In New York City, students are considered ‘college ready’ if they can score an 80 on the Algebra regents. Eighty is actually not a percentage, but a scaled score — a curve — that equates to something like 57% of the possible points.

We often hear about how the goal of education nowadays is for students to be college or career ‘ready,’ so I’ve been thinking about how Math, which I’ve taught for most of the past 23 years, fits into this, namely: Does learning Math (or at least faking Math enough to get 57% on some standardized test) have any correlation with being ready to succeed in college?

In a rare moment of wishy-washiness, I’m going to say ‘no’ and then ‘maybe’ and then ‘no’ again.

‘No’ because as a Math teacher and a Math lover, I do think that the ‘importance’ of Math is overstated. Like Music and Art, Mathematics is one of the most amazing creations (discoveries?) of mankind and, yes, there are aspects of it — even aspects of the horrible curriculum that has evolved in this country over the past 200 years — which truly expand the mind the way any great piece of Music or Art would.

But ‘No,’ it isn’t really that ‘important’ in the sense that people could get by in life very well with only knowing up to around 5th grade Math. Come on. Am I the kid who doesn’t know it’s taboo to point out the obvious when the Emperor has no clothes?

Even for future engineers and even Mathematicians, I think that people who truly love math and who demonstrate an aptitude for it, they should be offered higher math as electives in middle and high school and they would be in great shape to pursue it in college if it were needed for their degree or career.

But knowing Math, like Algebra, for example, isn’t really something anyone ‘needs’ for college the way that people would ‘need’ to be able to read.

The same, of course, applies to most subjects. People don’t ‘need’ to know Chemistry. It is good to get exposed to it in middle and high school and certainly learn about some of the great mysteries that mankind took thousands of years to sort out in developing the periodic table and things like that. But you can be ‘college ready’ even if you still think that the four elements are, like the ancient Greeks, Earth, Air, Fire, and Water.

Now I’m a Math teacher and I take my job very seriously and I hope to make kids like Math and get the Math ‘bug’ where they want to solve Math problems, not because they are practical, but because they are interesting. (See one of my most personal posts ‘The Death Of Math’ for more on this) I don’t always succeed at this, but I think I do succeed a decent amount of the time, and I’m very proud of this. When I present the right problem and I watch my students ponder it and then, one by one, they get that gleam in their eyes and then smile with the ‘aha’ moment — well, that’s the kind of thing that really keeps me going.

But I will be the first to admit that people could succeed just fine in college if they know very little Math — and the truth is that most people, even ones that managed to do well in their Math courses, really don’t ‘know’ Math at all, anyway. What they kind of knew, they ‘forgot’ soon after the tests that supposedly proved that they truly learned it in a way that they could never forget it.

But I’m going to waiver a bit here and upgrade my answer to ‘maybe.’ Because college is a lot of jumping through hoops, getting the proper requirements and all that. It is hard to succeed in college if you don’t have the ability to play by the rules, do stuff that you don’t want to do, pretend that you understand something that you really don’t, figure out how to pass tests proving that you understand something that you really don’t, etc. So in that sense, someone who can’t force themselves to do something that they really don’t want to do, like learn Math, may not fare very well in college.

Being ‘obedient’ is somewhat of a prerequisite for succeeding in college and since learning Math, unfortunately, has become highly correlated with being obedient, then not being able to fake your way through a Math course might mean that you won’t be able to fake your way through college requirements either.

But this concept is getting challenged by the Common Core. Now being ‘obedient’ isn’t enough. To pass, you have to, at least in an ideal world, really understand what’s going on. I do like the idea of challenging students to think more deeply about Math. This idea is something that most Math teachers have been striving for way before the Common Core. I have NCTM (National Council of Teachers of Mathematics) journals and yearbooks going back fifty years helping teachers to infuse this kind of learning into their instruction. And I do think that many students will not be able to deeply understand Math. Some won’t put in the necessary time to study it — just like without practicing the piano you can’t get that good at it. And some just won’t have the aptitude for it (this applies to rich kids and poor kids, so don’t attack me on this, thank you). Everyone, though, should get exposed to Math in school and get a chance to engage in thought provoking problems in it. But students who don’t master it to a deep level can certainly still succeed in college — they just won’t be Math majors.

In this case, Math will no longer be a euphemism for ‘obedience’ — something I’m OK with, but it also won’t correlate with ‘college readiness’ either.

In early arithmetic we talk about properties of a thing, abstracted from the thing. That first one, cardinality really, most everyone gets.

The next abstractions, now from cardinality, fractions first, negatives, algebra – all involve some knowledge, which may or may not be useful – but all involve further abstracted reasoning, which skill is useful in otherwise unrelated areas.

(Maybe probability is a special case, because lack of knowledge of probability can actually be harmful in a number of cases. It’s taught too early, and not well enough, but its value, I argue, is not in training the mind and disciplining the mind for more abstract reasoning.)

