"Learning mathematics forces one to learn how to think very logically and to solve problems using that skill. It also teaches one to be precise in thoughts and words. Practice doing that is obviously very useful in many different areas of life." - The Math ForumLet's make a bold claim. I'm going to claim that there is no evidence that learning math makes you better at other things unless those other things are
- math
- things that use math, (i.e. they're just math)
I'd love for someone to convince me that I'm wrong on this. Do mathematical habits or skills transfer to other contexts? Drop in the comments if you've got a good argument or some evidence.
Appendix:
Here are some other folks who make claims about the usefulness of math.
Appendix:
Here are some other folks who make claims about the usefulness of math.
"What of the claim that mathematics sharpens our minds and makes us more intellectually adept as individuals and a citizen body? It’s true that mathematics requires mental exertion. But there’s no evidence that being able to prove (x² + y²)² = (x² - y²)² + (2xy)² leads to more credible political opinions or social analysis." -Andrew Hacker, "Is Algebra Necessary?"
The argument for algebra rests on the transfer from math to other areas of life, something that has never been proven despite the claims of people such as University of Virginia cognitive scientist Daniel Willingham. -- Roger Schank, "No Algebra Isn't Necessary"
I think I have personal evidence of the converse. Being good at arguing (like lawyers) makes you good at math. Maybe it goes both ways.
ReplyDeleteI believe anyone who's skilled at mathematical reasoning can master computer programming pretty easily.
Anything that requires symbolic reasoning will benefit from math training. Hmm... I would bet any scientific discovery, even if it didn't need obvious math, would come more easily to someone skilled at mathematical reasoning.
I have no good evidence for any of them, though.
There's a difference between algebra and math. Being able to logically organize your thoughts and make a rational argument is not a skill that is solely taught in math, but it is certainly an important foundation of any quality mathematical education.
ReplyDeleteOne other things I've learned is that I often am surprised by what skills I need to handle a given problem. Something that doesn't seem mathematical turns out to be (of course, I love math, so maybe I look for it) - the same with writing, speaking, or any other critical thinking skill. So taking mathematical reasoning out of people's problem-solving arsenal, just because they won't often be faced with algebra problems, only restricts their ability to handle complex situations.
As I tell my students, every problem in life is a word problem. No one will ever walk up to you and say "7x+4=12, what's x?" They'll say "how many widgets can we buy this month?" or "show me how your proposal affects the budget." They'll have to make a case for something - a raise, a benefit, something like that - and the ability to do math - be comfortable with numbers, not be afraid of situations where a calculation is necessary, etc - will help them throughout their lives.
Sure. But "How many widgets" is a question about quantity, and so it's a mathematical question.
DeleteWhat I'm denying is that when somebody asks "Who's guilty in this case?" that there are some mathematical skills or habits that, if you know how to think mathematically, will bleed over into your ability to reason legally (or whatever).
Agreed! But I also think that making connections, reasoning, drawing conclusions, And representing ideas and solutions in multiple ways are the most important byproducts of taking any math class. Unless you're interested in pursuing a career in mathematics or a math-related field,the math itself is not as important as the other byproducts of studying it. And, those byproducts are most possible through excellent teaching!
ReplyDeleteJohanna, I'm claiming that there are no non-mathematical by-products of learn math. I'd argue that learning how to represent ideas in multiple ways is really learning how to represent mathematical ideas in multiple ways. What I'd love to see is evidence that learning how to represent ideas in multiple ways in math class helps you get better at representing historical, musical, or advertising ideas in multiple ways.
DeleteSo then my question becomes, "How does it not?" And I'm reminded of a saying I use a lot in my classes, which is, "There is always more than one way." I would argue that you would have a difficult time deciding what I taught based on that statement alone.
ReplyDelete"The relationship between mathematics preparation and conceptual learning gains in physics: A possible “hidden variable” in diagnostic pretest scores" -
ReplyDeletehttp://ajp.aapt.org/resource/1/ajpias/v70/i12/p1259_s1
Isn't that just because learning physics is easier if you have a meaningful mathematical framework to draw from, because physics involves a lot of math?
DeleteI'd file this under "things that use math."
Maybe. But when physics educators refer to "conceptual" physics they usually mean an understanding that does not require math or computation. Something like understanding that uniform motion does not require a force.
