Galileo once wrote “Mathematics is the language with which God has written the universe.” It is said that over the door of Plato’s Academy were the words “Let no one ignorant of geometry enter.” Indeed, the word ‘Mathematics’ comes from Ancient Greek meaning “that which is learnt.”
Few people love Mathematics more than I do. What music lovers get from listening to a Mozart concerto, and what poetry lovers get from reading a Shakespeare sonnet, I get from following through the steps of an Euler proof. In fact, the one book that I brought with me on my honeymoon six summers ago was William Dunham’s incredible, Euler: The Master of Us All.
When I’m teaching somebody about math, I feel like I’m enlightening them to some of the hardest earned secrets of the Universe. I take great pride in getting to be the one who shares these truths. While some religious people bring signs to football games saying ‘John 3:16,’ I’m tempted to bring my own sign that says ‘Euclid 9:20′ in honor of the greatest proof of all time.
Education ‘reformers’ often say that the average American student is not good at Mathematics. And, for once, I agree. But my evidence for believing this is completely different from theirs, what I think has caused this problem is completely different from what they think has caused this, and what I think we should do about this is completely different from what they think we should do about this.
‘Reformers’ believe this because we were ranked 25th out of 34 countries on the 2010 PISA tests. The PISA scores mean very little to me since I believe that the kids in the other countries ahead of us are not good in Mathematics either beyond a superficial understanding, so what difference does it make what place we’re in? ‘Reformers’ think that the blame is ‘ineffective’ teachers and ‘failing’ schools, while I think it has nothing to do with teachers and schools but, instead, a problem with the way Mathematics education has evolved over the past two hundred years, and which I will go into more detail below. ‘Reformers’ think that this is a ‘crisis’ and that we need to respond by spending billions of dollars making teachers more accountable and by increasing the ‘rigor’ of the content through the implementation of the common core standards and the accompanying assessments. I don’t consider it a ‘crisis,’ but something that is an unfortunate missed opportunity and one that can be ‘fixed’ by spending much less money on math education, but using that money in a much smarter way. I will expand on this below also.
Two hundred years ago, students who finished high school learned about as much Mathematical content as modern fifth graders learn today. And over the past two hundred years, topics were gradually added to the curriculum until the textbooks have become giant bloated monstrosities. And though the modern high schooler ‘learns’ Algebra, Geometry, Algebra II and Trigonometry, Statistics, and maybe even Precalculus and Calculus, the average adult still only remembers about as much as the adults from two hundred years ago did, or about what the average fifth grader is supposed to have learned.
I don’t know the exact figure, but I’d estimate that at least twenty percent of education spending on this country is somehow related to math. I always hear about how this school or that one is providing ‘double math’ blocks and, of course, the testing mania in which math has been elevated to half of an elementary school teacher’s rating and half of the school’s rating, there has been not just a lot of money spent on math, but a lot of equally valuable time. In terms that corporate reformers can relate to, we have not gotten a large return on investment with this. We have spent a lot of money and time on this and have very very little to show for it.
Earlier I wrote that I don’t really consider this a ‘crisis’ since I don’t agree with those who think that if students would do better on standardized math tests it would mean that they were more ready for college or ready to compete globally. I don’t think a large percent of students have ever been truly good at math in this country or in any other one, so to elevate this to a ‘crisis’ I think is exaggerating. But I do think it is sad when we dedicate all this money and time to a subject that is like a religion to me and that generally goes “in one ear and out the other.” It is not a ‘crisis,’ but it is a ‘problem.’ It is a shame and a waste of money and time, and a very unnecessary one, I think. For all we put into it, we get out of it a majority of people who ‘hate’ math and who feel that they were ‘bad’ at it.
The biggest problem with math education is that there are way too many topics that teachers are required to teach. Why has this happened? Over the years things have been added, often as a way to prepare students for something that is going to be needed for a future course. “They’re going to need to simplify complicated expressions in Calculus? Better drill them on simplifying complicated expressions in Algebra II.” Though things keep getting added, it is rare that anything ever gets removed from the curriculum. The common core was an opportunity to remedy this, but from what I can see they haven’t really allowed anything to be removed. If I were made ‘Math Czar’ I would gleefully chop at least forty percent of the topics that are currently taught from K to 12.
When teachers have to teach too many topics, they do not have time to cover them all in a deep way. The teacher, then, has to choose which topics to cover in a meaningful way, and which to cover superficially. It would be as if an English teacher was told to cover fifty novels with her class. Not being able to have her classes read all fifty books, she would pick some to read fully while having her class read excerpts or even summaries of the other ones.
I got to witness an extreme example of this decision making when I graded the Geometry Regents at the centralized grading center this past June. A huge part of Geometry, in my mind the most important part, is deductive proofs. I’d say that over half of a ‘true’ Geometry course would involve proving different theorems. Well, on the Geometry Regents these proofs are not a large percent of the test, less than ten points out of eighty. So on the June Regents the last question on the test, a six point question, was the proof question and I was assigned to grade about 200 papers from a school, I won’t say which one, to grade. As I graded I noticed that many of the students left the proof blank. By the end of my grading I realized that out of 200 papers, all that could have received up to 6 points for the proof — a total of 1,200 potential points to have been earned on this question, I had awarded only two points total. That’s two points out of a possible 1,200. I asked around and the consensus was that teachers, knowing that proofs would take months to cover but be only worth less ten percent of the points on the test, would be too risky to teach. All the time spent on this tough topic would only, at best, get the students a few extra points while they would lose all that time they could use to teach some of the easier topics that were more likely to be on the test.
