I an Adult Basic Educator. I teach HiSET (formerly GED.) I formerly taught Stastics to undergraduates at Boston University. As the HiSET has become more difficult the designers of the test have made a fatal flaw and this has to do with the Time restriction on the Test. Even myself, when given a fairly complex problem like to take my time and check my answers. When given a time restriction many questions become rushed and are given to educated guessing. This promotes anxiety. For people lacking basic arithmetic confidencde. Fighting the clock takes away from what they know and just causes plain old anxiety. If you want quality work from students on an ever increasingly complex test -DITCH THE CLOCK!
70
I have two key questions about math eduation:
1) WHY don't we emphasize the importance of math, not just for solving math problems, but also for logical thinking, in general? We use logical thinking, constantly, even though it is not mathematical, per say.
2) WHY does math have to be increasingly difficult? Why not focus more on easy math problems to encourage students to appreciate simple math ideas?
I say, easy does it, to increase interest in math in our schools.
==============================================
www.SavingSchools.org
1) WHY don't we emphasize the importance of math, not just for solving math problems, but also for logical thinking, in general? We use logical thinking, constantly, even though it is not mathematical, per say.
2) WHY does math have to be increasingly difficult? Why not focus more on easy math problems to encourage students to appreciate simple math ideas?
I say, easy does it, to increase interest in math in our schools.
==============================================
www.SavingSchools.org
21
I think the focus on math is all wrong. I am appalled at the lack of numeracy in the majority of the population. Being innumerate is as bad as being illiterate but in fact, most people are innumerate, including those with good math skills. Get our kids out of school being numerate and I will be happy. They may never use algebra, calculus et al in their entire lives but being numerate will serve them for their lifetime. https://en.wikipedia.org/wiki/Numeracy
13
Math has a liberal bias.
Why teach math? Is it to liberate and empower the thinking of our future free citizens? Or is to train them to be obedient, useful tools in capitalist marketplace? Math (as arithmetic) is the first subject we children that shows them that authority can be objectively wrong. But this only happens when math is taught as showing reality rather as some game of symbolic representation.
Why teach math? Is it to liberate and empower the thinking of our future free citizens? Or is to train them to be obedient, useful tools in capitalist marketplace? Math (as arithmetic) is the first subject we children that shows them that authority can be objectively wrong. But this only happens when math is taught as showing reality rather as some game of symbolic representation.
6
Some years ago I was a graduate assistant in the physics department at IU. I was the lab instructor for the class “Physics for Elementary Education Majors.” As test time approached, the professor told me to conduct a review session rather than a lab. He also asked that I introduce Archimedes Principle, since he hadn’t gotten to it in class yet. Dutiful grad student that I was, I wrote up some notes deriving the relevant equations. (Obviously a stupid approach, but I was studying physics, not Elementary Education.) As I delivered the lecture, I was met by a classroom full of blank stares. I asked if anyone was following this and one or two students managed a barely perceptible shake of their heads in response. Discarding my notes, I managed to get the idea across by having them weigh some aluminum cubes in and out of water. Fortunately, this seemed to work.
Afterward I asked one of the better students what she found so difficult about my earlier approach and she said it was all that math. So I wrote the equation A = B*C on the board and asked her if she could tell me what B was equal to. After some hesitation, and in a halting voice, she managed to say: “Well I can tell that it’s multiplied by C.”
Math, at any level, cannot be taught by teachers with no concept of what they’re teaching. It’s not that they’re bad teachers (many of the students in my class were intelligent responsible students) they’re simply math phobic.
Afterward I asked one of the better students what she found so difficult about my earlier approach and she said it was all that math. So I wrote the equation A = B*C on the board and asked her if she could tell me what B was equal to. After some hesitation, and in a halting voice, she managed to say: “Well I can tell that it’s multiplied by C.”
Math, at any level, cannot be taught by teachers with no concept of what they’re teaching. It’s not that they’re bad teachers (many of the students in my class were intelligent responsible students) they’re simply math phobic.
83
Mr. Phillips punts on the topic by ascribing the conflict over math education to politics and power. Also, he does not pose the question, "Why are we thinking 'either/or'"?
Math is a multiform language that requires skills. It is easier to understand algebra and think algebraically if you have arithmetic skills. Arithmetic and algebra go together. Similarly, it is easier to understand calculus and think "calculusly" if you have algebraic skills.
The only math program in America that thinks "both/and" rather than "either/or" is Kumon, a private after-school math education program. The only problem with Kumon is that it requires daily exercises. Mon Dieu! Homework!
The price of "either/or" is that American kids do not know how to do math well. The price of "both/and" is homework.
This is the choice that American parents have to make.
(Disclaimer: I am both a certified K-12 math teacher in Massachusetts and a certified Kumon instructor.)
Math is a multiform language that requires skills. It is easier to understand algebra and think algebraically if you have arithmetic skills. Arithmetic and algebra go together. Similarly, it is easier to understand calculus and think "calculusly" if you have algebraic skills.
The only math program in America that thinks "both/and" rather than "either/or" is Kumon, a private after-school math education program. The only problem with Kumon is that it requires daily exercises. Mon Dieu! Homework!
The price of "either/or" is that American kids do not know how to do math well. The price of "both/and" is homework.
This is the choice that American parents have to make.
(Disclaimer: I am both a certified K-12 math teacher in Massachusetts and a certified Kumon instructor.)
21
We fail to distinguish between the ways in which math is used. For some, it's a language that opens up relationships and wonders not readily visible in any other way. For most, however, arithmetic and a rudimentary statistical understand suffices. Yet our democratic spirit pushes us to demand of everyone a level of mathematical competence that is higher than 99% of us will ever need, and lower than what the engineers, scientists, statisticians, and other STEM people need.
25
Of course no math of any stripe makes supply side economics work, does it? The complaints about math seem correlated with the abandonment of science by the republican party.
24
As a retired elementary math teacher, this is a subject near and dear to my heart. It boils down to competent math teachers in the classroom who know how and when to use all the various strategies and approaches. A teacher who understands the value of computational fluency (memorization and process skills) and the value of experiential and problem solving approaches.
However, this takes a talented teacher, and we aren't willing to pay for that.
However, this takes a talented teacher, and we aren't willing to pay for that.
34
One thing that would help improve mathematical proficiency is to require prospective teachers to take more courses in math than in "educational theory" when they are in college. There is nowhere enough subject matter in the schools of ed.
14
As an Algebra Core instructor I can assure the course was very well considered and constructed. Many years ago the TIMMS study tried to identify areas responsible for a severe disintegration in American Mathematical ability. The study compared many factors and ultimately was able to quantify several areas where we could improve our instructional product.
Preparing to instruct the Algebra Core curriculum was an almost impossible task in such a short time (about 1.5 years of preparation). I was amazed at how much I didn't know mathematically or had to relearn in a more applicable manner. The learning curve, for me, was VERY steep. But, after having seen the ability of my students from a year of pretty good instruction, I'm absolutely certain every effort was MORE than worth it. And that the Algebra course has the potential to develop some of the best thinkers/problem solvers we've seen in generations!
Preparing to instruct the Algebra Core curriculum was an almost impossible task in such a short time (about 1.5 years of preparation). I was amazed at how much I didn't know mathematically or had to relearn in a more applicable manner. The learning curve, for me, was VERY steep. But, after having seen the ability of my students from a year of pretty good instruction, I'm absolutely certain every effort was MORE than worth it. And that the Algebra course has the potential to develop some of the best thinkers/problem solvers we've seen in generations!
12
When I think of the time I wasted before college learning basic arithmetic and algebra I could weep. Does the average mind really move so slowly that it takes 10 years before it can countenance simple simultaneous equations? And it takes another year before it can think about calculus? Shame on our educators for capitulating to the lowest common intellectual denominator (whatever teaching fad they settle on). But wait -- that's the American way!
8
I know of several people who majored in math during college but who did not go on to teach at the K-12 level. The major reason was they could get much better pay in private industry or even teaching at the college level. In general, math teachers are paid the same as any other teacher but the market for their skills leads to higher pay in places other than schools. This is particularly true for high school level math. In essence, we get what we pay for. A lot of "math" teachers are not really that good at math themselves.
13
I wonder why schools insist on algebra and calculus when statistics, the time value of money and probability make far more sense, both for rational thinking and everyday decision making.
Doesn't it make more sense to have kids going to college who already understand credit cards, mortgages, investments and the real cost of student loans? Who can't be tricked by dubious political math and who understand the odds of getting into Harvard versus obtaining an equally valuable, far less expensive, degree at a state university.
Doesn't it make more sense to have kids going to college who already understand credit cards, mortgages, investments and the real cost of student loans? Who can't be tricked by dubious political math and who understand the odds of getting into Harvard versus obtaining an equally valuable, far less expensive, degree at a state university.
24
This discussion reminded me of a short story I read years ago - for the life of me I can't recall the title or the author - in which a teacher gave a wonderful little problem to his students. The problem is sometimes called "The Monk and the Mountain", a discussion of which can be found here: http://bredemeyer.com/RequisiteVariety/?p=154. In the story one of the students complains to his father that he couldn't understand the problem and the teacher's explanation didn't make any sense. The father can't understand it either, and being an aggressive sort confronts the principle of the school to complain; the teacher is eventually fired. The point: Yes, sometimes mathematics education is quite political, especially when it encounters fear or ignorance.
5
Sputnik generated a lot of interest for science and math during the late 50's and because a very successful family friend was involved in the very early US Regulus missile program guidance system development. So Purdue it was and ended up going for a math degree.
Had some great instructors who showed us how to use math for decision making, particularly when lots of different variables were inter dependent. Our exercises and tests consisted of solving real world problems - our only tool was a K & E slide rule.
IMHO the real task is to show how math can be used in the real world to make better decisions. Years later I was able to change the direction of the world's largest advertising budget by using these concepts and with a large staff of GA Tech students helping. BTW GA Tech in my opinion is probably the best STEM school in the US.
Most people enter education for reasons of a secure job with summers off. Among my friends and neighbors, school is something for their children to do. If mom and dad are not interested in their children's education - why should they care.
