Poll: How much calculation work should be taught in schools?
13 comments
My elementary school had an interesting practice here, starting at the end of the 6th grade you could take a calculator drivers license test, which once passed allowed you to use your calculator on any assignment. The test had two components, the first required you to demonstrate an ability to do estimation, that you had a general understanding of the scale of numbers following various operations, and the second required you to perform the actual calculations.
It's really obvious to me - having dealt with people raised on calculators for several decades now - that if you can't do arithmetic in your head, you don't understand any math.
Calculator kids will type in 10 + 10, get 100, and then assume the answer is 100 because they have no intuitive understanding of the meaning of any of it. It's just abstract operations relating to button pressing.
Where this goes off the rails is the ineffective methods used in many schools, and the frequent methodology changes due to the latest teaching fad, which results in 12th graders practicing addition because they still haven't learned it. This has led some to conclude that they should skip the basics and move into advanced things. No. They should instead spend the very small amount of time it takes to teach the basics properly, and then move into advanced things.
College freshmen are arriving at schools and they can't make change from a dollar after 14 years of classes.
Calculator kids will type in 10 + 10, get 100, and then assume the answer is 100 because they have no intuitive understanding of the meaning of any of it. It's just abstract operations relating to button pressing.
Where this goes off the rails is the ineffective methods used in many schools, and the frequent methodology changes due to the latest teaching fad, which results in 12th graders practicing addition because they still haven't learned it. This has led some to conclude that they should skip the basics and move into advanced things. No. They should instead spend the very small amount of time it takes to teach the basics properly, and then move into advanced things.
College freshmen are arriving at schools and they can't make change from a dollar after 14 years of classes.
10 * 10 you mean?
However I do agree with you - I learned math not only by doing it by hand, but also by brute forcing it into my brain. I memorized my times tables from third through fifth grade until I knew them up to 25. I did the quadratic formula by hand until I knew the first five Pythagorean Triples.
These aren't necessarily the "right" ways to learn math - but it certainly made me more curious to find a deeper meaning of why mathematics works the way it does.
However I do agree with you - I learned math not only by doing it by hand, but also by brute forcing it into my brain. I memorized my times tables from third through fifth grade until I knew them up to 25. I did the quadratic formula by hand until I knew the first five Pythagorean Triples.
These aren't necessarily the "right" ways to learn math - but it certainly made me more curious to find a deeper meaning of why mathematics works the way it does.
I've confirmed my geekiness. I didn't even consider it could have been 10 * 10.. I assumed it was binary! :-)
The point of the example is that the student intends to type in 10+10, believes he typed in 10+10, and when he gets 100 as the answer he assumes that is the answer to 10+10 is 100 because the calculator gave the result and the calculator is not only never wrong, but it is also how we do math in the modern tech era. It doesn't occur to him that it might not be the correct answer because he doesn't know how to do calculation himself, has never drilled on it, and has no intuition for what the numbers or places mean. They are abstract symbols typed into a keyboard which then gives a result that is to be written down. This example is not made up, but happens every day in schools and life.
The reason you asked about 10*10 is because you either know how to do arithmetic in your head, or you have an intuition about numbers and how they can represent sizes and quantities. Which seems really common and normal to those of us who read a site such as HN. It seems really obvious to you what numbers are and what they mean. You might be surprised to know that many high school graduates don't have a working understanding of numbers and quantities that enables them to do simple calculations or detect when they have made grave and obvious errors.
This intuition about numbers is not present in purely calculator raised veal, er, I mean public school students. I taught math at a college some years ago and was astonished to find that the most enrolled math class started with tasks like how to count to 10 and hoped to work the students' way up to multiplication and division in only a semester. A lot of students fail this level of class because by the time you are an adult, if you don't know numbers, you're pretty much permanently disabled mentally on this stuff, in the same way that it is known that children can learn a language and speak it without an accent as a native in a way that adults who learn their first language will never be able to. So it is a really big deal that many grammar schools are not able to convey basic understanding of quantity to students in their time there. By the time they get to high school and schools are still trying to teach arithmetic it is too late.
It's not hard to teach math though and nearly anyone can learn arithmetic pretty quickly. There are many reasons why this doesn't happen for many people in the US. The reasons can vary from place to place even. Some reasons are math teachers who didn't major in math and hate math, people who hate children teaching, fad methods, reliance on calculators, and even too much of an academic focus too early which burns the kids out early. This last one is a big one. Finland has the best academic results on the planet and they don't even try to teach kids to read until age 7 because thy aren't ready for it until then. (http://online.wsj.com/article/SB120425355065601997.html)
For many kids it is better for them to not go to school at all, not even be homeschooled, but just do what they please each day. There are parents doing this as a schooling method and they consistently get substantially better results than public school educated kids. (One example http://en.wikipedia.org/wiki/Sudbury_school) Possibly the solution is to get rid of the schools, or at least what we think of as schools.