I teach at one of the top high schools in the country, Stuyvesant High School, where I have no trouble covering all the topics at a deep level. But when I taught in Houston in the beginning of my career I made the decision that given the choice between teaching everything at a superficial level or leaving out some topics, but teaching the others at a deep level, I would teach fewer topics at a deeper level. Fortunately for me, this was before the ‘accountability’ craze so if I didn’t finish the curriculum, I just left the topics I hadn’t gotten to off of my own final exam. So when I taught Geometry, you had better believe that I did not short change ‘proofs’ even though it might have meant that I ran out of time before getting to the tedious and mindless ‘coordinate Geometry’ unit.
So the first thing I’d do to ‘fix’ math is to:
1) Greatly reduce the number of required topics, and to expand the topics that remained so they can be covered more deeply with thought provoking lessons and activities.
I think that this was part of the original premise of the common core standards, but like a novel that gets optioned to be made into a movie, often a lot gets changed as more people get involved in a project. The common core standards has not reduced the number of topics, and has demanded that teachers teach all the topics to more depth, which is just not possible.
Most people think of math as a sequence of skills that each must be mastered before working on the next one, and the idea that topics can be skipped is surprising. But math is not a chain of skills. It is more like a tree where there are different branches, some of which can be pruned (I will address which ones in a future post). So by teaching fewer topics, but better, students still could be ready for Algebra by ninth grade.
The next prong of my reform of the way we teach Mathematics in this country would be very controversial. I would:
2) Make Mathematics, beyond eighth grade, into electives.
Much of the material from Algebra, Geometry, Algebra II and Trigonometry, Statistics, Precalculus, and Calculus would still exist, but only for students who want to take them. If you think nobody will want to take these, remember that you are imagining current 8th graders making this decision not the 8th graders who had just experienced a more interesting nine years of Mathematics. Yes, many students will opt out of these electives which would mean that there would be fewer high school math teachers, but for the students who did choose to continue, those courses could also be taught at a deeper level since the students in them will be the ones who like math, and are not, as is currently the case, forced to take a course that they despise. The billions of dollars saved by requiring less math could be used to expand other electives and bring back art, music, and drama to schools that have cut them in order to fund double block math test prep.
I know that many people would, at first, object to this. “Won’t this cause us to have a less educated population?” “Who will be our engineers and Mathematicians?” “Isn’t this the opposite of ‘more rigor’?” To answer these concerns, we need to examine what the purpose is of teaching Mathematics in the first place. If you think the idea is to maximize the number of students who become Math majors in college, I think this plan would produce more Math majors than we currently have. The same for engineers. I really think we would have more people interested in Math with the better K-8 courses so many people would choose those electives and they would graduate as much stronger Math students as the high school electives were not slowed down by students who had no interest in those courses. As far as whether having many students stop their math education before Algebra, remember that those students were ‘learning’ those topics in such a superficial way, if at all, and it was completely wiped out of their minds after, if not before, the final exam.
To me the purpose of learning math is very similar to the purpose of learning to play a musical instrument. I know that some math people might see this as an insult to math, comparing it to a lowly elective, but those people may not understand how highly I think of learning music. Engaging in real math is something that is as thrilling and suspenseful as any great mystery novel. A problem like ‘How many times would you have to fold a piece of paper until it becomes so thick that it reaches the moon?’ (Answer: Way less than you think. It is around 40 folds.) Causes a million times the excitement than a typical math question like ‘Write the expression: ‘5 less than a number’ in symbols’ (Answer: x-5, and NOT 5-x, ooh, trick question!)
To give you a sense of what I mean by ‘meaningful math,’ here is something I taught my 9th graders a few days ago. I first showed them a surprising number pattern that starts like this: 1=1*1, 1+3=4=2*2, 1+3+5=9=3*3, 1+3+5+7=16=4*4. The first question I posed was to find the next number in the pattern. (Answer: 1+3+5+7+9=25=5*5) The next question I posed was to determine the value of 1+3+5+7+…+95+97+99. (Answer: 2500=50*50 since there are 50 consecutive odd integers.) Finally, and you should know that this whole thing took over 20 minutes, and that was at Stuyvesant High School, so if I was teaching elsewhere this would be a 40 minute activity, I asked them to try to explain ‘why’ the sum of consecutive odd integers starting at 1 was always a perfect square. After giving them some time to try to figure it out themselves, I put on the screen an image, known as a ‘proof without words’ and waited a few minutes for all the various shouts of ‘I see it!’ and ‘Whoah!’ to die down followed by a general murmur of excitement as students explained to their neighbors who didn’t see it just yet. If you don’t consider yourself a ‘math person’, still give yourself a chance to revel in the beauty of this image, and I hope you’ll get to experience your own ‘aha’ moment that mathematicians live for.
If this image doesn’t excite you, you probably wouldn’t be one of the students who chose to take those elective math courses, but that’s OK, they are not for everyone.
