Is This the Key/Secret to Learning Math?

[chiro]

Also for Mark44.

You mention the ability of doing specific things and drills and the reply to this is that you have to specify what your goals are.

If you want to be able to play a couple of songs really well then you will allocate your activities accordingly. Perhaps learning specific chords and transitioning between those chords becomes a large point of focus on how you train to achieve those goals.

As a session musician you will probably have a very rigorous training program where you learn all the theory behind common genres, how music is built within those genres, timing, scales, and some common patterns in them. Effectively you train though to be able to read and play music effortlessly so that when a piece of sheet music is placed in front of you - you play it to specification.

But think about if you want to freely improvise - like say a jazz musician.

The training becomes completely different. The classical musician has a focus on certainty and for them it is not expected for them to improvise pieces. It would actually be extremely uncomfortable for many classical and session musicians to do improvisation because they have not trained themselves to do that.

You have to ask yourself what the goals of learning actually are. For the musician are you training to play a couple of songs? Become a session musician? Become a jazz musician with good improvisational skills?

The same questions exists with mathematics - are you trying to learn how to solve basic geometry problems? Trying to learn to build structures in carpentry?

Or do you want to be able to make sense of complexity in a more meaningful way where you apply your understanding of consistency and existing concepts in mathematics to make better sense of it - and more importantly when you haven't specifically trained beforehand to do so?

Answering that question will be necessary to decide what sort of answer exists for the individual and the collective regarding mathematics education.

You have to answer what the goals are and how you intend to reach them.

 

[Jeff Rosenbury]

And who wrote the plan, and according to what ideas?
My guess is that you might have been exposed to Bob Widlar's wake.

According to copyright page, all Air Force technical orders were written by someone named, "This page intentionally left blank". He was prolific.

Perhaps a nom de plume?

 

[Hornbein]

How come no one is looking at how those countries are teaching their kids in math?

This would contradict the "US has the best of everything" axiom.

 

[Hornbein]

With so little detail I can't say anything other than "if it works, it works."

 

[Hornbein]

According to copyright page, all Air Force technical orders were written by someone named, "This page intentionally left blank". He was prolific.

Perhaps a nom de plume?

I always thought that should read "this page intentionally not left blank." But I guess it is appropriate for Air Force technical orders to begin with a Catch 22.

 

[vanhees71]

I think the key of learning math is, indeed in some sense, the following (for me self-evident for many socalled "didactics experts" however seemingly a kind of heresy):

If a student doesn’t conceptually understand that they can’t add apples and oranges together, for example, how will they know what to do a few years later when they see different variables in an addition equation in algebra class.

My own learning history of math as a high school student is exactly confirming this trivial fact. You should teach math to student as what it is and not as a kind of cookbook of recipies to solve (often boring) standard problems in school textbooks: It's the science of formal structures. Anything follows from some basic assumptions or axioms, which you can introduce as a kind of rule of a game. Following these assumed rules, you can investigate by applying logical arguments the "universe" of theorems following from these axioms. If you don't understand a claim, i.e., if you cannot prove it, it's worthless in the sense of mathematics.

E.g., it doesn't make sense to teach kids to stupidly apply some "curve sketching" standard techniques to analyze a function if you do not have explained in a clear way to them what's the meaning of derivatives and what they tell you about the function. This should be done on all levels of abstraction. You can start with looking at graphs of functions and explain everything geometrically in terms of tangents as limits of secants, but then you also should make clear the formal way to understand such intuitive concepts and also the amazingly surprising exotics like a continuous function that is not differentiable anywhere (like Weierstraß's Monster, but that's to complicated for high school, but you can construct easily such a function geometrically).

My own experience with the subject called math in high school was pretty ennoying at first. I didn't understand much and had no fun and sometimes pretty bad marks in exams. I tried to study the school book we used to no success. I couldn't understand more compared to what my teacher told in school. Then I went to the local public library and borrowed a textbook called "Introduction to Mathematics for Engineers". It was written for engineering students at universities of applied sciences (in German "Fachhochschule"), and this book was a revelation for me. All of a sudden I could understand, why one introduces limits, derivatives and the like. It also introduced a bit to the applications in geometry and simple mechanics and so on. Of course, it took me a lot of time to get started and to really follow all arguments, but it became fun to do so for me. I also started to apply what read to the homework from school, and I could solve these problems much more easily than with the very superficial explanations in the school book.