The reason you asked about 10*10 is because you either know how to do arithmetic in your head, or you have an intuition about numbers and how they can represent sizes and quantities. Which seems really common and normal to those of us who read a site such as HN. It seems really obvious to you what numbers are and what they mean. You might be surprised to know that many high school graduates don't have a working understanding of numbers and quantities that enables them to do simple calculations or detect when they have made grave and obvious errors.
This intuition about numbers is not present in purely calculator raised veal, er, I mean public school students. I taught math at a college some years ago and was astonished to find that the most enrolled math class started with tasks like how to count to 10 and hoped to work the students' way up to multiplication and division in only a semester. A lot of students fail this level of class because by the time you are an adult, if you don't know numbers, you're pretty much permanently disabled mentally on this stuff, in the same way that it is known that children can learn a language and speak it without an accent as a native in a way that adults who learn their first language will never be able to. So it is a really big deal that many grammar schools are not able to convey basic understanding of quantity to students in their time there. By the time they get to high school and schools are still trying to teach arithmetic it is too late.
It's not hard to teach math though and nearly anyone can learn arithmetic pretty quickly. There are many reasons why this doesn't happen for many people in the US. The reasons can vary from place to place even. Some reasons are math teachers who didn't major in math and hate math, people who hate children teaching, fad methods, reliance on calculators, and even too much of an academic focus too early which burns the kids out early. This last one is a big one. Finland has the best academic results on the planet and they don't even try to teach kids to read until age 7 because thy aren't ready for it until then. (http://online.wsj.com/article/SB120425355065601997.html)
For many kids it is better for them to not go to school at all, not even be homeschooled, but just do what they please each day. There are parents doing this as a schooling method and they consistently get substantially better results than public school educated kids. (One example http://en.wikipedia.org/wiki/Sudbury_school) Possibly the solution is to get rid of the schools, or at least what we think of as schools.
Yeah it's all about getting the understanding of how something is working, then looking to move onto something that makes the repetition easy. It's the same in programming, it's great to learn how your code is compiled to assembly but once you have that understanding you can take it back up to a high level language which helps remove repetition.
I think it is impossible to get to the "higher orders of thinking" without slogging through some of the grunt work to see what is really going on. Unfortunately there is no magic number of cases or fixed level of calculation work that guarantees understanding. It varies by the nature of the maths being taught and the context, interest and intelligence of the student. Moreover, with today's dependence on calculators and computers many practicing engineers and scientists are often missing the meaning behind the numbers despite knowing the math.
I had this last point revealed to me when I gave my first engineering review lecture to an audience of about ~80 engineers. During the Q&A one of the best (and older) engineers asked a question that indicated I had made a mistake. However, the mistake could only be seen by connecting information from PPT slide 5 and slide 40. After the lecture I asked the guy how he ever saw my mistake, I was really curious. He said he tends to think in ratios and showed me that some (easy to calculate) ratios from early in my lecture did not agree with ratios calculated later in the lecture, which revealed the mistake. He explained that he acquired the habit from using a slide rule when he was in school. He noted that young engineers (me), with their over-dependence on calculators and computers, generally did not have this skill. He was not advocating using slide rules and banning computers but thought that using a slide rule had an epistemological value in understanding the meaning behind the numbers which was becoming a lost art.
I had this last point revealed to me when I gave my first engineering review lecture to an audience of about ~80 engineers. During the Q&A one of the best (and older) engineers asked a question that indicated I had made a mistake. However, the mistake could only be seen by connecting information from PPT slide 5 and slide 40. After the lecture I asked the guy how he ever saw my mistake, I was really curious. He said he tends to think in ratios and showed me that some (easy to calculate) ratios from early in my lecture did not agree with ratios calculated later in the lecture, which revealed the mistake. He explained that he acquired the habit from using a slide rule when he was in school. He noted that young engineers (me), with their over-dependence on calculators and computers, generally did not have this skill. He was not advocating using slide rules and banning computers but thought that using a slide rule had an epistemological value in understanding the meaning behind the numbers which was becoming a lost art.
Calculation is important, but not strictly because it contributes much to math, it's important because you ought to be able to do basic number manipulations in your head or you'll be taken for a ride in all kinds of ways. Bad sales pitches, obviously flawed political polemics, business plans that can't possibly fly, there's endless practical reasons to be able to do this stuff in your head rapidly and comfortably. Kids need to spend about the same time learning calculations, but it also needs to be somehow deprioritized so it is made more clear that it is in fact "calculation" and not "math".