Nobody ever consulted me when designing the common core, so I never got a chance to propose my two reforms. So instead of my ideas, we have ‘higher expectations’ with more ‘rigor’ and more ‘rigorous’ assessments. States that have started on these assessments, like in New York, have seen proficiency rates drop from 60% on the old tests to 30% on the new common core tests. The politicians assure us that when schools get used to the higher expectations, the scores will increase over the years. Those politicians, however, know nothing about teaching and learning. Higher expectations will not cause the scores to increase. Teachers are too constrained by the number of topics they have to teach and the number of students who hate math. So my prediction is that unless they change the tests or the cutoff scores to make it look like they were right, the percent proficient will remain around 30%. Maybe then they will go back to the drawing board and come up with a math education reform plan similar to what I just outlined.
In my title, I was very deliberate to write ‘math’ with a lowercase ‘m’ rather than ‘Mathematics’ with a capital one. The ‘math’ that clutters up textbooks nowadays is not, generally, worthwhile ‘Mathematics.’ So maybe an unintended consequence of the common core standards will be, as I wrote in my title ‘The Death of math.’ But maybe it will also be the rebirth of Mathematics.
And that’s what this ‘status quo defender’ thinks about that.
Note: This post was inspired by articles and blogs I have read over the years arguing for and against forcing students to take math. In the past two years there have been several articles in prominent publications calling for a reduction in the amount of math we require students to take. In July, 2012 there was The New York Times piece Is Algebra Necessary? and in the September 2013 issue of Harper’s, there was Wrong Answer. The case against Algebra II. Some of my brother Math bloggers have written some very persuasive responses to those articles, most notably Patrick Honner’s ‘Replace Algebra with … Algebra’ and Jose Vilson’s ‘Nobody Puts Algebra 2 In A Corner’. Before the articles in The New York Times and in Harper’s articles, I’ve read some even stronger cases against forcing students to take too much math by a university professor and by a high school teacher. Math professor Underwood Dudley’s ‘Is Math Necessary?’ argues that it isn’t, at least in the sense that most people think. And Paul Lockhart had a viral 25 page essay called ‘A Mathematician’s Lament’ in which he compares current math instruction to a fictional society where students are forced into thirteen years of music instruction without ever enjoying a melody.
I am a high school math teacher, following a career in the computer industry, business, and nonprofits. (My first love was math, and I have always wanted to teach math. I am realizing this dream in my later years. This article is EXACTLY what I have experienced and come to believe. Thank you for putting it all into a cohesive essay.
It is painful to love a subject and not be able to cover it in depth; it is painful to know that many will never truly understand what mathematics is.
Well said, Gary Rubinstein!
Gary, I agree with you on your assertion that math curriculum is too broad and greater focus would make it much more effective, but completely disagree on making math an elective. The implications of making math (and its cousins, engineering and economics) option are extremely damaging.
I think the key difference between math and music that stops this from working is that math is the foundation to a lot of other subjects, including virtually all sciences, economics, business and engineering. Moreover, unlike music, studying math (and its related fields) beyond high school is correlated with higher earning potential than studying something less quantitatively rigorous. The evidence couldn’t be more clear. Compare the average mid-career salaries of various math-intensive majors:
– Mathematics: $92k/yr
– Engineering: ~$100k/yr
– Economics: $98k/yr
vs liberal arts majors:
– English: $65k/yr
– History: $71k/yr
– Music: $55k/yr
By letting students self-select out of mathematics courses when they are 13-14 years old, you’re allowing them to permanently derail themselves from a number of future study and career options. Not everyone who ends up taking on a math-intensive major and working in a math-intensive field loved math their freshman year of high school. I certainly didn’t really like math that much until my junior year of high school, when I had an amazing Calculus teacher. That doesn’t mean I should have been allowed to self-select out of it like Spanish or Dance.
Once you fall off the math train, it’s hard to climb back on board. It’s not impossible, but if you realize senior year that you liked math after all, starting again at Algebra 1 is extremely demoralizing. It also means you’ll have to go through years of remediation before you can start on a math-intensive major, thus discouraging a lot of prospective students from pursuing those majors.
The socio-economic implications are incredibly disturbing. Consider the following:
1) Think about how likely it is that high school boys will self-select into math/science electives relative to high school girls. Women are already heavily under-represented when it comes to STEM, will making math optional after age-13 close this gap or broaden it?
2) Think about how likely it is that parents who are highly involved in their children’ education will let their kids self-select out of math? Let me be more specific:
2b) Think about how likely it is that parents who are middle and upper class will let their kids self-select out of math relative to lower-class parents. No, wait, let me be even more specific.
2c) Think about how likely it is that white and asian parents will allow their kids to self-select out of math relative to hispanic and black parents. Really, honestly reflect on this question. How many students did you know whose parents pushed them to go into law or medicine when you were in college? (I knew a ton). What did those students look like?
Gary, as you’ve mentioned you, teach at Stuyvesant High School. Stuyvesant High School is 59% male 41% female and a school that is 72% asian and 23% white. You have to admit that almost no one in your school is going to be self-selecting out of math any time soon. Your students won’t be impacted. If you applied this same optionality to all schools, the impact would differ from community to community, and the impact on the racial gap in opportunities and outcomes would only get worse.
I have to agree with Mr. Vartanov.