All that said, the Common Core is about testing, not about learning, and after a few rounds of tests teachers will figure it out, and, under the shadow of perverse evaluation systems, teach kids to do well on the test. I’d rather that we concentrated on teaching kids to think.

Jonathan

I agree with the thrust of your post. As the parent of middle-school girls, I would love for them to get the math bug and have as much fun with the Martin Gardener books as I did at their age. I think you left building Self-Efficacy in math off the list of things that you should be doing as a teacher. (Pajares has a study that shows that it was as important as general intelligence in predicting math outcomes).

On the other hand, as the teacher of a graduate introductory Statistics course, I am disappointed with the ability of my students to do any kind of mathematical reasoning. As an example, consider what happens to the width of a confidence interval as the sample size increases. As the denominator of the expression for the standard error has a square root of n in it, the standard error goes down as the sample size goes up. As the standard error drives the width, the width of the confidence interval decreases. I pose this as a question in class and have about a 50/50 shot of getting the right answer. When I explain my reasoning, about half the students look at my like I’m from another planet (things are slightly better with the foreign students, who through training and/or selection bias, have better training).

I think you are right in the sense that overreliance on the standardized tests drives out working with this kind of thinking. Mathematical sophistication of this kind is an ill-defined construct, so it is neither part of the standards nor well-tested. The washback effect of all the testing is that these things are no longer emphasized. In my opinion the common core doesn’t change the situation that much (it is better than the older state standards it was replacing in some cases, worse in others). The problem is that the tests focus on what is easy to test and not what it important.

I really do consider elementary statistics to be an essential college (and indeed even high school) skill. Unless you understand the basics of statistics it is too easy to get misled by a statistical argument that is sloppy or dishonest (or both). But the prerequisite there is mathematical sophistication and not the ability to solve a quadradic equation.

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I’m a high school math teacher too, and I always tell the kids our job is to introduce them to lots of subjects to some degree so that they have an idea of what they might want to pursue in college (or not in college). I don’t think 5th grade math gives an accurate enough picture of what math is, nor is nine or ten old enough for a person to judge their own interests, for that person to have a good opinion on what they want to pursue later in life. Is high school the right age and is what we teach there the right math? Another question entirely…

Hi Hedi, Maybe 5th grade is pushing it, but I think that most adults have about a fifth grade understanding of math. Maybe if they stopped in 5th grade, they’d only have a second grade understanding of math and that would not be good. In ‘The Death of math’ I suggested mandatory math until 9th grade algebra, maybe that is better.

But if they never see proof in geometry they have no idea about the beauty…! The power…! I know, I know, not every school teaches kids that stuff and it’s naive to think that everyone will see that much in high school. I just don’t think algebra is the best agent to represent why math is awesome.

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It’s not so much the content but a way of thinking that makes math through high school–regardless of future major–so important. You can go back to at least as early as Plato to find how math makes us think in realms of that which cannot physically be shown, providing a glimpse into a world of abstractions. It also requires logic, reasoning and a flexibility of the mind–all things that the older educational traditions believed could be obtained by learning Latin and Greek.

Back on May 12th I posted the following similar thoughts to my FB Group.

Is There Really a Common Core? I’m Having Second Thoughts.

I had a strange epiphany the other day. One of the joys of my life is the opportunity to tutor my granddaughters. As Elsa and I were practicing finding the square root of a series of expressions, I couldn’t help thinking, “Why are we doing this?” It is a question that students ponder every day in many subjects, not just math. I can find examples where some people might have an occasion to use some of the things they are taught; but in general, I had to ask myself, “Is there really a Common Core?” I think the answer is, “Yes, but!”; and the ‘but’ is huge compared to the ‘yes’. Part of the problem with the standards is they make an implicit assumption that learning stops when school stops, so you have to learn all this stuff while you are in school. WRONG! Learning is a life-long endeavor. School is where you need to learn to learn. You have your whole life to learn stuff when you need it.

In previous postings, I have tacitly accepted having a set of academic standards while deploring the testing regimen that has been attached to the standards. My earlier acceptance was based on the concept of standards, without thinking about the breadth and depth of the application. As I sat there with Elsa, I had time to reflect on the material she was studying in Algebra. When did you last have to solve a quadratic equation? How often do you need to find the upper or lower limit of a parabola? The list goes on…. Having said that, there are some valuable Algebra topics and principles, including:

• Properties – Associative, Commutative, Distributive, Identity, etc. (The names are not as important as their application.) In general, regardless of the subject area, fundamental principles and properties are good to be aware of.

• Abstract thinking – Solving for ‘x’. Working with unknowns and variables.

• Estimating and validating assumptions.

There is a longer list of items that will be useful to many people who pursue technical and professional careers. At the same time, there are many professions that will never require any more algebra than those three topics I mentioned earlier. If you have to take Algebra, then, by all means, learn as much algebra as you can. But why so much Algebra? And, it’s not just Math.