You need comfort with numbers to learn math, but as any number of math professors can demonstrate on demand, you can get an awfully long way with single-digit multiplication and addition and not being able to do much else. It's the rare algebra problem that has you multiplying 37 by 23 if it isn't explicitly a fraction problem.
I'd support splitting the terminology up.
Or I would support a terminology split, if I had any confidence there was anybody left in the education system that actually has any clue about what "math" is anymore. Until you solve this problem it's all moving deck chairs around on the Titanic. And I mean the Titanic of today, sitting there long sunk, not the usual metaphorical sinking Titanic.
You need comfort with numbers to learn math, but as any number of math professors can demonstrate on demand, you can get an awfully long way with single-digit multiplication and addition and not being able to do much else. It's the rare algebra problem that has you multiplying 37 by 23 if it isn't explicitly a fraction problem.
I'd support splitting the terminology up.
Or I would support a terminology split, if I had any confidence there was anybody left in the education system that actually has any clue about what "math" is anymore. Until you solve this problem it's all moving deck chairs around on the Titanic. And I mean the Titanic of today, sitting there long sunk, not the usual metaphorical sinking Titanic.
> ...if I had any confidence there was anybody left in the education system that actually has any clue about what "math" is anymore
Yes, I think you are right. Reminds me of Richard Feynman's views of the education system as expressed in his book, Surely you're joking, Mr. Feynman.
http://books.google.com/books?id=7papZR4oVssC&lpg=PA60&#...
Yes, I think you are right. Reminds me of Richard Feynman's views of the education system as expressed in his book, Surely you're joking, Mr. Feynman.
http://books.google.com/books?id=7papZR4oVssC&lpg=PA60&#...
The way calculation is taught to kids I think could use a whole lot of work. Calculation and mental arithmetic are pretty interesting and quite handy, even for adults. Check out Vedic and Trachtenberg techniques:
http://en.wikipedia.org/wiki/Vedic_math
http://en.wikipedia.org/wiki/Trachtenberg_method
Doing 3-digit multiplication in your head with those techs is like doing a mental pushup. Practicing calculation this way helps keep your mind active and your concentration sharp. Here's a fun clip of Scott Flansburg, Guinness world record holder for being the fastest "mental calculator", teaching a class of kids - http://www.youtube.com/watch?v=fjSPMfBoV0w
Teach math to kids like that and I bet you'll a foster a stronger appreciation for the field early on.
http://en.wikipedia.org/wiki/Vedic_math
http://en.wikipedia.org/wiki/Trachtenberg_method
Doing 3-digit multiplication in your head with those techs is like doing a mental pushup. Practicing calculation this way helps keep your mind active and your concentration sharp. Here's a fun clip of Scott Flansburg, Guinness world record holder for being the fastest "mental calculator", teaching a class of kids - http://www.youtube.com/watch?v=fjSPMfBoV0w
Teach math to kids like that and I bet you'll a foster a stronger appreciation for the field early on.
Depending on the area, both...
Some things, like solving simple algebra and simultaneous equations are done to death, get the students to understand the concept, do a few different examples and move on. I hate when students are given say 20 equations to do which are all the same rules with different numbers.
Something like differential equations, if you are plugging these into a calculator at first, you probably won't really get exactly what is going on. It requires more examples and exercises (there are a lot more different cases to give to) before a student could plug it into a calculator rather than going through the repetition of doing it by hand.
I just missed some of the newer calculator heavy course at the end of high school, my brother got it the year after I left I guess, wasn't a big fan of it by the looks of it though.
Something like differential equations, if you are plugging these into a calculator at first, you probably won't really get exactly what is going on. It requires more examples and exercises (there are a lot more different cases to give to) before a student could plug it into a calculator rather than going through the repetition of doing it by hand.
I just missed some of the newer calculator heavy course at the end of high school, my brother got it the year after I left I guess, wasn't a big fan of it by the looks of it though.
Calculation is important for understanding math. It should not be the only thing taught, but it should definitely be in the curriculum, and in a substantial amount, at least in the first few years. I think it lays a lot of ground work for further mathematical reasoning. Probably the most compelling reason, though, is that in mathematics you're assumed to have a certain degree of competency with computation. If you can't add or multiply small numbers in your head, you would be out of place among the current crop of mathematicians. While not everyone is trying to be a mathematician, it seems that it's not a good idea to remove something so fundamental that is essentially taken for granted in certain fields.
The wording I used for the fifth choice probably played played a role in why it got chosen the most.
