<p>This is one of the most interesting and affecting pieces of education journalism I have ever read: an argument that American theorists have taught the world how to teach math better, only to see a near-complete failure to take advantage of their insights here in the U.S. The main culprit: an almost complete disregard for the idea that teachers have to be taught how to teach, and have to upgrade continuously their understanding not only of their subject matter but also of how to teach it effectively. One of the things that distinguishes successful education systems from ours is that in the successful systems teachers spend much less time teaching and much more time learning -- in Finland, teachers spend only 600 hours/year on classroom instruction, vs. 1,000+ hours here, despite their higher pay. In math, the teaching corps is deficient in its understanding of the subject matter, but absolutely clueless about how to teach math effectively. </p>
<p>The result is tragic: people who are capable of solving complex math problems efficiently in their daily lives cannot solve the same problems accurately if they are presented as math problems, and we perpetuate a myth that math is unnatural and hard to learn, and that students should be expected not to get math. Parents, administrators, and politicians, like teachers, also misapprehend what math instruction should be, and also need to have their expectations changed.</p>
<p>Politicians and school administrators ought to be required to pass a test on this in order to retain their jobs. (That was a joke, btw.)</p>
[QUOTE=JHS]
The result is tragic: people who are capable of solving complex math problems efficiently in their daily lives cannot solve the same problems accurately if they are presented as math problems, and we perpetuate a myth that math is unnatural and hard to learn, and that students should be expected not to get math.
[/QUOTE ]
And, of course, this extends to all the math applications, like Physics. Very interesting article.</p>
<p>I read the article and thought it was more of the same. More defense of teaching kids to “think” and “understand” instead of mastering at least one way of solving problems. It gets a little traction because it hangs its hat on an asian example. </p>
<p>You can find all you need to know about math education in the US if you go to a fairly good school and take, say fourth or fifth graders, and smilingly ask them (individually) “what’s 7 times 9, or 4 times 7”. If you have the experience I had some years back when my children were around that age, even the brightest kids were remarkably slow at simple multiplication facts. They’re all doing some mental form of counting on their fingers instead of recalling the (only) correct answer. </p>
<p>The result we have in math instruction is analogous to what we’d have in sports if we taught throwing and catching baseballs by having the kids “understand it” instead of just repeatedly doing it. (I think the Mets used that method the first year they came into existence.) </p>
Most of the article I’ve seen before, but this was new to me. Our school seemed to come up with a new math curriculum every few years, the last of which in elementary school incorporated most of these ideas. They spent a lot of time training the teachers and also training the parents. No one gave up completely on getting kids to memorize their times tables - though I will admit, that my younger son was never very good at memorizing math facts. (He might well say - it’s 7x10 - 7) That said, he’s much better at doing math in his head than I am, and his pre-calc teacher in a college letter of recommendation commented on how well he thought mathematically. (I think after he couldn’t remember the Pythagorean theorem and had to figure it out from scratch on an exam.) </p>
<p>Sorry to say this, dadx, but that is a profoundly wrong statement. Mastering one way of solving problems isn’t “mastering” anything at all, especially if it’s the one way your teacher has taught you. And memorizing times tables isn’t mastering anything, either. We seem to devote more time to that than systems that do a better job teaching math, precisely because so many parents, teachers, and principals think it means students are learning something, and it doesn’t.</p>
<p>All I know is that SOMETHING is not working. Our district has tried several math curriculums. And the 9th grade teachers are still tearing their hair out. Kids are coming to high school not knowing how to work with fractions or even divide numbers! The teachers have to take a break from what they are SUPPOSED to be teaching and devote time to remedial math. It’s nuts.</p>
<p>The teaching/learning model is an obvious part of the issue. The basic issue in the United States is that we expect people to teach subjects they do NOT master fully. The changes in a country such as Finland are not only based on overhauling the model of education but mostly on having changing drastically how they select, recruit, and train their aspiring teachers. They start with the BEST students and let them develop into generalists or specialists to cover DISTINCT parts of their curriculum. We, in turn, mostly start with the worst students, feed them pedagogical science of dubious value, and hope they will acquire the necessary skills to teach at the lowest levels, and then grab a Master’s degree at one of those awful schools of education and become a leader. The mastery of subjects is an afterthought as there are no quality requirements of any value. Nor much professional testing in a world where tenure and longevity are the yardsticks of choice. </p>