Perhaps it's not the right way for everybody, but I've made the experience also when helping out my class mates that it helped them when I could explain a bit what's behind the "recipies", i.e., about what's the actual meaning of the operations we had to do, and it's of course the very nature of math to have clear concepts and structures in terms of axioms, definitions, and theorems. The mechanical solution of boring problems without having this understanding is pretty pointless.

 

[Mark44]

For the reply to Mark44.

Capturing variation is literally that - capturing complexity. This is what mathematics does - it captures complexity and tries to organize it in the best way possible to make sense of it. You can call it variation, complexity, abstractness - even entropy - they all mean the same thing.

Mathematics has variation through variables primarily. The whole point of mathematics is allow one to take this complexity and do two things - organize it effectively and make it consistent. This is all mathematics is - an attempt to take higher and higher levels of complexity and make it consistent.

The way it's done depends on the field. Probabilities do it for probability - calculus does it for derivative and integrals, geometry does it for distance and angle, topology does it for continuity, and other fields do it in their respective way.

It's the ability to be able to translate between different situations and relate that information to something consistent and organized that facilitates understanding and critical thinking.

Knowing sines, cosines and tangents is useful for geometry and I would expect physicists and engineers to be able to recite results and apply it.

But knowledge by itself and the recall of facts means nothing with the relations that come with it.

A student in high school is not going to see that mathematics helps with understanding complexity. They are going to see a bunch of random examples from two dimensional geometry, some algebra, polynomials and they are going to not appreciate that mathematics is used to make sense of complexity - something everyone faces in their daily lives.

I agree to some extent with what you're saying, but if what you're saying about understanding complexity comes at the expense of having a solid foundation in basic skills such as the ability to do arithmetic and solve simple equations in algebra and trig, then I strongly disagree. "Capturing complexity" and "capturing complexity" are ideal notions, but for students in K-12 grades, these ideals are worthless if the students aren't able to summon the skills to complete a problem of the type I describe.

They don't see that complexity entails all sorts of information whether that information be about what home loan to get, how to make sense of lies and evaluate them for consistency, or being able to accurately read statistics, fractions, and other claims via information.

My concern is for US high school graduates who are completely unable to do arithmetic, or perform calculations with fractions, let alone do calculations that involve statistics.

I taught mathematics for 21, of which 18 years were at a community college. Of the math courses we offered, the vast majority in any given quarter were remedial classes for students who had graduated from high school, but were not proficient in high school-level mathematics, and too many weren't even proficient in the type of arithmetic that is usually taught in 5th grade or so. Your talk of "capturing complexity" would be completely lost on most of these students.

The ability to extract the useful attributes of complexity and make sense of the consistency is what real mathematics is about.

Any bozo can memorize things and learn to recite them - it's the ability to deal with uncertainty that is more impressive.

Are you asserting that it's not important to memorize anything? Being able to make sense of uncertainty is important, I agree, but being able to do so without having a set of fundamental skills is not possible. This is like a carpenter showing up at a job site without his tool box.

It is a survival skill to be able to face uncertainty and deal with all of the BS that is faced in the world for some reason and another.

The best thing that anyone can do for any set of logical relations when it comes to evaluating them is to evaluate them for consistency - and mathematics is the primary area of knowledge that evaluates consistency.

You may think I'm talking about the axiomatic pure mathematic logic stuff but I'm not.

The ability to show inconsistency whether it's via predicate logic, statistics, sets of linear equations or inequalities, optimization or any other thing is something that allows a person to deal with uncertainty in the best way possible.

The arguments of science use it and certainly engineers use it for the same purposes - but this is not the only forte of using mathematics and consistency to evaluate things.

