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  • Are we being taught mathematics badly?

    My brother is a pretty smart guy, taking a lot of mathematics courses and majoring in computer science in college. Bored, i typed his name into google and found a really interesting essay he wrote about how our current way of teaching math is flawed as well as an alternative that could have the average 9th grader learning calculus!

    http://patrick.lioi.net/formal-systems/

    The first fourth of it is why our education system is flawed, and the rest of it is about his alternative. Its a long, but good read, and i was wondering what everyone else thinks?

    From personal experience, ive gone through all my math courses learning mathematics like a cookbook. This is what you do here, and this is how you do this. Every time we started a new concept, id treat it as a new discipline and of course this makes things very difficult. Now that im taking Calculus, im starting to see whats going on behind the numbers, but its still difficult to visualize it because ive been taught mathematics in such a counter intuitive way.

  • #2
    Re: Are we being taught mathematics badly?

    This is some good stuff Santa! I am someone who has struggled to really understand math my whole life (I could never even hack pre-calc) and now watching my nine-year old struggle with it, I have often thought there must be a a better way to learn math. I have given this one quick run through and dig what he is saying...I will read it again tomorrow for sure (ironically, I have to help my nine-year old with her math homework so no time for a second read now!)
    LoyalGuard

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    • #3
      Re: Are we being taught mathematics badly?

      There are a lot of logical inaccuracies in GEB, and the book is based on a poor analogy between formal systems and brains.

      Anyhow, I'm not sure teaching young students principles of formal logic--basically what your brother is describing in his blog--would be a good alternative to math. Most students, I suspect, just wouldn't be able to get it.

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      • #4
        Re: Are we being taught mathematics badly?

        I think we should take a look at the english classes, first! IRONIC!

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        • #5
          Re: Are we being taught mathematics badly?

          I just don't think math is that important in everyday life for most people. Communication skills are so much more important. So I agree with section-8, more English classes (and a couple of other languages) would better serve the majority.

          And I hate saying that.

          Because I abhor grammar and spelling. And I don't like it because I suck at it. It does not come easy to me.

          Math I get. Math I like. Math is relatively easy for me. But I rarely use any math at all.

          The best math program I have seen is Saxon math. It simply admits that some people have, and always will have, difficulties learning math. So it just forces the student to practice, practice, practice. No new fancy learning methods. Just doing it over and over and over until it finally sinks in. Anything more complicated would probably just serve to confuse the situation.

          I also have a daughter that has a very difficult time with math. She breezes through the language arts. I have tried to show her different ways to "understand" what is going on but that only makes it worse. In the end I wrote a program that would just drill her on the basics and it worked. Still only gets C's in the subject but that is better than F's.

          What your brother describes should be available to those that can handle it but I can't see it helping most kids.
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          • #6
            Re: Are we being taught mathematics badly?

            There's a fine reason for symbolic logic being primarily a college-level course [everywhere I've had occasion to see it]. It's hard as a nut to wrap your brain around some of those concepts at times.

            Do I doubt that some kids could benefit from it earlier in their curriculum, particularly those struggling with the status quo? No, not at all... but I'd view this as an alternate method and by no means a replacement for the primary system currently in use.
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            • #7
              Re: Are we being taught mathematics badly?

              I've re-read your brother's article a couple of times now. I have far too little mathematical or educational institution background to judge whether introducing "formal systems" in an elementary school setting would enable greater numbers of students to be able to handle complex math at an earlier age, but, I do have some general comments that I think are applicable.

              As Gringo was hinting at, there are two fundamental different types of needs for mathematical knowledge:

              1) To be able to withstand the daily trials we face that require basic math (make change from a $20, balance your checkbook, measure a 2x4, calculate the interest on a 30-yr mortgage).

              2) Application of mathematics for scientific purposes such as computer science, plotting a spaceship's course to the moon, building a nuclear bomb, etc.

              I think most of the world fits into the first type. This is not to say that we all shouldn't be able to have the potential to learn advanced math, but there I think it is impractical for everyone to know it.

