math of addition and subtraction, let alone multiplication without a calculator! There are enough scientific studies out there to prove that doing upper level mathematics forms new pathways in the brain, thereby, making it possible for the student to learn new things easier. Seeing as how the national standard already has our average high school graduate being barely proficient in Algebra I skills, I do not see how we can lower the bar anymore without hurting our country more. The detail, thinking, logic, etc. taught in mathematics is beneficial to more than just the math major or engineering student! I was far more bored in high school English and history classes when I was growing up. I now, as an adult, recognize the necessity for both of them to be taught, despite my personal likes or dislikes that I had as a teenager. Being a ‘best seller’ author does not make this professor an authority in all things! As with all things, be careful of your source of information and what you believe. His subjective experiments he references are just that, subjective.
The reason why so many liberal arts majors are working at minimum wage is that they avoided the type of classes that required learning of new concepts and understanding how to think through tough problems. Also, most STEM majors take a full set of social sciences and liberal arts, philosophy, literature, history, writing and comprehension. Few Liberal Arts majors take even basic math or science. The ability of politicians to manipulate the public on critical decisions that involve science is proof that our college system DOESN’T produce better citizens.
Sorry Gender Studies major, basic astronomy class for Football players doesn’t qualify as real science. The freshman ‘ecology’ indoctrination class is not real science.
Those of my generation, college educated in the 70s, claim they ‘succeeded’ without math. The working world they entered had no foreign competition, little use of computers, and few people dealt with complex finances on a daily bases.
I would say things have changed a bit. Listen to workers who are in their late 20s not retired workers…
Exactly. Visit a school that focuses on Diesel mechanics some time. Look at the complex analyzers, computer measuring and test systems. Don’t think you need math? Try putting an engine back together these days.