More than currently - let students calculate almost everything by hand; let them use calculators/computers once they've done it by hand
Sounds the most traditional to me, and it seems as if people like traditional. However, I'd have thought most people would have wanted more reform. So it'd be nice to hear the views of the people who chose that option and why they thought it was the best.
More than currently - let students calculate almost everything by hand; let them use calculators/computers once they've done it by hand
Sounds the most traditional to me, and it seems as if people like traditional. However, I'd have thought most people would have wanted more reform. So it'd be nice to hear the views of the people who chose that option and why they thought it was the best.
I don't think there's such a dichotomy. I certainly believe more focus should be placed on "higher order" thinking but the calculation should still be covered in depth in order to solve those problems.
Essentially teaching in the form "have a problem, learn some techniques" instead of "learn a technique, get some problems that will use it." The latter is how my math education went and it doesn't teach students how to choose the right techniques to use in the first place.
Essentially teaching in the form "have a problem, learn some techniques" instead of "learn a technique, get some problems that will use it." The latter is how my math education went and it doesn't teach students how to choose the right techniques to use in the first place.
Please add "bring back the slide rule". I've never used one, but I think graphical techniques (including the use of slide rules - multiplication can be done by addition if you use logs) helps you get a better feel for maths.
Programming is a separate subject. Computer driven data-analysis is also a separate art to mathematics, though a maths background helps understand it. But these aren't core maths.
"Computers are useless. They can only give answers" - Pablo Picasso
Programming is a separate subject. Computer driven data-analysis is also a separate art to mathematics, though a maths background helps understand it. But these aren't core maths.
"Computers are useless. They can only give answers" - Pablo Picasso
If I recall correctly, they tried teaching calculation-light math in Washington and it turned out poorly.
This is really interesting.
Do you remeber more details , or can give a link to the study ?
This was one of the things I saw that made me think it turned out poorly.
http://www.youtube.com/watch?v=Tr1qee-bTZI
http://www.youtube.com/watch?v=Tr1qee-bTZI
It would have saved me a ton of frustration and wasted effort if it were a lot less than currently. In high school I remember hating math - it was obvious some people were far better at grinding through computations than I could ever hope to be, and it made me resentful being forced to go through what seemed a pointless exercise.
In university I made a late switch to computer science, and took real analysis thinking it would be very painful, but would make up for having no calculus. I actually enjoyed so much I switched to math, and so did sort of a math crash course over a year to prepare for grad school. Some of the math was hell. Studying differential equations, complex variables (computation oriented), and bits of differential geometry, etc. would leave me with so frustrated, wondering why I was putting myself through this. On the other hand, measure theory, complex analysis, functional analysis, algebra, and galois theory were so illuminating. I could just sit there for hours working through proofs, and not burn out.
However, this would not work for everyone (or most even). It shocked me that everyone else didn't see it my way, so it was very interesting observing how others studied/thought about math. I had a friend in all my pure math classes whom I worked on assignments with. He was a computational wizard who could flawlessly plow through pages of computations. My rate of errors - flipping a sign, carelessly misapplying a rule, etc. - was so much higher than his I had to conclude our brains were just wired in a totally different way. I noticed, when trying to prove something, we'd proceed very differently. He'd take what he knew to be true, and just begin enumerating some logical consequences of that, and go in the direction which seemed to have the smallest blowup in data. I'd usually assume the statement was false, and think and think and think about why that would be so absurd. Then, when I came up with the abstract explanation in my mind, in my mind I'd shape it into something concrete enough to write down (or even describe).
I identify a lot with the mathematician Alexander Grothendieck (at least before he became a little eccentric), because he's one of the rare examples (I know of) of an outstanding mathematician who seems to have approached math the way I do, and derives value from it for similar reasons. AG was reknowned for thinking about math in an extremely abstract way: many people learn by taking specific examples and then playing around with them mechanically until they get a general 'feel' for what's going on. Instead AG would describe the phenomenon being observed in the most abstract and general way possible, sometimes building an entire new theory of which the solution to the original problem was merely a trivial consequence.
Here is a two part piece biographical essay that, regardless of your interest in math, you'll probably find very interesting. He led an extraordinary life:
http://www.ams.org/notices/200409/fea-grothendieck-part1.pdf
http://www.scribd.com/doc/35435936/As-If-Summoned-from-the-V...
EDIT: Hmm, I sort of went off on a general rant instead of making the point I intended (this tiny text field makes that so easy, haha). I wanted to point out it's a common distinction in math made between the "problem solvers" and the "theory builders" (AG, mentioned above, epitomizes the latter). Most subjects in pure math are populated by people in one field or the other, and I believe that by forcing a computational approach on people early in their development, you're completely turning people off of math who could have fallen in love with the abstract theory. How prevalent this is, I don't know.
