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To play the devils advocate: IMO, it's the focus on the micro steps that's the problem. HS level math is vary simple. You can write a single textbook that cov

To play the devils advocate: IMO, it's the focus on the micro steps that's the problem. HS level math is vary simple. You can write a single textbook that covered the full range of math from preschool to calculus, but the focus on 3 days of instruction, a day of review, and then a quiz or test slows things down. It can be easier to skip ahead and then go back and review than try and approach math in tinny nibbles. It's like spending a full year going over cement foundations before you mention that the goal is to put a house on top of that flat slab. Personally, I used to do other classes homework assignments in my math classes. I can recall getting in about 4 seconds a new topic that the class spend a full week going though in minute detail. It was so bad I once accidentally did the next chapters review vs the assigned homework and did not even notice at the time. PS: I am all for better instruction, but perhaps we could consider going a little further. We could probably get the average 10th grader to really understand Calculus, but I think the goal should be to dive into DifEq and number theory etc.

tokenadult 15y ago
IMO, it's the focus on the micro steps that's the problem. It would be very problematic, and this happens, if teaching consisting of showing micro steps is followed up by homework and assessments that also focus narrowly on micro steps. That serves up exercises for students, but it doesn't allow students the learning opportunities developed by working on actual problems. The rest of this comment is a FAQ file I send to families of new students in my math classes about the distinction between "exercises" and "problems." PROBLEMS VERSUS EXERCISES I frequently encounter discussions among parents about repetitive school math lessons, so a few years ago I prepared this Frequently Asked Question (FAQ) document about the distinction between math exercises (good in sufficient but not excessive amount) and math problems (always good in any amount). Most books about mathematics have what are called "exercises" in them, questions that prompt a learner to practice the concepts discussed in the mathematics book. By reading one mathematics book, and then several more, I learned that some mathematicians draw a distinction between "exercises" and "problems" (which is the terminology generally used by the mathematicians who draw this distinction). I think this distinction is useful for teachers and learners to consider while selecting materials for studying mathematics, so I'll share the quotations from which I learned this distinction here. I first read about the distinction between exercises and problems in a Taiwan reprint of a book by Howard Eves. "It is perhaps pertinent to make a comment or two here about the problems of the text. There is a distinction between what may be called a PROBLEM and what may be considered an EXERCISE. The latter serves to drill a student in some technique or procedure, and requires little, if any, original thought. Thus, after a student beginning algebra has encountered the quadratic formula, he should undoubtedly be given a set of exercises in the form of specific quadratic equations to be solved by the newly acquired tool. The working of these exercises will help clinch his grasp of the formula and will assure his ability to use the formula. An exercise, then, can always be done with reasonable dispatch and with a minimum of creative thinking. In contrast to an exercise, a problem, if it is a good one for its level, should require thought on the part of the student. The student must devise strategic attacks, some of which may fail, others of which may partially or completely carry him through. He may need to look up some procedure or some associated material in texts, so that he can push his plan through. Having successfully solved a problem, the student should consider it to see if he can devise a different and perhaps better solution. He should look for further deductions, generalizations, applications, and allied results. In short, he should live with the thing for a time, and examine it carefully in all lights. To be suitable, a problem must be such that the student cannot solve it immediately. One does not complain about a problem being too difficult, but rather too easy. "It is impossible to overstate the importance of problems in mathematics. It is by means of of problems that mathematics develops and actually lifts itself by its own bootstraps. Every research article, every doctoral thesis, every new discovery in mathematics, results from an attempt to solve some problem. The posing of appropriate problems, then, appears to be a very suitable way to introduce the student to mathematical research. And it is worth noting, the more problems one plays with, the more problems one may be able to pose on one's own. The ability to propose significant problems is one requirement to be a creative mathematician." Eves, Howard (1963). A Survey of Geometry volume 1. Boston: Allyn and Bacon, page ix. I have since read about this distinction in several other books. "Before going any further, let's digress a minute to discuss different levels of problems that might appear in a book about mathematics: Level 1. Given an explicit object x and an explicit property P(x), prove that P(x) is true. . . . Level 2. Given an explicit set X and an explicit property P(x), prove that P(x) is true for FOR ALL x [existing in] X. . . . Level 3. Given an explicit set X and an explicit property P(x), prove OR