In my twenty years of teaching high school I have come across only a handful of ninth and tenth graders who could be counted upon to make all of their decisions for themselves (as in, they were making mature, adult decisions all of the time). Allowing kids to “opt out” of mathematics at the 8th grade (or even the 9th or 10th grades) would certainly be devastating. Sure, I get Gary’s point, and understand that it is predicated upon students receiving optimal math instruction from the start, but I still think Gary’s off base with this.
I don’t have the same passion for math that Gary does. But, as a physics and chemistry teacher who has taken courses in things such as partial differential equations and complex analysis, I certainly see the utility of it.
For most of us, math is a tool. Perhaps the real observation is that we should place more importance on math instruction in earlier grades (particularly junior high), and that MANY juniors and seniors in high school MAY be better served by taking classes that are more relevant to a REAL, VIABLE CAREER.
You know, when they have a clue.
Perhaps the answer is not to allow students to opt out in 8th grade, but rather when they have achieved mastery of a basic curriculum. I don’t usually teach math, but I did teach remedial math to a bunch of inner-city 10th graders. These students didn’t know their perfect squares, but I was told to teach them algebra — a pointless exercise. Some students will need more time to complete the basic curriculum. After they’ve done that, let them opt out if they wish. Others, who have been able to master the material by 8th grade can be induced to continue by the usual means — graduation requirements, mainly, the requirements of state tests, and the need for math in the sciences and other fields. It’s really important to realize that college is only one destination for kids, and for the rest, an 8th grade level of math is quite sufficient; when it isn’t they can go the remedial route later.
He’s right, if only because math instruction in a low-income high school is no more than training students to use a calculator effectively enough to pass the End of Course test. At least, knowing some teachers, I know for a fact that this is what math is now, in my city.
That type of “instruction” is never going to be useful to students, either in college or in any career, and you might as well get rid of it.
I love and see the beauty of math, too, by the way (I love Kurt Gödel), and I hate the way it is being completely destroyed by these accountability measures. You couldn’t teach it well now if you wanted to, because you would lose your job the next year.
You guys did not absorb (because you’re not addressing) his speculation that the 13/14 year olds won’t opt out if their math experience is what he wants it to be. It’s a package deal. I’ll add that if you have lousy math instruction in the early days you are much less likely to be sufficiently delighted with the upper level courses you are forced to take and then unlikely go on to major in it in college. Taught RIGHT, it will encourage the people who love that square-of-dots thing, and quit wasting the time of the people who don’t, and shouldn’t that be what we can all agree on would be a good thing?
You have completely neglected to consider that, if math were being taught as an in-depth subject instead of superficially as it’s being taught now, more students WOULD choose it as an elective. Additionally, by 8th grade students would have developed a far better understanding of math and its concepts, rather than just how the numbers work; they’ll have a better grasp of the logic, the practical uses, and the creativity behind it. This will drive FAR more students to WANT to explore it.
And while I understand your point of how hard it COULD be to go back to math after a few years, your comment of a high school senior finding Algebra I “demoralizing” is only true in our current curriculum, where we require students to have taken it much earlier. It would be demoralizing for a CURRENT senior because s/he would be “stuck” in a class of lowly freshmen, at best. If we removed these barriers by making it an elective, that senior would be taking Algebra I with a much greater spread of students, would still be easily understanding the concepts because of the deep understanding gained in earlier schooling, and would be happy to be there instead of showing up because s/he has no choice.
The FULL suggestion is to teach math in-depth on fewer subjects to help students gain a TRUE understanding of it. This absolutely provides incentives for students to take further classes, because far more students will have reached the point of enjoying math by the time they have to make that choice. It also means that students who opt-out will have a much easier time opting in later in life, because the knowledge they have of math will be deep; we’re more likely to forget the superficial things we currently teach. Finally, we will have a LOT more students interested in pursuing math-related fields in college and beyond, because it will be a subject more students are truly passionate about. The current generation of students are pursuing subjects like English, history, and art because those are things they became passionate about in high school when they discovered they were really good at it, really enjoyed it, really understood it. Let’s give students a greater chance to feel that passionate about math, by fostering a love for it instead of a begrudging necessity.
What would happen if, instead of requiring ever more classes in mathematical algorithms (which is really all current high school classes are), we instead required courses in economics and engineering? We spend so much time teaching the “how” rather than the “why”. Of course our kids think they’re “never going to use this stuff,” because we never shown them how it is useful.
Amen is all I can say. I am not a math teacher, though I “taught” some algebra in summer school and one geometry semester for a pregnant colleague, but you have put into words my own ideas. I almost lost a friend during that Harpers period when I advocated making math into hs electives.
I think people are overreacting a bit here. Just because Gary wants math to be an elective, does not mean that a flood of students will “opt out.” In many states, just Alegbra 1 and Geometry (with no proofs) are required to graduate. However, most colleges require that you take 3-4 years of math, making it to at least Algebra 2 and Pre Calc. Knowing that I was going to my (god forbid) local state school which required that I take 4 years, that is exactly what I did. So, I guess you can say that myself and thousands of others around the country took several years of math as “electives” in order to be accepted into college. Just because high schools change the requirement, doesn’t mean that colleges will, so I guess I don’t get why so many think what Gary is suggesting is so atrocious. What we really should be talking about is how we can get more counselors into schools so kids have someone advocating for the best high school and post graduate options for them.
