math obvious

Teaching Math and the Obvious

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My #1 goal, when I teach a math class, is to convey a certain way of thinking about math. It’s quite different from what my students have done before, and many of them find it difficult and frustrating. But in the end, many of them start to see math the way I do–and I count that as a major success even if they eventually forget all the details, facts, and techniques we learned along the way.

I begin on Day 1 by writing this on the board:

3 + 7 = 7 + 3

This is called the “Commutative Property of Addition,” but I have another word. I call it: obvious. If you have three apples and I have seven…or, if you have seven apples and I have three…the total we have together is the same either way, isn’t it?

Now, I put something else on the board:

3 × 7 = 7 × 3

You can guess that this is the “Commutative Property of Multiplication.” But I call it: not obvious.

Let’s think about what that’s saying. 3×7 means you have three groups, each of which has seven items. Let’s count that out: 7, 14, 21.

7×3 means seven groups of three each. Let’s count that out: 3, 6, 9, 12, 15, 18…21. Hey, it came out the same. It worked!

But is it just a coincidence? Will it also work for 6×9, and for 12×137, and for every other possible multiplication we could do? What I’m asking is, can you think of a reason that will make it obvious that these two things–three groups of seven, and seven groups of three–had to come out the same?

This is not a rhetorical question. I give the class some time to think about it, and I get a variety of answers. The simplest explanation I know is this picture:

3-by-7
Can you see it? You can look at this as three groups (rows) with seven squares each: 7+7+7. Or, you can look at it as seven groups (columns) with three squares each: 3+3+3+3+3+3+3. Either way, you get the same number of squares. That picture convinces me that it will also work for 12×137: I don’t have to count it out, and I don’t have to take anyone’s word for it either. It’s…well, obvious.

So I’m not using the word “obvious” the way that most people do. When I say something is “obvious” I do not mean “anyone would figure it out quickly.” Sometimes I spend hours and hours trying to make something obvious to myself. But if I succeed, I eventually get to the point where I can “Well, of course it’s that way. It couldn’t possibly be any other way.” That, to me, is what math is all about.

 

Math is often compared to a foreign language. I’ve even heard math teachers say “math is just another language.” I think this is a very misleading analogy.

Math has a language. Certain words (“root”) are used very differently from how they are conventionally used, and other words (“polynomial”) are never used outside of math. But these words express the math: they are not the math itself. If we started saying “fizbot” instead of root, the math (“you can’t take the square fizbot of a negative number”) would be the same.

What math has in common with a foreign language is that you have to memorize rules of how things fit together. “Je vais, tu vas, il va…” “Negative b plus or minus the square root of b-squared minus…” Apply these rules scrupulously or you will go in the wrong direction.

But here’s the key difference. If you ask your French teacher “Why does it go Je vais, tu vas, il va?” the only answer you might expect is historical (“Here’s how it evolved from the Latin roots”). You’re not looking for “why it makes sense” because it isn’t really supposed to. It’s just how they talk in France.

In math, on the other hand, there is a reason why you can’t take the square root of a negative number. There is a reason why 12÷4 is smaller than 12, but 12÷(1/4) is bigger than 12. And that reason has to go deeper than saying “When we divide by a fraction, we flip-and-multiply.” It has to go to the heart of what division means, until you say “Well of course 12÷(1/4) has to be 48, and nothing but 48 would make sense. It’s obvious!” (Hint: what does this problem mean in terms of pizza?)

To put it another way: if all the English speakers in the world decided tomorrow that we would use the word “thurple” to mean “lunch,” they would all be correct–by definition they would be correct, simply because they agreed on it–and the world would go on just as before. But if all the mathematicians in the world decided that 37 was the same thing as 73, they would all, unanimously, be wrong. If they convinced the engineers, bridges would fall down. No one decided that there is no “Commutative Property of Exponents”: someone figured it out. You can figure it out too.

 

So why don’t people get that? Why do my students look terrified and cry out “Just tell us the answer!” when I suggest they figure something out?

For the most part, I blame their elementary school teachers. Very few people go into elementary school teaching because they love math. They go because they love children, they love playing games and telling stories, or (in some cases) it was the only professional job they could possibly get and keep.

So the teacher learns the rules of math from the book, and teaches them from the book. And if a student asks “Why do we need a common denominator when we add fractions, but not when we multiply them,” the teacher has no clue. Rather than risk embarrassment in front of the class, she glowers and says “Because that’s how we do it.”