              Most of our primary and seconday educational systems only have 12 years to teach us what we need to know to get through life. Would introducing formal systems increase the time required by the educational system (extra hour per day, or a month per school year, or an additional 1-2 total years of school) or would it take more time in the beginning and actually increase effeciency as a student progresses thereby taking the same amount of time but fitting more in? This is an important question to consider.

              So, even though I cannot say that teaching formal systems is the alternative to the way we learn math now, I definitely do think that there needs to be a variety of alternative curricula for all educational disciplines. I have two reasons for this as well:

              1) People who need more than basic knowledge in a particular discipline need as many of the mental tools as possible required by that discipline in order to perform optimally. If I want to be a computer scientist (and more importantly I have the potential to be a computer scientist) if there is an alternative curriculum available that will help me undertand lambda calculus or whatever else will help me excel in my chosen field then that is the curriculum that should be available to me. Of course, it is probably hard to find a large number of 2nd-graders that know they want to be (and have the potential to be) a computer scientist at age seven when they should begin an alternative curriculum.

              2) Different people learn better via some teaching methods and poorly from others. Take learning languages for an example. There are many methods of teaching a language: The Grammar Translation Method, The Direct/Natural Method, The Immersion method, etc. Depending on how one learn's best, one may be better for you and a different method may be better for someone else. The best method for you may also vary as you get older.

              So, in order for each one of us to meet our full intellectual potential, whether it be math, or language, or other discipline, I think alternative curricula are crucial. The problem is then how to implement alternative curricula in a school system that at best case only has the resources to ensure its students pass standardized educational requirements and at worst case may struggle to teach people even the most basic of reading and writing skills. (Not to say all school systems are like this, just that systems like this do exist).
              Last edited by loyalguard; 09-12-2007, 12:37 PM.
              LoyalGuard

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              • #8
                Re: Are we being taught mathematics badly?

                I believe that learning is a matter of motivation. Gene Hackman put it best in Crimson Tide when he said "put a cattle prod up a horse's *** and you can make it dance. it's a simple matter of voltage." It's quite true. Having been a teacher myself, I actually got the opportunity to ask students what it would take to make them learn better (especially the lazy ones and kids that other people would brand as being "dumb" or "stupid"). The standard response was a measure of force. If someone forced them to learn and pass a subject with a definitive "or else" afterward, they'd have no problem doing it. However, since there are no consequences for a sub-par education, people slip into the role nice and easy. As a result, the teachers become disenfranchised with the notion of trying to teach kids who are unwilling to learn and as such the students who do want to learn get a lesser grade of education. It's a vicious cycle.

                As far as the correctness of the mathematics we learn, I cant really comment. I have always found that the mathematics that I learned are accurate and although I dont agree with the structure of teaching as far as mathematics goes, the same can be said for any subject matter in the current educational system.

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                • #9
                  Re: Are we being taught mathematics badly?

                  Motivation is a factor in every human endeavor. That factor is, i believe, implicit during any discussion of teaching and or learning.

                  I think the focus of this thread is if there might be better ways to teach a particular subject to motivated kids.

                  Success is a great motivator for many and failure can crush a kids desire to learn pretty quick as well. The method of teaching can influence the level of success a child experiences thus increasing motivation. This can be a vicious, but welcomed, upward spiral.
                  Iím not racists, I have republican friends. Radio show host.
                  - "The essence of tyranny is the denial of complexity". -Jacob Burkhardt
                  - "A foolish consistency is the hobgoblin of little minds" - Emerson
                  - "People should not be afraid of it's government, government should be afraid of it's People." - Line from V for Vendetta
                  - If software were as unreliable as economic theory, there wouldn't be a plane made of anything other than paper that could get off the ground. Jim Fawcette
                  - "Let me now state what seems to me the decisive objection to any conservatism which deserves to be called such. It is that by its very nature it cannot offer an alternative to the direction in which we are moving." -Friedrich Hayek
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                  • #10
                    Re: Are we being taught mathematics badly?

