Oct 19 2011

Readin’ > ‘Rithmetic

Something that nearly every adult has wondered about, but is scared to say it out loud is “Is knowing math really that important?”

They don’t say it out loud because they fear that someone, possibly a math teacher like me, or someone else who knows better, will put them in their place with a long explanation about the importance of math in everyday life, in the sciences, and in the development of minds.

I love math. I realized I was good at in when I was in 6th grade and have progressed through the levels, majoring in it as an undergraduate. I’ve spent most of the past 20 years as a math teacher. So when I write what I’m about to write, it will certainly shock most people. Here it is: math is overrated.

Nowadays, with high stakes testing, we are told that the only two things that really matter, in equal amounts, are reading and math. Schools are praised or shut down for their scores (or ‘gains’) on these subjects. Some schools, particularly charters, have realized that if they focus on math, at the expense of reading, they can get better combined results than if they try to focus on both, equally. This implies that proficiency in math is somehow as important as proficiency in reading, and I’m here to say that this is completely absurd. I’m one of the few people who is not scared to say this since I will hold my own in a debate against anyone on this topic, as I’ve been pondering it, nearly daily, for about twenty years. In this post, I’ll try to summarize why I think this.

First let me distinguish between two types of math, which I’ll call Mathematics and ‘math’. When I say I love math, I’m talking about Mathematics. The word Mathematics is derived from the Greek, meaning ‘knowledge.’ It doesn’t say anything about adding fractions with unlike denominators or about the area of a triangle. I study Mathematics in my spare time, subscribe to a journal called Mathematics Magazine, and really pursue it in the way that people pursue art or music. I will give an example in a bit about what I mean by Mathematics and how it relates to its distant, and inferior, relative which I’ll put in quotes and start with a lowercase letter, ‘math.’

‘math’ is what is taught in school. Kids are subjected to twelve years of boredom, memorizing math facts and formulas, and churning out solutions using algorithms, without the slightest idea of why the formulas or algorithms work. Students are told (lied to really) that they will ‘use this’ someday, when they are building something or balancing their checkbooks or shopping. For those students who manage to get a year ahead, they are rewarded, after twelve years, of taking Calculus in their senior year. Unfortunately, what was once regally known as The Calculus could now only be described as ‘calculus’ since it is also a mindless set of algorithms which would make Isaac Newton flip in his Westminster Abby grave.

The high stakes ‘state tests’ are about ‘math’ which is too bad since doing well on such state tests does not mean the students know anything about Mathematics. It is more a measurement of how obedient they are. This is something that most thinking adults probably suspected, but didn’t feel they could defend these thoughts against someone who is much better than them in math. The fact is that the ‘math’ that students learn will not help them much in later life. Over the years topics kept getting added and added until poor ‘math’ teachers had no choice but to teach each topic in a superficial way in order to ‘get though’ them all. As a result, all the Mathematics was stripped from the ‘math’ leaving something that, for the most part, is a waste of time. What they ‘learn’ about ‘math’ is quickly forgotten after the test. That which isn’t will rarely be used again, not by scientists, not by engineers, not by bankers.

But true Mathematics, I think, is worth studying. I still don’t think that it is nearly as important as the skill of reading, but real Mathematics, when experienced properly, is something uniquely human which makes the mind flex in ways it cannot in any other discipline.

Here’s an example of what I mean by Mathematics. And, believe it or not, I’m going to pose it as a multiple choice question, but one that requires an explanation.

Here it is, and I’m going to ask the reader to spend a few minutes thinking about this.

3 is an odd number and 7 is an odd number, but when you add them together you get 10 which is an even number. Which of the following do you think is correct? Justify your answer.
A) An odd plus an odd is never even.
B) An odd plus an odd is always even.
C) And odd plus an odd is sometimes even and sometimes odd. It depends what the numbers are.

Take a few moments and see if you can experience a mini ‘Aha’ moment.

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The correct answer is B.

A beautiful ‘proof ‘ of this can be done with something called a ‘proof without words.’ Look and ponder for another minute.