It doesn’t take long for the students to get the idea: don’t try to think, just learn the rules. Or, to put it another way, all of math is glorified long division. We divide, multiply, subtract, bring down, divide, multiply, subtract, bring down, and some day when we’re in high school we’ll do pages and pages of dividing, multiplying, subtracting, bring downing.

Is it any wonder that they come out thinking math is pointless and boring?

 

In five years, my current crop of Common Core students will not remember the laws of logarithms, and my Calculus students will not remember the quotient rule. I don’t mind that a bit. If they ever need those things, they can look them up.

What I do hope they remember–and many of them have told me that they do–is that math is not at all what they used to think it was. Math is not a foreign language, or a set of rules that you learn and apply, or glorified long division. Math is the most perfect elaboration of common sense. It has no rules except the rules that we were all born with, built into our brains. And any time anyone tells you “This is the way it is” in math, you have the right–even the obligation–to ask why, and to keep asking why. You’re not done when you can say “I can do it now.” You’re done when you say “Of course. Now that I see it this way…it’s obvious.”

 

Want to read more? Here are a couple of links from excellent educators. At the level of details, their messages are quite different from mine (and from each other’s). But in terms of the big pictures, I think we all have the same core message: it’s more important to get students actively engaged in formulating the math, and applying their own thinking, than working through given processes to find a right answer.


Kenny Felder

http://www.felderbooks.com

20 replies
  1. RUber
    RUber says:

    When we talk about building a "number sense" in students, I think this is exactly what we are referring to, the common sense of numbers–making certain things obvious by teaching our children to put numbers in a context that makes sense to them–just like Siri does when you ask her what zero divided by zero is. Nice article.

  2. Amrator
    Amrator says:

    Fantastic article, Kenny Felder! It is so upsetting that mathematics isn't taught intuitively but rather it is taught through rote learning. I love math because I like puzzles. Whenever I'm given a new math concept or problem, I try to look at it as a puzzle. Where do the numbers and/or symbols go and why do they go there? That's one way math could be taught.

  3. Dr. Courtney
    Dr. Courtney says:

    We (Air Force Academy Math Department) would always hear from the engineering departments when upper level students were no longer adept at computing derivatives and applying log properties.There is a truth to what you are saying, but it needs to be expanded more broadly to quantitative problem solving.  My goal was to impart quantitative problem solving confidence and prowess.  If they don't have those (or lose those later), they can't look it up like simple mechanical steps in the process.

  4. jerromyjon
    jerromyjon says:

    Very well stated! I was blessed with the insight to "figure" this out at a very young age, and I know many adults for whom this is still not "obvious". I believe one's "faith" and ability to use math as the powerful tool that it is has very broad implications throughout the entire human intellect.

  5. Fooality
    Fooality says:

    "So I’m not using the word “obvious” the way that most people do. When I say something is “obvious” I do not mean “anyone would figure it out quickly.” Sometimes I spend hours and hours trying to make something obvious to myself. But if I succeed, I eventually get to the point where I can “Well, of course it’s that way. It couldn’t possibly be any other way.” That, to me, is what math is all about."That's a really good point. I used to tutor math, and I'd see people almost get embarrassed when they'd finally see through a problem to a simple solution. It was as if they felt that since the solution was simple, they were silly for struggling for so long, they should have seen it right away. Of course, in reality its as you say: The problem is complex until its really solved/understood, at which point it always becomes simple in some way.

  6. snowman_
    snowman_ says:

    Good article, but the jab at elementary teachers was very uncalled for…I'd say it (lack of interest in learning the why behind things) has more to do with lack of parent involvement in a child's early schooling than it does the teachers. Your attitude towards elementary teachers is rude and insulting.

  7. Bystander
    Bystander says:
    snowman_

    the jab at elementary teachers was very uncalled for

    Should "Mom and Dad" turn off the tube and attend school board meetings? Yes.
    Should kids be held a little more accountable? Yes.
    Are there motivated, competent teachers out there? Yes.
    Are teachers as a group motivated and competent?

  8. symbolipoint
    symbolipoint says:
    Bystander

    Should "Mom and Dad" turn off the tube and attend school board meetings? Yes.
    Should kids be held a little more accountable? Yes.
    Are there motivated, competent teachers out there? Yes.
    Are teachers as a group motivated and competent?


    They (those teachers) are over-worked.

    Some kids take longer to reach the stage that some things in arithmetic or algebra become obvious. What I saw as a student, was that there were several SMART kids who had trouble with what is obvious.