Had some great instructors who showed us how to use math for decision making, particularly when lots of different variables were inter dependent. Our exercises and tests consisted of solving real world problems - our only tool was a K & E slide rule.
IMHO the real task is to show how math can be used in the real world to make better decisions. Years later I was able to change the direction of the world's largest advertising budget by using these concepts and with a large staff of GA Tech students helping. BTW GA Tech in my opinion is probably the best STEM school in the US.
Most people enter education for reasons of a secure job with summers off. Among my friends and neighbors, school is something for their children to do. If mom and dad are not interested in their children's education - why should they care.
7
"As long as learning math counts as learning to think, the fortunes of any math curriculum will almost certainly be closely tied to claims about what constitutes rigorous thought — and who gets to decide."
You can substitute any subject you care to in place of "math". The basic premise of teaching is to get people from the unknown to the known. There will always be some "new and improved" way to do this and their adherents will trumpet said new method as the best thing since sliced bread. Common Core is just the latest in a long line of "new and improved".
What math has in it, though, is the element of logic. Most student find that in a geometry course, learning about inductive and deductive reasoning. What follows is most students' worst nightmare - memorizing axioms and theorems and applying them to proofs. It shows how to take a big problem and logically work your way through it.
Knowing how to do proofs and using logic translates into other areas. I wish it was emphasized more.
You can substitute any subject you care to in place of "math". The basic premise of teaching is to get people from the unknown to the known. There will always be some "new and improved" way to do this and their adherents will trumpet said new method as the best thing since sliced bread. Common Core is just the latest in a long line of "new and improved".
What math has in it, though, is the element of logic. Most student find that in a geometry course, learning about inductive and deductive reasoning. What follows is most students' worst nightmare - memorizing axioms and theorems and applying them to proofs. It shows how to take a big problem and logically work your way through it.
Knowing how to do proofs and using logic translates into other areas. I wish it was emphasized more.
15
Socioeconomic status largely determines success. Students from higher socioeconomic status households do much better in the US. Early exposure to abstract thinking, homework support at home, expectations of academic performance, and living in a household that is not disordered produces students who do well in school and particularly in arithmetic and mathematics. A one hour per day class is not going to overcome this as most of the learning happens outside the classroom if it is going to happen at all.
13
Math requires mastery learning, unlike other subjects.
Getting MOST of the steps correct in a solution to a problem will result in total garbage as the answer. EVERY step must be correct.
The standard of mastery is clearly possible, as many people achieve it, but it is mostly lacking in our school system and in our culture today, where sloppy thinking is the norm, if not something to wear proudly.
For that reason I believe that, for most people, math education should be simply abandoned as an unreachable and unrealistic goal. Either do it right, or stop wasting time and money pretending.
There are other approaches to life where being mostly correct is sufficient to make a valuable contribution to a team. These still require being psychologically capable of admitting that one is "wrong" some of the time, which, again, for some reason, seems culturally prohibited in the US, as is a process by which intelligent people sit around a table and try to figure out jointly what the truth is and where each of them is mistaken.
Still, learning how to operate as part of a team of faulty components that hope to produce a correct result is possible, and FAR more likely than retroactively teaching people who have been saturated with the concept of sloppy behavior and covering up mistakes how to do "math".
I discuss this idea more at: http://newbricks.blogspot.com/2014/06/why-is-math-so-hard-what-can-we-do...
Getting MOST of the steps correct in a solution to a problem will result in total garbage as the answer. EVERY step must be correct.
The standard of mastery is clearly possible, as many people achieve it, but it is mostly lacking in our school system and in our culture today, where sloppy thinking is the norm, if not something to wear proudly.
For that reason I believe that, for most people, math education should be simply abandoned as an unreachable and unrealistic goal. Either do it right, or stop wasting time and money pretending.
There are other approaches to life where being mostly correct is sufficient to make a valuable contribution to a team. These still require being psychologically capable of admitting that one is "wrong" some of the time, which, again, for some reason, seems culturally prohibited in the US, as is a process by which intelligent people sit around a table and try to figure out jointly what the truth is and where each of them is mistaken.
Still, learning how to operate as part of a team of faulty components that hope to produce a correct result is possible, and FAR more likely than retroactively teaching people who have been saturated with the concept of sloppy behavior and covering up mistakes how to do "math".
I discuss this idea more at: http://newbricks.blogspot.com/2014/06/why-is-math-so-hard-what-can-we-do...
3
Without a basic understanding of math, I would never have achieved my life goals. I didn't need calculus, my brother the engineer did, but I needed to understand that an increase of 2% to 4% is 50% and not 2%. Most of the people who took adjustable mortgages at 4% were surprised when their monthly payment increased upon adjustment to 6% was 30% and were often unable to meet the new requirement.
34
In 1000 hours a year in school, a teacher can not defeat 8000 hours of parenting. All of the great liberal reforms have failed because they fail (or refuse) to address the problem. My kids have excellent math scores, not because I passed them the genes for brilliance, not because I'm a brilliant teacher, but because I cared enough to make them do the homework.
Liberals funded by the publishers keep pushing new curriculums. Liberals funded by the teacher's unions keep pushing the union's political agenda. Obstructionist conservatives continue to knee-jerk against these. However, the only significant improvement I've seen is tying ADC to attendance. But, what do I know, I'm a teacher who studies teaching, not a politician, ivory tower soothsayer or right-wing nut job.
Liberals funded by the publishers keep pushing new curriculums. Liberals funded by the teacher's unions keep pushing the union's political agenda. Obstructionist conservatives continue to knee-jerk against these. However, the only significant improvement I've seen is tying ADC to attendance. But, what do I know, I'm a teacher who studies teaching, not a politician, ivory tower soothsayer or right-wing nut job.
13
It is not just teaching high school math that is deficient. It is also applying math to physics and chemistry. Students are not taught to arrive to a solution in the algebraic form to understand the behavior of the dependency (linear, square, exponential, etc.). Rather, they plug numerical values immediately both in math and scientific problems. Not good.
Another point. Non decimal bases that seemed exotic in the 60s are the must knowledge for anyone designing or programming digital devices.
Another point. Non decimal bases that seemed exotic in the 60s are the must knowledge for anyone designing or programming digital devices.
16
Mr. Phillips has performed a service by showing the ideological roots of both the "back-to-basics" and Common Core (thought by some to be "Progressive") approaches to math education, but I think his statemet that learning math is learning how to think is a bit too general. More modestly, learning math is learning how to think mathematically. The other way of putting the problem actually encourages a continuation of the cycling between the two extremes of "back-to-basics" and "Progressive education". FWIW, I am a high school dropout who has an MA is Mathematics, an MS in Computer Science, an MS in History (from CMU) and a PhD in Education (Social and Comparative Analysis).
6
I learned my arithmetic and math a long time ago in the traditional way. I have used advanced math over the years as a PhD physicist. The new math always struck me as a method that was not conducive to understanding, especially for a young student. The concepts are deep, not intuitive, and the results show that most students do not progress well (not to mention that fact that most of the teachers have no clue at all). I have used the concepts of new math in my work and am comfortable with them, but I cannot endorse them for most kids.
58
There are ways of thinking that are logical without being mathematical. Math is a bit too abstract for people with a more pragmatic inclination. There is nothing wrong with that. There will always be people with a great natural affinity and aptitude for math. Let them make good use of it.
A mistake we make in education is to assume everyone needs to know stuff that is only interesting or useful to academics. I think Common Core attempts to focus on the essentials of math, science and literacy that are needed to function as a citizen and worker in our society.
However, higher education should be available and affordable for anyone who shows the interest and aptitude to benefit from it. The way this country punishes the non-rich who seek higher education is disgraceful.
A mistake we make in education is to assume everyone needs to know stuff that is only interesting or useful to academics. I think Common Core attempts to focus on the essentials of math, science and literacy that are needed to function as a citizen and worker in our society.
However, higher education should be available and affordable for anyone who shows the interest and aptitude to benefit from it. The way this country punishes the non-rich who seek higher education is disgraceful.
2
I appreciate this article highlighting the ideological issues that have been at play in the debate over various mathematics standards and approaches. As a teacher I worked to help children conceptualize math as thinking. I have friends who are mathematics educators and assure me that the Common Core math standards are actually quite strong in this regard. However, what the author leaves out is the way the Common Core is linked to the current drive to privatize education and turn it into a playground for venture capitalism. My opposition to the Common Core has more to do with its ties to Pearson and less to do with the standards themselves.
3
I had a breathtakingly eye-opening experience last night, helping my second grade son with his math homework. He is going through Everyday Math curriculum for Common Core standards. The last problem of his homework was to add any three two digit numbers together. We decided to have a little fun with the teacher by choosing three products of 13, factoring out the primes, and then multiplying out the sum. As we was finishing up (correctly, his proud pa may say!), he looked at me and said:
"So, Dad, it doesn't matter how I get the answer as long as it makes sense to me?"
EXACTLY! My seven year old articulated in one sentence what it took me a B.S. in Mathematics to figure out. The point of elementary math is to provide the basic tools kids can apply to complicated problems. That is exactly what advanced mathematicians and scientists do every day - try to imagine interesting applications of core principles to complex issues. In other words, figure out what works in a way that you understand.
The whole current pedagogical trend of forcing kids to mimic particular strategies or algorithms to get to answers is such unbelievably wasteful garbage. The people who have truly made history (whether it be Galileo, Newton, Einstein, Feynmann or whoever) were the ones who were able to unlearn the strategies and algorithms common to their day to come to a new understanding based on basic principals.
"So, Dad, it doesn't matter how I get the answer as long as it makes sense to me?"
EXACTLY! My seven year old articulated in one sentence what it took me a B.S. in Mathematics to figure out. The point of elementary math is to provide the basic tools kids can apply to complicated problems. That is exactly what advanced mathematicians and scientists do every day - try to imagine interesting applications of core principles to complex issues. In other words, figure out what works in a way that you understand.
The whole current pedagogical trend of forcing kids to mimic particular strategies or algorithms to get to answers is such unbelievably wasteful garbage. The people who have truly made history (whether it be Galileo, Newton, Einstein, Feynmann or whoever) were the ones who were able to unlearn the strategies and algorithms common to their day to come to a new understanding based on basic principals.