DeleteBut results like those presented in the paper indicate that math preparation is important for this "conceptual" understanding. Now it's possible this is direct evidence against your claim. Or, it's possible this just shows that even "conceptual" physics is fundamentally mathematical.
I've often wondered about the latter but nevertheless this cordoning-off of "concepts" is common in the field.
My guess was actually that the physics courses themselves were heavily mathematical, and so a student's comprehension of the conceptual stuff was greater if they came in with math, because then they could understand more of what's going on in class. They could also connect the material to more things. There would be less of a cognitive burden, etc.
DeleteI just looked at the paper again and I think you are correct (that the course is itself heavily mathematical).
DeleteThis suggests an obvious follow-up study: Do this in a truly "conceptual" physics course (we teach such things) that does not use math, but look to see if math preparation predicts performance (while trying to control for overall GPA or something)
Sue and Johanna,
ReplyDeleteI love the pushback that you're giving. I don't really have any way to respond since I'm not sitting on any studies. What I do know is that transfer is tricky and knotty. We know that transfer doesn't even occur between classroom math and everyday math without serious, explicit connections made during instruction time. So why should we think that there is a transfer of skills from math to other domains?
My inadequate response: the burden of proof is on the claim that there is transfer.
Have you taught any other subjects other than math? If you have, you could certainly draw from your experience to serve as a study. I'm entering my 19th year of teaching and my 7th year of teaching middle school math. I've seen much transfer of mathematical thinking and "byproducts" of mathematical thinking to other subject areas in a multitude of elementary grade levels. Good elementary teachers show the connections between the thinking used to "learn" about various subject areas and encourage their students to draw upon these various ways to think in many kinds of situations. I refer to middle school as elementary school as well, btw. Mathematical thinking is not reserved for mathematics alone.
DeleteFrom math, I learned that if you look at a situation and you know what you want to get to (the answer to a question, if it's a math problem), but you just can't see the connection and it really, honestly, looks ***ing impossible, that you can ask yourself "what *do* I know about" either the question or the answer, and then see if something about that gets you to something closer to the problem... and you end up looking like you're brilliant when all you did was mess around in the right direction.
ReplyDeleteI also learned that if a problem looks unsolvable (not necessarily math), to ask "okay, but what do I *wish* the problem were? Now, can I do something to make this problem that problem?"
Big howevers: a person could learn that stuff in a different context, and ... most people didn't and don't learn that from math. (On the third hand, prob'ly anything that could really apply completely outside of a math context could thus also be learned there. I think math really lends itself to that 'how to rework a problem' strategy better than most things.)
Math teacher and former lawyer here. Math ways of thinking definitely transfered to the study of the law. I imagine the same skills might transfer to philosophy, or any subject where it's very important to define your terms very carefully and use them in a system that's close to formal logic, even if it does use more plain language.
ReplyDeleteMore pertinent to legal skills: learnign to peruse a mess of information and suck out what's useful and relevant for a specific problem.
ReplyDeleteThink about those people you know who DON'T "do math," or haven't studied math (and I'll include science here.) Do you find their lack of reasoning ability frustrating? I know I do! "Vaccines cause autism, because Jenny McCarthy cured her son." "I really like that politician. He's good looking, and my husband says our taxes will go up if the other guy gets in." "We should implement this (totally unproven) new educational pedagogy because it's really popular right now." I find those of us in the math/science department have a tough time with outsiders. Yep. Being on a committee with members of Guidance or English or the art department? Bad stuff. I sometimes wonder how they get through the day.
ReplyDeleteLove it, Susan! So true! :)
DeleteI totally agree with you, and it drives me crazy. But those people did "learn" math in school, and it apparently didn't do them any good. If they'd learned logic in the context of what they enjoy and are good at, (such as writing) wouldn't they be a lot better off?
DeleteI love all the disagreement here. Thank you all so much.
ReplyDeleteSince we're talking about careful reasoning here, let's be clear about what would and would not be evidence of math skills or reasoning bleeding into non-math reasoning.
The fact (if it's true) that people who study math tend to be better all-around reasoners would NOT constitute evidence that math helps general reasoning skills. It could very well be that people who study math are better all-around reasoners while actually studying math doesn't do much for your general reasoning ability.