<p>Repeating the above comes with perils as most do NOT like to see criticisms hurled at our teachers. And for good reasons as most of them are decent people who are victims of a system that has deteriorated for six or seven decades. Yes, the profession USED to be full of very smart and well-educated teachers who happened to have the wrong gender to enter different professions. That has changed and our system never adapted to replace the next generations of teachers. </p>
<p>The lacking “think” model is also a fruit of the degradation of teaching “excellence.” Again, one cannot teach what he or she does not do well. Our system is based on regurgitation of facts and rewards the students who go through the motion. Take a look at our school books and at the Teacher’s Manuals … without the latter, chances are that the teachers might not grade the homework properly. Reasoning and thinking skills are rarely taught as they do not fit well in our curriculum. And that is why tests such as the SAT befuddle so many of our teenagers who require specific “delearning” and relearning of basic concepts. </p>
<p>In the meantime, the US will remain a country where the very best professors and students are world leaders in their field, but where the rest is hopelessly behind the ROW in terms of education, erudition, and culture. </p>
<p>But we will always retain the highest self-esteem in the entire world. And that, by itself, is the biggest handicap we have to understand how deep and wide our failures really are. </p>
<p>If I may interject something here … part of the problem is that many believe memorization and thinking are not related and come from different fields. In order for one to think and reason “well” he or she needs to have learned the basics and memorized the important ones. This is necessary to apply logical skills to determine which basic knowledge has to be applied. This is where long term memory and short term memory differ. </p>
<p>The above is evident when presenting a problem in a form of an equation to some and the same problem with WORDS to others. Almost all students in the US find it easy to solve the equation and usually can show the step by step approach. On the other hand, the addition of words that describe the same basic math causes many problems with many abandoning or guessing a possible answers. </p>
<p>The problem is not that students had to memorize “stuff” like formulas or tables. The real problem is that they never understood when and where that information would have to be used in the future. They are often allowed to forget and move to the next set of information after the final exam! </p>
<p>It is nuts because districts and teachers believe that there is a solution via the latest fad. That is why they jump onto the “Singapore” or “German” new math and hope it will stick. The problem of your 9th grade teacher is not the system … it is the 8th and 7th grade math teachers who did not plug the holes in the knowledge and sent the students upwards in dire need of remedial classes. And chances are that remedial classes were needed in 5th and 6th grades as well. </p>
<p>Obviously, it is not the sole fault of the teachers to struggle with changes. When districts change direction, they might consider a simple step: testing their own teacher and see if they could pass the basic requirements of the material they are expected to teach. </p>
<p>All in all, the teaching of subjects such as math never required drastic changes, Children had few problems learning math a century ago. Take a look at the entrance exams for Harvard in the 1900s and compare that to our current SAT or ACT. It is laughable. Students have not gotten dumber, The system has simply forgotten what it takes to be able to teach the young. </p>
<p>One of the most serious issues we face is that principals, supervisors and administrators don’t understand these curriculum changes enough to make appropriate placement decisions. My college roommate was a veteran first grade teacher, the teacher everyone hoped that their children would be assigned to. Several years ago, she left school at the end of the semester only to be told during the summer that she had now been assigned to be the match specialist in her building, very large elementary school (K-6) with over 700 students.</p>