DISPROVE that P(x) is true for for all x [existing in] X. . . . Level 4. Given an explicit set X and an explicit property P(x), find a NECESSARY AND SUFFICIENT CONDITION Q(x) that P(x) is true. . . . Level 5. Given an explicit set X, find an INTERESTING PROPERTY P(x) of its elements. Now we're in the scary domain of pure research, where students might think that total chaos reigns. This is real mathematics. Authors of textbooks rarely dare to pose level 5 problems." Graham, Ronald, Knuth, Donald, and Patashnik, Oren (1994). Concrete Mathematics Second Edition. Boston: Addison-Wesley, pages 72-73. This digression becomes the subject of a, um, problem in Exercise 4 of Chapter 3: "The text describes problems at levels 1 through 5. What is a level 0 problem? (This, by the way, is NOT a level 0 problem.)" "First, what is a PROBLEM? We distinguish between PROBLEMS and EXERCISES. An exercise is a question that you know how to resolve immediately. Whether you get it right or not depends on how expertly you apply specific techniques, but you don't need to puzzle out what techniques to use. In contrast, a problem demands much thought and resourcefulness before the right approach is found. . . . "A good problem is mysterious and interesting. It is mysterious, because at first you don't know how to solve it. If it is not interesting, you won't think about it much. If it is interesting, though, you will want to put a lot of time and effort into understanding it." Zeitz, Paul (1999). The Art and Craft of Problem Solving. New York: Wiley, pages 3 and 4. ". . . . As Paul Halmos said, 'Problems are the heart of mathematics,' so we should 'emphasize them more and more in the classroom, in seminars, and in the books and articles we write, to train our students to be better problem-posers and problem-solvers than we are.' "The problems we have selected are definitely not exercises. Our definition of an exercise is that you look at it and know immediately how to complete it. It is just a question of doing the work, whereas by a problem, we mean a more intricate question for which at first one has probably no clue to how to approach it, but by perseverance and inspired effort one can transform it into a sequence of exercises." Andreescu, Titu & Gelca, Razvan (2000), Mathematical Olympiad Challenges. Boston: Birkhäuser, page xiii. "It is easier to advance in one topic by going ahead with the more elementary parts of another topic, where the first one is applied. The brain much prefers to work that way, rather than to concentrate on ugly technical formulas which are obviously unrelated to anything except artificial drilling. Of course, some rote drilling is necessary. The problem is how to strike a balance." Lang, Serge (1988), Basic Mathematics. New York: Springer-Verlag, p. xi.
kenjackson 15y ago
* To play the devils advocate: IMO, it's the focus on the micro steps that's the problem.* I don't think so. Everyone builds on a solid foundation and microsteps. I think what tends to happen though, and is discussed in the article, is that by the time kids reach HS they either have a very solid foundation, or an extremely poor one. The kids with the poor foundation will have trouble understanding things regardless of how long you spend on it. They've developed gaps over the past nine years of math education. I think if we address the gaps upfront in elementary school that in HS math will move much quicker. Not because the kids are smarter, but simply because you aren't compensating for 30 kids in a class who all struggle with a variety of concepts they should have learned in elementary school.
- simpleTruth 15y ago
  Math is not a single linear path. EX: You don't need to know long division to get Algebra. You can even go the other way. If you say 4a + 8b + 4 = x, and a = 10 * 10; b = 10. Now divide by 2b. Want to explain another base, say a = 16 * 16; b = 16. etc. PS: My father even taught a first grader long division that way. (Note: I think he used x1; x2, x3 and he soon move to the traditional form.) Edit: My point is if you want to build a house you can start with a foundation, or the windows. Keep studding math and there are plenty of opportunity's to review older concepts with new insights. Buy trying to focus on each little nibble in isolation it's harder to link concepts. EX: The sign rules are really just the associative property of addition and multiplication in another form. {-b + a = a + -b = a - b} {-b + -a = - ( a + b) } {-a * -b = (-1 * -1) * (a * b) = 1 * a * b}
  - hackinthebochs 15y ago
    >My point is if you want to build a house you can start with a foundation, or the windows. This is exactly the reason why most people don't get math. If you start with the windows, the student asks "whats the point in learning this?". They can't see the house being built when you start with the windows. Without the foundation there is no intuitive understanding thus everything seems like meaningless rules. And in fact they are just that-meaningless.
  - 15y ago
hackinthebochs 15y ago
>IMO, it's the focus on the micro steps that's the problem. No, the micro-steps are the solution, not the problem. Those who "get" math intuitively break problems down into smaller steps, often unconsciously. For those that can't intuitively do this (most students), they need to be taught explicitly the micro-steps. The micro-steps build the foundation for the higher level steps and the problem solving. Learning the micro-steps well is like learning an abstraction in a program. Once you learn the abstraction to the point that your understanding of it is unconscious, then your thought processes are lifted to a higher level. This is true understanding of math.
- happy4crazy 15y ago