People are bombarded by claims, information, results, and they often have no real way to deal with it. If they understood what mathematics actually is as opposed to what they think it is they would probably be a bit more interested because they would realize how useful it is to making big life decisions and being able to find a way to apply consistency to an unfamiliar situation - something I think many parents would want their children to be able to do.

 

[Mark44]

Also for Mark44.

You mention the ability of doing specific things and drills and the reply to this is that you have to specify what your goals are.

If you want to be able to play a couple of songs really well then you will allocate your activities accordingly. Perhaps learning specific chords and transitioning between those chords becomes a large point of focus on how you train to achieve those goals.

As a session musician you will probably have a very rigorous training program where you learn all the theory behind common genres, how music is built within those genres, timing, scales, and some common patterns in them. Effectively you train though to be able to read and play music effortlessly so that when a piece of sheet music is placed in front of you - you play it to specification.

But think about if you want to freely improvise - like say a jazz musician.

The training becomes completely different. The classical musician has a focus on certainty and for them it is not expected for them to improvise pieces. It would actually be extremely uncomfortable for many classical and session musicians to do improvisation because they have not trained themselves to do that.

Do you actually play an instrument? I ask because what you say suggests to me that you don't. Musicians of all kinds have to spend a lot of time on learning basic skills before their playing becomes fluid.

You have to ask yourself what the goals of learning actually are. For the musician are you training to play a couple of songs? Become a session musician? Become a jazz musician with good improvisational skills?

You seem to be saying that classical or session musicians are unable to improvise, which further convinces me that you do not play any musical instrument.

The same questions exists with mathematics - are you trying to learn how to solve basic geometry problems? Trying to learn to build structures in carpentry?

Or do you want to be able to make sense of complexity in a more meaningful way where you apply your understanding of consistency and existing concepts in mathematics to make better sense of it - and more importantly when you haven't specifically trained beforehand to do so?

My claim is that you will never be able to make sense of complexity without a solid foundation on the basic concepts of mathematics: arithmetic, algebra, geometry, trigonometry, calculus, statistics, linear algebra, and differential equations.

Answering that question will be necessary to decide what sort of answer exists for the individual and the collective regarding mathematics education.

You have to answer what the goals are and how you intend to reach them.

 

[bballwaterboy]

For those who spoke of other countries doing better at math, are you referring to PISA scores?

If so, aren't those scores misleading, because of the ways that some countries track poor performing students into vocational education? Those who go to trade school don't end up taking those PISA tests.

I thought I read a few years ago that the U.S. is actually doing just fine in math after you factor this into the equation.

In any case, I have also heard that something called Singapore math is supposed to be very effective at teaching math to kids. Don't know anything about it, but just remember seeing it on TV on some episode of like 20/20 or whatever.

 

[vanhees71]

Well, for Germany I can only say that the math education at our high schools is on a down-hill slope for decades now. In our physics curriculum at the university we try to help students as good as we can with an introductory crash course and special additional math lectures in parallel to the theoretical-physics lectures. Other faculties do the same. It's really strange that particularly math is covered so badly at our high schools!

 

[Mister T]

This would contradict the "US has the best of everything" axiom.

That's a far too simple explanation of why the US doesn't adopt teaching styles used in other countries. First, it's understood that the other countries are doing a better job, otherwise we wouldn't be asking the question.

The reasons vary, but usually what's happening in other countries is a sorting process where under-performing students are placed into other programs of study where they have a better chance of succeeding. In the US we used to do that. We called it tracking. An attempt at re-introducing such a scheme in the US is immediately met with fierce resistance. There is no way a school board in the US would introduce such a scheme. We have a culture that promotes "education for all" and our society simply will not tolerate the tracking of students into different programs of study based on their performance. It is, however, the only way we'll ever promote the sense of personal and parental responsibility necessary for improvements to the system. When students fail we don't blame it on the students or the parents, we blame it on the teachers. And our solution is to micro-manage the teacher to the extent that very little real teaching occurs in far too many classrooms.

 

[gleem]

Google "teaching math in other countries" There are some answers there.

 

[chiro]

I play a few musical instruments.

