Over at Pharyngula, PZ Myers has commented on one of the periodic issues that occurs in mathematics education and that is what mathematics should form part of the general education of everyone. This time the discussion is over whether algebra should be a requirement for a basic general education. Those who argue for its removal say that it is not a skill that most people need in everyday life and that in addition, students seem to find it very hard and fail in large numbers.
PZ argues that “Algebra is a kind of minimum standard for elementary numeracy” but despite my love for the subject I am not convinced, mainly because that statement depends on how much algebra we are talking about. While some algebra can have practical benefits in daily life, at what level of depth is really the contentious issue. I suspect that most people get through life without using any algebra at all or just minimally. The question of whether the manipulative techniques that one learns to use in algebra are generally applicable is also an open one. I am also leery of the argument that students should learn some subject purely because it ‘strengthens the mind’ in some vaguely defined way. Those arguments were used formerly to require students to learn Latin and Greek but we no longer do so.
But that is a different debate. I want to focus on a different issue because it has long puzzled me and that is this question of why algebra is seen as so hard. I can understand people struggling with calculus because the concepts of infinitesimals, limits, differentials, integrals, and why dy/dx is a single entity and cannot be reduced to y/x by canceling the d, are all kind of tricky. The same is true for trigonometry. One starts with the concepts of sine and cosine and tangent defined in relation to right angle triangles and that seems fine. But these three things then break free of their triangular roots and take on a life of their own and go on a wild spree, forming many complicated relationships with each other that seem to have little concrete meaning. While I loved manipulating trigonometric identities (I viewed them as essentially logic puzzles that I enjoyed solving) I can see why their appeal could be mystifying to those who did not love puzzle solving.
But algebra seems straightforward to me and I am sure it was so for some readers of this blog too. Is that because we were weird nerds? Is the teaching of algebra in the US really lousy so that only a few spontaneously grasp it? Note that schools in Sri Lanka start teaching algebra in the 6th or 7th grade, a much younger age than they do here, and I don’t recall that the general student body found it so hard that it created some sort of crisis. Were students to the left of me and students to the right of me falling away in large numbers and I just did not notice? I don’t think so but memories of long ago can be notoriously unreliable. Even if many of them did not ‘get it’ quickly, the fact that it was taught at such a young age suggests that it was not seen as being a particularly difficult subject to learn, given the right teaching. So while the merits of whether it needs to be a basic component of general education can be debated, arguing that the level of difficulty is an issue is surprising to me.
It should be noted that the system in Sri Lanka was to teach all the math and science subjects every year so that we got it regularly in smaller doses over a long period, like vaccines, unlike in the US where there is a tendency to teach a topic in a single year. The latter requires much faster processing of new information.
There are many problems that students encounter in learning algebra. One is the common belief that for any function f, f(a+b)=f(a)+f(b). Hence students tend to think that (a+b)2=a2+b2, sqrt(a+b)=sqrt(a)+sqrt(b), and so on. You would think that by simply inserting numbers into those equations as a test, they would quickly realize that those relationships are not true. But that does not happen, because many students do not instinctively check to see if their reasoning is correct but march along. Some also lack the kind of intuitive sense of numbers that will enable them to realize when they are doing something wrong. While it may well be true that the lack of a number sense is due to the widespread use of calculators, there are three things that the educational and cognitive science literature has exposed that compound the problem, and all are based on the idea that the brain tends to take the intuitive easy way out (System 1 thinking) when learning something.
The first is that given any number of alternative schemas for explaining something, people seize the first plausible one that comes along without stopping to weigh possible alternatives that might be better.
The second is that the human brain has a tendency to take any plausible belief that is learned in a specific context and then disregard the context and apply it universally as a general rule. In this case, students learn early the distributive rule that c(a+b)=ca+cb where a,b,c are numbers. This makes intuitive sense and is simple. They then think that this also applies for f(a+b) even when f is a function and a+b the argument of the function, maybe because a function is a more complicated beast and it is easier to treat f as if it were just a number too. After all f and c are both just letters, no?