What I’d like to see (and what we can probably all agree with) is greater emphasis placed upon, and better appoaches taken for, math instruction in elementary school and junior high.
Most would agree that kids are capable of much more than what is being asked of them…particularly in junior high.
Please click here for my comments http://mrdunk.blogspot.com/2013/10/the-death-of-math-my-response-to-gary.html
The educational establishment would not like it. There is money to be made, tenure achieved, and pensions can be earned. The more useless stuff being taught, the more there people achieve for themselves. We have so many young people in debt in excess of $50,000, unable to find a job to sustain them. It’s OK, because their professors are doing well.
Take a look at the educational system where I grew up (I graduated in the late 80s) in Germany:
The students with the lowest academic aptitude are able to graduate after 9th grade (they are spending 3-5 years with students of their academic level). These students are tracked into qualified trades after a 3-year apprenticeship, 2 or 3 days a week at the job site and the remaining days of the work week at a school, taking academic classes targeted to completing their diploma. They receive a stipend from their employer.
Students with an average aptitude graduate after 10th grade to start into a 3 or 3 1/2 year apprenticeship/training program for either skilled trades or administrative jobs. Just as the previous level’s students, they are using the dual track to give them job skills, as well as the academic skills specifically for their job field, and receive a stipend from their employer. Nursing is one of the tracks at this level, as are many others that require a bachelors degree in the US.
The students with the highest academic aptitude will graduate after (new 12th) 13th grade, and are able to immediately enter a university. They can not necessarily choose their major. My nephew was chosen to study Chemistry, based on his grades.
Some of the highest aptitude students decide to take the mid-level track, especially for very competitive job fields. There is the possibility for dedicated students to take the path from 9th or 10th grade grad to the university or technical college level, after completing their academics at night school.
In this setup valuable resources are not wasted for students who are simply set up for failure. Instead they are able to succeed and find a well-paying job that’s in demand, rather than loosing hope, dropping out, and being stuck in a minimum wage/government assistance life, unless they are willing to borrow lots of $$$ or receive a Pell grant using government funding to help them achieve their educational goals.
The system is completely broken, but the reasoning of teaching the students that they can achieve everything if they just work hard is insane. Either we need to lower the standards to help make everybody successful, just so the majority has to take remedial classes at the community college after graduation, or we need to give our young people valid alternatives. Personally, I believe it’s much healthier for somebody to be productive and successful at their level, than to feel like a failure, and to simply give up.
You skipped the educational pathway for future politicians. You know, the one where you’re allowed to plagiarize your phd….
“The common core standards has not reduced the number of topics, and has demanded that teachers teach all the topics to more depth, which is just not possible.” Well said. This is a problem with the ELA standards, too.
About math, wouldn’t it be possible for high schools to offer (let’s say) a seven-year course sequence, beginning with middle-school math (for students whose prior schooling was, for whatever reason, inadequate to teach them this), through prealgebra, to algebra, and so on until calculus, and allow students to a) start where they are, and b) move on to the next course when they demonstrate readiness? I think that doubling-up of algebra periods has to do with our standards-obsessed society’s insistence that all students be in the same place at the same time. But human beings develop in different ways and at different rates; the student who is badly frustrated by prealgebra may discover a year or two later that he or she loves math. Sometimes it just takes time — not class time, as in a doubling up of math periods, but human, developmental time — to grow into our capabilities and interests.
I’ll throw my blogpost on the Harper’s article out there as well for your consideration, for a orthogonal take:
“Nicholson Baker, Algebra 2, and Equity”
My impression is that your Common Core Math prediction will be true, based on the results of creating a “asprirational” math assessment — the NECAP. Rhode Island, NH, and VT are all still in the 30’s on that one.
We’ve had high schools in Providence that went up 40-50 points in reading proficiency after various re-structurings, and did not move a single point in math. Central Falls HS put a huge focus on math and went from 7% to 14%, etc. etc.
In doing some research on the NECAP I also found that Minnesota managed to give themselves a test that they can’t get out of the 30’s on, despite being in the top five for math performance.
So… yes, without knowing much about the particulars of any of these tests, it seems likely that we’ll end up with tests for which the scores don’t magically go up every year.
Certainly in RI, that’s just an article of faith at this point. Hold the line and the scores will go up because… SCORES GO UP!
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The sad thing is the the sales pitch for Common Core was that it would do exactly what this essay recommends: Cover let material at greater depth. As noted above, it does no such thing. Sadly.
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My high school required everyone to take four years of English but offered some half- or full-year “electives” in addition to the traditional junior and senior classes. What do you think about doing the same for math? After students take Algebra II, they could choose the traditional path to trig or take electives in probability and statistics, computer science (!), linear programming, Rubix cubes…
Sorry Mr. Rubik.
Loved the example, nothing like the combo of the graphical representation and the algorithm. We only saw equations when I was in school. While tutoring, I thought the investigative discovery math texts were a little cryptic, but I really enjoyed the pictures of quadratics or the visual layout of the double angle formula in trig. That blew me away.