We also can't use anecdotes about student growth without being careful. That's because our students are both learning math and learning reasoning skills as they progress through school. The danger is that we'd mistakenly look at that growth and infer a causal connection when there isn't any.
The kind of evidence that we'd need -- either for or against my claim -- may not be out there. But what we'd really want is a study that isolates the impact that learning math might have on general reasoning, taking into account where a person is starting and where we would have expected them to get without learning math.
Have you taught any other subjects other than math?
DeleteComputer programming.
DeleteA math-related subject, to be sure! :) I think I'm starting to understand where you're coming from, however. (Tell me if I'm way off base!) I believe you're suggesting that all of the byproducts of studying math could be learned WITHOUT ever having studied math--is that right? If that's the case, I agree with you (as I did in my first post).
DeleteOkay, for argument's sake, let's take the premise as stated: "I'm going to claim that there is no evidence that learning math makes you better at other things unless those other things are
ReplyDeletemath
things that use math, (i.e. they're just math"
Now let's use Hacker's argument, as quoted: "But there’s no evidence that being able to prove (x² + y²)² = (x² - y²)² + (2xy)² leads to more credible political opinions or social analysis." -Andrew Hacker, "Is Algebra Necessary?"
I submit that Hacker's article would have been a whole lot MORE convincing if, indeed, (x² + y²)² = (x² - y²)² + (2xy)². But it's not. In other words, if he had studied more math (or perhaps if he had studied math better), he would be a better writer.
A little tongue-in-cheek, perhaps, but that non-identity has been bothering me since I read the column.
It's late and I'm very tired, but I think that equation is correct, Susan.
DeleteIt is correct...
DeleteDarn it! I've stared at that all summer and stewed over it - I forgot to square the 2 in the (2xy)^2. I need to shake out a few cobwebs before next Monday! Old age is setting in fast here. (Hanging head in shame...)
DeleteNo shame, ever, in putting yourself out there! :)
DeleteIt happens to the best of us. The worst is when it happens in front of a class that's itching for a chance to criticize!
DeleteTwo examples, both personal anecdotal evidence to be sure but...
ReplyDeleteA few years ago I did a battery of IQ, personality and critical thinking tests for a job application. The critical thinking test had a section on syllogisms about political issues and other non-math stuff, and I had to complete them and find mistakes in them. Apparently I scored a perfect score, something that testing office had never seen before in the many years they'd had of testing people for highly qualified jobs. But for a math teacher who knows the difference between if and iff almost any syllogism is obvious, isn't it? I also did very well on the other critical thinking parts - and I remember myself using a lot of venn diagrams to represent the information in the question and draw conclusions about it. What did give me trouble were questions asking me for an estimate of how sure ("reasonably sure", "completely sure") we could be of some information. I had a hard time with such vague language, most likely because most of my training is in the crisp language of mathematics.
In daily life, I find myself using mathematical concepts frequently. "My sleep is still bad but at least the derivative is positive. However the state of my sleep might not be a differentiable or even continuous function, so I'm not sure what to expect from tonight." Mostly calculus concepts, but even stuff such as chaos theory provides me with analogies to better understand my life. Very abstract concepts such as those in higher algebra and integration theory constantly inform me that there are possibilities out there, whole generalities which include innumerable other ways of viewing and doing things than what I'm used to. I carry that insight with me in everything I do.
Reasoning-wise, mathematics (and maybe programming, philosophy and law) provides the BEST way to learn not just the laws of logic but also possible applications of these laws. We learn the value of clear definitions, what makes a good definition, that it's possible to create definitions (they're not given by "god" or "nature"), that there are many forms of propositions and their relationships to each other. By proving/disproving things we learn to use direct proof, proof by contradiction, proof by induction, and the value of proof by construction compared to other forms of proof. All of these are then applicable to understanding the reasoning of oneself and other people. You may say that reasoning is not math, but I disagree. Mathematics IS clearly defined axioms combined through the laws of logical reasoning to much more complex ideas. In school we focus on one or just a few manifestations of mathematics, but the possibilities are endless.
One other thing: I highly suspect, but have no research evidence for, that training in mathematics is an important influence in helping students achieve Piaget's "Formal Operational Stage" (if one might be forgiven for referring to Piaget's stages nowadays). In that stage people can handle a whole world of hypothetical situations, which is important in too many ways to list here.