                    Originally posted by Ferris Bueller View Post
                    I believe that learning is a matter of motivation. Gene Hackman put it best in Crimson Tide when he said "put a cattle prod up a horse's *** and you can make it dance. it's a simple matter of voltage." It's quite true. Having been a teacher myself, I actually got the opportunity to ask students what it would take to make them learn better (especially the lazy ones and kids that other people would brand as being "dumb" or "stupid"). The standard response was a measure of force. If someone forced them to learn and pass a subject with a definitive "or else" afterward, they'd have no problem doing it. However, since there are no consequences for a sub-par education, people slip into the role nice and easy. As a result, the teachers become disenfranchised with the notion of trying to teach kids who are unwilling to learn and as such the students who do want to learn get a lesser grade of education. It's a vicious cycle.

                    As far as the correctness of the mathematics we learn, I cant really comment. I have always found that the mathematics that I learned are accurate and although I dont agree with the structure of teaching as far as mathematics goes, the same can be said for any subject matter in the current educational system.

                    This is why I condone educational violence. Particularly the cattle prod idea....
                    ---Bellicosity---

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                    • #11
                      Re: Are we being taught mathematics badly?

                      Originally posted by El_Gringo_Grande View Post
                      I just don't think math is that important in everyday life for most people. Communication skills are so much more important. So I agree with section-8, more English classes (and a couple of other languages) would better serve the majority. be available to those that can handle it but I can't see it helping most kids.
                      Perhaps the specifics of mathematics aren't relevant to most people; I'm an engineer, but I have friends in other majors and other walks of life, so I can accept that, as horrifying as it seems. However, the current way public secondary schools teach math (I didn't go to public school, but I can see the way my friends who did think) is geared towards making them understand the specifics, not the problem-solving aspects. What they were being taught isn't relevant to them in the least, but they could easily have benefited from being taught how to solve problems. In fact, they themselves admit that their other (completely unrelated) courses are made much easier after taking an entry-level Calculus course taught by one remarkable professor at this university.
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                      • #12
                        Re: Are we being taught mathematics badly?

                        If I can use this is an excuse of why I'm so bad at Calculus, then I'll take it.


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                        • #13
                          Re: Are we being taught mathematics badly?

                          It always felt absurd to me that we are thought math (in Belgium at least), yet without explaining what math is to the children. This under the assumption that they cannot understand the underlying principles.

                          It was only on university in a "logics" course, that I got some explanation on how math is a logical, closed system that is be based created by a set of strict assumptions. During that course I suddenly understood why I was fooling around with "venn-diagrams" and that funny ruler to draw them all that time.

                          The idea puzzles me of teaching something without even trying to teach the basics of it. 1+1 is NOT 2, it only is that as part of a strict system that people attempt to use to describe things in reality.

                          I see that it can be hard to try and teach such nearly philosophical matters to young children, but people don't even try. I for one do not believe at all that a child would be less able then a grown-up to understand even the deepest philosophical matter. They simply haven't learned to/how to communicate about it.

                          Trying to teach how math ties into reality, would perhaps boost students interest in seemingly irrelevant exercises.

                          *At this point I actually felt I needed to read the article:).
                          Santa's brother seems to have had the same experience as I did.

                          El gringo' comment is interesting in this light. I actually think teaching more of the basics can be a motivator, instead of for motivated students. I know I was never motivated by math, where I certainly find it interesting now.

                          I think a child around 8-10 years old should be able to understand these basics. Off course there will always be children that don't get anything out of a deeper explanation, it still should be tried imo. You don't teach a basketball player to dunk before he can dribble.

                          The problem may be that the teachers that teach at that level, probably never had a logics course in their life, probably haven't the faintest idea of the basics of math, and the interconnection between the different skills thought in their text books. I know this is the case in Belgium. Becoming a teacher for 6-12yr old range does not require any serious math/logics course. The teacher's schooling is more focused on pedagogical aspects then actual knowledge and insight.

                          I guess I'm just a fanboy for the "Homo Universalis" principle.

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                          • #14
                            Re: Are we being taught mathematics badly?

                            Originally posted by Santa View Post
                            My brother is a pretty smart guy, taking a lot of mathematics courses and majoring in computer science in college. Bored, i typed his name into google and found a really interesting essay he wrote about how our current way of teaching math is flawed as well as an alternative that could have the average 9th grader learning calculus!

                            http://patrick.lioi.net/formal-systems/

                            The first fourth of it is why our education system is flawed, and the rest of it is about his alternative. Its a long, but good read, and i was wondering what everyone else thinks?