What is a complicated piece of math that should be taught for a week? (like the Gettysburg Address) I would say paying off a loan with interest. What does that look like graphically (some kind of dragons back) Graphing loan balance versus time. The steps are all even with even payments but the slopes gradually decrease as the interest paid is less on each installment. Or how about comparing home loan points versus interest? What’s the optimal loan for a planned stay?
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I agree so strongly with this post it’s hard for me to put in words. One example (among so many this past year) from this week:
I am giving a standardized, short cycle assessment to my students in order to gauge how prepared they are for the Common Core assessment at the end of the year. Yes, I’m giving up A WEEK of instructional time to test whether my students are on track to pass another test at the end of the year, and this is only the first time out of three I have to administer this test.
Much of the material on this test they haven’t even seen yet because we are only a quarter of the way through the year. So, my students feel unsuccessful and hate math even more.
The true tragedy is that I could be using this time do something so much more interesting. I had a student ask me what binary was at the end of class last week, and I spent the last 10 minutes of class doing a quick overview of binary, base 10, hexadecimal and what they are used for. The students in my remedial Pre-Algebra class were enthralled. They had no idea that math could be interesting or break out of their expectations. But now I’m back to the death march of our pacing guide (which my assistant principal insists we follow to the day on pain of a bad evaluation).
Most of the students in this class are not interested in going to a traditional college and are never going to need to use Algebra II in their lives, so why is it a graduation requirement? Why do they need to spend three more years slogging through classes they hate when they could be taking ‘electives’ like auto body or taking classes at the community college that can actually help them get the jobs they want?
I would go a little bit further and say that a solid grounding in what I think of as Algebra (solving and graphing linear equations and inequalities) is where students should end up, but yes, let’s make everything after that an elective.
The community college will give them placement exams in ELA and math, as you well know. When your former students place back in to Arithmetic (the lowest level math course offered at junior colleges), Pre-Algebra, or Algebra 1, then what?
They will have to slog through four semesters of mathematics aimed at 10 year olds before they can even qualify to enroll in Geometry or Algebra 2. They will be allowed to enroll in technical career paths, or move into a collegiate transfer track AFTER they pass Algebra 2.
Sorry to say, but it is better that they receive exposure when they are younger. Thus, the buck stops at you.
In comparing learning math to learning to play a musical instrument, I think you get at something important… but miss something else. Part of the purpose of education is to give people the tools to be active, informed citizens. In this era of standardized testing, this side of education seems to me to be neglected — and where it’s recognized, it seems to be confined to subjects like history and civics. However, I think these days that it’s hard to be an active and informed citizen without a basic understanding of statistics — of the importance of sample sizes, of effect sizes and confidence intervals, and so on — and how to apply that understanding to issues of the day. That’s not a class either of my kids was even offered in high school. It seems to be too math-y for social studies teachers, but the fact that it would have to focus on understanding the news of the day rather than on math in a pure sense means it doesn’t get taught in math departments either. We are all doubly poorer as a result — poorer because we and our neighbors don’t have the tools to understand our world, and poorer still because the same is true of the folks in the media on whom we rely to help us make sense of the world.
I feel like the main problem with math education is that we don’t have a good use case to drive it. Kids love learning how to DO things, but they don’t love learning the process only for the processes sake.
Fortunately, we have entered a world where one of the things that kids are the most passionate about (video games) is also one of the fields that really uses math (all the way up to calculus) in just about ever part of the process. I think a new curriculum, pairing math with computer science, focused around building games (which in reality is a great deal about building good simulations) would totally revitalize math education.
I gave a talk to a group of fifth graders last year where I just talked about the role that triangles play in the process of creating video games, how we study the way that light bounces off triangles and how we build a whole world out of millions of them, and the reaction was extremely good.
We live in a time now when a child around 5-7th grade could easily start making games, and using a huge amount of math to make the game they have in their heads. And I think it could really create a new generation of people who use and appreciate math at a deep level.
By the way, were there high schools 200 years ago? I thought high schools weren’t common until after 1900.
By the way, were there high schools 200 years ago? I thought high schools didn’t become common until after 1900.
No, high schools were not common. Most students, in fact, were home schooled. Some use this as an argument against public schooling in general, but I think the proposal in this argument takes one of the greatest advantages of home schooling and brings it to the classroom: the ability to delve into a topic in depth, find out its backgrounds and applications, and really get a chance to learn and love the content.
Great visual “proof!” Good math teachers educate by providing sparks, nuggets that clarify and ignite the imagination. The resulting explorations create a mental web of relationships that enhance and establish a long lasting understanding.
What’s new today is the learning is undergoing disintermediation. The resources now available, near instantly, to explore concepts and relationships are light years ahead of that available 50 years ago when I was in high school. At that time pursuit of my interests in math meant taking a 1.5 hr public transportation ride to the Museum of Science in Boston where I would spend a few hours finding books on subjects of interest. Amongst others, this included an abacus which I checked out and became quite good at using. Combined with logarithms, I used these tools to develop a good number sense without the use of much paper. Over that summer and subsequent weekends I explored trig, algebra, complex numbers, and some calculus. By the end of the 7th grade I had pretty much learned the math that I was forced to sit though the rest of high school.
Today it is far easier to learn and educators should focus on igniting the sparks.
Want to learn logistical regression, singular value decomposition, Galois math for satellite communications, or apply Monte Carlo analysis to a thought problem? The resources are right there on the web. No long bus ride required.