I understand that you have to learn scales, do drills with finger work, learn about chords, keep precise timing and do a range of things. It takes a very long time to have these things become automatic so that you are thinking more about the structure of the music and the overall groove rather than where to put your fingers. After a while you don't need to remember chords because you aren't thinking about chords and I would imagine many people who have done something for a very long time are the same way.

It is like many things - including driving a car. At first you focus on the pedals and the gears and eventually you are able to navigate the road and plan your destination (and way of getting there).

The thing though is that high school is limited in its time for particular subjects and becoming good at playing an instrument takes many years of concentrated effort and often a lifetime to attempt to master (and there is usually some element you won't master). Engineering takes four years for basic education and then you actually have to learn to do it in practice.

In contrast to the above, high school allocates a small section of time and energy to mathematics and combined with the problems that teachers face (like babysitting students as many make their lives hell) you don't end up doing all that much. It's not like a craft where there is continuity and specific direction like you would find in a serious music education, engineering education, or other similar sort of training.

I'm not saying that memorization doesn't have its place at all. I'm also not implying that you don't need a foundation for things - everything has a foundation. I wholeheartedly agree that a lot of boring stuff needs to be done before more complex tasks are done in any endeavor.

The problem though, in my opinion is that their is a big focus on specifics without that ability to capture variation in the long term - and one reason for this has a lot to do with the time people have and the motivation they have to learn it.

I agree that arithmetic should be prioritized over the complicated notions of looking at complexity - even though it is actually one of the best ways to understand complexity since so many things can be expressed as one dimensional quantities.

I also agree that engineers and scientists should learn what they learn because it is specific to their job and likely a requirement to be able to do their job in a reasonable capacity. This education for these people is a specialist education since they are performing specific tasks.

What I don't agree with though are the classes that focus on lots of unconnected things (in the eyes and minds of many students they are boring and don't make sense).

For those who don't go into science, engineering, mathematics and analytic fields I think these people would be better off understanding how arithmetic captures variation than learning about the quadratic formula or even Pythagoras' theorem.

The example of arithmetic is actually the most important form of complexity there is. The fact that the basic laws all balance out is critical for people to do commerce and make sense of many natural phenomena.

Even putting fractions, averages, standard deviations and other simple things in context would in my opinion be far better than looking at quadratic formula's, locus and directrix, Pythagoras' Theorem and even the introductory calculus.

This ability to make sense of this sort of information - which is a lot more basic than the stuff mathematicians, scientists and engineers often study (and have to study because of their specialization) is a useful life skill since people are bombarded with information.

 

[Student100]

I'm perplex about what "laws" will change or not be true.

The majority of them? Especially true in physics, in which nothing is absolutely true and merely an approximation of truth; as such, anything is only "shown to be true" down to the constraints on the model or experimental accuracy.

I don't believe that changing the word "law" to "property" or "principle" (or whatever) would decrease the confusion. The problem is, I believe, in not listing the conditions under which the law can be applied. Newton's Second Law of Motion is F = ma, which is not valid for an object whose mass is changing.

In any case, there are many laws that are named after the persons who discovered them. See the wiki article for a long list of such laws: https://en.wikipedia.org/wiki/List_of_scientific_laws_named_after_people. It makes no sense to me to revise history by renaming, say, Kirchhoff's Law or Boyle's Law.

It seems to me that your disagreement is due to the way these "laws" are presented (or misrepresented) rather than with the use of the word "law" per se; i.e., without any "fine print" giving the limitations. One example is saying that $\sqrt{ab} = \sqrt{a}\sqrt{b}$ without also listing the restrictions on a and b.

Also, do you have any evidence that she is inaccurately conveying these laws? Just because some teachers and some references are sloppy doesn't mean you can extrapolate this sloppiness to every teacher.

I don't have any evidence, I was merely making an assumption. My mistake.

That is my major gripe, I think we agree to some extent, but disagree that simply changing the word might ease confusion. My opinion is that it would - at least - not lead to a false sense of security or misunderstanding that the "laws" of science and mathematics are somehow universally applicable or somehow fundamental to nature itself.