One caveat: I do not think that the mathematical training most students get in school develops the kinds of skills I've mentioned above. Two reasons: the emphasis in school is usually on procedures rather than concepts or reasoning, and student attitude that math is something to survive, rather than engage with, also make it less likely that transfer to other areas of life will be made possible.
Also, couldn't you make the same argument about most subjects taught in high school? I mean, once you've grasped the basics of reading/writing/'rithmetic by about age 10, what's the point? Have I ever needed to parse a sentence or recognize iambic pentameter or discuss the molecular weight of silver or, or, or... Yet I love my math with a passion, and have a greatly enriched life because of it. I didn't know I'd get to this point when I was in 10th grade (I started life as a French major, of all things) so I'm thrilled that I was "forced" to study math until I got to this point!
ReplyDeleteI wasn't forced to study math in high school, I wanted to. (I took two classes my sophomore year.) And the courses I was forced to take in school did damage. Art and music class were uncomfortable, sometimes embarrassing, and I felt like I was terrible at both. I enjoy doodling and drawing (once in a great while), and I love to sing. But it took years to recuperate from the damage done when I was younger.
ReplyDeleteIn my high school in the early 70's, from 10th grade on we got to pick our English mini-courses (switching every 9 weeks). I took Shakespeare, mythology, debate, forensics, college writing (I wrote a stupid paper), speed reading (silly class), a 2 more. If they'd offered science fiction, I would have taken that.
We had lots of choice in high school, and I'm grateful for it.
As much as I value math, it disturbs me that we require students to take so many classes in high school and college. It should be more about offering what interests them.
This! All my art and phys ed classes were damaging. Every one. Most of my English and Social Studies was damaging, but fortunately I didn't let that stop me from reading and writing outside of school. Music wasn't damaging, but it was a waste for me -- when I started playing around on my own I got somewhere.
DeleteMost of the useful, valuable things I know/remember/use, I learned outside of school. And everything I've ever learned, it's because I wanted to. Learning is simply not something you can force.
I've been thinking about this a lot this summer. I agree with SiouxGeonz. Unfortunately, I can't post from my phone, or I would have been here sooner. I wish I could interact with y'all more!
ReplyDeleteLogical thinking, problem solving, and precise explanations are VERY useful in computer programming, theology, scientific research, journalism, engine troubleshooting, law... reasoning about ANY subject.
But why do we assume that you have to learn the skills in math to use them in other fields? Why couldn't you learn them in other fields to use them in math? Or learn them in another field and not learn math if you don't feel the need?
Many many years ago, it was considered important for every educated person to learn rhetoric. The word "logic", unqualified, meant logic in the context of verbal argument. I wonder how many of the statesmen who wrote our constitution could do algebra?
If I were going to require one subject for the purpose of learning logical thinking, it wouldn't be math. It would be logic itself: methods of reasoning and types of fallacies, with students writing and debating about subjects of interest to them.
I've gradually come to the conclusion that only two of the skills taught in school are universally necessary: literacy and numeracy. (By numeracy, I mean understanding of basic math for daily life.)Personally, I learned more about those two outside of school than in school.
Logic is even more necessary than literacy and numeracy, but I've rarely seen it taught effectively in any school setting. If it were, I'd be looking for a new job. (I'm a math/science tutor.)
I think I need to make this a post.
Hacker really chose the wrong example --- (x^2-y^2)^2 + (2xy)^2 = (x^2+y^2)^2 is a highly interesting identity. You can produce all kinds of cool right triangles from it! Use x=2 and y=1 and you get 3^2+4^2=5^2, the Pyth. Thm. for a 3-4-5 right triangle. Use x=3 and y=2 and you get the 5-12-13 triangle. Use x=4 and y=1 and you get the 8-15-17 right triangle. etc. etc. And this of course forces us to the question, "Are all right triangles with integer sides obtainable from this formula?"...
ReplyDeleteWith regard to your question, exploration and forming guesses about what is observed is a ubiquitously useful skill, but nothing comes close to mathematics for the simple and systematic way in which these can be done.