Formal education should primarily be for facilitating the acquisition of knowledge for those interested subsequent to high school and college. Learning, like never before, is less something that ends after 12, 16, or more years.
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Grew up in England where in the seventies one could elect to drop math for last two years. Not knowing about SAT requirement, I did so. Although I still managed a 690 a year later, I always regretted the decision, which kept me from statistics. But it was bad maths teachers that led me to the decision.
I’ve taught SAT math, and students (and parents) are always shocked to find out that it doesn’t actually test any math content beyond what’s taught in middle schools. The questions my students struggled with the most were those that took normal math and used it in an unusual way. They find ways to take even elementary grade math and turn it on its head–my favorite topics to teach were remainders and factorials. The SAT is not a math test, it’s a critical thinking test.
Speaking as an elementary school math teacher (albeit in Canada rather than the US) I like the way you think!!!
Steve, community colleges are moving away from that. As part of our high school redesign, we are working with the local community college to create pathways that combine classes at the high school with classes at the community college so that students can graduate with a certificate in a career field or into an apprenticeship with a local business. The community college has seen the same problem with students dropping out because of the higher steps of math that they don’t need, and they are moving to address it even more aggressively because their stats as a school go down when people don’t finish programs.
More and more people are realizing that getting a good education means different things to different student and I welcome more options for these kids.
I like your thoughts and agree with you. I especially appreciate the attention to programs like music and art being underfunded. (I would also include PE.) I’m nothing close to a mathematician, and didn’t become an engineer or anything of the sort, but I loved math in high school and went all the way to calculus. I even considered architecture or engineering, but was more attracted to social science instead. I am a huge fan of education in general, so perhaps because of this bias, my opinion is affected; nevertheless, I believe some concepts in algebra and geometry (for example) should still be required, as well as basic stats and budgeting. Perhaps I’m talking about “applied” math. Geometry proofs are the foundation for philosophy, and of course discussion and debate. Any individual ought to be able to think logically so they can make informed choices. There should be a basic understanding of volume, mass, and area. There should be an understanding of variables and constants; of the bell curve. And of course financial planning so we can bring our children up right. Perhaps you don’t disagree, but I just felt the need to say it since I didn’t see it specifically addressed in your post. Thanks again for your thoughts. I’m encouraged that you aren’t the only one out there sharing your ideas on the subject.
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I highly recommend this read on how Math was taught in Russia over the years with very successful results
(proofs approach is used widely). The paper also discusses experiments with teaching math in fun and meaningful way to non-math majors – less topics in depth.
Karp A., Vogeli B. (eds.) Russian mathematics education.. Programs and practices (WS, 2011)(ISBN 9814322709)(O)(514s)_MSch_
Click to access Karp%20A.,%20Vogeli%20B.%20(eds.)%20Russian%20mathematics%20education..%20Programs%20and%20practices%20(WS,%202011)(ISBN%209814322709)(O)(514s)_MSch_.pdf
For Geometry proofs lovers I suggest getting a Russian 7-9 and 10-11 grade geometry books brilliantly translated by UC Berkeley math professor Givental:
This is an English translation of a classical Russian grade school-level text in plane Euclidean geometry. (The solid geometry part is published as Kiselev’s Geometry / Book II. Stereometry ISBN 0977985210.) The book dominated in Russian math education for several decades, was reprinted in dozens of millions of copies, influenced geometry education in Eastern Europe and China, and is still active as a textbook for 7-9 grades. The book is adapted to the modern US curricula by a professor of mathematics from UC Berkeley.
The book is an English adaptation of a classical Russian grade school-level text in solid Euclidean geometry. It contains the chapters Lines and Planes, Polyhedra, Round Solids, which include the traditional material about dihedral and polyhedral angles, Platonic solids, symmetry and similarity of space figures, volumes and surface areas of prisms, pyramids, cylinders, cones and balls. The English edition also contains a new chapter Vectors and Foundations (written by A. Givental) about vectors, their applications, vector foundations of Euclidean geometry, and introduction to spherical and hyperbolic geometries. This volume completes the English adaptation of Kiselev’s Geometry whose 1st part ( Book I. Planimetry ), dedicated to plane geometry, was published by Sumizdat in 2006 as ISBN 0977985202.
Both volumes of Kiselev’s Geometry are praised for precision, simplicity and clarity of exposition, and excellent collection of exercises. They dominated Russian math education for several decades, were reprinted in dozens of millions of copies, influenced geometry education in Eastern Europe and China, and are still active as textbooks for 7-11 grades. The books are adapted to the modern US curricula by a professor of mathematics from UC Berkeley Givental.
I hope these will help you to understand some successfully tried methods of math education in Russia.
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I am home schooling my child, and this is exactly the sort of thing I would like to expose him to. Can you recommend any books containing mathematical treasures like this, suitable for teaching to children?
I also want to say that this problem extends well beyond mathematics. Every year the core curriculum is adding another 3-ish million adult Americans to our population who have all read the exact same books, been sung the exact same songs, danced the same dances, worked the same problems, and watched the same films. Diversity is a strength, not a weakness!