People can't read and look up the nuance when they get more mathematically mature?

Sure, considering they go on to become mathematically mature. Many don't. If I could, I would replace the words "law" and "theory" with principle and model in the sciences. The former words have too much baggage to be salvageable.

 

[Inventive]

My father had a PHD in mathematics and he would explain by concept and bring previous math concepts into the current level of math he was talking to me about. That helped me learn math by concept and not by memorization. How many students can take a problem in one form that is difficult and see that an equivalent form is easier? For example isn't that why Laplace transforms are used in some differential equation problems? I am not sure about anyone else, but has anybody gone to a math class and got the sense that the instructor is teaching right out of
A book? Those folks who tried to teach me this way made me feel less confident in what they know vs the teachers who made up problems as
they went even if they made a mistake and corrected it. Thats human nature and i feel confident in those types of instructors

 

[Jeff Rosenbury]

My father had a PHD in mathematics and he would explain by concept and bring previous math concepts into the current level of math he was talking to me about. That helped me learn math by concept and not by memorization. How many students can take a problem in one form that is difficult and see that an equivalent form is easier? For example isn't that why Laplace transforms are used in some differential equation problems? I am not sure about anyone else, but has anybody gone to a math class and got the sense that the instructor is teaching right out of
A book? Those folks who tried to teach me this way made me feel less confident in what they know vs the teachers who made up problems as
they went even if they made a mistake and corrected it. Thats human nature and i feel confident in those types of instructors

Unfortunately most people don't seem to think that way. This leaves a severe shortage of good teachers. Deep understanding often seems lacking, and it is hard to test for (at least honestly). Combine this lack with the societal desire to give every child an equal education and we run into roadblock. It is (was?) simply not possible to teach everyone that way. (Thank your dad for me, BTW.)

Perhaps AI will solve this problem? AlphaGo claims deep understanding.

 

[gleem]

How come no one is looking at how those countries are teaching their kids in math?

I suppose you mean why don't our educators look at other countries. Apparently they do but is seems to me that their methods are not amenable to our teaching culture.
I looked into Finland for example. since they are recognized as the best or one of the best European countries for math, science, reading as judged by PISA scores. The best in the world are Asian countries. Finland's system is not compatible the current US educational culture

Give her the support, the respect, the tools and she becomes a life changer for these kids.

I doubt it has much at all to do with her teaching style, and more to do with the fact that she has dedicated, one-on-one access with the under-performing students

Finland's teachers held in high esteem and are paid well . It gives them extraordinary control of their teaching methods including one on one teaching solutions, and curriculum development.

We attempt to hold our teachers accountable by dissecting their subjects into pieces, followed by measurements of how well students perform each piece.

Finland's teachers are extraordinarily qualified each teacher is required to have a MS. Regarding teacher accountability it is not an issue. they are given great leeway in their teaching techniques, student evaluation and school autonomy.and thus great responsibilities which they have accepted for they have a vested interest.

"There's no word for accountability in Finnish," Pasi Sahlberg, director of the Finnish Ministry of Education's Center for International Mobility, once told an audience at the Teachers College of Columbia University. "Accountability is something that is left when responsibility has been subtracted."

Finland has focused on equity in education making all schools equally strong accessible to all instead of using the US model of having schools compete against one another and creating a few good schools that most children cannot get into.

I have also heard that something called Singapore math is supposed to be very effective at teaching math to kids.

Singapore in recent years has revamped its educational system. In the beginning it introduced "tracking" during the first six years of education placing students in programs suitable to their academic abilities. more recently they emphasize creativity and school autonomy.