<p>No additional training provided in how to “work” along with classroom teachers on multiple grade levels as well as supervise testing for all incoming students to determine their skill levels, provide individual tutoring to those in need as well as create supplemental lessons for those on the low end/skill levels as well as the few truly gifted who were capable of middle school math. Had she been a math specialist or received any extra math training… no. Her principal’s response was that her license in NYS was K-12 and he could place her where most needed. While she did have to quick-learn a good deal about elementary math in order to accomplish her job, her expertise in creating an outstanding first-grade classroom/learning environment were not utilized. She had to work alongside colleagues who were happy to collaborate and others who were happy to sit back and let her take over. She wasn’t a supervisor but all of sudden she was responsible for outcomes that in many cases were beyond her control… disorganized classrooms, indifferent faculty - many who she had worked beside for close to 20 years. She couldn’t run to the principal with complaints all day long as that would not have earned her either willing close cooperation among her peers. It was quite stressful and in her case led her to retire a year or two earlier than she initially intended. </p>
<p>I am willing to be enlightened about what it is about math that needs to be “mastered” by the overwhelming majority of students beyond being able to do computations (correctly and quickly). To me, being quick with your brain in math is as valuable as not having to carry around a dictionary to understand speech. We know that counting on your fingers works (within limits), and so do the “manipulatives” or whatever they call the counting sticks that are used. But producing kids who can’t quickly answer multiplication problems, or add columns of numbers on paper, or do a seven digit subtraction problem means that those children are partly handicapped. </p>
<p>My personal experience is at odds with the idea that schools are spending time doing too much drill. On the contrary, you see very little of it. The kids who are born with the highest math ability can probably succeed in spite of the methods. But too much of what passes for math class in elementary school should be properly termed “math appreciation”, and it cheats the vast majority of students. </p>
<p>In most countries, developed and underdeveloped, this type of question:
"You can find all you need to know about math education in the US if you go to a fairly good school and take, say fourth or fifth graders, and smilingly ask them (individually) “what’s 7 times 9, or 4 times 7”.<br>
would not be addressed to 4th or 5th grader, but rather to a 1st grader. If you start teaching them at 7 and not at 5 when they should spend their time palying, and absolutely forget about stupid idea of “head start”, if you start teaching them at 7, they are capapble of learning that table of multiplication byt the end of 1st grade and they are capapble of understanding the concept of representing any number by a letter which gives them a tiny glimpse at algebra. Then in 5th / 6th grade, they actually should start algebra, and they should laugh at anybody who is asking them about product of 7X9 and 4X7. They will send to a bunch of the first graders to ask this question. They will have no difficulties telling you what 30% or 1/3 means (ask any sales person at the counter for 1/3 lbs of something, you will receive an empty stare that will prompt you to “correct” yourself asking for half a pound, that is how low it got).<br>
Well, in most countries, the 7x9 and 4X7 is not called math, it is called arithmetics. Study of percentanges, proportions, decimals and so forth are still arithmetic which is part of math that teach one about numbers without representing them as letters to bring the level to more abstarct form. And then, of course geometry. OMG, everybody is so scared of it and god forbid if they are required to proof anything, forget that one. And that is where the real critical thinking begins. Her you can see how variables are functionning together. That is when this critical thinking ability is killed because this process is not taught properly.<br>
Exactly right, "The problem is not that students had to memorize “stuff” like formulas or tables. " Math actually requires very little memorization. Anything that you had ve memorise actually could be picked up on internet easily. Absolutely vast majority of math you can derive from very few formulas and facts…that is if your brain is trained how to do it, if you see easily how one formula is leading to several others. This function is killled in American k -12, one reason is that if critical thinking is there, it is not that easy to brainwash such a brain, it will not believe anything that does not make any sense, which is not logical, like a current economic concept of money that grow on trees. Whatever does not add up mathematically is only obvious to those who can see that it does not add up and most would not question it. I guess, not knowing, being blind to the facts actually can make a person much happier. So, if the happy blind is the goal, then it explains very well everything about teaching math in k -12.</p>
<p>I taught “business math” (some forecasting, modelling, stats, decision theory) to a class of college students who were business students precisely because they didn’t like math. By and large, they approached a word problem (all of my exam questions were word problems) as “how do I plug some numbers into a formula I sort of remember seeing a problem about in the homework”. They had clearly never, in their 12+ years of schooling, been given the tools or techniques to see the logic behind what a question was asking. Math was foreign, strange, arbitrary, and “hard”. The common core has plenty of detractors, but I get what they are trying for. There is a subset of kids (maybe 10 percent), who seem to know how to do this intuitively, but I think the majority need to be helped to that intuition. </p>