This discussion is kind of depressing me. Maybe we should throw in the towel, resign, and not show up for school next week. Are we really just baby-sitting? I'm actually of the mind that we can't just teach kids what interests them, the same way we have to make them eat their vegetables. Hopefully we're sewing the seeds in at least some of them. There's time to pick and choose when you're "old enough to know better," as my mother always said. (Although I agree, gym was torture for me, too. But now I'm physically active and could probably take down a gym teacher or two, and I can certainly out-crunch my students!!)
ReplyDeleteSusan, I have felt like that in the past (depressed, wondering if I was just babysitting). But I love teaching math. I teach math not for its usefulness, but for its beauty.
DeleteStudents don't need to be forced, but they do need mentors. This is not baby-sitting (unless you take your babysitting gigs very seriously). What do you love learning? Have you taken a course in it, or gotten a teacher or trainer?
I went to a folk music camp years ago. Of course there were no grades, no one took attendance, and no one checked homework. But we practiced between class (I think it was called class...) sessions. I adored it, and practiced a lot, but I was still the slowest student in my penny whistle class. I was able to record all the songs we worked on, and practice after the camp ended. I never believed I could memorize music, but I did it. I play the penny whistle still, and would thank that teacher for changing my life if I knew his name.
A philosophy TA at the University of Michigan helped me learn to write better. She went out to a coffee shop with me and we discussed the topic I would write about. I recorded that conversation, and used it to help me. I wish I knew her name.
A good friend helped me learn to sing. We practiced one phrase at a time of Country Roads, over and over, until I got it.
Can you mentor students in a class they've been forced to take? If they're willing to put the effort in, yes. But the situation is complex in part because they haven't chosen it.
Forcing children to eat their vegetables isn't a good strategy either.
Sue, I was being facetious. I love teaching, and I love my math. I just think it gets into muddy water if we start to ask if everything we learn (or teach) is "useful." Maybe not for everyone, but I do think we should teach it - because we might be the spark for a particular student who falls in love with it, too. And analyzing a spreadsheet, or creating a spreadsheet, may be important stuff even to those kids who don't go into math/science fields. And I do believe kids need to eat their veggies whether they like them or not. They need fiber and vitamins, darn it! You can disguise them, embed them in a sauce, make fun shapes out of them (my kids never fell for that - a cucumber is a cucumber) or find those veggies that they like, but they do need some vegetables. And hopefully, by the time they move out of the house, they opt for healthy foods on their own, at least sometimes. (My son leaves for college on Tuesday, and I'm encouraging him to choose the salad bar at least once in a while!!)
DeleteAnyway, I'll be back teaching mid-level Algebra 2 kids next week, crossing my fingers and hoping they grow a little bit between now and June.
My answers to this question are here:
ReplyDeletehttp://mathmaine.wordpress.com/2010/04/10/broadly-useful-skills/
The history of math is very useful. The concept of zero had to be invented. Also negative numbers,
ReplyDeleteirrational numbers, and imaginary numbers. For many centuries people simply had no concept of these
things. I find this very useful in social analysis; the fact that people can't conceive of something doesn't
mean it doesn't exist. We are prisoners in the thinking of our time.
People might argue that the history of math is not math, but I think actually understanding these concepts
is essential to understanding this idea.
What is an example of an "Other Thing" that you believe math does not make you better at?
ReplyDeleteMath is a way of describing the world/universe around us. It enables us to communicate. I cannot think of something that Math does not apply to.
As Richard Feynman said: http://www.youtube.com/watch?v=ZbFM3rn4ldo It enables us to see more ... can only add, I do not see how it subtracts.
I appreciate so much that you as a teacher want to teach math in a way so that the usefulness of math shines. It is such a welcome contrast against those teachers who tell you to just learn math and see later, what it is good for.
ReplyDeleteThat math is useful for all science is obvious. Why do we have to defend this as mathematicians? Ask the physicist, or anyone using statistics for an empirical science , not me!
But maybe the question was another one: Does algebra form the brain and helps for better thinking? Then I say no! Rather it may get in the way for clear thinking. It is a tool, much like learning to read notes to make music, necessary, but not the core of it.
It may be so that the structural thinking that we base math on, including the abstraction to cover a wider range of applications for our theorems, helps to better understand real life problems. This happens if we are able to formulate a math model for the problem. But my idea is that the math model and the real life problem remain two different things, which we both need to understand.
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