This is what I mean by Mathematics. This is way mathematicians call a succinct proof in Mathematics, ‘elegant.’

So how is this concept covered in ‘math’? Well students are asked to memorize the facts: even+even=even, odd+odd=even, even+odd=odd, even*even=even, even*odd=even, and odd*odd=odd. Some justification might be presented, but there is not enough time to really allow students to think about why these rules are true because we have to get to the way these are presented on the standardized test. This is a question from a practice test for the SSHSAT, which is the test eighth grade students in New York City have to take to qualify for one of the specialized high schools.

If x is an integer, which one of the following must be odd?

A) 3x + 1
B) 3x + 2
C) 4x – 1
D) 4x – 2
E) 5x – 2x

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The answer to this one is C, but if there was a choice that said ‘Who Cares?’ that would also be right.

I hope you’re beginning to see the difference between Mathematics and ‘math.’

One reason we are stuck with ‘math’ instead of Mathematics is that there are not enough teachers who are qualified to teach Mathematics properly. And of course I’m not attacking teachers and demand they be fired for not being trained properly. It’s just that there is a real shortage of highly qualified math teachers so they have to do the best they can. But the bloated curriculum is the main culprit. Each year new topics get added, but things rarely get taken out. The only topic I can think of which used to be popular and now is out of the curriculum is taking square roots by hand.

If it were up to me, I’d gleefully scrap about 40% of the topics required by most curricula. Out would go the word problems where people work together to paint a house and we figure out how long it will take, or the famous train leaving Chicago at 80 miles per hour …, and a whole lot more.

The Common Core Standards in math have, in many ways, made the situation worse. They added many more topics and removed very few so that some of the topics had to be moved down to lower grades to make room for the new topics in the upper grades. So topics that kids are not ready for developmentally are now taught too soon.

But the crazy thing about all this is that so many policy makers believe that this ‘math’ is so important that schools need to be closed and teachers fired when students are not proficient in it. This is also why I cringe, knowingly, when I hear of the success of the ‘no excuses’ charters of getting math scores up, despite having little or no improvement in reading. Most recently, Houston tried to take the KIPP model and create something called the Apollo 20 program, which pumped a lot of money into math tutors, so they got their math scores up but nothing in reading.

As you suspected long ago, ‘math’ is not very important. It doesn’t measure intelligence. It doesn’t make countries ‘globally competitive.’ It’s mostly a waste of time.

As a professional ‘math’ teacher for nearly twenty years, I’ve found ways to sneak Mathematics into my ‘math’ course. Sometimes this means that I give a very cursory treatment of certain topics so I can have time for what I consider important. I’m fortunate to teach one section of something called ‘math research’ where I get to choose the topics so I can ensure that it’s all Mathematics all the time. Sometimes I feel like I’m a trained Chef who is forced to work in a McDonalds. I have to do my job and make those Big Macs and fries, but I search for opportunities, even within my constraints, do things the way they are supposed to be done. It isn’t easy since there are just way too many topics to be done in so little time.

Feel free to browse my youtube channel if you want to experience a bit of what I consider Mathematics.

I am not alone among math educators about the sorry state of the imposter that we currently call ‘math.’ Just a day after I started working on this post, the cover story in the magazine American Educator written by professor Hung-Hsi Wu identified the same contempt for what he calls Textbook School Mathematics or TSM. He, like me, thinks that the current math curriculum has developed into something that serves little purpose. He feels that things will improve if four things happen: 1) If the Common Core Standards for math are good, 2) If textbook companies get behind them, rather than just slapping a sticker on their old books and calling them ‘compliant with Common Core.’ 3) If schools of Ed get on board training math teachers properly, and 4) If the mathematical community invests some brain power on the problem. He believes that the Common Core was well created so there is a chance that things will get fixed. Though I agree with his assessment of the problems that have led to this math crises, I don’t agree that the Common Core were appropriately developed. His defense of the Common Core includes a lengthy description on page 7 of the new way to teach that -1*-1=+1 for 7th graders, and it will make your head spin if you read it. I agree that IF Common Core was good, that would be a good, and necessary, start. For the same reason that ‘math’ is in the state it is in, the Common Core fell victim to the same problems — too many people with too many favorite topics.