  Learning the micro-steps well is like learning an abstraction in a program. Once you learn the abstraction to the point that your understanding of it is unconscious, then your thought processes are lifted to a higher level. This is true understanding of math. Exactly. The argument is not to "do things in micro-steps", it's to do things in appropriately-sized steps. As your understanding of math improves, you can (and should!) take bigger and bigger steps. The point is that, for presumably a variety of reasons, a really large percentage of students don't drop down a level in abstraction when they get stuck--they get depressed and give up. I see a large part of education's role being the inculcation of good habits, and this is just a sucky, sucky habit.
- simpleTruth 15y ago
  There are plenty of useful things to memorize that let you flat out skip steps. Some of this stuff might feel like party tricks, but don't assume people are using the same steps subconsciously.
  - hackinthebochs 15y ago
    I don't mean to say that people use the micro-steps all the time. What I mean is that, on initially learning something, those that "get" it subconsciously break it down into micro-steps that are just small enough for them. This builds the bridge from what they know to what they're learning. Once the abstraction or technique makes sense, then you apply it as a whole on further usage. I think one of the main things that separates the quick learners from the slow learners is how much breaking steps into micro-steps can be done unconsciously. The data lends weight to this. Teaching a concept in micro-steps doesn't help the smartest people learn it better, but it brings the slow learners up to speed. The difference is how small the steps have to be for the students' unconscious to make the connections and thus reach "understanding".
mattdeboard 15y ago
Congratulations, you internalized the micro steps by repetition and critical analysis. Not everyone gets there on their own for a million different reasons. I do not think there should be a 'focus' per se on 'micro steps' but there does need to be encouragement to kids to think about this and find solutions on their own. Case in point, I was made a very proud dad today. My first-grade daughter and I were going over her math homework, which was flash cards with addition equations on them. This week the focus in her class is the numbers 4 and 5, so on each flash card, at least one of the terms was 4 or 5. I was drilling her on them, and every time there was an equation the sum of which was >10, she'd mutter under her breath something about subtraction that I couldn't quite discern. I finally stopped at the "5+7" flash card, and asked her to walk me through her process for solving that particular problem. Her response: "Well, I know 7 is minus three [from ten], so I do minus 3 from 5 which is 2, then I put a 1 in front of that, and that's 12." In other words, she's doing | (10 - 7) - 5 | + 10. Or, basically, 7+3 = 10, 5-3 = 2, 10+2 = 12. I asked her, "Emma, is that how they teach you to add at school?" "No," she said, "I just think about it like that." Is that the absolute easiest way to solve that problem? Well, maybe; I don't know. I'm like 99% sure, however, I do that exact same process when I do mental calculations. That doesn't matter though. What matters is that she's analyzing problems, identifying and generalizing patterns, then applying those patterns to NEW problems. She is numerically literate! (http://en.wikipedia.org/wiki/Numeracy http://en.wikipedia.org/wiki/Numeracy) This is especially gratifying for me, since we spend a few hours at home every week going over more advanced math topics like fractions and multiplication. She doesn't quite grok it yet, but at least when she sees it in school it won't be a total surprise. In the end, she figured out a system to solve problems on her own that works 100% of the time. Critical thinking skills are paramount. Learn those and everything else comes eventually.
dataduck 15y ago
I've been tutoring maths at primary to A-level in the UK for more than ten years now. In that time, the most critical problem I've seen is when children (usually in the 12-16 range) see mathematics as a plethora of small, arbitrary techniques to solve specific problems, and can't see that they all are related together under a few key concepts. This means they hit a brick wall when they are assumed to have mastered and internalised the previous techniques. For instance, many fail to connect multiplication, division, addition, subtraction, integers and scale in order to construct an intuitive sense of rational number. This makes further work with rationals an uphill struggle. I've found going back to the beginning and putting together whatever concepts are lacking using proof and demonstration creates an almost digital change in confidence and proficiency: before it's a mystery, after it's simple. I'm pretty sure that the way schools break down mathematics into tiny portions and call each one a separate topic is to blame here. While breaking things up into micro steps is the easiest way to solve a particular mathematical exercise (and important in solving problems), joining things up into a single overarching entity is how you actually understand the subject. So breaking stuff up is really useful within a problem, but on its own doesn't give you the long-term preparation you need, and is inappropriate over an entire curriculum. I think any mathematics curriculum needs to have both, but it's the joined-up approach that's more often lacking in schools. I haven't been able to get the JUMP curriculum (site down?), but I'd really like to see how it deals with these issues.