<p>Are you sure you meant to say that kids should learn algebra in 5th or 6th grade? I attended a school I think was pretty advanced and I do not think anyone started with algebra until 7th grade. I also had the chance to study the curriculum of schools in Belgium (that has a pretty advanced math curriculum) and I did not see much algebra until 7th or 8th grade. </p>
<p>You might have experienced different math or a different definition of algebra, but I do not think that Algebra is or should be the domain of 5th and 6th graders. If they covered Algebra in 5th grade, what would they study in 9th grade? MV Calculus? </p>
<p>I believe that we ought to be happy if middle schoolers graduate with a good understanding of arithmetic, a bit of basic geometry, and the ability to solve basic word problems, and perhaps some pre-algebra. </p>
<p>I have no idea what to call what – I was never a “math person,” and math always seemed fundamentally stupid to me because so much of it consisted of memorizing formulas. In my school, we did arithmetic and word problems through sixth grade, and started on two-variable simultaneous equations in 7th grade. By the end of 8th grade, we were supposed to be (and I was) good at two-and three-variable simultaneous equations, basic geometry, basic probability, and basic financial math involving time-value. For the life of me, I can’t remember a thing I learned in math in high school, because my 8th grade math has been completely sufficient to handle fairly sophisticated business problems. (I got As in math and a 5 on the Calculus AP of my day, which was probably equivalent to AB; I just can’t remember anything I learned in that class. I remember the names of trigonometric functions, just not what they are or what you do with them.)</p>
<p>The problems described in the article are endemic to American education, not just math education. It’s not like English teachers actually know much about language or literature, and they don’t get ongoing training in how to teach, either. Lots of people are content with their children’s English classes if their children can spell and use standard grammar most of the time, know a bunch of SAT vocabulary, and write a formulaic 3- or 5-paragraph essay. And,. indeed, that would be a huge improvement for lots of kids. But that sure isn’t “mastering” communication and interpretation in English.</p>
<p>Fwiw, the article quoted by JHS does point out to the problems that still exist, but also seems to confuse the creators of the problems with the possible bearers of solution. There are no tangible proofs that the processes observed by Akihiko Takahashi were beneficial and that the US method would solve a problem. One could easily conclude that the findings of that “National Council of Teachers of Mathematics” have led to more problems (if not outright disasters like their 1989 suggestions) and derailed the creation of effective methods. New math, reform math, and similar changes have all led to the abominations found in our school books. 600 to 800 pages full of images and stories, and little substance. Perhaps we ought to copy of few pages from the Kumon books! </p>
<p>From the days of George and Frederique Papy to today’s new fads via that Japanese “guru”, we are none the wiser. And still trying to develop a system that requires little work, no continuous efforts, and no teachers who understand the material! The American dream! </p>
<p>Poland was recently cited as a country which gets outstanding results from its public school kids. Not only are the times tables introduced in 1st grade (although the kids are a year older than their American counterparts, by 1st grade most are 7) – so is the idea of X as an unknown. Algebra absolutely begins in fifth/sixth grade. As does physics. </p>
<p>xiggi, I don’t read many middle-school math books, but at least according to Green the problem isn’t that we have 800 pages of images and stories – the majority of the content is the same as it was 40 years ago with the mildest gloss of the latest fashion to allow salespeople to sell new books. What she identifies as the problem, at least, is the near-total disengagement of legislatures, school districts, principals, ed schools, parents, and teachers themselves from anything approaching an effort to learn how to teach new material. Which, of course, includes understanding the new material you are teaching. </p>
<p>Just understanding the material isn’t enough, though. At the college level, we are all thoroughly familiar with the cliche of the Nobel Prize winner who is a completely ineffective teacher. It’s not because he or she does not understand the material; it’s because he or she doesn’t know how to communicate that understanding effectively, and doesn’t care enough to learn. What stuns me a little is learning that no one in the education establishment is working systematically to develop and to teach better teaching methods. Teacher training is still done fundamentally on a sink-or-swim basis.</p>