13
Education must be accepted as something that is needed to ensure a good earning, good future and in general to provide you with the tools to make good decisions in life. There must be no need to assure students every step of the way that everything that they are taught in the classroom has immediate use for something that they will be doing that night. We all need to do things in life that we do not always enjoy but there may be rewards at a much later time. often the attitude that we encounter is that there will be a gadget to perform all these tasks and so I need not learn all these boring topics. It is a monumental task to convince students of the intangible benefits such as analytical and reason based approach to solving problems, be it a mathematical problem or otherwise. I should know! I have gone from looking forward to going to class everyday and sharing my knowledge with a room full of eager learners who will challenge me to become a more knowledgeable teacher, two decades ago, to dreading the constant need to apologize for teaching them difficult concepts that take serious concentration and focus on their part, forcing them to turn their attention away from their cell phones. No, it is not always the student's fault. Nor is it always the teacher or his/her way of teaching. It is also the societal attitude about doing difficult things that may take time to yield benefits. Short cuts are not always the answer and instant gratification should not be the expectation.
14
I agree with the author formal schooling is a battle ground for ideaology.
It does not follow that this has to result in lack of math skills.
Most teachers come from the bottom 40% of SAT scorers. Most math teachers have less than two years of math schooling, from second rate colleges. Those who do are constrained by public officials and unions in how they should teach.
All the structure for algebra can be summarized in a few Wikipedia pages. Yes, it takes more time to *understand* it. To do so requires working with students, reacting to them. I do not see how someone who has not actually taken a formal class in abstract algebra can properly react -- often I hear a student being told "wrong" by a school teacher when what they do is actually uncover a deep and subtle point.
My son's Algebra 1 text book has 7 authors. Not one a mathematician. The book is over 500 pages long with all the supplemental material, color images of happy children distracting the reader, and tons of special case / made up /redundant rules meant to "help". No wonder students start to see it as just diconnected facts. This to be covered in one year.
We do not expect that the average school music teacher will produce even average orchestra players. We are not surprised that few students can play an instrument and most drop out of lessons.
We do not invest in math teachers and schooling to the extent we have expectations from the process.
Yet we are surprised by the results.
It does not follow that this has to result in lack of math skills.
Most teachers come from the bottom 40% of SAT scorers. Most math teachers have less than two years of math schooling, from second rate colleges. Those who do are constrained by public officials and unions in how they should teach.
All the structure for algebra can be summarized in a few Wikipedia pages. Yes, it takes more time to *understand* it. To do so requires working with students, reacting to them. I do not see how someone who has not actually taken a formal class in abstract algebra can properly react -- often I hear a student being told "wrong" by a school teacher when what they do is actually uncover a deep and subtle point.
My son's Algebra 1 text book has 7 authors. Not one a mathematician. The book is over 500 pages long with all the supplemental material, color images of happy children distracting the reader, and tons of special case / made up /redundant rules meant to "help". No wonder students start to see it as just diconnected facts. This to be covered in one year.
We do not expect that the average school music teacher will produce even average orchestra players. We are not surprised that few students can play an instrument and most drop out of lessons.
We do not invest in math teachers and schooling to the extent we have expectations from the process.
Yet we are surprised by the results.
2
America's actual relationship to mathematics?
America is fond of speaking about the value of mathematics--math, science, STEM education. But America's actual relationship to math, how much politics influences math and science? Well, that should be easy to determine. What you do is take the most naturally gifted mathematicians and make a national, even Presidential announcement (which shouldn't be difficult because math is oh, so important) that our most mathematically gifted minds will apply math to every aspect of our society, tell us what they really think about us, America, and that their results will be broadcast widely, nationally to all Americans.
But can we imagine such a thing occurring? No. Why? I leave that up to the reader to guess. But if the most important, nationally declared endeavor (which mathematics is), cannot be freely, brutally applied to all aspects of American society and widely announced, what does that mean? The obvious: Not only is our most important method to knowledge compromised (most people consider math the most important method), it is probably censored, twisted this way and that--and if math is subject to this problem, imagine the fate of other methods toward knowledge, the written word, writers...
Interestingly, to ask exactly how serious America is about math is to simultaneously expose how much America by both major political parties and of course institutions ranging from education to NSA is a place of lack of freedom of thought.
America is fond of speaking about the value of mathematics--math, science, STEM education. But America's actual relationship to math, how much politics influences math and science? Well, that should be easy to determine. What you do is take the most naturally gifted mathematicians and make a national, even Presidential announcement (which shouldn't be difficult because math is oh, so important) that our most mathematically gifted minds will apply math to every aspect of our society, tell us what they really think about us, America, and that their results will be broadcast widely, nationally to all Americans.
But can we imagine such a thing occurring? No. Why? I leave that up to the reader to guess. But if the most important, nationally declared endeavor (which mathematics is), cannot be freely, brutally applied to all aspects of American society and widely announced, what does that mean? The obvious: Not only is our most important method to knowledge compromised (most people consider math the most important method), it is probably censored, twisted this way and that--and if math is subject to this problem, imagine the fate of other methods toward knowledge, the written word, writers...
Interestingly, to ask exactly how serious America is about math is to simultaneously expose how much America by both major political parties and of course institutions ranging from education to NSA is a place of lack of freedom of thought.
Not all humans are created equal. Not all children and young adults will understand advanced math not matter what methods are used to teach it whether it is "new math' or old school math. I learned quadratic equations in high school by rote. I never 'got' word problems until I learned by rote by recognizing what form of a word problem it was. I still didn't understand it. Not all of us will grow up to be engineers. Nor should we. Having a father and dad as an engineer, I can testify that growing up as a child of an engineer can be a lonely experience. As commentator Richard Luettgen states math is a form of communication using logic. Math in and of itself does not 'teach' logic for those that relate to their outside word intuitively(emotionally). They will never understand higher math. Philosophy teaches logic using real world experience which we all can understand. I went back to college at 60 years old. I am a creative person adept at the communication arts. But because some board of educators thought I should learn higher math I went from a 4.0 grade level with a full load of classes to a, well lets just say math is not my forte. Until we understand that all students will not be rocket scientists the better we will be able to concentrate on what our students strengths are and what they can contribute to society. Only by assessing and guiding our students instead of using our teachers precious time force feeding brains that weren't designed to be engineers will we succeed.
1
What a great column. Math is a language, the only truly universal language in the world today. It transcends the spoken word precisely because it requires abstract thought and reasoning.
The issue is that we have a bell curve of learners and we have a bell curve of teachers and a bell curve of practitioners.
The challenge is how to segment the three categories of math participants so that the learners are matched with the appropriate teachers that will in turn enable the practitioners to apply math in their daily lives.
There is a huge cultural challenge here. In my view, the challenge goes beyond the political state / federal dimension cited by the author which in no way diminishes that aspect. The bigger view, alluded to by the author is the "elite" vs "everyday man" tension, which has been co-opted by many in the political arena.
The simple reality is that too many in our society shy away from the reality that intelligence and work ethic are not distributed evenly across the population. Math is the best exemplar given the intellectual dimension of the subject. Ironically, "everyman" understands the bell curve in terms of physical prowess, but pushes back against the reality when it comes to intellectual prowess.
The fact that we have just as much contention over the "nature" vs "nurture" aspect as we do over the "state" vs "federal" aspect of teaching, learning and outcomes is one of the biggest inhibitors to raising the bar in America.
The issue is that we have a bell curve of learners and we have a bell curve of teachers and a bell curve of practitioners.
The challenge is how to segment the three categories of math participants so that the learners are matched with the appropriate teachers that will in turn enable the practitioners to apply math in their daily lives.
There is a huge cultural challenge here. In my view, the challenge goes beyond the political state / federal dimension cited by the author which in no way diminishes that aspect. The bigger view, alluded to by the author is the "elite" vs "everyday man" tension, which has been co-opted by many in the political arena.
The simple reality is that too many in our society shy away from the reality that intelligence and work ethic are not distributed evenly across the population. Math is the best exemplar given the intellectual dimension of the subject. Ironically, "everyman" understands the bell curve in terms of physical prowess, but pushes back against the reality when it comes to intellectual prowess.
The fact that we have just as much contention over the "nature" vs "nurture" aspect as we do over the "state" vs "federal" aspect of teaching, learning and outcomes is one of the biggest inhibitors to raising the bar in America.
1
The lynchpin of Western higher education was, at one time, the study of rhetoric, or how language works. It was replaced by mathematics, the "queen of the sciences," when cross cultural contact, the metaphoric Tower of Babel, gave rise to a need for a "universal language" that would allow for greater cooperation between disparate cultures. The mastery of mathematics comes at an early age. The mastery of language is more complicated and requires more time, sometimes decades. The priveleging of rhetoric historically led to an elite culture that fostered insularity and discouraged cross cultural contact. The privileging of mathematics has made us far less literate and, basically, dumber. Perhaps a better way to look at it is how do we strike a balance between the study of mathematics and language, rather than a balance between the political forces within the study of math.
1
Paul Lockhart wrote a devastating essay about math classes that fail to hint that mathematics is beautiful (*A Mathematician's Lament*). His essay riffs on a “what if?”: What if schools taught music the way they teach mathematics? Lockhart describes a music curriculum in which students never, ever hear music.
"Music class is where we take out our staff paper, our teacher puts some notes on the board, and we copy them or transpose them into a different key. We have to make sure to get the clefs and key signatures right, and our teacher is very picky about making sure we fill in our quarter-notes completely."
Lockhart imagines how many "music" students would love music when they've never heard any. Mathematics is beautiful too, he says, but K-12 students never see the beautiful parts.
K-12 math conceals many, many astonishing and beautiful ideas. For example: Every quantity has a number--the grains of sand in a sandbox, the blades of grass in a field. Another example: Every two numbers have a sum, including numbers no one has ever thought of.
Most K-12 music teachers show students that they love music. Very few K-12 math teachers show that they love mathematics. Bored teachers produce bored students. And bored students don't want to learn.