From http://www.oecd.org/countries/singapore/46581101.pdf a OECD report © OECD 2010 Strong Performers and Successful Reformers in Education: Lessons from PISA for the United States

. The Singapore approach to mathematics is distinctive and has become well-known because of Singapore students’ success. Developed in the 1980s from reviews of mathematics research around the world, and refined several times since, the Singapore national mathematics curriculum is based on the assumption that the role of the mathematics teacher is to instil “maths sense”. In a Singapore classroom, the focus is not on one right answer; rather the goal is to help students understand how to solve a mathematics problem. The Singapore “Model Method” also makes extensive use of visual aids and visualisation to help students understand mathematics. The concrete-pictorial-abstract model used is based on an understanding of how children learn mathematics rather than on language considerations. Teachers cover far less material than in many other countries, but cover it in depth: the goal is to master mathematics concepts (Hong et al., 2009). The level of mathematics in the Primary School Leaving Examination (grade 6), is approximately 7 Singapore: Rapid Improvement Followed by Strong Performance Strong Performers and Successful Reformers in Education: Lessons from PISA for the United States © OECD 2010 169 two years ahead of that in most US schools (Schmidt, 2005). Singapore mathematics also blurs the distinction between algebra and geometry. These concepts are integrated into basic mathematics instruction before students reach high school. Singapore teachers are all trained in how to teach the national mathematics curriculum and meet regularly to fine tune exercises and hone lessons..

Finland and Singapore are small compared to the US and they tend to have a more homogeneous culture as well as a more progressive attitude toward education. It would seem to be easier for them to revamp their educational system. But Finland population of 5.4M is as big or bigger than 30 of our states and some way bigger. It would seem from what I have read that state control of the education systems would be a good start. that does't look like it could happen any time soon.

 

[ohwilleke]

First, a side point. "Copyrighting" something is pretty much meaningless. A patented idea must be useful and novel. You can't copyright an idea, only an expression of an idea and pretty much anything you write is automatically protected by copyright and merely has to be registered before you bring a lawsuit based upon it.

Second. If you have a kid who is underperforming academically into late elementary school or beyond, it is much easier to bring that kid up to grade level in mathematics than it is in most other subjects because almost all other subjects (not just English, but social studies and science as well) mostly involve reading and writing.

Reading and writing are skills that are optimally learned as first languages at a very young age and involve very large universes of knowledge. Grammatical rules, for example, in practice, have far more exceptions than either teachers or students consciously realize and are actually learned mostly by example and not by logically and consciously applying grammatical rules to potential sentences. You can perfectly master every rule taught in a typical English grammar textbook according to its terms and still be incapable of communicating fluently in the idiomatic English of a native speaking of standing middle class American (or British) English.

Moreover, a large share of students who are underperforming in these areas either grew up learning a language other than English as a first language, (31% of students in the Denver Public Schools that my children attend) or learned a dialect of English other than the standard middle class white (Northern) American or (Southern) British dialect of English (e.g. African-America, South Asian, Southern, Appalachian, urban Scottish, some New York City dialects, or working class London dialects like cockney) (at least another 20% of students in the Denver Public Schools). So, getting a kid up to grade level in these subjects involves not only learning new information but also unlearning the child's native dialect or language in a way that also drives a wedge between parent and child culturally.

In contrast, the core set of information that you have to learn to master mathematics at the very modest levels considered to be grade level in late elementary school, middle school or early high school is much smaller, has far fewer exceptions to the general rules, and is learned in a part of our brains that is not so extremely biased towards acquiring information at a very young age.

Furthermore, while an underperforming math student may have to learn new things, it is very rare that an underperforming math student will have affirmatively learned many incorrect mathematical principles that must be rejected with cultural consequences before that student can learn standard ways of doing math. The rare student who has learned an alternative approach to the mainstream one to doing math will almost always be performing at above grade level, rather than below it.

Also, because effective math instruction is so sensitive to the order in which concepts within the field of math are taught in a way that most language based subjects (even science subjects) are not, the negative educational impact of being placed in the wrong level of math class for a kid's abilities are much much worse than the negative impact of being places in the wrong level of a language based subject. A kid who is a year behind his peers in an English class will still learn a lot from the class readings and discussions even though it won't be optimal. A kid who is a year behind his peers in a math class will learn virtually nothing because he or she doesn't have the necessary foundation to learn the concepts that are being taught.