tokenadult 15y ago

IMO, it's the focus on the micro steps that's the problem. It would be very problematic, and this happens, if teaching consisting of showing micro steps is followed up by homework and assessments that also focus narrowly on micro steps. That serves up exercises for students, but it doesn't allow students the learning opportunities developed by working on actual problems. The rest of this comment is a FAQ file I send to families of new students in my math classes about the distinction between "exercises" and "problems." PROBLEMS VERSUS EXERCISES I frequently encounter discussions among parents about repetitive school math lessons, so a few years ago I prepared this Frequently Asked Question (FAQ) document about the distinction between math exercises (good in sufficient but not excessive amount) and math problems (always good in any amount). Most books about mathematics have what are called "exercises" in them, questions that prompt a learner to practice the concepts discussed in the mathematics book. By reading one mathematics book, and then several more, I learned that some mathematicians draw a distinction between "exercises" and "problems" (which is the terminology generally used by the mathematicians who draw this distinction). I think this distinction is useful for teachers and learners to consider while selecting materials for studying mathematics, so I'll share the quotations from which I learned this distinction here. I first read about the distinction between exercises and problems in a Taiwan reprint of a book by Howard Eves. "It is perhaps pertinent to make a comment or two here about the problems of the text. There is a distinction between what may be called a PROBLEM and what may be considered an EXERCISE. The latter serves to drill a student in some technique or procedure, and requires little, if any, original thought. Thus, after a student beginning algebra has encountered the quadratic formula, he should undoubtedly be given a set of exercises in the form of specific quadratic equations to be solved by the newly acquired tool. The working of these exercises will help clinch his grasp of the formula and will assure his ability to use the formula. An exercise, then, can always be done with reasonable dispatch and with a minimum of creative thinking. In contrast to an exercise, a problem, if it is a good one for its level, should require thought on the part of the student. The student must devise strategic attacks, some of which may fail, others of which may partially or completely carry him through. He may need to look up some procedure or some associated material in texts, so that he can push his plan through. Having successfully solved a problem, the student should consider it to see if he can devise a different and perhaps better solution. He should look for further deductions, generalizations, applications, and allied results. In short, he should live with the thing for a time, and examine it carefully in all lights. To be suitable, a problem must be such that the student cannot solve it immediately. One does not complain about a problem being too difficult, but rather too easy. "It is impossible to overstate the importance of problems in mathematics. It is by means of of problems that mathematics develops and actually lifts itself by its own bootstraps. Every research article, every doctoral thesis, every new discovery in mathematics, results from an attempt to solve some problem. The posing of appropriate problems, then, appears to be a very suitable way to introduce the student to mathematical research. And it is worth noting, the more problems one plays with, the more problems one may be able to pose on one's own. The ability to propose significant problems is one requirement to be a creative mathematician." Eves, Howard (1963). A Survey of Geometry volume 1. Boston: Allyn and Bacon, page ix. I have since read about this distinction in several other books. "Before going any further, let's digress a minute to discuss different levels of problems that might appear in a book about mathematics: Level 1. Given an explicit object x and an explicit property P(x), prove that P(x) is true. . . . Level 2. Given an explicit set X and an explicit property P(x), prove that P(x) is true for FOR ALL x [existing in] X. . . . Level 3. Given an explicit set X and an explicit property P(x), prove OR DISPROVE that P(x) is true for for all x [existing in] X. . . . Level 4. Given an explicit set X and an explicit property P(x), find a NECESSARY AND SUFFICIENT CONDITION Q(x) that P(x) is true. . . . Level 5. Given an explicit set X, find an INTERESTING PROPERTY P(x) of its elements. Now we're in the scary domain of pure research, where students might think that total chaos reigns. This is real mathematics. Authors of textbooks rarely dare to pose level 5 problems." Graham, Ronald, Knuth, Donald, and Patashnik, Oren (1994). Concrete Mathematics Second Edition. Boston: Addison-Wesley, pages 72-73. This digression becomes the subject of a, um, problem in Exercise 4 of Chapter 3: "The text describes problems at levels 1 through 5. What is a level 0 problem? (This, by the way, is NOT a level 0 problem.)" "First, what is a PROBLEM? We distinguish between PROBLEMS and EXERCISES. An exercise is a question that you know how to resolve immediately. Whether you get it right or not depends on how expertly you apply specific techniques, but you don't need to puzzle out what techniques to use. In contrast, a problem demands much thought and resourcefulness before the right approach is found. . . . "A good problem is mysterious and interesting. It is mysterious, because at first you don't know how to solve it. If it is not interesting, you won't think about it much. If it is interesting, though, you will want to put a lot of time and effort into understanding it." Zeitz, Paul (1999). The Art and Craft of Problem Solving. New York: Wiley, pages 3 and 4. ". . . . As Paul Halmos said, 'Problems are the heart of mathematics,' so we should 'emphasize them more and more in the classroom, in seminars, and in the books and articles we write, to train our students to be better problem-posers and problem-solvers than we are.' "The problems we have selected are definitely not exercises. Our definition of an exercise is that you look at it and know immediately how to complete it. It is just a question of doing the work, whereas by a problem, we mean a more intricate question for which at first one has probably no clue to how to approach it, but by perseverance and inspired effort one can transform it into a sequence of exercises." Andreescu, Titu & Gelca, Razvan (2000), Mathematical Olympiad Challenges. Boston: Birkhäuser, page xiii. "It is easier to advance in one topic by going ahead with the more elementary parts of another topic, where the first one is applied. The brain much prefers to work that way, rather than to concentrate on ugly technical formulas which are obviously unrelated to anything except artificial drilling. Of course, some rote drilling is necessary. The problem is how to strike a balance." Lang, Serge (1988), Basic Mathematics. New York: Springer-Verlag, p. xi.