16 Responses

  1. Hi Gary, I’m a 2011 CM teaching math in a high school with very low math achievement (even by TFA standards). I agree with your assessment that most curricula teach mathematics rather than Mathematics (to use your formalism). However, in your criticism of this fact, I feel as though your background as a Mathematics major may actually limit your perspective. Even among generally “successful” individuals, there are few who, at least initially, appreciate Mathematics for the reasons that you do–the beauty of elegant proofs, the foundations of number theory, etc. As an astronomy and physics major, I certainly didn’t need to understand integrals down to a fundamental level in order to apply them in answering very important questions about the universe. Econ majors don’t need to understand derivatives beyond “mindless algorithms” in order to figure out marginal rates of return for real businesses (though in the process, they might just pick up a conceptual understanding). Social scientists don’t need to understand the details of how statistical analysis software works in order to draw valid and valuable conclusions from data (though it would probably help a little).

    In short, I think it’s rather unfair to say that lower-case mathematics is a waste of time. Certainly, the more rigorous and conceptual, the better, but that doesn’t mean it’s impossible (or even particularly difficult) to show students that the more mechanical stuff they’re learning can be very useful and relevant in the real world, and not just for buying groceries or paying bills.

    Perhaps I’m just arguing semantics?

  2. Cal

    Most students care neither about Math or math, and any Math teacher who spent his time trying to get students to find Math beautiful and fascinating will spend a wasted and unhappy life.

    “If schools of Ed get on board training math teachers properly,”

    There’s no way to “train” math teachers. They either have the knowledge or they don’t, and that assessment is derived from tests, not ed school classes. What most Math folks mean when they say this is that they want math teachers indoctrinated into teaching Math instead of math. And, I say this politely, you are nuts if you think you will ever get most math teachers interested in teaching Math. You will get fewer math teachers, and we already have a shortage. So before you go round wantonly changing the profession, some evidence would be useful.

    And for such an evidence-based guy, you know full well that you have no evidence that teaching Math instead of math would improve instruction or results. You teach really, really smart kids now, as I understand it. Cast your mind back to your early days and confront the reality that many of those kids would still not be interested–and the math would even be further beyond them than the skills-based work we try to teach them now.

    ” If the mathematical community invests some brain power on the problem.”

    Why do we want them involved? We aren’t turning students into mathematicians. Maybe 3% of the population has both the capacity and interest for that. We could maybe ask engineers, mechanics, and plumbers, but mathematicians are the last people whose opinion we should care about for other than the most advanced students. I’m tired of people like Wu (and you, for that matter) sneering at applied math. It’s absurd, elitist, and totally unrealistic.

    “‘This implies that proficiency in math is somehow as important as proficiency in reading, and I’m here to say that this is completely absurd. ”

    I am a credentialed math, English, and history teacher, and I’m here to say that giving students a solid ability in problem solving through rules and processes at their level of ability is essential as a signal of that student’s ability. Research routinely finds that students who do well in applied math are stronger than students who have strong verbal skills alone.

    It is slightly less important than reading, but kids *should* have the reading ability they need for a life skill by 8th grade. At that point, yes, applied math is as important, if not more important, than teaching students to analyze Shakespeare or Sophocles, which is what most high school English involves.

    I do agree that teachers should not tell them that they will use math in “real life”. I never do. It’s a lie. However, learning how to apply rules and processes to problems is essential, and that’s a very good description of applied math.

    And the reason math instruction is dismal has nothing to do with what we teach but our determination to ignore the ability of who we teach.

    Ask yourself–whether math or Math is important–do you really think that the ability to do math isn’t a sign of competence and intellectual ability? You really think there’s no difference in intelligence and ability between a kid who grasps algebra and a kid who takes algebra but has never been able to understand fractions or negative numbers? If you do, you’ve been spending too much time in your rarefied middle school and not enough time in a title I high school.

    • Gary Rubinstein

      Those first two quotes were not from me, but from the author of the article I was citing at the end.