Accurate diagnostic tests and fine grained tracking of mathematics instruction is critical because if you can accurately place an underperforming child at exactly the right point in the mathematics curriculum (which often won't be shared by many of his or her peers) you bring learning per session from 0% to normal almost instantly. Learning also drops to almost 0% when material that has already been mastered is taught.

Drilling, concepts or anything else, in math, can be fruitful, but only if you are in the sweet spot of material that has not been mastered but does not have any prerequisites to mastery that have not been mastered. People hate drilling mostly because they were drilled on topics after they had already mastered them (resulting in near zero learning), or because they were drilled on topics that they only learned by rote and never really understood (resulting in lots of errors and near zero learning because the student doesn't understand the problem). Drilling on math concepts that are at the right level of instruction for the student is still work, but it isn't awful drudgery, and because it assures mastery later on, reduces the need for time consuming review later on and makes learning subsequent topics go more smoothly, so it can be efficient in the long run.

Because the amount of information necessary to go from say the 4th grade level to the 7th grade level in mathematics is pretty modest, it doesn't take a hell of a lot of focus and discipline to make progress once the student experiences the joy of being taught at a level that the child can understand but hasn't already mastered. The sum total of what that kid needs to learn to catch up from being three grade levels behind in mathematics fits in three late elementary school sized textbooks (which aren't very big and have big print and lots of pictures). Once you consolidate that material to remove review of the previous year's studies and repetition of concepts due to imperfect coordination of the curriculum at different grade levels, you are down to one and a half or two elementary school sized textbooks worth of material that you must teach to the kid for him or her to advance three grade levels.

This can be done simply by spending 1.5x to 2x times as much time on math as a typical student at that grade level. Since a typical student at these grade levels is spending perhaps 100-200 hours a year on mathematics instruction and homework, it takes only about 50-200 extra hours a year (2-8 hours a week) to make multi-year progress in one's grade level in mathematics if a student has the necessary staff support and is being taught at the right grade level. The is manageable without totally changing the rest of the kid's life, through before or after or free period tutoring and a few hours a week of summer school.

Also, because mathematics is such a focused area of inquiry, the amount of vocabulary and grammar that must be learned to profit from mathematics instruction is a very modest percentage of the total body of language knowledge that is needed to perform at grade level in reading and writing. So, even if a kid is lagging in language skills, he or she can still learn math. You do not have to be fluent across the board in standard middle class American English to understand mathematics instruction, and lots of the vocabulary and grammar that is pertinent to mathematics instruction (e.g. understanding the words for numbers) doesn't vary all that much between widely differing dialects of English.

In contrast, in language based subjects, lots of what you need to know isn't found in a few textbooks, because a lot of language mastery comes from immersion in an environment where people are speaking, reading or writing much more than half of their waking hours every day, so spending 1.5x to 2x times as much time on language mastery as a typical student simply isn't possible.

The flip side of mathematics, however, is that pretty much the only way that it can be learned at even the modestly advanced levels of middle school mathematics is through intentional instruction from someone who has already mastered the subject-matter being taught.

Also, it is easy to get off track in mathematics and once you get off track you are doomed to learn almost nothing at all until intensive personalized instruction gets you back on track. If you change schools mid-year to a school that teaches math subjects at your grade level in a different order, you may miss critical skills needed to understand instruction in the following year. If you are sick for a couple of months (or miss class due to disciplinary issues or to care for a sick sibling or parent or extended family member) and don't make a concerted effort to catch up immediately, you will have to repeat the year of instruction in misery with younger smarter at math kids who aren't your peers, while spending most of the year on boring and useless review, or will advance with your peers and learn nothing in the next year because you lack the prerequisites. If you only earn a C or D in a math class, you are pretty much guaranteed to fail the next math class that builds on your current math class; gaining anything less than complete mastery from a math class is basically worthless. If you have ADHD which makes it hard for you to sit still and listen during lectures in a math class, you will fail that math class and every math class that follows. If you have a lousy teacher in a math class who advances you to the next level with a tolerable grade without actually teaching you the material, you are doomed to struggle and probably fail the next year no matter how good the teacher who is trying to teach at grade level is in the following year. For example, changing schools midyear harms a student's math performance more than it harms a student's language arts performance. http://uknowledge.uky.edu/cgi/viewcontent.cgi?article=1180&context=gradschool_diss