"Music class is where we take out our staff paper, our teacher puts some notes on the board, and we copy them or transpose them into a different key. We have to make sure to get the clefs and key signatures right, and our teacher is very picky about making sure we fill in our quarter-notes completely."
Lockhart imagines how many "music" students would love music when they've never heard any. Mathematics is beautiful too, he says, but K-12 students never see the beautiful parts.
K-12 math conceals many, many astonishing and beautiful ideas. For example: Every quantity has a number--the grains of sand in a sandbox, the blades of grass in a field. Another example: Every two numbers have a sum, including numbers no one has ever thought of.
Most K-12 music teachers show students that they love music. Very few K-12 math teachers show that they love mathematics. Bored teachers produce bored students. And bored students don't want to learn.
4
Math is a way to teach logic and thinking; unfortunately, most elementary teachers don't see it that way. That may be because they were not particularly good at math and therefore teach without the enthusiasm for it that comes from being assured of one's content knowledge. Further, when you aren't confident you cannot establish meaningful relationships with the class and with individual students that will make them better at it. How can a teacher help a student improve if he/she can't accurately diagnose what it is that the student(s) is not understanding?
This isn't just about new math or common core (curriculum issues), it's about university requirements in general education that allows aspiring elementary teachers to graduate with one college algebra course and a math methods class. First, elementary teachers don't really teach algebra, and second, their college algebra class gives them no sense of how numbers work. Without that, most of their teaching will be making their students memorize algorithmic processes and not teaching the students to reason through logic. The problem of math student achievement is complex and so are the solutions.
This isn't just about new math or common core (curriculum issues), it's about university requirements in general education that allows aspiring elementary teachers to graduate with one college algebra course and a math methods class. First, elementary teachers don't really teach algebra, and second, their college algebra class gives them no sense of how numbers work. Without that, most of their teaching will be making their students memorize algorithmic processes and not teaching the students to reason through logic. The problem of math student achievement is complex and so are the solutions.
4
The fundamental problem with math education is that students are allowed to proceed with establishing a strong foundation. There are many methods that can and do work, but none of them will work if students are socially promoted rather than forced to master each concept. Student who are not masters of addition and subtraction are doomed to be weak at multiplication and division, leading only to frustration and disinterest. With all the technology out there, surely someone can devise a system that encourages mastery, diagnoses errors, provides alternate instruction, and guides teachers to those students struggling with certain concepts. One size should not fit all, and each student needs to be provided the tools they need and be required to build a strong foundation before proceeding to the next level. Forget grades, there is only mastery or not yet mastered, anything else is doomed to ultimately fail.
4
It has always been my experience that too many (maybe most) teachers of math are very good at understanding math for themselves. However, they do a very poor job of communicating these concepts in words. Too many of them assume that math is easy, as it is for them, when in fact math concepts are very much more abstract than the ideas of other grade school subjects. Furthermore, young children are unable to think in those abstract terms.
Words and sentences, the common language of communication used by people, are difficult for these math teachers to organize in a way that explains mathematical concepts to other people, especially children, in a way that enlightens and makes mathematics fun or even interesting. The "fault" is mostly in themselves -- not in their students.
Words and sentences, the common language of communication used by people, are difficult for these math teachers to organize in a way that explains mathematical concepts to other people, especially children, in a way that enlightens and makes mathematics fun or even interesting. The "fault" is mostly in themselves -- not in their students.
2
I see math curricula as still rooted in social systems that are decades old. Sure teaching geometry will help some students learn to think logically, but the emphasis on geometry seems to me to come from an agricultural past where laying out fields and determining their ownerships was an important social institution.
In contrast, what is one thing that most Americans see all the time in the news? Polling results. Do we teach students enough probability and statistics in high school to understand what the "margin of error" is and why it is or is not meaningful? Can they understand why the error on one candidate's lead over another is larger than that supposed margin of error?
As a social scientist, I use statistics every day. I never use trigonometry and rarely use calculus. I can't think of the last time I needed to determine if two triangles are congruent. I see high-school mathematics as giving students tools that they will largely never use in adult life, and ignoring mathematical applications that arise every day.
In contrast, what is one thing that most Americans see all the time in the news? Polling results. Do we teach students enough probability and statistics in high school to understand what the "margin of error" is and why it is or is not meaningful? Can they understand why the error on one candidate's lead over another is larger than that supposed margin of error?
As a social scientist, I use statistics every day. I never use trigonometry and rarely use calculus. I can't think of the last time I needed to determine if two triangles are congruent. I see high-school mathematics as giving students tools that they will largely never use in adult life, and ignoring mathematical applications that arise every day.
3
This is an interesting article; I found the political analysis of the conflict over math education very insightful.
However, to me, it seems like something is ignored by this analysis, which is that at least in the case of the Common Core, much of the (at least, grassroots) opposition to the changes have been driven by the simple fact that many parents who are not themselves very strong at math have trouble understanding the new material.
In short, if the parents can't do the homework, there will be opposition, regardless of how those parents might feel philosophically about the merits of different ways of reasoning.
I wonder whether this simple effect might overwhelm others in the Common Core debate. And, if it does, it begs the question of how Math education in the U.S. can ever be expected to improve. Should we really expect today's average parent to be able to do the math homework of an American 8th grader from the year 2100?
However, to me, it seems like something is ignored by this analysis, which is that at least in the case of the Common Core, much of the (at least, grassroots) opposition to the changes have been driven by the simple fact that many parents who are not themselves very strong at math have trouble understanding the new material.
In short, if the parents can't do the homework, there will be opposition, regardless of how those parents might feel philosophically about the merits of different ways of reasoning.
I wonder whether this simple effect might overwhelm others in the Common Core debate. And, if it does, it begs the question of how Math education in the U.S. can ever be expected to improve. Should we really expect today's average parent to be able to do the math homework of an American 8th grader from the year 2100?
10
This article misses the big picture: the computer revolution has changed math at all levels -- even the purest of pure math -- profoundly. Not only because we can compute and visualize better and faster, but because mathematicians can ask fundamental questions that were literally unthinkable before. The New Math of the mid-20th century, which predated this revolution, was a filtered-down product of extreme post-war abstraction (google Bourbaki). No one, especially not the math "elites," would design such a curriculum today. The question for math education is not whether to go back to basics, but what the basics actually are.
4
Teaching a foreign language in even the poorest public school requires a teacher who has an objectively proven reasonably proficiency, right?
Math is equally a foreign "language", yet public schools have always permitted the subject to be taught by math illiterates.
So why is anyone surprised that American kids do so poorly in the subject?
Math is equally a foreign "language", yet public schools have always permitted the subject to be taught by math illiterates.
So why is anyone surprised that American kids do so poorly in the subject?
10
As an instructor of the Algebra Core Curriculum I can assure you it is outstandingly considered and well written. Many years ago the TIMMS study was conducted and identified many of the "flaws" in American mathematical instruction versus some of the highest achieving countries. The Core Algebra Curriculum addressed a majority of those issues/concerns quiet effectively.
It took an almost impossible effort to prepare to teach the course and I'm shocked at how much I had to learn, or better aquint myself with, to be able to properly deliver its content. The learning curve to be able to effectively deliver the Algerba Core curriculum is steep, but from what I've seen the results and subsequent rewards are MORE than worth the effort!
It took an almost impossible effort to prepare to teach the course and I'm shocked at how much I had to learn, or better aquint myself with, to be able to properly deliver its content. The learning curve to be able to effectively deliver the Algerba Core curriculum is steep, but from what I've seen the results and subsequent rewards are MORE than worth the effort!
5
How many hour do kids spend on math and at what age do they start? I would bet that early and often exposure is a key factor. If students spent as many hours learning math when they are young as they do learning to read I think we would see higher competency.
5
With our factory model approach to education, students are merely the product and uniformity is required for any assembly line. Not until we learn to address each learner as a self creation with a great many ways to rationality will math and other academic areas be considered in school in their full range of rationallity
1
As children grow up, their brains develop, and they become able to do different things at different stages of their growth. While general trends have been observed, adolescent brain development can't be standardized.
That said, I believe there is a time for general instruction of individual students, and I think "memorization . . . and rapid recall . . ." has a place. I don't need "militaristic discipline", but I do need compliance. Trust me, I know what I'm talking about. As a 7th-grade teacher in Los Angeles attempting to manage a classroom under the "Restorative Justice" system, I am regularly defied and cussed out by 12-year-olds.
The fact that the new Common Core Standards were put together with no attempt to make them age-appropriate is already creating big problems. This column does not address that aspect of math education.
That said, I believe there is a time for general instruction of individual students, and I think "memorization . . . and rapid recall . . ." has a place. I don't need "militaristic discipline", but I do need compliance. Trust me, I know what I'm talking about. As a 7th-grade teacher in Los Angeles attempting to manage a classroom under the "Restorative Justice" system, I am regularly defied and cussed out by 12-year-olds.
The fact that the new Common Core Standards were put together with no attempt to make them age-appropriate is already creating big problems. This column does not address that aspect of math education.
4
What about changing to metrics? Base 10 is essential and if children at an early age understood how numbers represent quantities that fit within each other, it would make the rest much easier to fathom. Or does the imperial system still represents what the Common Core is all about?
3
Even in these comments there is disagreement about the "thinking" that goes into math. As a student, I was terrible in math--it simply made no sense to me although I was a good student in other areas. But as a professional educator, I was asked to develop Common Core curriculum materials, and suddenly math made sense because I had found the logic that was its foundation. I agree completely with the reviewer who said we need to teach logic in schools--logical understanding can make our students better writers, better scientists, and better mathematicians.
38
If "learning math counts as learning to reason," then the current mode of teaching math has failed. Math curriculum is constantly changing, about every two years. I think that the textbook industry is the one that has profited the most with the least guidance to teachers who are stuck teaching whatever new version comes along. Adding, subtraction, multiplication, and division should be taught in its simplest, concrete form. If learning to reason is the goal, then philosophy should be taught at the HS level but basic math should not be the form used to do so. The frustration of elementary aged children will carry through to later grades and adulthood. Basic math should not be that hard to teach and to learn.