  9. MidgetDwarf
    MidgetDwarf says:
    symbolipoint


    They (those teachers) are over-worked.

    Some kids take longer to reach the stage that some things in arithmetic or algebra become obvious. What I saw as a student, was that there were several SMART kids who had trouble with what is obvious.

    Being overworked should not be an excuse to not execute one's job at full capacity all the time.. Let's be honest here. Are the parents responsible? Yes. Should mom and dad be more involved? Yes.

    It comes down to economics and parents level of education. People in low- income areas: have less money for tutoring services, lack the education themselves, or may not see the value of education. A recent google search of tutoring jobs in my state of California revealed some amazing results. In the Malibu, Brentwood, Palos Verdes, or even middle class areas, such as, Culver City, people are paying at minimum 60,000 yearly salary for a personal teacher for their kids. I have seen postings for as much as 120,000. Won't the kids whose parents can afford to higher a private instructor have the greatest chance of success? Or even middle class households were at least one parent completed the educational system, will have a the know how to get their children to college.

    Another factor are the textbooks used in the American public school system. Many inner city kids do not have books. Most do not get math books until 6 or 7th grade. Therefore, a child has not practiced how to read mathematical books until the age of 12 or 13 in most cases. Kids are given atrocious work sheets by their teacher. Even if the kids received textbooks, they are of inferior quality. Richard Feynman mentions, "In Surely You're Joking," how the schools adopt the textbooks. Im sure everyone is familiar with this book on physics forum so I will cut that part short. Books are extremely dumbed down.

    Language barrier of immigrant groups also plays a factor in students success. In the USA hispanic are the 2nd largest population. Caucasians are first place and third are African-Americans. With the influx of asian and middle-eastern immigration groups, this is becoming a greater problem. A lot of immigrant parents, both the wife and father, work 2 jobs to support their family structure. Oftentimes, children rarely see their parents thoughtout the day. Kids are usually taken care of by an older brother or sister.

    The most important in my opinion are the teachers. Many teachers had dreams of grandeur while attending college. Once completing their education and getting a degree, most but not all, failed to live out their dreams. Instead, the only jobs they could get was teaching.

    I have a friend who recently got her masters in English about 2 yrs ago. She got a degree that she could not find employment with. Her highest math course taken was statistics.
    We took pre-calculus together. The reason she took pre-calculus was that she wanted to go into the medical field and she lacked the science/math for the MCAT etc.
    She could not hang dropped the course. Today she teaches algebra to students. She is not qualified to teach children, but the state of California says she is. Instead of prepping her lesson plan/ lectures she goes bar hopping….

    I know many other stories like this.

  10. symbolipoint
    symbolipoint says:

    MidgetDwarf, you give enough information that suggests both that teachers are overworked, and that some of them are less qualified than necessary to do some parts of their jobs. Maybe both things occur. Part of the problem is how the determination of who should teach what is made. Also, administrators have ways of ignoring rules (but they don't always want to).

  11. snowman_
    snowman_ says:

    MidgetDwarf, even if you work full-time, if you care about your child's education then you'll take part in it…there are more than 40-50 hours in a week. You don't need a tutoring service to help your child with their math/reading homework. A lot of people just have poor time management skills and then blame it on teachers. You get out of education what you put into it. Teachers go above and beyond a 40 hour week with many working beyond the "clock" with no compensation. In districts with very little money the teachers are even expected to buy all the school supplies for the students…when you get paid so little it's rough buying 30 composition notebooks, folders, pens and pencils etc.. It's insulting to imply that teachers don't put all their effort into their job…I've met very few teachers who are that lazy, many are incredibly passionate and put in an extreme amount of effort.

    In my experience it's not that the parents are too busy, many simply don't get a crap…you get a lot of kids living with their grandparents (because parents ditched out) who most of the time participate very little in the child's education.

    Also if you get a degree in elementary education you actually do have to take a math class which explains all the "whys" behind things…it also covers math topics at a very intuitive level, definitely not memorization or "do this because that's the way we do it".

    Just because you know a handful of slackers doesn't mean that elementary education in general is like that. The teacher is the last thing in the chain of issues that is a child's early education…the administration policies (which includes stricter hiring policies), parent involvement, and district funding are much bigger issues.