      I think some competency in math is a good thing, but to force kids to take 12 years of it is silly. So there would be fewer math teachers, and that would be fine with me.

      No, I’m not advocating that we just teach the ‘beauty’ of math and no skills. I said that I’d cut 40% of the topics, not 100%.

      My odd + odd example was just a taste of the kind of thinking that we need to encourage more. Some topics are better for encouraging this kind of thinking (call it problem solving if you like to) and other topics are not.

  3. Math Teacher

    I’m the kind of math teacher you probably hate. I do not have a degree in math. I have no love for the “beauty of Mathematics.” When students ask “When will I use this skill in my life,” I offer one of the following answers:

    1) You will never use it if you don’t learn it.
    2) I am not a psychic. Neither of us has any idea what you will pursue professionally in 10 years, and you may or may not ever need this knowledge.

    But my favorite answer is…

    We learn math to make our brains strong, in much the same way that we lift weights to make our muscles strong. In daily life, you will never need to lift a dumbbell for 2 sets of 12 reps. You lift the dumbbell so that you have strong muscles. You may never calculate the volume of a sphere in your daily life. But learning how to do so makes your brain more able to learn how to learn things you will need.

    • Gary Rubinstein

      Well, I don’t ‘hate’ any teachers, let alone any math teachers. Your three reasons to ‘when will I use this’ are pretty good stock teacher responses. I used to post a lot of advice to teachers, and there is a power to saying some things that are very teacher-like.

      Also, I do think there is some value in the discipline of math — the memorizing of the times tables and using them to do more complicated calculations. The process of just writing and thinking with patience is something that may build concentration powers, which will be useful later.

      The jury is out on whether the mind really works like a muscle and if learning one skill makes your brain stronger so you can learn another skill more easily. Even if it is not true, it is an OK lie to tell students since it might be a little true.

      As far as the ‘beauty’ of math, I don’t expect everyone, or every math teacher, to share what I feel. Kids don’t need every one of the 12 math teachers they get in school to have this, maybe one or two are sufficient.

      I admit that I think that being a math teacher when you don’t have some love for the beauty of it is a bit depressing. It would be like if I were forced to teach music when I don’t have a love for it. It doesn’t mean I wouldn’t be a good teacher, but I’d imagine that the day would go by pretty slowly.

      But, to repeat, i definitely don’t hate or have any contempt for you. If you do a good job doing what you were hired to do, which is to teach certain basic skills and you are doing that, you should keep up the good work.

  4. Alohagirl

    I really like how you posed the examples of different “types” of math. I am not a math teacher, have avoided math for the last 20 years, and although i did get as far as Calculus have never really enjoyed math. But your first question made me think and consider, and the second led me to start plugging in numbers until i came to the answer – never once did I consider why I got that answer. I imagine it’s like that for the kids. The first one wasn’t painful, it was kind of pleasant, and I fully agree that’s what math should be about. Too bad, with all of our middle schoolers at my school now taking 2 class periods of it at the expense of our reading classes….argh, venting for another day. Thanks for the post.

  5. EB

    Well, here’s one person who loves math because I do find it useful, and doesn’t love Math because it’s too abstract for me. And I scored 760 on the math SAT. I never went beyond Trig (which I like most do not use) but I do use arithmentic, algebra, and geometry in mly daily life and sometimes at work. People think they don’t use that type of math, but the parts of it that you do use, become instinctual, just like reading becomes instinctual.

  6. Cal

    “I admit that I think that being a math teacher when you don’t have some love for the beauty of it is a bit depressing. It would be like if I were forced to teach music when I don’t have a love for it. It doesn’t mean I wouldn’t be a good teacher, but I’d imagine that the day would go by pretty slowly.”

    You’d imagine incorrectly. And insultingly. Which is why Math Teacher thinks you hate people who don’t teach for precisely the same reasons you do.

    In a way, you’re a priest, not a teacher. The teaching is incidental. You’re worshipping at the altar of math, looking for the occasional acolyte.