The third is perhaps the most problematic and that is that even showing in a few instances that these simple beliefs lead to manifest contradictions is not sufficient to change these entrenched beliefs because it is so much easier to retain an incorrect belief, especially if it works on a few occasions, because that reinforces the idea that it is correct (the familiar confirmation bias problem).
This is a problem in physics education too. Students tend to acquire an Aristotelian view of dynamics early in life because it makes so much intuitive sense. They absorb it based on their everyday experiences so that one does not need to be even taught it. Even when they learn Newtonian dynamics, misconceptions about (say) the Third Law are very hard to dislodge. Similarly, the idea that electric current is ‘used up’ when a battery powers a device makes so much intuitive sense that getting students to realize that the amount of current leaving an electrical device is the same as the amount entering is difficult, even if they learn to manipulate the laws of electricity.
One needs a systematic educational strategy for combating these features of the brain that lead to entrenched wrong ideas.
I personally did not find algebra easy at the beginning but there were specific reasons in my case. I left Sri Lanka in the middle of 6th grade for medical treatment in England and missed a whole year of schooling due to being in hospital. But despite this, when I returned, I was promoted to the middle of 7th grade. I was able to catch up fairly easily on the missed year of work in all subjects except for algebra that they had started teaching in 7th grade and thus I had missed the first six months of it. My algebra teacher was also terrible, simply assigning one problem after another from a huge textbook named Hall’s Algebra as I recall. I remember struggling and being baffled by this mysterious entity called ‘x’ that we were asked to ‘solve’ for and being tremendously frustrated at my inability to grasp the subject. But I do recall that at some point, maybe after a month or so, it dawned on me what the basic idea of algebra was and after that I quickly caught up and loved it.
So I can sympathize with people who have difficulty with algebra at the beginning. My point is that if I had started with the rest of my peers, the dawn of understanding would likely have come earlier, and having a good teacher would have helped a lot. So is the problem just bad teaching of algebra that makes it so hard for so many students here? What was your experience with learning algebra? And how much algebra do you use in your life?
It doesn’t help. It’s a depressingly common thing to hear otherwise intelligent people say “I was never any good at maths”. I usually find that, if pressed, such people usually relate some early formative bad experience with a maths teacher, which put them off the subject for life. They never had that epiphany you had, where it clicks and they start to enjoy it. I was lucky to have some excellent teachers. Learning algebra was made simple, intuitive and enjoyable, at least as I remember it. It didn’t hurt that it was something I was able to do more easily than practically everyone else in my school, so the positive reinforcement was strong.
As for how much algebra I use in my life, not a fair question. I’m an engineer by inclination as well as profession. If I can’t find an excuse to use algebra, I’ll manufacture one.
One problem I noticed in my son’s education was that in early grades the kids were introduced to problems that can be easily solved with algebra, except the students haven’t been introduced to it yet, so one common method they are taught is to solve numerically by guessing and checking. This leads to students developing assumptions about math problems and how to solve them that are counterproductive when the solution is not a whole number or when there are multiple possible solutions.
Insufficient practice with a variety of problem types and de-emphasizing mental math leads to poor number sense.
Many students do not develop an intuition for the subject, but instead are trying to game the problems with the tools they have, regardless of relevance -- hence situations like “How old is the shepherd?” — The problem that shook school mathematics
And one of the biggest problems with how my son was taught middle and high school math, from algebra to calculus, was not teaching the reasoning behind the formulas. When I was in school, I learned to solve quadratic equations by completing the square, which is then generalized to the quadratic formula. My son was taught to memorize the formula. I was shown how to derive polynomials and trigonometric functions directly from the definition of the derivative as a limit of the ratio of the change in y and the change in x, my son memorized the formulas. (And yes, I was taught some algebra and some geometry every year from 7th grade to 10th. And introduced to the basics of trigonometry and calculus in 10th, to be expanded in 11th, by which I took the matriculation exam. My friends in the physical science concentration had an extra year of more trigonometry, calculus and analytical geometry.)