In a typical central city school district like the Denver Public Schools which my children attend, the percentage of students who face one or more of the impediments above or are simply assigned to the wrong level mathematics class by school officials and can't promptly correct the inaccurate assignment is a very large percentage of the total student body. For example, seventy three percent of DPS students qualify for Free and Reduced Lunch, and poor students are much more vulnerable to occasional disruptions in their educations at some point over twelve years than more affluent students. Eight percent of students change schools midyear in DPS as a whole, but in some schools in DPS as many as one in six students do. And, the percentage of students who change schools at least once or twice during their school careers is much higher.

In contrast, you can totally screw up iambic pentameter during a Shakespeare unit and still perform just as well in the short story unit that follows, or completely fail to grok the Russian Revolution and still not be at a disadvantage in learning the causes of the American Civil War.

If your parents were also bad at math, you also can get almost no help from them with your homework, which is rarely the case in language based subjects.

So, prompt school initiated intervention when a kid stumbles in math in an environment where parental instruction or parent funded and driven tutoring and stable lifestyles aren't like to make math missteps rare and prone to self-correction is also crucial. Poor performance in math needs to be addressed decisively the very week or month that it happens, not once final grades for a semester or even midterm grades are assigned, or the long term costs will be huge.

You can master reading and writing through osmosis from your peers and pleasure activities without ever having to really consciously study after the early elementary school years, and this is precisely how a lot of kids do master reading and writing and other subjects in practice. But, you can't do this with math.

So, a kid who is not connecting with teachers in the course of the classroom experience not only will fall behind grade level in math, but will learn absolutely nothing and furthermore be rendered incapable of learning math in ordinary classroom settings in future years as a result. But, a kid who is not connecting with teachers in the course of the classroom experience in language mastery based courses will just fall a little behind each year because he or she will pick up a lot of the concepts and vocabulary from his or her peers unconsciously by interacting with them, even if he or she gets nothing out of the classroom experience, and will be able to pick something up if that child reconnects with the classroom experience in later years.

Similarly, if a high school graduate is admitted to an open admissions college but is not ready for college level reading and writing, it is almost impossible to bring that student up to speed in enough time to prevent that student from dropping out of college out of the frustration of not being able to take any of the college level classes being taken by his or her peers. It could easily take several years for a student needed remedial reading or writing upon entering college to reach freshman in college level reading and writing skills, if the student ever acquires them. And those skills are needed in almost every class.

In contrast, if a high school graduate can read and write at a college level, but is a year or two behind the college level in mathematics (i.e. finished only Geometry or Algebra II in high school), it is much more doable to get that student to master a year or two of high school math while otherwise taking college level course work with his or her peers, over 12 months or so. And, since most college students outside STEM fields only take one or two years of mathematics in college itself, an incoming student who is a year or two behind in math upon entering college can still complete those years of math at the ordinary high school pace, plus one year of college level math at ordinary college level pace by the end of two or three years (or less with summer school).

Because math is much easier to make progress in than other subjects for students who are behind grade level is also a natural place to being to instill earned confidence based on real achievement and not just hype, and to develop hope in academically underperforming kids that they are capable of functioning academically. This, in turn, can help them summon the drive to take on the much more difficult and incremental task of trying to progress in reading and writing at more than one grade level a year, even though it may take several years of disciplined work there to catch up by even one grade level in that part of their studies.

In short then, a math first approach that starts with intensive mathematics instruction for kids who are academically below grade level, is an excellent approach to take for reasons that are deeply and fundamentally related to the intrinsic differences between learning math and learning other subjects.

Also, a math first approach teaches otherwise academically underperforming kids something that will be useful to them in life, even if their academic performance in other subjects never catches up to their peers and they don't continue their schooling past high school (as will often be the case). While one can doubt the practical benefit of learning calculus or more advanced mathematics outside STEM fields that require college educations, the practical benefits of lower level middle school and high school math skills in daily and professional life are significant.