  12. MidgetDwarf
    MidgetDwarf says:
    snowman_

    MidgetDwarf, even if you work full-time, if you care about your child's education then you'll take part in it…there are more than 40-50 hours in a week. You don't need a tutoring service to help your child with their math/reading homework. A lot of people just have poor time management skills and then blame it on teachers. You get out of education what you put into it. Teachers go above and beyond a 40 hour week with many working beyond the "clock" with no compensation. In districts with very little money the teachers are even expected to buy all the school supplies for the students…when you get paid so little it's rough buying 30 composition notebooks, folders, pens and pencils etc.. It's insulting to imply that teachers don't put all their effort into their job…I've met very few teachers who are that lazy, many are incredibly passionate and put in an extreme amount of effort.

    In my experience it's not that the parents are too busy, many simply don't get a crap…you get a lot of kids living with their grandparents (because parents ditched out) who most of the time participate very little in the child's education.

    Also if you get a degree in elementary education you actually do have to take a math class which explains all the "whys" behind things…it also covers math topics at a very intuitive level, definitely not memorization or "do this because that's the way we do it".

    Just because you know a handful of slackers doesn't mean that elementary education in general is like that. The teacher is the last thing in the chain of issues that is a child's early education…the administration policies (which includes stricter hiring policies), parent involvement, and district funding are much bigger issues.

    Believe it or not, there are people whose parents still can not read in this day and age. Stop looking at from a first world perspective. It is not just one factor but many, just how symbo pointed out. I had a friend growing up who was from the middle east. His parents brought him here to escape a tyrannical government. His parents could not read or write english. My friend had to repeat grades several times because of the language barier. Or even the immigration problem that is facing the United States. Like I said many parents have to work multiple jobs to provide for their families. You do understand that the 2nd generation American has a better chance at success than the first? Remember it is not the kids fault, rather the adults who placed them on this earth or are running things. What teachers have you seen going above and beyond 40 hour a week? Are you talking about teachers in middle class and afluent areas or poor areas? Like i said, the major problem is economic. People in better areas will have better resources, because they can afford to do so. For every 1 good teacher there are 10 that should not be teaching.

    I do not speak in absolutes. I was playing devil's advocate to start a discussion. You have to be aware that not everyone grew up in the same circumstances you did. You may or may not have had people who took an interest in your education. If you did, then had a blessed child hood.

    You cannot just isolate one factor or factors. Rather, you have to look at the picture as the whole. It is a chicken and egg argument. Improving textbooks and getting them into the hands of younger students is a must. Look at what the Soviet Union did. (lets not get into the whole civil rights violations argument).

    Truth be told. A lot of people do not care about others if it does not effect them directly. Not that they should. It is the sad reality in which we live.

  13. snowman_
    snowman_ says:
    MidgetDwarf

    You cannot just isolate one factor or factors. Rather, you have to look at the picture as the whole. It is a chicken and egg argument. Improving textbooks and getting them into the hands of younger students is a must. Look at what the Soviet Union did. (lets not get into the whole civil rights violations argument).

    Truth be told. A lot of people do not care about others if it does not effect them directly. Not that they should. It is the sad reality in which we live.

    I unfortunately did not have my parents help whatsoever in my primary education. I was at risk of repeating every grade from 6th to 12th, early on it was because my parents were too busy doing things that they shouldn't and then later on it was due to poor habits I had developed. If anything I speak from experience seeing what happens with lack of parent involvement.

    Ok, so parents can't read…then why the hell are they having kids if they can't help facilitate an education? So maybe parents aren't to blame, but why the heck are we placing the blame with the teachers?

    The fault is definitely not with the kids…it's with the parents who brought them into this world when they are unable to fully provide for their children along with other factors. As for the teachers working over 40 hours, it's about every school in my district (not a well off district) that has plenty of teachers doing so.

    I bolded what I did because my point was specifically that there are multiple things at play and the teacher is the least of our concerns. The kids are only with the teacher a few hours each day. I don't care to derail this thread, but I just wanted to point out how stupid the comment on elementary teachers was in the article. If anything it distracts from real issues.

    btw I appreciate the polite reply.

  14. MidgetDwarf
    MidgetDwarf says:
    snowman_

    I unfortunately did not have my parents help whatsoever in my primary education. I was at risk of repeating every grade from 6th to 12th, early on it was because my parents were too busy doing things that they shouldn't and then later on it was due to poor habits I had developed. If anything I speak from experience seeing what happens with lack of parent involvement.

    Ok, so parents can't read…then why the hell are they having kids if they can't help facilitate an education? So maybe parents aren't to blame, but why the heck are we placing the blame with the teachers?