kenjackson 15y ago

* To play the devils advocate: IMO, it's the focus on the micro steps that's the problem.* I don't think so. Everyone builds on a solid foundation and microsteps. I think what tends to happen though, and is discussed in the article, is that by the time kids reach HS they either have a very solid foundation, or an extremely poor one. The kids with the poor foundation will have trouble understanding things regardless of how long you spend on it. They've developed gaps over the past nine years of math education. I think if we address the gaps upfront in elementary school that in HS math will move much quicker. Not because the kids are smarter, but simply because you aren't compensating for 30 kids in a class who all struggle with a variety of concepts they should have learned in elementary school.

simpleTruth 15y ago
Math is not a single linear path. EX: You don't need to know long division to get Algebra. You can even go the other way. If you say 4a + 8b + 4 = x, and a = 10 * 10; b = 10. Now divide by 2b. Want to explain another base, say a = 16 * 16; b = 16. etc. PS: My father even taught a first grader long division that way. (Note: I think he used x1; x2, x3 and he soon move to the traditional form.) Edit: My point is if you want to build a house you can start with a foundation, or the windows. Keep studding math and there are plenty of opportunity's to review older concepts with new insights. Buy trying to focus on each little nibble in isolation it's harder to link concepts. EX: The sign rules are really just the associative property of addition and multiplication in another form. {-b + a = a + -b = a - b} {-b + -a = - ( a + b) } {-a * -b = (-1 * -1) * (a * b) = 1 * a * b}
- hackinthebochs 15y ago
  >My point is if you want to build a house you can start with a foundation, or the windows. This is exactly the reason why most people don't get math. If you start with the windows, the student asks "whats the point in learning this?". They can't see the house being built when you start with the windows. Without the foundation there is no intuitive understanding thus everything seems like meaningless rules. And in fact they are just that-meaningless.
- 15y ago

hackinthebochs 15y ago

>IMO, it's the focus on the micro steps that's the problem. No, the micro-steps are the solution, not the problem. Those who "get" math intuitively break problems down into smaller steps, often unconsciously. For those that can't intuitively do this (most students), they need to be taught explicitly the micro-steps. The micro-steps build the foundation for the higher level steps and the problem solving. Learning the micro-steps well is like learning an abstraction in a program. Once you learn the abstraction to the point that your understanding of it is unconscious, then your thought processes are lifted to a higher level. This is true understanding of math.