The common core recognized that children were not receiving an adequate education and turned the process into a factory driven creator of cookie-cutter citizenry. It’s fine not to know everything — we should be learning deeply and narrowly, and joining together in teams to accomplish great things.
even at mit or caltech, people who take first year courses ie ad. elective high school math-phys courses are clear majority NOW, if i am not mistaken. even though colleges put free resources for courses up to grad level online, there are many forums vs. graduation requerements clearly shows this: no electronics, programing, int analysis or even int qm but a hum or ss course every term. will they say you are not prepared for science based university-major, take some time off, study at community college or use online sources then try again or will universities be forced to admit and teach them high school courses themselves, taking resources futher away from interested students. that is what differs US from other fortunate countries ( and conservatories! ) that tracks people purely based on interest and academics, which is the main reason discussions about education are so complicated in the us, in my opinion
I would really like some clarification on this phrase: no electronics, programing, int analysis or even int qm but a hum or ss course every term
I think you make a good point but your writing is hard to follow…
i am sorry i hate writing essays. i was talking about mit-caltech graduation requerements. caltech had a version of qm course for non physics majors required etc until recently as every caltech grad should know this because it is for science people like conservatories for music people so all should know say bach or something. people said mit had it as well in the past i am not sure. i meant when high school expectations get lower, then would not college expectations for science universities or majors get lower too, because people do not like the idea of tracking people out.
When I was a techer one could major in English but qm was still a required subject as were chemistry, physics, and the associated math. The first two years had mandatory classes for everyone in those subjects no matter the major one eventually chose. That was a long time ago though and no doubt things are different these days.
Oddly, I found that most of the math I needed for work was learned subsequent to graduation.
The best outcome of formal education is teaching students how to learn after that formal process is completed.
Here are some book recommendations from Berkeley math circle
Thanks for this excellent post. One major issue, however, that I see is that if we want K-8 mathematical instruction to be, as you argue it should be, more interesting and in-depth, then we must completely revise the teaching of elementary education majors. Nearly every el-Ed major I have ever met is absolutely terrified at the fact that they have to teach math… And generally they are among the worst performing students i ever encounter in the subject themselves in addition to being phobic. How do you see this happening? It would be WONDERFUL if we could have elementary school teachers who grasp mathematics well enough to teach it in the way required by your proposal – with which I agree fully – and this is a MASSIVE sea change and cultural reversal.
This is absolutely true, but I think a lot of this problem would be solved by raising the bar for education students at the college level. Countries whose students are doing well on the PISA are the countries who are being highly selective in the students they allow to train to be a teacher, even at the elementary level. In Finland, it’s as hard to get into teacher training programs as it is to get into MIT, according to Amanda Ripley’s new book.
1) We’ve been tutoring some local high school kids, and we agree that modern math courses try to teach too much. This leads them to neglect the important stuff. All too many of our students would remark that just as they were starting to understand fractions or variables or parabolas, the course would move on. They’d have to start learning something else instead. An awful lot of the course seems to be digressions, wanderings down the alleyways rather than along the main avenues. There is a huge amount of time spent on scaling functions, but exponentials and logarithms were dismissed in a few days. There was almost no time spent on exploring the proof structure of geometry. Instead, they learned the lots of facts about the rhombus, as opposed to that a rhombus is a specialization of stuff they already knew. In fact, there was surprisingly little, learn a few general principles and then apply them, as opposed to learn a hundred applications.
The math curriculum could be seriously simplified if it were more focused on the important stuff. Maybe the kids would even get the time to learn some of it.
2) As for letting kids opt out of math early, I’ll just quote from John Scalzi’s famous essay about “Being Poor”:
“Being poor is having to live with choices you didn’t know you made when you were 14 years old.”
I completely agree with gkm001 drawing attention to this sentence: “The common core standards has not reduced the number of topics, and has demanded that teachers teach all the topics to more depth, which is just not possible.” As a math instructional coach in a district transitioning to the common core this year, one of my biggest frustrations has been this fact. And after I read “Lockhart’s Lament,” I looked back on my experience loving math as a student and realized Lockhart was completely right: those of us who love math growing up do so in spite of an educational system that takes away so much of what makes math interesting.
I’d be interested, Gary, to hear your thoughts on this article: http://www.salon.com/2013/08/26/school_is_a_prison_and_damaging_our_kids/. I have never been to a Sudbury Valley School, and they seem to serve a mostly privileged portion of the population, but I think the model is an interesting way to apply the ideas you wrote to more than just mathematics education. I’m curious about how a modified version of this type of school might work in a low-income neighborhood.
It is interesting that this “avoidance conversation” seems only relevant in the USA. Most folks in other countries would shake their head and wonder: What happened to America? High obesity, low math!? Hmmm – Critical thinking certainly suffers from both! (And a few other things…)
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The common core writers I know absolutely believe that we needed fewer topics in greater depth. Of course, the were politics, this and that, blah blah blah, and they didn’t get to see all of their ideas come through. I actually do think that the high school standards are much less overwhelming than the Pre-calculus standards I had in TFA.
The common core writer I know said that Precalculus is the place where standards go to die. Anyone with power who has a pet idea that they believe needs to be in high school math gets to lump it into Precalc until there are so many standards the class would take 2 or 3 years to teach well. I really think the common core is an improvement on what Nevada had because it cut out a lot.
Is there anyone out there who believes that the problem with high school math isn’t really high school math, but elementary school math education?
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