As for whether algebra is an essential part of one’s education -- I think some concepts are. The concept of abstracting a mathematical relationship in the form of a function. Different kinds of growth -- linear vs polynomial vs exponential (as exemplified by compound interest), decline/decay. Unfortunately many of the problems that are used to teach the subject are obviously artificial and unrelateable -- who cares how long it would take to fill the tub with the various plumbing adding and subtracting water from it?
Over the years my tutoring work in four different school systems ranged across a wide swath of subjects and grades, but in recent years I have been called upon more and more to teach high school mathematics and, thanks to Ohio’s insane legislative requirement that ALL students must pass Algebra II in order to graduate, probably 75 percent of my work is now focused on that single topic.
There can be no pedagogical justification for such a requirement.
What I have observed about the challenge of Algebra is pretty straight forward. Ask a student to solve x + 5 = 12 or 4x =16 and they will find x using inverse operations without blinking. Complicate the problem with more than one operation, however, such as a linear equation y = ax + b, and the car careens off the road. I blame the rigid drill of PEMDAS (order of operations) which works fine once you know the value of all the variables, but is all but worthless in reconfiguring an Algebraic equation to isolate the variable.
A student, convinced of the dogma of PEMDAS, looks at the equation 3x + 4 = 32 and thinks that the first step is to divide by 3x (yes, you can solve for x that way, but the problem is unnecessarily messy) instead of subtracting 4 from both sides and then dividing both sides by 3.
Beyond solving two, very basic, kinds of Algebraic equation (x + 5 = 12 and 8 / 12 = x / 4), Algebra is of no use to 99.9 percent of the population. (Kind of like Greek and Latin.)
I highly commend the works of mathematician Paul Lockhart—A Mathematician’s Lament and Measurement—for your consideration and what I think is pure genius on the topic of why we’re getting maths all wrong.
Pierce R. Butler says
Though an A & B student in almost everything else, I really struggled with algebra and geometry -- and never could figure out why.
It may have had something to do with never attending the same school more than two years in a row, with each grade’s lessons plans based on assumptions of what went into previous classes -- especially during the “new math” curriculum experimentation period in the US.
Lockstep schedules certainly made it worse: fail to grasp something in one week’s classes, and you slip further behind in the next week’s, the overload and confusion accumulating throughout the year and beyond.
hyphenman, in Washington state 3 years of high school math are required, however in some cases Algebra II can be replaced with an alternative ‘practical algebra’ or similar courses. However 4 year colleges do not accept such substitute classes. Such substitutions are only allowed following a consultation of the parent/guardian with school representatives and signing of a form by the parent/guardian (presumably acknowledging awareness of the implications of such a choice).
If you want schools to stop requiring Algebra II you need to start with colleges.
I do notice that 4 year colleges offer college credit for courses that appear to be lower than pre-calculus and may at least overlap with Algebra II. My son’s college offers a course in mathematical reasoning intended for students in the humanities -- it seems to be about functions, understanding data display, basic probability, decision making based on quantitative information. There is also a class on algebra and functions intended for students who will continue to pre-calculus. If such classes are offered to students who are required to have high school credit for Algebra II, what happens if Algebra II becomes optional? Do we add more college algebra classes or do we need to start with even lower college math classes?
Algebra is similar to religions and should appeal to American children. It is full of unknowns, and people who study it finally figure out the answers, although their conclusions might be heretical to somebody who gets different results. The square root of minus one is definitely satanic, too.
require students to learn Latin and Greek but we no longer do so
I don’t know about Greek but the loss of Latin in schools has led to a sad decline in writing skills in English. Learning to construct a rolling periodic sentence in Latin does wonders for later English composition.
I have not had any contact with the provincial school system since just after Trudeau became Prime Minister (Pierre, not Justin) so I am completely out of date on what and how it does things. Still, a little thing like that never stopped me from commenting.
Back in my day we did not start algebra until Gr. 9 but a quick look at some materials on line suggests that some algebra is introduced at least as early as Gr. 4. As are Venn diagrams and why not? They have to be more fun than extensive long division practice.