                            From personal experience, ive gone through all my math courses learning mathematics like a cookbook. This is what you do here, and this is how you do this. Every time we started a new concept, id treat it as a new discipline and of course this makes things very difficult. Now that im taking Calculus, im starting to see whats going on behind the numbers, but its still difficult to visualize it because ive been taught mathematics in such a counter intuitive way.

                            I know what you mean, I take pre-calculus now and I struggle. This year I understand it better because my video teacher goes over each lesson for like two or three days at a time ;0 [just that means tests are monsters]. I was formerly a math superstar until, I had started my tenth grade year with my science teacher turned math teacher- she is semi-crazy and she does not truly know what she is teaching. :(

                            What should I expect for Meteorology?
                            Last edited by BeSiege82; 09-14-2007, 12:14 AM.



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                            • #15
                              Re: Are we being taught mathematics badly?

                              I've read through the article, and there are a number of really good points that have been brought up. Overall I think the main issue in teaching is the objective of what is being taught, and of course many of the posters here are dead on that for most it is simply being able to do tasks such as balancing your accounts, working out your tax etc. I think that we should ignore this for 2 reasons: firstly, on a possibly ethical level I think it is sad that we should just give up on those who are less able and not bother with trying to teach them something because they probably won't understand (note that this is different from not finding it useful); secondly there is a proven relationship between increasing your intelligence by teaching abstract theory - the understanding of why certain things work the way they do goes on to help you form other relationships in your brain (in fact one of the most hotly debated subjects in intelligence is the existence of original thought).


                              What most strikes me about the ideas presented by Santa's brother (hereby abbreviated to SB) is that what he is trying to teach is not so much mathematics as logic. And this is where the big flaw in his argument lies. Yes I agree that it would be simpler to teach about the underlying theory of something before going to describe the details, but there are 2 large barriers, and all to do with how the brain works.

                              The first of these problems is rooted deep into our neurological development, and as such is completely unassailable. Barring the odd case of child genius humans do not develop the capacity for logical thought until they are around 12 years old. Although they can make up for this by blindingly fast executive processing the strategy that they undertake is far more trial and error than logical inference. The irony here (and please check that this is irony) is that your intelligence at this age, concerning the ability to build up mathematical laws that have not been hardwired into you, is really no better than that of the computer who's science you are attempting to understand. This is why younger chess players, although often very tactically aware, are often strategically naieve, and tend to only undertake long term good moves because they have been told that it's a good thing to do, without understanding why.

                              The second problem comes from experience itself, and the problem solving nature of the human mind. We, as a species, are concerned with overcoming difficulties, and our mental capabilities are fully geared to achieving this issue. Many associated characteristics of our mental fortitude can be seen as examples (von Restorff effect, extreme limitations of non-metaphor based human memory) of this. Simply put unless there is a driving need to solve an issue (or at least the ability to trick a young mind into believing the necessity) we're really not interested. But this is not to do with a lack of focus, or restlessness on the part of the student. I'm talking about the brain (a natural problem solver) finding a situation pressing enough to devote resources into solving it. Now this is a lot easier than might be thought because your brain loves to make sense of its environment (it is very important what sounds animals make, for example), but when a problem becomes highly abstract to a mind that has little or no experience of abstractness then you hit a wall.

                              Mathematics itself developed from counting (or accounting might be a better choice of word), because it is important to know how much of something you have. Prior even to accounting (as defined by appraising something to its exact number) it was important to have an understanding or appreciation for something's size (starting with big & small, and developing to few & many) due to the ancestral history our species has with agriculture. Knowing how many goats you had in your herd, and how large a tribe that could support, and then even on to how many you could eat and not deplete your stock to an irrecoverable or unsustainable level was extremely important to the survival (and thus development) of our species.

                              What this means is that without some sort of environmental factor, or metaphor, these concepts are really tricky to pick up. It should be no surprise therefore that young children have difficulty at first adding just numbers together. What does 2 plus 2 equal can be confusing becaue the child will be wondering "2 what?". This is why 2 apples plus 2 apples is a great starting point - even better if you actually use apples as a counting aid. Whilst you teach your child the basics of counting apples you are teaching your child's brain about how numbers form relationships. Once they have grasped this you can move on to drop the associated items, as they understand that "2", as a number by itself has meaning (because it can be manipulated by the processes that you have taught them).

