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 is 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.