I suspect that much of the problem with mathematics teaching in general including algebra is that the very basics in the lower grades is poorly taught. I am generalizing from some general news articles, CBC Radio programs and a few more academic discussions of math teaching read casually over several years.
I think first problem is that many primary school teachers are poorly grounded in math and are possibly a bit math-phobic. I may be wrong but I doubt that many university math majors teach in the primary grades.
This does not auger well for their ability to teach and inspire enthusiasm for mathematics in their students. It seems more likely to encourage more math-phobia and an attitude that math is hard so we shouldn’t expect all the students to understand it.
Secondly, I seem to have heard that, in some provinces, no idea about other countries, old techniques such as memorizing the times tables is discouraged. Some of the replacement exercises seem amazingly complicated and often pointless. And weird. My math was never great but I survived a couple of university courses, in Calculus and Algebra and I was totally confused at what some of those exercises were trying to get at. It was not developing number sense!
IIRC, CBC Radio had a program on subversive parents (aka university math profs) teaching their children and other children the times tables.
I think we, overall, have suffered from a number of math teaching fads, starting way back in the 1960’s with the New Math and it does not seem to have gotten better, overall. I had a math teacher back in paleolithic times who commented that anyone who survived the chaos of the introduction to New Math clearly had learned math in spite of rather than because of the changes.
As I recall it, the New Math was not a bad program all told but apparently it was different enough from the “old” Math to cause real consternation and confusion among teachers. That may have been why after I did my homework I had to do my mother’s homework before she sallied forth the next morning to teach math to her class.
An indication of how badly mathematics is taught at the lower grades can, I think, be seen in the popularity of the JUMP Mathematics program https://jumpmath.org/ developed, almost accidentally, by John Mighton, a Toronto actor, playwright and mathematician. From my cursory examination of a few of the teaching materials it looks like a well designed program bases on sound learning theory principles and with good teaching materials.
One gets the idea that some teachers are grabbing it like one would grab a life jacket as one was being swept out to sea.
# 3 anat
who cares how long it would take to fill the tub with the various plumbing adding and subtracting water from it?
The teacher? I was arguing something about the amount of work that went into pumping water into one of those municipal water towers we see around here one day in first year university Calculus and after two or three minutes of back and forth a water pipe overhead started dribbling water onto my head. I never did figure out how Dr. R did that!
I do agree that in many cases a somewhat more realistic problem/example would help.
# 7 grasshopper
The square root of minus one is definitely satanic
I am not sure about actually satanic but that renowned publishing house, ABeka, has reservations about some math:
‘Unlike the “modern math” theorists, who believe that mathematics is a creation of man and thus arbitrary and relative, A Beka Book teaches that the laws of mathematics are a creation of God and thus absolute….A Beka Book provides attractive, legible, and workable traditional mathematics texts that are not burdened with modern theories such as set theory.’ — ABeka.com
The square root of -1 is definitely not satanic, it’s just imaginary!
I honestly don’t know how people live their lives without a knowledge of algebra, or even a basic grasp of the concepts of integral and differential calculus. I’m not being facetious.
To be clear, I’m not referring to the specific mechanics of calculus, but rather, the big ideas. I remember talking to a colleague many years ago about math courses. I was in my mid to late 20s and lamenting the fact that I had already forgotten much of the detail I had learned in calculus classes as I didn’t use it regularly on my job. He said, “That’s not important because you can always pick that up again. The important thing is that taking calculus changes the way you think about things.” I wholeheartedly agree.
Pierce R. Butler says
jimf @ # 11: … taking calculus changes the way you think …
So I have heard from just about every math student & teacher -- none of whom could answer my “So why don’t they teach it that way from the start?”
I’m a business school graduate so I needed basic algebra for Stats101 and business mathematics.
Unreal numbers weren’t a problem at High School, however calculus and trigonometry were almost brick walls, I just scraped through. Probably most people can learn basic algebra, however it’s a matter of aptitude beyond a certain level of expertise. We can’t all be theoretical physicists or learn foreign languages easily.
How much, I wonder, is any maths discussion hindered by survivor bias?