happy4crazy 15y ago
Learning the micro-steps well is like learning an abstraction in a program. Once you learn the abstraction to the point that your understanding of it is unconscious, then your thought processes are lifted to a higher level. This is true understanding of math. Exactly. The argument is not to "do things in micro-steps", it's to do things in appropriately-sized steps. As your understanding of math improves, you can (and should!) take bigger and bigger steps. The point is that, for presumably a variety of reasons, a really large percentage of students don't drop down a level in abstraction when they get stuck--they get depressed and give up. I see a large part of education's role being the inculcation of good habits, and this is just a sucky, sucky habit.
simpleTruth 15y ago
There are plenty of useful things to memorize that let you flat out skip steps. Some of this stuff might feel like party tricks, but don't assume people are using the same steps subconsciously.
- hackinthebochs 15y ago
  I don't mean to say that people use the micro-steps all the time. What I mean is that, on initially learning something, those that "get" it subconsciously break it down into micro-steps that are just small enough for them. This builds the bridge from what they know to what they're learning. Once the abstraction or technique makes sense, then you apply it as a whole on further usage. I think one of the main things that separates the quick learners from the slow learners is how much breaking steps into micro-steps can be done unconsciously. The data lends weight to this. Teaching a concept in micro-steps doesn't help the smartest people learn it better, but it brings the slow learners up to speed. The difference is how small the steps have to be for the students' unconscious to make the connections and thus reach "understanding".

mattdeboard 15y ago

Congratulations, you internalized the micro steps by repetition and critical analysis. Not everyone gets there on their own for a million different reasons. I do not think there should be a 'focus' per se on 'micro steps' but there does need to be encouragement to kids to think about this and find solutions on their own. Case in point, I was made a very proud dad today. My first-grade daughter and I were going over her math homework, which was flash cards with addition equations on them. This week the focus in her class is the numbers 4 and 5, so on each flash card, at least one of the terms was 4 or 5. I was drilling her on them, and every time there was an equation the sum of which was >10, she'd mutter under her breath something about subtraction that I couldn't quite discern. I finally stopped at the "5+7" flash card, and asked her to walk me through her process for solving that particular problem. Her response: "Well, I know 7 is minus three [from ten], so I do minus 3 from 5 which is 2, then I put a 1 in front of that, and that's 12." In other words, she's doing | (10 - 7) - 5 | + 10. Or, basically, 7+3 = 10, 5-3 = 2, 10+2 = 12. I asked her, "Emma, is that how they teach you to add at school?" "No," she said, "I just think about it like that." Is that the absolute easiest way to solve that problem? Well, maybe; I don't know. I'm like 99% sure, however, I do that exact same process when I do mental calculations. That doesn't matter though. What matters is that she's analyzing problems, identifying and generalizing patterns, then applying those patterns to NEW problems. She is numerically literate! (http://en.wikipedia.org/wiki/Numeracy http://en.wikipedia.org/wiki/Numeracy) This is especially gratifying for me, since we spend a few hours at home every week going over more advanced math topics like fractions and multiplication. She doesn't quite grok it yet, but at least when she sees it in school it won't be a total surprise. In the end, she figured out a system to solve problems on her own that works 100% of the time. Critical thinking skills are paramount. Learn those and everything else comes eventually.

dataduck 15y ago

I've been tutoring maths at primary to A-level in the UK for more than ten years now. In that time, the most critical problem I've seen is when children (usually in the 12-16 range) see mathematics as a plethora of small, arbitrary techniques to solve specific problems, and can't see that they all are related together under a few key concepts. This means they hit a brick wall when they are assumed to have mastered and internalised the previous techniques. For instance, many fail to connect multiplication, division, addition, subtraction, integers and scale in order to construct an intuitive sense of rational number. This makes further work with rationals an uphill struggle. I've found going back to the beginning and putting together whatever concepts are lacking using proof and demonstration creates an almost digital change in confidence and proficiency: before it's a mystery, after it's simple. I'm pretty sure that the way schools break down mathematics into tiny portions and call each one a separate topic is to blame here. While breaking things up into micro steps is the easiest way to solve a particular mathematical exercise (and important in solving problems), joining things up into a single overarching entity is how you actually understand the subject. So breaking stuff up is really useful within a problem, but on its own doesn't give you the long-term preparation you need, and is inappropriate over an entire curriculum. I think any mathematics curriculum needs to have both, but it's the joined-up approach that's more often lacking in schools. I haven't been able to get the JUMP curriculum (site down?), but I'd really like to see how it deals with these issues.