1
Math is universal, not local.
2 + 2 = 4 everywhere on Earth. Advanced math is harder to learn than simple math, everywhere on Earth.
So-called "local standards" for math education are a farce. Most localities in America settle for "standards" that leave their own kids far behind students in many other countries.
To some, local is better than national in every possible way, including in math education. But it's only true for math education if it's okay for most localities in the U.S. to provide inadequate math education.
2 + 2 = 4 everywhere on Earth. Advanced math is harder to learn than simple math, everywhere on Earth.
So-called "local standards" for math education are a farce. Most localities in America settle for "standards" that leave their own kids far behind students in many other countries.
To some, local is better than national in every possible way, including in math education. But it's only true for math education if it's okay for most localities in the U.S. to provide inadequate math education.
6
Problem of Common Core is not testing. It suffers from the fundamental problem of past US math education - trying to teach math applications without teaching math itself.
When John Dewey started a ‘new education’ in University of Chicago more than one hundred years ago, he did not include arithmetic in his elementary school curriculum, because he decided symbolic knowledge such as arithmetic is too abstract for young children to learn and is not amenable to his doctrine of ‘learning by doing’. Instead, he introduced 'figuring' as an activity of solving daily problems involving numbers, quantities, and measurements. His three R’ were ‘reading, writing, and figuring’ He tried to teach the application of mathematics without teaching mathematics. Since then, 'figuring' (problem solving in today's term), not mathematics, has been the core subject of US math education.
I taught computer science in an engineering school for more than thirty years. We need students who know mathematics - arithmetic, algebra and calculus. We can develop capabilities of problem solving, critical thinking, creativity, and so on, after students enter engineering school. K - 12 math education should focus on teaching mathematics first, then 'figuring'. Otherwise, US scientists and engineers would be dominated by people who learned mathematics in other countries such as India and China.
The Common Core teaches 'figuring', not mathematics.
When John Dewey started a ‘new education’ in University of Chicago more than one hundred years ago, he did not include arithmetic in his elementary school curriculum, because he decided symbolic knowledge such as arithmetic is too abstract for young children to learn and is not amenable to his doctrine of ‘learning by doing’. Instead, he introduced 'figuring' as an activity of solving daily problems involving numbers, quantities, and measurements. His three R’ were ‘reading, writing, and figuring’ He tried to teach the application of mathematics without teaching mathematics. Since then, 'figuring' (problem solving in today's term), not mathematics, has been the core subject of US math education.
I taught computer science in an engineering school for more than thirty years. We need students who know mathematics - arithmetic, algebra and calculus. We can develop capabilities of problem solving, critical thinking, creativity, and so on, after students enter engineering school. K - 12 math education should focus on teaching mathematics first, then 'figuring'. Otherwise, US scientists and engineers would be dominated by people who learned mathematics in other countries such as India and China.
The Common Core teaches 'figuring', not mathematics.
2
Mathematics describes and predicts - models - behaviors and outcomes. Those behaviors and outcomes can be and are politicized, but the subject matter itself is immutable and indifferent: two plus two equals four, regardless of how you feel about it. Growing up in a family devastated by terminal illnesses, I found that fact so comforting that I majored in mathematics.
Sorry, posted this as a reply rather than free-standing comment. For those who are complacent that our best is great though our average may be poor, etc.
http://cue.caltech.edu/documents/22-core_with_tables_appendix.pdf
"The transition from high school to college presents problems for all students, but for some students it is particularly challenging. At Caltech, many newly admitted students lack the background in mathematics that is necessary to succeed in Ma 1a. Unfortunately, few of them are even aware that their background in mathematics is deficient. This is not their fault. The mathematics curriculum in high schools is less rigorous than it was even a few decades ago. In conversations with Caltech students who have struggled with freshman mathematics, most report that they were star math students in high school, which of course is a major reason why they were offered admission to Caltech in the first place. Many of them, however, have never seen mathematics as it is taught at Caltech. "
"In order to understand the specific reasons why many of our freshman struggle in Ma 1a, the undergraduate Academics & Research Committee conducted ... survey ... the most common area of weakness that students identified was that of formal reasoning, writing proofs, and common proof techniques. The results thus corroborate what most people connected with Ma 1a have known anecdotally—that many Caltech freshmen, though computationally skilled, struggle with basic proof concepts. .... "
http://cue.caltech.edu/documents/22-core_with_tables_appendix.pdf
"The transition from high school to college presents problems for all students, but for some students it is particularly challenging. At Caltech, many newly admitted students lack the background in mathematics that is necessary to succeed in Ma 1a. Unfortunately, few of them are even aware that their background in mathematics is deficient. This is not their fault. The mathematics curriculum in high schools is less rigorous than it was even a few decades ago. In conversations with Caltech students who have struggled with freshman mathematics, most report that they were star math students in high school, which of course is a major reason why they were offered admission to Caltech in the first place. Many of them, however, have never seen mathematics as it is taught at Caltech. "
"In order to understand the specific reasons why many of our freshman struggle in Ma 1a, the undergraduate Academics & Research Committee conducted ... survey ... the most common area of weakness that students identified was that of formal reasoning, writing proofs, and common proof techniques. The results thus corroborate what most people connected with Ma 1a have known anecdotally—that many Caltech freshmen, though computationally skilled, struggle with basic proof concepts. .... "
3
Sorry but if Caltech students are "deficient" in math, it may be very well be because their professors are expecting way too much of college freshman. Chances are very high that those same professors simply don't want to teach beginning college level students. It's an old problem: instead of working with the students you have (and at Caltech they are likely to be very smart) and the knowledge they bring, the students are blamed because they aren't yet at the level of the grad students that the professors really want to teach.
6
What an insightful and thiught-provoking article. It might be good to remember, however, that while math curricula were left in the hands of the mathematicians, Adlai Stevenson was successfully campaigned against as an "egghead." America's disdain for rational thought and academic excellence runs very deep. "Elitism" on the left, "arrogance" on the right - we are politically more fearful of "unequal outcomes" than of losing the intellectual basis of our civilization.
25
Here’s the logic in math: Americans aren’t good in math but look how successful we are. How many parents apply the same logic to their families? “I was never good at math...” sends a message of low expectations to children. A love of learning and of intellectual enrichment is not part of American culture and that fact is reflected in our approach to education.
42
And get rid of the notion "learning to reason is a political matter." Make some room for logic.
2
Let's begin the argument with a clear definition of terms, starting with the word "conservative". Are we talking about people with respect for tradition and a cautious outlook on change, or are we using the word in its modern American form as a euphemism for religiosity and ignorance.
Let's also be clear on what we mean by "mathematics". Up until grade 5 or so, school deals mostly with arithmetic, with perhaps some discussion of basic concepts like geometric shapes, graphs and the like. The abstraction and symbolic reasoning of mathematics comes much later, sometimes in the form of coding and algorithm development, but mostly in the algebra-calculus sequence that everyone sees in high school.
The political issues arise because informed parents understand that early arithmetic can be taught in a way that makes later mathematics easier to learn. Parents whose understanding never got beyond arithmetic fall into two categories: those who want their children to learn a useful intellectual skill, and those who euphemistically call themselves conservatives or (in the most hopeless cases) "reformers".
Let's also be clear on what we mean by "mathematics". Up until grade 5 or so, school deals mostly with arithmetic, with perhaps some discussion of basic concepts like geometric shapes, graphs and the like. The abstraction and symbolic reasoning of mathematics comes much later, sometimes in the form of coding and algorithm development, but mostly in the algebra-calculus sequence that everyone sees in high school.
The political issues arise because informed parents understand that early arithmetic can be taught in a way that makes later mathematics easier to learn. Parents whose understanding never got beyond arithmetic fall into two categories: those who want their children to learn a useful intellectual skill, and those who euphemistically call themselves conservatives or (in the most hopeless cases) "reformers".
25
American mathematics could do itself a favor by looking for and listening to leaders among the ranks of teachers of the subject in community colleges. These folks deal with elementary and secondary student achievement shortcomings; with challenging talented students to master calculus and differential equations for careers in science and engineering and, yes, mathematics; and with teaching reasoning in the context of philosophy, logic, history and social sciences.
4
For me, my understanding of math came from memorizing and being told how to solve equations. Through practice I started to see patterns emerge. I started seeing the connections between the different approaches to finding solutions. It was through my own math journey and my own ability to find the connections did I learn to appreciate math. It gave me a sense of accomplishment. Trying to "teach" students to make the connections removes any sense of accomplishment and therefore, in my opinion, takes away any excitement about being able to bring together all of the information you have "memorized" and finding solutions to the problems.
I have taken courses on how to teach students based on the common core principles and what I found is that I was being taught to remove any of the self discovery from math. People look at teaching English as an opportunity to enhance student creativity, but that is after learning all of the basic rules. Why do we not look at math similarly? We need to first teach the basics (as boring as they may be) so that students can be creative and find ways to make connections on their own.
I have taken courses on how to teach students based on the common core principles and what I found is that I was being taught to remove any of the self discovery from math. People look at teaching English as an opportunity to enhance student creativity, but that is after learning all of the basic rules. Why do we not look at math similarly? We need to first teach the basics (as boring as they may be) so that students can be creative and find ways to make connections on their own.
27
Practice, practice, practice. It can be a boring, boring, boring exercise... however, I will also say that I, too, began to see patterns, in order to make the practice easier or quicker. Remember "spelling bees"? Well, maybe we should have "arithmetic bees"... adding, subtracting, multiplying, dividing, addition of fractions - using NO calculators. I teach in a college and I find that the significant difference between today's students and those of yesteryear is that today's students rely on the calculator. Add 2/3 and 4/7. Try it on paper, then try it mentally. Stop using that d*** calculator for trivial stuff.
4
" inadequate teacher training"?
How about stupid teachers?
A decade ago a friend of mine, a brilliant professor, and teacher, (n.b. the two are not necessarily identical) of mathematics in a fairly exclusive, expensive, private university was sentenced to teaching math in the education department.
Her students were about to become the people teaching Johnny to count.
She devised three tests, intended to determine the future teacher's ability to comprehend the subject that teacher was going to be communicating to others.