In addition, here’s a thought to make your head hurt: In Great Britain, they talk about about Maths, while in the United States we say Math, while at the same time we teach Mathematics.
@Pierce R. Butler, No. 12 and rjw1, No. 13…
That’s true, you’re absolutely correct.
Marcus Ranum says
It’s just construction, or machining, it’s just you’re machining numbers. Code is machining code using more code. You explain to a kid “ok, you’re going to make something, and the material it’s made of is ideas but just like electrical circuits or a car engine or a brick wall it has rules for how it works.”
My math teacher in high school knew me well and said “algebra is how you calculate the ballistics …” and we were off..
Marcus Ranum says
Algebra is useful for calculating how long it will take a brick falling out a window to hit the ground. It’s exactly the same problem as calculating how long it’ll take a company with a certain burn-rate to run out of money.
WMDKitty -- Survivor says
The problem for me has always been that there are TOO MANY STEPS, VARIABLES, and other fiddly bits to keep track of. I find Y, I lose X, I get numbers flipped around, and if there’s not a way to represent the equation physically, with manipulable objects, I’m completely lost.
It’s not that I’ve had bad teachers — they all went above and beyond trying to help me understand the how and why of algebra. It’s that my brain is wired differently.
Words, on the other paw, words, I’m good with. At least when I can string them together in coherent sentences. But only in writing. Speaking is… well… I can be selectively verbal.
In the U.S., Algebra is hard for a host of nuanced reasons but the general answer is its hard because we say it is.
In the U.S. we revere the brilliance of people in STEM professions. We praise students in advanced math & sciences but aren’t similarly impressed by students who excel at other subjects.
Adults joke about bad math skills being passed down to their kids. Teachers say “I didn’t get algebra either, that’s why I teach history.”
We talk as if Albegra is different and “hard” and sure enough it becomes just that. Its a self fulfilling prophecy.
Reginald Selkirk says
The Problem Isn’t Algebra
Reginald Selkirk says
Now here’s someone with some math problems:
Dads are three and a half years older than they were in the 1970s
I suspect any dad who was around in the 1970s is about 40 years older now.
The problem is more than just algebra, it’s a stifling of curiosity in many areas. (Excuse me for quoting Isaac Asimov, but his statement about “That’s Funny, not Eureka” still holds true.) Kids should be encouraged to think, ask questions, and learn about things that won’t help them immediately. I was fortunate to have a few teachers who always took advantage of “teaching moments” and asked questions that had no benefit to grades but plenty to making students think.
The US system seems geared towards training employees rather than thinkers. And with an system of funding designed to make schools in poor areas worthless (local property taxes), many capable and curious minds are lost. Canada pools its federal income taxes and funds school districts on a per student basis. Larger districts may have more bulk purchasing power, but even the smallest school district in a poor rural area of Canada can provide an education that will prepare kids for college. Most large city schools in the US can’t do that.
Very little, since my job involves language, not mathematics. But I devour figure logics (also called cross figures) and other math puzzles, many of which involve the skills of algebra. They have no practical value, but they are a lot of fun.
Krazy Dad Cross Figures
Puzzle Choice, Cross Figures
bluerizlagirl . says
If there is one thing in maths that ought to be consigned to the dustbin of history, it’s mixed integers and ratiometric fractions. For intermediate calculations where you can’t afford to sacrifice precision yet, write 15 / 4; give your final answer as 3.75; but for the love of all that is sane and wholesome, please just let 3 3/4 die already.
The thing is: Real Life is decimal. Money is decimal. Measuring instruments are decimal. And there’s really no good reason to express fractions any other way, except deliberately to make calculations more difficult than they need to be.
@ 20 Reginald Selkirk
Re: The Problem Isn’t Algebra
One of the points I was trying to make earlier in the thread it is not so much that algebra is difficult but that students don’t have the basics.
Coincidentally there was an interview this morning on CBC Radio about falling math standards in the Ontario education system.
Among things the education professor mentioned was a lack of mathematical knowledge among teachers in the primary grades—in her teacher-training classes 4/5ths of the students had no post-secondary mathematics courses and the majority had never taken a math course beyond Gr. 11.