In university I made a late switch to computer science, and took real analysis thinking it would be very painful, but would make up for having no calculus. I actually enjoyed so much I switched to math, and so did sort of a math crash course over a year to prepare for grad school. Some of the math was hell. Studying differential equations, complex variables (computation oriented), and bits of differential geometry, etc. would leave me with so frustrated, wondering why I was putting myself through this. On the other hand, measure theory, complex analysis, functional analysis, algebra, and galois theory were so illuminating. I could just sit there for hours working through proofs, and not burn out.
However, this would not work for everyone (or most even). It shocked me that everyone else didn't see it my way, so it was very interesting observing how others studied/thought about math. I had a friend in all my pure math classes whom I worked on assignments with. He was a computational wizard who could flawlessly plow through pages of computations. My rate of errors - flipping a sign, carelessly misapplying a rule, etc. - was so much higher than his I had to conclude our brains were just wired in a totally different way. I noticed, when trying to prove something, we'd proceed very differently. He'd take what he knew to be true, and just begin enumerating some logical consequences of that, and go in the direction which seemed to have the smallest blowup in data. I'd usually assume the statement was false, and think and think and think about why that would be so absurd. Then, when I came up with the abstract explanation in my mind, in my mind I'd shape it into something concrete enough to write down (or even describe).
I identify a lot with the mathematician Alexander Grothendieck (at least before he became a little eccentric), because he's one of the rare examples (I know of) of an outstanding mathematician who seems to have approached math the way I do, and derives value from it for similar reasons. AG was reknowned for thinking about math in an extremely abstract way: many people learn by taking specific examples and then playing around with them mechanically until they get a general 'feel' for what's going on. Instead AG would describe the phenomenon being observed in the most abstract and general way possible, sometimes building an entire new theory of which the solution to the original problem was merely a trivial consequence.
Here is a two part piece biographical essay that, regardless of your interest in math, you'll probably find very interesting. He led an extraordinary life:
http://www.ams.org/notices/200409/fea-grothendieck-part1.pdf
http://www.scribd.com/doc/35435936/As-If-Summoned-from-the-V...
EDIT: Hmm, I sort of went off on a general rant instead of making the point I intended (this tiny text field makes that so easy, haha). I wanted to point out it's a common distinction in math made between the "problem solvers" and the "theory builders" (AG, mentioned above, epitomizes the latter). Most subjects in pure math are populated by people in one field or the other, and I believe that by forcing a computational approach on people early in their development, you're completely turning people off of math who could have fallen in love with the abstract theory. How prevalent this is, I don't know.
someone once described the different areas of study as a wheel. You've got
The Arts -> The Humanities -> Life Sciences -> Physical Sciences -> Mathematics -> The Arts
We often view The Arts and Mathematics on opposite ends of the spectrum, but as I've gone deeper into mathematics, I've learned that an elegant proof is far more similar to Poetry than to the sciences.
If you approach mathematics from an engineering/applied science angle, it seems like doing it by hand is useless.
Another great analogy I've heard is that the language of mathematics requires about 12 years of "spelling and grammar tests". Most students then stop, considering themselves savvy enough. After those twelve years, those who go on experience the poetry of proof, where the basic building blocks are twisted and interwoven in new and creative ways. Math at this level is often more like creative writing, exploring new areas and explaining exciting things.
I really feel that most students get so turned off by the calculations that they never get to see how creative and elegant math can be.
The Arts -> The Humanities -> Life Sciences -> Physical Sciences -> Mathematics -> The Arts
We often view The Arts and Mathematics on opposite ends of the spectrum, but as I've gone deeper into mathematics, I've learned that an elegant proof is far more similar to Poetry than to the sciences.
If you approach mathematics from an engineering/applied science angle, it seems like doing it by hand is useless.
Another great analogy I've heard is that the language of mathematics requires about 12 years of "spelling and grammar tests". Most students then stop, considering themselves savvy enough. After those twelve years, those who go on experience the poetry of proof, where the basic building blocks are twisted and interwoven in new and creative ways. Math at this level is often more like creative writing, exploring new areas and explaining exciting things.
I really feel that most students get so turned off by the calculations that they never get to see how creative and elegant math can be.
Should you let the student do calculations because it gives them an understanding of the bigger concepts?
Or should you let the computers do the calculations and not waste the students' time?
To learn more about what I'm talking about, read the debate going on over here: http://news.ycombinator.com/item?id=2213258
Also, please watch this video that includes some of my views: http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_math_with_computers.html