One test on whole numbers, one test on fractions, one test on decimals.
Multiply 386 times 64.
Reduce 9 over 63 to its lowest terms.
Which is larger .11 or .09?
Not exactly Fermat's Last Theorem.
Each test had ten questions, the passing grade was 70% (thats 7 out of 10, for you teachers in the crowd) and the students could continue to take additional tests until passing.
A week before graduation more than half the class had not yet achieved these modest requirements.
And a week later, they were capped, gowned and "qualified" to make America great, again.
Do the math, if you can.
How about stupid teachers?
A decade ago a friend of mine, a brilliant professor, and teacher, (n.b. the two are not necessarily identical) of mathematics in a fairly exclusive, expensive, private university was sentenced to teaching math in the education department.
Her students were about to become the people teaching Johnny to count.
She devised three tests, intended to determine the future teacher's ability to comprehend the subject that teacher was going to be communicating to others.
One test on whole numbers, one test on fractions, one test on decimals.
Multiply 386 times 64.
Reduce 9 over 63 to its lowest terms.
Which is larger .11 or .09?
Not exactly Fermat's Last Theorem.
Each test had ten questions, the passing grade was 70% (thats 7 out of 10, for you teachers in the crowd) and the students could continue to take additional tests until passing.
A week before graduation more than half the class had not yet achieved these modest requirements.
And a week later, they were capped, gowned and "qualified" to make America great, again.
Do the math, if you can.
66
I can assure you that to be a good teacher of kids below the 6th grade level, the factoids you present are completely irrelevant. By contrast, many smarty pants could do these problems within nanoseconds and yet would not have a clue how to manage and motivate a classroom full of children-- not widgets-- but 20 or 30 individuals with different skills and drivers.
2
When you pay/treat teachers like we do in this country, you get teacher shortages. Also, if your friend thought his students' understanding of math was so abysmal, it's curious that so many passed. I've heard this kind of flippant, mean-spirited, anecdotal argument a million times. It's meant to direct frustrations at under-payed over-worked teachers instead of the institutional failings and corruption that is really at the heart of the problem.
5
You are so right! Teacher education in this country is abysmal. It needs to be totally overhauled, although I don't see that happening.
1
The basics should include logic and critical thinking and should start in grade school. How to think is more important than memorization, It might frighten some of our politicians, if the content of their speeches started getting deconstructed by the average voter.
12
Higher level mathematics is built from lower lever arithmetic. If a child does not have the facts at their fingertips they will not be able to go on to a higher level. This means memorization, especially the multiplication tables, and lots of drilling in the basics or arithmetic.
1
I believe it was Richard Feyman that roughly said, "I am such a poor teacher that I have to try everything to reach my students. Bu, if I can reach them once, they can usually take it from there on their own."
8
All I know is that the way math is taught today in elementary and middle school is perplexing to student and parent alike. It certainly doesn't help anyone improve their power of reasoning.
6
The "new mathematics" designed by real mathematicians emphasizes the fact that mathematics is a mental construct and, as such, one needs to look carefully at the premises to determine if there is a clear "right answer". A quick example: in non-Euclidean geometry two points define a curve and not a line. By changing that one premise, all of the premises of the Euclidean geometry we learned in HS become worthless. One's faith in the order of the world can be challenged when you realize that the "rules of geometry" can be rendered useless… and that lack of faith might compel one to question the fundamental premises of a political system.
Those who think it is plausible that a Democratic Socialist could become President, like the new math. Those who want to reinforce the dominant paradigm are more comfortable with Euclid.
Those who think it is plausible that a Democratic Socialist could become President, like the new math. Those who want to reinforce the dominant paradigm are more comfortable with Euclid.
4
Most of the GOP voters would not know the name Euclid if they saw it in print. What nonsense.
Do carpenters use plane euclidean geometry or non-euclidean geometry when they are building a house? And - do they us some principles of three-dimensional geometry in their work?
The cacophony of voices distracts from the teaching, but that's America!
Rather why not develop a series of standard year-end tests, instead of the nearly monthly hodgepodge that now exists, then hold principals and teachers accountable for the results, based on a reasonable metric that takes into account a myriad of social factors?
But the fanatics and the autocrats have adopted the sledgehammer approach; 'Pass or die'. It's bullying, from top to bottom.
Where's the love? No where to be found. But that's America!
Rather why not develop a series of standard year-end tests, instead of the nearly monthly hodgepodge that now exists, then hold principals and teachers accountable for the results, based on a reasonable metric that takes into account a myriad of social factors?
But the fanatics and the autocrats have adopted the sledgehammer approach; 'Pass or die'. It's bullying, from top to bottom.
Where's the love? No where to be found. But that's America!
3
It saddens and worries me that children's mathematical education is buffeted by politics and profits, even as this article says that is inevitable. Teachers and students pay the price for waves of changes and counterchanges. Language arts and mathematics learning are sequential in nature, so the backlash against Common Core (the guidelines, not the PARCC testing) will mean students who move from state to state may be penalized by missing some pieces and repeating others. Military dependents are among these.
Moreover, it is not factually correct that American students are bad at math. That doesn't stop politicians and others from saying it of course. Uri Treisman at UT Austin and others have shown that American math scores are competitive with international scores except at those schools that serve economically challenged students, which we ought to address by working with those schools.
The politics also leads to one-size-fits-all "solutions" and single number summaries of complex data. These discussions are often silly. For instance, the Times published "who needs algebra". Algebra is the way quantitative information is communicated. There are very few ways to get computers to do something quantitative that don't use algebra, so lack of algebra knowledge means missing out on nearly all quantitative computing, which surely has lifelong ramifications.
Having highlighted the politics of math education, I hope Prof. Phillips will show us a way to minimize its damage.
Moreover, it is not factually correct that American students are bad at math. That doesn't stop politicians and others from saying it of course. Uri Treisman at UT Austin and others have shown that American math scores are competitive with international scores except at those schools that serve economically challenged students, which we ought to address by working with those schools.
The politics also leads to one-size-fits-all "solutions" and single number summaries of complex data. These discussions are often silly. For instance, the Times published "who needs algebra". Algebra is the way quantitative information is communicated. There are very few ways to get computers to do something quantitative that don't use algebra, so lack of algebra knowledge means missing out on nearly all quantitative computing, which surely has lifelong ramifications.
Having highlighted the politics of math education, I hope Prof. Phillips will show us a way to minimize its damage.
10
"Who needs algebra?" is a legitimate question if one's goal is to become a curator of an art museum, a buyer for a clothing store, a salesperson of almost anything, a teacher of English Lit, etc. Than heavens the world is full of people who want to do things other than technical skills. Diversity of people is what makes for an interesting society: a dull world indeed if everyone is an engineer!
2
I had a funny/sad experience a few years back. My daughter was being taught to derive algebraic expressions through some obtuse methodology that her school used.
I looked at one of her answers and said it was incorrect. She assured me that it was correct. I then pointed out to her we could prove it was incorrect by plugging in numbers from a real life example. And sure enough, her equation provided the wrong answer for a clearly obvious situation.
Remarkably, her teacher insisted in class that the answer my daughter derived was the correct answer, *despite the fact* that my daughter also showed her and asked her why it didn't fit the real world. No answer.
When teachers are that far down the pedagogical rabbit hole, it's time to rethink.
I looked at one of her answers and said it was incorrect. She assured me that it was correct. I then pointed out to her we could prove it was incorrect by plugging in numbers from a real life example. And sure enough, her equation provided the wrong answer for a clearly obvious situation.
Remarkably, her teacher insisted in class that the answer my daughter derived was the correct answer, *despite the fact* that my daughter also showed her and asked her why it didn't fit the real world. No answer.
When teachers are that far down the pedagogical rabbit hole, it's time to rethink.
51
As someone who suffered through the New Math in the 60s our math instruction was not good. Years later I learned that part of the problem was teacher education and the fact that we often have teachers teaching in subjects that they are not expert in. I learned that many textbook publishers edit out the connecting information which leaves those of us who are not mathematically gifted, in the dark about how to understand the reasons why one answer or method is better than another. Memorizing is important for multiplication, division, and other beginning operations in math. Once we progress past the basics it's reading comprehension and good teaching that matter the most. Aptitude is one part, good teaching, programs, books, and patience are even more vital. Our world operates on numbers and processes that are based on numbers and formulas as we express them. How we can have a first class educational system if we don't have a first class way to teach math, at any point in time, to our students?
I have felt the handicap of poor teaching since the 60s. We had ill thought out classes because teachers didn't have the time to understand what they were supposed to do. We were taught concepts that we could not comprehend until we had memorized the basics. You don't teach first graders about syllabication while they are learning to recognize their first words. Yet we were "taught" about various mathematical properties before we understood addition and subtraction.
I have felt the handicap of poor teaching since the 60s. We had ill thought out classes because teachers didn't have the time to understand what they were supposed to do. We were taught concepts that we could not comprehend until we had memorized the basics. You don't teach first graders about syllabication while they are learning to recognize their first words. Yet we were "taught" about various mathematical properties before we understood addition and subtraction.
23
I remember the 'new math' too. Horrible. I had to have tutoring in fractions the neXt year at a new school because I never understood how they worked. The math they have out now is even worse. In the 4th and 5th grade my daughter was bringing home a print-out with 4 problems on it. They had to solve it with tic marks in a little window. We supplemented, but what about kids whose parents didn't? When we moved in middle school she was still behind. Took her a year of work to get back up.
2
Before learning "math" students need to first learn 'ARITHMETIC'
Students need to learn their addition and multiplication tables.
They need to be able to add, subtract , multiply and divide without a calculator.
Students who can not do ARITHMETIC with numbers will be unable to learn algebra with variables.
For example a student who can not add 2/3 + 3/4 by hand will be unable to combine a/ b + c/d.
Forget about the terms commutative and associate and distributive laws.
They are just terms. I didn't know what these terms meant until I was in college but I knew how to use them in grade school.
How many people who can spout the associative law of multiplication know whether any of the following is true.
( a b) / c = (a /c) (b/c) ?