    Teachers who find that form of priesthood a tad offputting often find more value in helping their students feel competent and increasingly hopeful about their ability to take on challenges, without creating worshippers at the great altar of Math (or whatever the subject matter of the priesthood).

    And if you think that description of your motives is insulting, then snap, as the kids say.

  7. Andrew

    Cal,

    If one doesn’t see the beauty in Mathematics (as Gary put it), you really don’t get Mathematics, and probably have no business teaching it past an elementary level. That being said…

    Gary,

    You are just totally right. I’m not sure why studying economics, philosophy or biology is any less important than studying ‘Mathematics’. I studied ‘Mathematics’ in college, got certified to teach, and have been doing ‘Mathematics’ it on/off for most of my computer programming career, but that is specialized work and specialized training that 98% of people never need (and even most computer programmers rarely need). People can learn what they need on the job if need be — not everything needs to be learned in a classroom. I simply can’t tell my children that Mathematics is more important than, say, clear writing. It’s not.

    Students need to be freed to follow THEIR interests. The current system is coercive. Why should anyone be forced to sit in a room for an hour every workday for a year listening to a teacher talk about a subject that they aren’t interested in. It’s a disgrace. The model of the self-contained classroom with captive students is a huge problem that nobody seems to want to acknowledge.

    • Dylan

      I agree Andrew. I’d like to know if there’s a term for that educational policy of letting students decide their curriculum starting at adolescence. It needs a name to build support.

      Early education should be and already is designed for the subjects that are needed for every job, which are reading, writing, and arithmetic. What it should do but currently doesn’t do is to introduce kids to the jobs people do in the world, so that students have enough information to develop a job goal by adolescence. Once students have a job goal, they should go to a magnet program with a curriculum designed to directly reach that job goal and where they would learn from people capable of the job they want.

      • Andrew

        Dylan,

        Search for “Unschooling.” Also take a look at the Sudbury Valley Model Schools.

        I have to disagree with the job goal thing. I think people should learn to satisfy their interests and let jobs come where they may. The problem for most is that more and more careers require schooling even if it really shouldn’t be required. Engineers used to be able to apprentice and then take a tests to get the P.E. stamp. No more. Heck, even the people who cut your hair have to go to school. It makes it more difficult and expensive to change careers – another example of shifting costs from employers to employees.

        More apprenticeships, please.

  8. Andrew

    EB,

    The point is that the essence of Mathematics is the abstraction of ideas and the beauty in that abstraction. Many have trouble with that abstraction, but that’s what Mathematics is. The rest is essentially formula and computation — what Gary is calling ‘math’. If you haven’t done Mathematics beyond trig, perhaps you miss to a large extent what Mathematics really is. It’s OK if we decide that teaching ‘math’ is important in it’s own right, but it is a different thing that ‘Mathematics’.

  9. Dylan

    I didn’t retain much beyond basic arithmetic and algebra from my high school education. I had to teach myself everything again when I finally fell in love with engineering and needed to learn math. I didn’t need any instructors, just decent self-instruction books whose existence I only discovered by browsing bookstores, not because of any teachers. I suggest that every ounce of education for adolescents without job goals should be about getting them interested in a job goal, whether flipping burgers or building space shuttles. Then the system can train them for their interest.

    For preadolescents, forced arithmetic instructions makes sense as it is needed in every job and that age group will generally learn and be interested simply because the teacher wants them to.

  10. Sarah

    Thank you Gary, my most recent reading adventure (Isis Unveiled) led me to further investigate the difference between math and Mathematics. The problems you presented made me laugh, especially the second one. The first one I figured out the principle rather quickly and the second one I laughed at, said who cares, and then scrolled to see you felt the same way. I agree with you in that it is more important to understand why these rules are the rules. It is my belief that understanding the rules make solving problems that much easier by simply determining the answer combinations of the rules. I also enjoyed giving thought as to why those rules make sense and how they work. I am sure as I learn more and continue my reading, the principles and rules of Mathematics will become increasingly clearer as to the importance of them in a deeper sense and on a universal level, as well as when I start to approach reading and understanding Pythagoras. Again, thank you for your informative and simple explanations.

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