                              Now this might seem quite fuzzy, and possibly not too different from the first reason, but I'd like to highlight it with two further examples to show how keenly human brains associate numbers with objects (ie counting). I'll use one historical fact and one anecdote from my early schooling, and both of these will hopefully fix the idea that abstractness of value and dissociation with objects are part of human nature. The first is one I have used quite often as it really is a great example of how mathematics has developed. The Ancient Greeks were pretty clever chaps by all accounts but they had no concept of negative numbers. The mathematician Diophantus described the equation 4x+20=0 as absurd, because no example of its solution existed in the real world. This is quite hard rooted into counting - if you have 2 apples how can I take 3 of them away? In modern mathematics we treat negative numbers as more like a vector, with it having a modulus (size of value) and a direction (positive or negative of 0). When described in this state it makes more sense, as two of the early examples of the use of negative numbers show. In Ancient China they would use different coloured rods to represent values (understanding the cancelling nature of positive and negative numbers whilst avoiding the difficulties of comprehending less than nothing). In India around the 7th century negative numbers would be used to represent debts, and this latter point shows the vector nature of the system as well as it's desire to cling to real objects.

                              The second example is a little more subtle, and as I have no proof you'll just have to accept this as having happened. When I was 5 the school class I was in had a play shop. In this shop many plasticine items were for sale, and you could trade these pretend items for even more pretend plastic coins. We would all take it in turns being the shopkeeper, which basically meant adding up the values of the items bought, asking for the amount and then calculating the change. One of my classmates came up to buy a delicious plasticine cake, at the cost of 3p. When I asked for the amount he gave me 3 coins: 10p, 5p and 1p. I asked why he was giving me all that as it was 16p and he only needed to give me 3p. He replied that it was 3p and counted the coins in his hand to prove that there were 3. It was not that he was not clever (later years proved he was a capable student) it was just at that point I had understood that each coin could have an associated value attached to it, whereas his brain had not got that far. For him a coin was a coin, and the idea that each coin had an abstract value placed upon it was absurd (even though he could recognise that the coins were different).


                              Points 1 & 2 have very much focused on the development of understanding, and I have deliberately targetted young children (or earlier civilisations) to prove that teaching abstract concepts or underlying theory from day 1 is a flawed strategy. You need a certain amount of experience and brain development to be able to understand such theories. That said I still think SB is onto something important. We may need to define some boundaries and develop younger minds in the existing experience-repitition way, but what about older students? At 14 we should all have developed the mental capabiities to keenly understand abstract issues as well as having the experience to give them good grounding. Well the good news is that it is only sociopolitics that stand in the way here. The bad news is that, well, its sociopolitics.

                              I was lucky enough to go to a good school, and as with the other keen mathematicians there was given a pretty accelerated course (even though the drudgery of a national curriculum in my opinion held many of us back). Many of the discussions that SB brings up I was actually able to experience first hand because we were encouraged to think for ourselves. We also had really good teachers that took the effort to frame a problem, but even here framing was evident. I learnt how to integrate by looking at the relationship between acceleration, velocity, and distance. This not only brought in the concept that geometry and calculus were linked (distance after all is just the area under the velocity line) but also gave an example of how theoretical (pure) mathematics can be applied to physics (or mechanics).


                              The problem here should be self evident. I appreciate the fact that I was given the educational freedom to develop thought not by starting off with a blank slate and trying to fill it, but by investigating just why things are going on the way they are. At no point did I consider any part of mathematics to be wholly separate disciplines and this was almost entirely down to my teachers introducing new topics in relation to old ones. Even in statistics (a soul-less branch of mathematics in my opinion) we were not taught just to use certain functions because they looked good, but we were introduced to them from a solid grounding. I learnt why certain tests in statistics are actually 'proof' and not just a good guess because our teachers spent time deriving them from first principles - but those first principles had been built up over many years.



                              A well written article, which is a good read and very thought provoking, despite a couple of logical inconsistencies.

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