(a b) / c = ( a/c) b ?
(a b) /c = a ( b/c) ?
Students can learn how to solve quantitative problems with mathematical reasoning without ever knowing about sets, or different bases or rotational symmetry.
While these topics might be interesting they must not be allowed to substitute for the student learning ARITHMETIC and basic algebra.
The Common Core expresses admirable goals but its pedagogy is total nonsense.
In a subsequent post I will give examples.
Students need to learn their addition and multiplication tables.
They need to be able to add, subtract , multiply and divide without a calculator.
Students who can not do ARITHMETIC with numbers will be unable to learn algebra with variables.
For example a student who can not add 2/3 + 3/4 by hand will be unable to combine a/ b + c/d.
Forget about the terms commutative and associate and distributive laws.
They are just terms. I didn't know what these terms meant until I was in college but I knew how to use them in grade school.
How many people who can spout the associative law of multiplication know whether any of the following is true.
( a b) / c = (a /c) (b/c) ?
(a b) / c = ( a/c) b ?
(a b) /c = a ( b/c) ?
Students can learn how to solve quantitative problems with mathematical reasoning without ever knowing about sets, or different bases or rotational symmetry.
While these topics might be interesting they must not be allowed to substitute for the student learning ARITHMETIC and basic algebra.
The Common Core expresses admirable goals but its pedagogy is total nonsense.
In a subsequent post I will give examples.
74
The following appeared in the Journal News a Gannett Newspaper of Westchester. Rockland, Putnam NY Counties.
"During the 90-minute session, educators demonstrated different ways to solve basic math problems.
“How would you add these two single-digit numbers?” asked one, as she writes 7+6 on a whiteboard.
The teachers went on to explain ways to get to the answer: “doubles,” “count on” and “bridge to 10.”
In doubles, the student finds the closest double and works from there. So, 7+6 becomes 6+6=12+1=13.
In bridge to 10, students break one of the other numbers up to form a combination that makes 10. In 7+6, break up the 6 to make 3+3. Then, 7+3=10. 10+3=13.
In count on, the student finds the largest number and adds to it, one by one. In 7+6, the student might count on 7, 8, 9, and reach 10. Add the remaining 3 to get 13."
I think this is totally nuts and counterproductive.
If students can memorize that 6 + 6 = 12 and 7 + 3 = 10 they can memorize that 6 + 7 = 13
What is gained by this type of Common Core nonsense. This just detracts from students learning basic ARITHMETIC.
How is this bridge to ten nonsense going to be used to add a a bunch of numbers.
How do you add 35 + 79 + 95 or
35 + 792+ 2354
Those pushing this Common Core nonsense may have made a pile of dough but that does not give them any knowledge of what is of importance for students to learn or how best to teach that material.
"During the 90-minute session, educators demonstrated different ways to solve basic math problems.
“How would you add these two single-digit numbers?” asked one, as she writes 7+6 on a whiteboard.
The teachers went on to explain ways to get to the answer: “doubles,” “count on” and “bridge to 10.”
In doubles, the student finds the closest double and works from there. So, 7+6 becomes 6+6=12+1=13.
In bridge to 10, students break one of the other numbers up to form a combination that makes 10. In 7+6, break up the 6 to make 3+3. Then, 7+3=10. 10+3=13.
In count on, the student finds the largest number and adds to it, one by one. In 7+6, the student might count on 7, 8, 9, and reach 10. Add the remaining 3 to get 13."
I think this is totally nuts and counterproductive.
If students can memorize that 6 + 6 = 12 and 7 + 3 = 10 they can memorize that 6 + 7 = 13
What is gained by this type of Common Core nonsense. This just detracts from students learning basic ARITHMETIC.
How is this bridge to ten nonsense going to be used to add a a bunch of numbers.
How do you add 35 + 79 + 95 or
35 + 792+ 2354
Those pushing this Common Core nonsense may have made a pile of dough but that does not give them any knowledge of what is of importance for students to learn or how best to teach that material.
75
However, for the student who does not always recognize 7 + 6 = 13, the bridge to 10s makes a lot of sense. It also makes it a lot easier to add a long column of numbers.
It all has to do with the wild and wonderful ways the brain can mix things up. This is especially true when there are many numbers to add.
Blessed are those who simply remember the total of each combination, they will get an A in bookkeeping. For those of us who are of a more theoretical bent, learning to figure out how to deal with the numbers is great.
It all has to do with the wild and wonderful ways the brain can mix things up. This is especially true when there are many numbers to add.
Blessed are those who simply remember the total of each combination, they will get an A in bookkeeping. For those of us who are of a more theoretical bent, learning to figure out how to deal with the numbers is great.
15
Being a teacher myself I appreciate your several examples of how to add. We learned in our pedagogy math classes that mastery comes when we know at least three ways to do something. Teaching children to do this in their heads leads to "formal" development of their cognitive abilities (Piaget). Few of us become mathematicians but all of us need to develop critical thinking skills. This ability transfers to virtually all areas of their future lives. Seriously "nonsense?"
3
"sets, nondecimal bases and formal definitions"
As a Boomer product of the new math, I found these subjects to be excellent preparation for a career in computer science. These topics are foundational and are often useful in every day practice.
Sure, one may disagree about teaching methods, to wit, how to teach and how students learn mathematics. However, I would not discard these subjects in a world that emphasizes information representation and processing.
As a Boomer product of the new math, I found these subjects to be excellent preparation for a career in computer science. These topics are foundational and are often useful in every day practice.
Sure, one may disagree about teaching methods, to wit, how to teach and how students learn mathematics. However, I would not discard these subjects in a world that emphasizes information representation and processing.
29
Yes sets, different bases and formal definitions are important.
The question is when should these concepts be taught.
Students need to FIRST learn ARITHMETIC, their addition and multiplication tables.
Sets etc. must not be allowed to replace the learning of arithmetic and basic algebra.
Knowing about sets etc. does not help the student learn arithmetic and algebra but takes time away from teaching arithmetic and algebra.
Sets etc. can be taught to interested students at the college level.
The question is when should these concepts be taught.
Students need to FIRST learn ARITHMETIC, their addition and multiplication tables.
Sets etc. must not be allowed to replace the learning of arithmetic and basic algebra.
Knowing about sets etc. does not help the student learn arithmetic and algebra but takes time away from teaching arithmetic and algebra.
Sets etc. can be taught to interested students at the college level.
30
Knowing about sets does help students learn algebra. Set theory is useful for solving basic world problems. If students think math is just a process of memorizing formulas, and they don't understand the conceptual reasoning underlying the algorithms, by the time they get to college, they will have already lost interest in math. I'm glad my 7th grade algebra and 8th grade geometry teach taught some of the basic concepts of sets. Teachers can use fun tools like Venn diagrams to make the material come alive, and explain the abstractions.
3
David,
No matter how many times you capitalize ARITHMETIC--the internet equivalent of shouting--it won't automatically make you right. My children have been learning math in the way you disapprove of, and so far it seems to be going extremely well for them. Certainly, my 7th grade daughter's math skills, especially her grasp of mathematical concepts, is far superior to mine at that age. For me at least, treating math as simple rote memorization was a big part of what I loathed about it. By the time I got to the point where I realized that math was something far more interesting, that it is a way to think and conceptualize, it was too late. I hadn't learned my arithmetic adequately because simply memorizing my multiplication tables bored me to tears. Consequently, I didn't do it very well.
No matter how many times you capitalize ARITHMETIC--the internet equivalent of shouting--it won't automatically make you right. My children have been learning math in the way you disapprove of, and so far it seems to be going extremely well for them. Certainly, my 7th grade daughter's math skills, especially her grasp of mathematical concepts, is far superior to mine at that age. For me at least, treating math as simple rote memorization was a big part of what I loathed about it. By the time I got to the point where I realized that math was something far more interesting, that it is a way to think and conceptualize, it was too late. I hadn't learned my arithmetic adequately because simply memorizing my multiplication tables bored me to tears. Consequently, I didn't do it very well.
5
"What constitutes rigorous thought" and mastery of mathematical skills that can be used in science and everyday life are largely separate things. Actually mathematics as a tool has typically developed faster than the logical underpinnings. Society needs people who can use math, not necessarily philosophers who can explain why it is rigorous. At any rate the mastery of skills is something that can be evaluated fairly precisely, as can the methods for teaching it. Teaching methods for math should be judged mostly on development of the skills, not on the underlying logic or even less on political grounds.
4
Wrong. The underlying logic is necessary for developing skills. People often declare, "I'm not a math person," because they believe math is a bunch of formulas without context. They were either never taught or never interested in learning the conceptual reasoning underpinning the abstractions of mathematics.
6
Great article Snake oils salesman are behind common core. They sold it as a secret sauce They pretended that htey had the recipe to teach critical thinking, problem solving, creativity, ...
Common core is basically having a teacher who doesn't understand math try to explain the same idea in several different ways to a student who can't even understand one way.
Every time they change the curriculum it is just more wasted money and wasted time. Students fall behind as teachers learn the new curriculum. But the consultants and administrators get rich.
Common core is basically having a teacher who doesn't understand math try to explain the same idea in several different ways to a student who can't even understand one way.
Every time they change the curriculum it is just more wasted money and wasted time. Students fall behind as teachers learn the new curriculum. But the consultants and administrators get rich.
48
Bill Gates is a snake oil salesman?
12
He sure is when it comes to education. He may know about computers but he is clueless when it comes to educational policy, a subject in which he has no background or training. For that matter, neither do any other of these pompous hedge fund managers and business people that are driving policy these days.
4
Is that so hard to believe? Lol. Seriously though it seems you're making an appeal to authority. What makes Bill an expert?
3
Math is magic! Pure and simple. When we start teaching it that way, "scores" will soar.
29
Math--to me--is not magic. It's boring. I stopped taking math courses in high school after algebra. Not a day in my life, and I'm 80, have I missed math. I've grown old and semi-prospered without math. To be accurate, this article should have started with "SOME American children have been bad at math...". And I think this is the problem with American education: we try to make every child above average in everything. Despite our endless experimenting, or perhaps because of it, some children will never get math, just as some children will never get Spanish or even English. Let them learn to live with it.
