Teaching Math as Obvious Thinking | Make Math Click
Table of Contents
My goal when I teach math
My #1 goal, when I teach a math class, is to convey a way of thinking about math. It’s quite different from what most students have done before, and many of them find it difficult and frustrating. But in the end, many begin to see math the way I do — and I count that as a major success even if they eventually forget details, facts, and techniques.
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 for 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.
3 × 7 = 7 × 3
You can guess that this is the “Commutative Property of Multiplication.” But I call it not obvious.
Seeing why multiplication works
Think about what that statement means. 3 × 7 means three groups with seven items each: 7, 14, 21. 7 × 3 means seven groups with three items each: 3, 6, 9, 12, 15, 18, 21. It came out the same — it worked.
But is it just a coincidence? Will it also work for 6 × 9, 12 × 137, and every other multiplication? Can you think of a reason that makes it obvious that three groups of seven and seven groups of three must produce the same total?
This is not rhetorical. I give the class time to think, and I get a variety of answers. One convincing picture is a grid: three rows of seven squares (7 + 7 + 7) or seven columns of three squares (3 + 3 + 3 + 3 + 3 + 3 + 3). Either way you count, the number of squares is the same. That picture convinces me it will also work for 12 × 137: I don’t have to count it, and I don’t have to take anyone’s word for it. It’s obvious.
What I mean by “obvious”
I’m not using “obvious” the way most people do. When I say something is obvious, I do not mean “anyone would figure it out quickly.” Sometimes I spend hours trying to make something obvious to myself. If I succeed, I get to the point where I can say, “Of course — it couldn’t possibly be any other way.” That, to me, is what math is all about.
Math compared to language — why the analogy misleads
Math is often compared to a foreign language. I’ve even heard math teachers say that “Math is just another language.” I think this analogy is misleading.
Math has a language. Certain words (“root”) are used differently than in everyday speech, and other words (“polynomial”) are mostly found in math. But these words express the math; they are not the math itself. If we called a “root” a “fizbot,” the math (“you can’t take the square fizbot of a negative number”) would still be the same.
What math shares with language is that you must memorize rules for how things fit together: “Je vais, tu vas, il va…” and “negative b plus or minus the square root of b-squared minus…” Apply these rules exactly or you’ll go in the wrong direction. But here’s the key difference:
If you ask a French teacher, “Why does it go Je vais, tu vas, il va?” the answer is usually historical. It’s not meant to “make sense” in a deeper way; it’s just how the language evolved.
In math, on the other hand, there is a reason behind statements. There’s a reason you can’t take the square root of a negative number; a reason why 12 ÷ 4 is smaller than 12 while 12 ÷ (1/4) is larger than 12. That reason goes deeper than “when we divide by a fraction, we flip and multiply.” You can explore what division means (hint: think pizza) until you say, “Of course 12 ÷ (1/4) has to be 48 — it couldn’t be anything else.”
Rules versus reasons
To put it another way: if all English speakers tomorrow decided that “thurple” meant “lunch,” they’d be correct by definition because they’d agree. But if all mathematicians decided that 37 equals 73, they’d be wrong. Engineers and structures rely on mathematical facts; no one gets to decide them. Someone discovered them, and you can discover them too.
Why many students resist thinking
So why don’t students get that? Why do many cry out, “Just tell us the answer!” when I suggest they figure something out?
For the most part, I blame early schooling. Very few people go into elementary teaching because they love math. They go because they love children, telling stories, playing games, or because it was a stable job.
The teacher learns rules from a textbook and teaches them from the book. If a student asks, “Why do we need a common denominator to add fractions but not to multiply them?” the teacher may have no clue. Rather than risk embarrassment, she may say, “Because that’s how we do it.”
Students learn the lesson: don’t try to think — just learn rules. Math becomes glorified long division: divide, multiply, subtract, bring numbers down; divide, multiply, subtract, bring numbers down; and someday in high school we’ll do pages and pages of dividing, multiplying, subtracting, and bringing numbers down.
Is it any wonder they leave thinking math is pointless and boring?
What I hope they remember
In five years my current Common Core students won’t remember the laws of logarithms, and my calculus students will probably forget the quotient rule. I don’t mind. If they need those things later, they can look them up.
What I do hope they remember — and many tell me they do — is that math is not what they used to think. It’s not merely a foreign language, a set of rules to memorize, or glorified long division. Math is the most perfect elaboration of common sense. It builds on reasoning that we all have in our brains. Whenever anyone tells you “This is the way it is” in math, you have the right — and the obligation — to ask why, and keep asking until you can say, “Of course. Now that I see it this way, it’s obvious.”
Further reading
Want to read more? Here are a couple of excellent resources. Their details differ, but the big picture aligns with my message: it’s more important to get students actively engaged in formulating math and applying their own thinking than to have them follow procedures to get the right answer.
- Lockhart’s Lament — https://www.maa.org/external_archive/devlin/LockhartsLament.pdf. Longer than this essay and highly recommended; read the opening “musician’s nightmare” section if you don’t read the whole piece.
- Dan Meyer’s TEDx talk — http://www.youtube.com/watch?v=BlvKWEvKSi8. I’ve shown this to my students.
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.
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.
Maybe I did not make my point firmly enough: Do not blame the teachers. They work hard.
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.
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.
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.
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.
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.
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).
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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.
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.
the jab at elementary teachers was very uncalled forShould "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?
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.
"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.
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.
Thanks Kenny! Very nice!
Excellent and enjoyable read. Thank you.
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.
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.
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.