    The fault is definitely not with the kids…it's with the parents who brought them into this world when they are unable to fully provide for their children along with other factors. As for the teachers working over 40 hours, it's about every school in my district (not a well off district) that has plenty of teachers doing so.

    I bolded what I did because my point was specifically that there are multiple things at play and the teacher is the least of our concerns. The kids are only with the teacher a few hours each day. I don't care to derail this thread, but I just wanted to point out how stupid the comment on elementary teachers was in the article. If anything it distracts from real issues.

    btw I appreciate the polite reply.

    It has been proven that American education is a joke. So the schools are a major contributor to lack of student success, but not the sole factor. Are you aware that most schools do not even textbooks that children can take home and look at? Even if they did have textbooks, the books are of such low quality. No major mathematics is done at a young age compared to other nations. Even developing nations have a stronger educational system than the US.

    Are you an educator or have family members that are? You are also aware, that unlike other nations, such as: Japan, Sweeden, Denmark, China, Costa Rica etc. are homogenous populations. Race is also a contributing factor in America.

    Are you American? Not naturalized American but actually born and raised in the US.

  15. snowman_
    snowman_ says:
    MidgetDwarf

    It has been proven that American education is a joke. So the schools are a major contributor to lack of student success, but not the sole factor. Are you aware that most schools do not even textbooks that children can take home and look at? Even if they did have textbooks, the books are of such low quality. No major mathematics is done at a young age compared to other nations. Even developing nations have a stronger educational system than the US.

    Are you an educator or have family members that are? You are also aware, that unlike other nations, such as: Japan, Sweeden, Denmark, China, Costa Rica etc. are homogenous populations. Race is also a contributing factor in America.

    Are you American? Not naturalized American but actually born and raised in the US.

    I agree with what you're saying. The article however specifically called out the teachers as a main issue, not the school in general. From the article "For the most part, I blame their elementary school teachers. " That's what I was commenting on, I do realize the school system is far less than ideal…I just think it's silly to partially single out teachers as the problem.

    I don't think I made myself clear, I apologize. The school system is a joke, but the teachers in my experience do their best to make do with what resources they have available. Blaming the teachers and not the system itself or the parents (who imo carry much more responsibility in a child's education) is just wrong imo.

    Yes I was born and raised in the US.

    Btw I was taught algebra in elementary school in the 3rd grade, non-gifted classes took it in 4th. Standard run of mill public school in an ok district. Then again, teachers don't set curriculums so if I didn't have "advanced math" it certainly wouldn't be the teachers fault.

  16. Guapa
    Guapa says:

    I think every person has it very own way to understand the world that sourrounds him/her. I studied law some years ago, not my desire but by default, due to some circumstances. Along with highschool I took English, then allong with college I took French, German and I could not finish Iitalian due to family and personal circumstances. My native tongue is Spanish. A few years ago,  having some spare time (or Sabbatical), I decided to learn Math as a language, I defined that the numbers is the substance and the other symbols as plus or minus signs are the procedure ordered or requested to be performed, I also defined that de the actual problem is telling me what it wants or  what is need it to be done, by means of observation, analysis and practice I could understand how to solve from simple to complex equations. so after all this learning, I got my certificate on Calculus Derivatives and soon I hope given time I should be taking Integration.  I love being able of  seeing math simbols and understand them in the way I do, it is a great skill or knowledge. And above all it is fun because it is challenging.

  17. 256bits
    256bits says:

    Is it any wonder that they come out thinking math is pointless and boring?Well math is boring if one is not interested in math. I do wonder though if that is the teacher's fault for not making it exciting for all students, or if it is rather an innate feature of a student him/herself. Or does it become boring because the student is not excelling and receiving positive reinforcement by encouragement and marks. Large classrooms do not allow as much time spent per child as necessary for detection of problem areas that a student can have, and they fall behind and this becomes only noticeable at quiz time or examination, when it is surely sometimes too late. Every student does not learn at the same rate, nor ready to learn the same material at the same age. Yet, the standard is to group all students by age and give to them a curriculum to infuse into their brains. One can definitely see the  deviation from the "norm" in physical appearance of young people, so why does it not seem also obvious that the mental maturity and intellect also has a variance amongst the young people. The simplest explanation I know is this picture:A part I especially like. Knowing that, a student can feel "one up" on the subject. Other tiny tidbits placed in strategic locations surely should, I think, feel that the student is being let in on little secrets ( and what student doesn't like secrets, which they will remember better than the main textual explanations ) of math.

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