7
Back in the late 1950s I failed the 3rd grade because of "arithmetic." In 7th grade I suffered through "word problems." In 8th grade I walked into an algebra class and panicked; I got myself transferred to "business" math which was even worse.
Later I made it through 2 semesters of Algebra OK. In college I took that again and, then, signed up for trig because I didn't want to suffer through "liberal arts" math. Toward the end of the year, I had a realization. I thought to myself, "If they had told me this in the beginning, I would have eagerly pursued the basics in order to achieve this little nirvana." I earned a "C" for the year of which, even today, I am still very proud. It is magic.
Later I made it through 2 semesters of Algebra OK. In college I took that again and, then, signed up for trig because I didn't want to suffer through "liberal arts" math. Toward the end of the year, I had a realization. I thought to myself, "If they had told me this in the beginning, I would have eagerly pursued the basics in order to achieve this little nirvana." I earned a "C" for the year of which, even today, I am still very proud. It is magic.
21
I don't know about the math programs in all schools but I do have a 9th graders view of the curriculum offered in a public school vs a private school. Earning a scholarship my son moved from a pretty well regarded public middle school to a private high school. A few weeks in I asked him how it was going because it seemed like he had a lot more homework; he never had homework in 8th grade. My son's response was telling. 'Dad it is really tough. The class time is the same. In middle school we messed around for the first 10 minutes and we were given time for homework the last 15 minutes. Now we take our seat and the teacher starts teaching and he teaches to the bell. There is no time for homework at school. Maybe public schools would be better if the teachers were teaching the whole time.'
I have looked at the middle school math program here over a 10 year period and our local schools' approach has been extremely disjointed and way too fluid. The private school was using books that had been in the school for over 20 years and supplemented them with training in the latest technology (calculators and spreadsheets).
The private school also spends less money per student. Go figure.
I have looked at the middle school math program here over a 10 year period and our local schools' approach has been extremely disjointed and way too fluid. The private school was using books that had been in the school for over 20 years and supplemented them with training in the latest technology (calculators and spreadsheets).
The private school also spends less money per student. Go figure.
58
I find what you say interesting and true. But the money spend per student will always be higher for public schools than private because public has to take every student (no matter what the expensive needs) and is not mandated by law to take care of so many social problems.
7
No, private schools do NOT spend less per pupil. It seems that way until you factor in special ed (which they for the most part do not participate in), disabled students, and behavioral issues (which they eliminate though expulsion). This is just the tip of the iceberg. Does your private school have disabled students who need 24 hour nursing aide? This all comes out of the public school budget Staffing for special ed alone in a given district will account for much of the perceived difference in per pupil spending.
9
"latest technology -- calculators and spreadsheets"
Hilarious.
Hilarious.
7
As long as mathematics teachers are unprepared to teach whatever program or strategy or philosophy is currently embraced, most students will struggle and fail to thrive. When I was in secondary school in the early 1960s, we had hastily-printed paperbound texts from the "School Mathematics Study Group" and wrestled with set theory and base two while our teachers tried to stay a page or two ahead of us. It was interesting and challenging but not taught with insight or confidence.
Again and again, teachers are given new standards, new materials, new methods -- but little opportunity to upgrade their own skills or prepare their students for the transition. "Common Core" is a case in point -- with exemplary goals and content, but completely lacking in effective plans for incremental change.
Add to that the fact that many elementary school teachers -- in whose classrooms essential foundations are laid (or not) -- are far more knowledgeable in and enthusiastic about reading and writing than about mathematics, and the result is a system built for failure.
Again and again, teachers are given new standards, new materials, new methods -- but little opportunity to upgrade their own skills or prepare their students for the transition. "Common Core" is a case in point -- with exemplary goals and content, but completely lacking in effective plans for incremental change.
Add to that the fact that many elementary school teachers -- in whose classrooms essential foundations are laid (or not) -- are far more knowledgeable in and enthusiastic about reading and writing than about mathematics, and the result is a system built for failure.
112
Whatever you do in America to redress this problem (that is if you do anything at all), then please get it right, because there is every likelihood that we in Australia will slavishly follow you regardless of how sensible or otherwise your reforms are.
13
Math is not the only discipline through which students, or people in general, can learn to think. Logic and philosophy have the same purpose. Why are these two subjects not given more emphasis in current curricula, either in elementary, middle or high school?
65
Many of us can testify that the charge is compelling by our own observations while growing up. Math doesn’t come naturally to a lot of us. It never came naturally or effortlessly to me – I could navigate it right up through integral and differential calculus, but it required a degree of concentration and intense intellectual effort that I didn’t need to apply in ANY other field to understand it and be successful at it.
That said, though, we seem to produce enough people and always have to lead the world in highly technical patents, excelling in chemistry, physics, computer technology, genetics, communication – all disciplines that require, to one extent or another, some facility with math. We also have our fair share of Nobelist economists whose arcane relations are nothing if not mathematically expressed. Then, we’ve never been shy of engineers to turn ideas into working mechanisms and formulations, and a solid grounding in math is required for good engineers.
But learning math does NOT count as “learning to think”. Thinking is far more abstract than that and math often is merely the means of expressing the thought: it’s a means of communication. To me, effective math education has always meant defining a meaningful level of math proficiency that serves the vast bulk of our math-handicapped society and how to convey it, then provide the support for those with a talent at it to dive deep. Common Core can only be a narrow aspect of those charges.
That said, though, we seem to produce enough people and always have to lead the world in highly technical patents, excelling in chemistry, physics, computer technology, genetics, communication – all disciplines that require, to one extent or another, some facility with math. We also have our fair share of Nobelist economists whose arcane relations are nothing if not mathematically expressed. Then, we’ve never been shy of engineers to turn ideas into working mechanisms and formulations, and a solid grounding in math is required for good engineers.
But learning math does NOT count as “learning to think”. Thinking is far more abstract than that and math often is merely the means of expressing the thought: it’s a means of communication. To me, effective math education has always meant defining a meaningful level of math proficiency that serves the vast bulk of our math-handicapped society and how to convey it, then provide the support for those with a talent at it to dive deep. Common Core can only be a narrow aspect of those charges.
93
http://cue.caltech.edu/documents/22-core_with_tables_appendix.pdf
"The transition from high school to college presents problems for all students, but for some students it is particularly challenging. At Caltech, many newly admitted students lack the background in mathematics that is necessary to succeed in Ma 1a. Unfortunately, few of them are even aware that their background in mathematics is deficient. This is not their fault. The mathematics curriculum in high schools is less rigorous than it was even a few decades ago. In conversations with Caltech students who have struggled with freshman mathematics, most report that they were star math students in high school, which of course is a major reason why they were offered admission to Caltech in the first place. Many of them, however, have never seen mathematics as it is taught at Caltech. "
"In order to understand the specific reasons why many of our freshman struggle in Ma 1a, the undergraduate Academics and Research Committee conducted an online survey that asked a series of specific questions about the difficulties they encountered in Ma1a. From the survey results, the most common area of weakness that students identified was that of formal reasoning, writing proofs, and common proof techniques. The results thus corroborate what most people connected with Ma 1a have known anecdotally—that many Caltech freshmen, though computationally skilled, struggle with basic proof concepts. .... "
"The transition from high school to college presents problems for all students, but for some students it is particularly challenging. At Caltech, many newly admitted students lack the background in mathematics that is necessary to succeed in Ma 1a. Unfortunately, few of them are even aware that their background in mathematics is deficient. This is not their fault. The mathematics curriculum in high schools is less rigorous than it was even a few decades ago. In conversations with Caltech students who have struggled with freshman mathematics, most report that they were star math students in high school, which of course is a major reason why they were offered admission to Caltech in the first place. Many of them, however, have never seen mathematics as it is taught at Caltech. "
"In order to understand the specific reasons why many of our freshman struggle in Ma 1a, the undergraduate Academics and Research Committee conducted an online survey that asked a series of specific questions about the difficulties they encountered in Ma1a. From the survey results, the most common area of weakness that students identified was that of formal reasoning, writing proofs, and common proof techniques. The results thus corroborate what most people connected with Ma 1a have known anecdotally—that many Caltech freshmen, though computationally skilled, struggle with basic proof concepts. .... "
6
" .. math does not count as 'learning to think' ". True enough, in the sense that neither does drawing, or writing, or physics or music or any other discipline. Each discipline has it's own mental challenges. If we adopted the teaching & learning protocols of countries whose students do well in math, it would also likely improve outcomes in the other disciplines as well. Except, to change such protocols would entail major changes to our culture, economy and society. Doubtful that will happen soon, if ever.
22
Unfortunately, this is simply untrue.
American mathematics and science education simply do not produce enough competent individuals to fill our employment needs. We don't even produce enough to get into graduate school.
A very significant proportion of the students we are training in graduate science, mathematics, and engineering programs are from other countries. In research laboratories, graduate students are, by far, the people who do most of the actual research - and we don't prepare enough people well enough to do this work. In graduate school, we were surrounded by people from other countries who had a superior education and who spoke multiple languages. It was a privilege to interact with such intelligence and eye-opening to difficulties in other parts of the world. But it was embarrassing that our own, enormous country produced so few people that could match my peers from other parts of the world.
American mathematics and science education simply do not produce enough competent individuals to fill our employment needs. We don't even produce enough to get into graduate school.
A very significant proportion of the students we are training in graduate science, mathematics, and engineering programs are from other countries. In research laboratories, graduate students are, by far, the people who do most of the actual research - and we don't prepare enough people well enough to do this work. In graduate school, we were surrounded by people from other countries who had a superior education and who spoke multiple languages. It was a privilege to interact with such intelligence and eye-opening to difficulties in other parts of the world. But it was embarrassing that our own, enormous country produced so few people that could match my peers from other parts of the world.
8
Attitudes to math and any other aspect of education are determined more by the mirror on the wall than any other factor. Researchers and parents have difficulty recognizing anything other than their own reflections. New math arose in an age when parents sought more for the children and expected education to improve their children's lives. Those parents were succeeded by the Me generation, whose narcissism dominates everything.
11