Her research also indicates that many primary school teachers suffer from math anxiety, a much nicer term than my “math phobia”, and that teachers can and do transfer this anxiety to their students.
The prof mentioned, in the Ontario context, a need for improved professional training and support plus (yet another) curriculum reform, reducing the overly broad range of subjects and returning, in some cases, to more traditional techniques such as memorization of the times tables.
1. Ontario Morning -- Thursday August 31, 2017 -- Part 2 perhaps 2:45 minites in. http://www.cbc.ca/news/canada/toronto/programs/ontariomorning/ontario-morning-thursday-august-31-2017-part-2-1.4269861
Sunday Afternoon says
I routinely use algebra for my work and in my side projects. My background is in the physical sciences. In a broad sense, anyone who writes computer code is using algebra, though some of the rules of code do not follow those of algebra.
makes sense using coding rules, but breaks the rules of algebra.
On learning algebra -- I don’t recall having a struggle. I do have a vivid memory of an “aha” moment in early secondary school in Scotland (around age 12). We were going through plotting exercises along these line: draw y=2, x=4, y=x etc.
The first two were obvious, a horizontal line, and a vertical line. For the third, I drew another horizontal line passing through y=4 as x was previously defined as 4. The aha was having the diagonal line pointed out to me as the answer to y=x.
My highschool math consisted of Algebra I, Geometry, then Algebra II and Trigonometry. I found early on that I wanted to make sense of the formulas, which mostly helped. I could have rotely memorized them all and forgotten them a couple months later but instead they stuck with me and helped me learn more complicated math. It helped me make intuitive sense of more complex math because I was training my intuition. Geometry was actually a problem. It was so easy to grasp I was bored out of my mind and I couldn’t find any way to make it more interesting. I felt like I could have condensed a year’s worth of it into a quarter and come out fine.
When we got to Algebra again I was fine all the way through to trigonometry. I could never make sense of it. There must be some basis for it that would make it all click for me or it never would have been discovered. But whatever it is, I never heard about it and that effectively walled off calculus as well, which also walled off physics. If trigonometry made sense to me I would have at least 200 level college physics knowledge and at least enough calculus to use it. As it is, I mostly just bombed out of calculus classes.
@bluerizlagirl, No. 23
That’s true you’re absolutely correct. Mixed numbers only make sense when you’re cooking or shipping products.
My students, however, can be too quick to convert to decimals and this allows rounding errors to creep in.
I tell them that only the final answer should be converted to a decimal.
bluerizlagirl . says
Why would you need mixed numbers for cooking? Weighing scales and measuring jugs go up by thousands …..
Peter B says
It might have been 1961 my sister in high school (age ~17) did well in “New Math”. Among much else, her advanced math class defined the logarithm of whatever as the area under the unit hyperbola, x*y = 1, from 1 to whatever. Indeed, they plotted the curve and counted tiny squares from x=n to n+1. The guy that got zero to one pointed out that the square counting was already done by the rest of the class and he shouldn’t have to count what could be done by reflection. He was allowed to use the work of others. My sister ended up with a Masters in math from Berkley.
Someone verbally proved 4=2. I had him write it down.
Multiply both side by a: a2 = 2a
Subtract 4 from both sides: a2-4= 2a-4
Factor: (a+2)(a-2) = 2(a-2)
Cancel common multipliers: a+2 = 2
But we started with a = 2
Therefore 2+2 = 2
And… 4 = 2
All of the steps looked good at first glance but something just had to be wrong. I restated the above replacing each “a” with “2”. Spotted the error in 10 seconds. I might have been 15 at the time. It took me several minutes to explain to several guys age ~30 what was wrong with that proof.
While in high school I spent a lot of time trying to solve this:
A 4 foot square box sits on the floor pushed against a wall. Everything is right angles. A 10 foot ladder sets on the floor and the wall such that the ladder just touches the outside corner of the box. How high up on the wall is the top of the ladder? Somehow my quadratic equations failed me.