I did some volunteer work in a middle school science classroom earlier this year. Looking back, students were learning about the periodic table, balancing chemical equations, and other very interesting activities – to some of the students. Once again, there is much more material they are studying that they will never use than material that will be valuable to them later in life.

The standards have this lofty stated goal of getting the students ‘college and career ready’. I’ve already pointed out that many careers require a minimum of Algebra and Science. Different careers or professions require different topical expertise. Accountants and architects both use math extensively in their work, but very different elements of math. Do architects need to know about amortization and actuarial tables? Do accountants need to be able to apply differential equations to analyze stresses and force vectors? Most careers have their own set of knowledge standards that vary significantly from one profession to another.

There is also a body of knowledge needed to be a good citizen. I recognize that it is important to be an informed citizen on climate change, environmental, energy, social and other policy issues. You don’t need to balance chemical equations or know what a co-valent bond is to be a scientifically literate voter. If you don’t need the knowledge for a career or to be a good citizen, what’s left? College!

If you are planning to go to college, you need to take Algebra ! & II and Geometry plus some other higher math courses because you won’t be able to pass the college courses without those fundamentals. Learning something simply to be prepared to learn more of the something you will never need doesn’t make a lot of sense. One of my favorite Peter Drucker quotes is, “There is nothing quite so useless, as doing with great efficiency, something that should not be done at all.’ (Not that schools are doing anything with great efficiency!) Instead of having general math courses in college, they should offer math courses tailored to the professions.

So what should students be doing with all that school time?

I have some thoughts I will share with you in a coming post; but first, I would like to hear from you. Do you agree that schools today are teaching a bunch of stuff that students will seldom use? If so, what should schools be doing with the students? If you disagree, please help me understand.

I was a middle school math teacher in the 80’s and 90’s….I tend to agree with you that the math requirements today in K-12 education are way too much. I would think high school teachers would be THRILLED if 9th graders came to them with a very good conceptual understanding of and facility in using basic arithmetic. Think how much faster they could move through algebra 1 and 2 and geometry if they didn’t have to reteach basic arithmetic all the time. I think college science and social science professors would be thrilled if they got freshman students who were well versed in algebra 1/2 and geometry. Even math professors would be thrilled to get freshmen students well versed through trigonometry…they could then really teach calculus at a truly college pace and level. I think the public, in general, would be thrilled if students came out of high school being fluent in basic math, basic statistics and having an understanding of basic financial math (check books, student loans, mortgages, credit card rates and payments, and such)…things that now, there is no time for (although, my son’s high school is requiring an on-line financial education course for graduation…I don’t really like this idea, I’d rather they have a class in it).

I am not sure I agree that it is wise to reduce the number of years of required math in high school, just the pace at which it is taught. I don’t think slowing down the pace (and thus the level attained) is a bad thing, it is a way of accepting that the pacing of math instruction over the last few decades (if not longer) has not led to better math achievement, but that allowing students to develop their math abilities at the pace they need might just work better for more students, and society as a whole.

I am not a math teacher, but I rewrote/summarized my son’s geometry textbook to help him pass the course. Years later when he became interested in carpentry, he regretted his dismissal of the subject. I could go on with further anecdotes, but I think you get the point. While I understand the math teacher’s desire to awaken fascination with playing with math, the practical applications are probably more important than you think. Those practical, concrete applications have driven a much more sophisticated understanding of the beauty of math in my children. They use it as a tool quite naturally just as most people probably use reading as a tool. While my own math education supports your reasoning, having to teach struggling learners awakened me to the fun of math.

What is this “college-ready” hogwash? Kids are “college ready” when they are ready to go to a place where they can learn as much as they can about a subject that really grabs them. When I was eighteen I was sad I could not live forever because there was so much interesting stuff to learn. My college experience did not include jumping through hoops or just meeting prerequisites. I couldn’t wait to get to the next course of a never-ending feast. I majored in math and taught high school math and science for 35 years.

I told my students to “Find out what grabs you and go as hard as you can for as long as they’ll let you!”

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I’m not sure I agree with you as far as Algebra being needed for college (disclaimer: I’m a mathematician); I can see your case for non-STEM majors… What I’ve been saying for years, though, is that part of the problem is that we confuse mathematics with numeracy. We don’t make that mistake with English: we understand the difference between literacy (a needed skill that every citizen should have, and that is an absolute must for any higher education) and literature (not really a needed skill, needed only for a few majors in college; should everyone know about Shakespeare’s plays and sonnets? Well, it is part of the general heritage, but not *needed*). There are certain topics that we teach as “Math” because they are numeracy: basic arithmetic, basic algebra; trigonometry used to be numeracy, but now it is not and is taught mainly as needed for calculus (we should drop trigonometry until college, as far as I’m concerned; and I’ve never had to “solve a triangle” except when teaching precalculus). We should also teach basic probability and basic statistics (at the level of, say, “How to Lie With Statistics”) as numeracy for everyone. On the other hand, there are topics that we teach as “Math” but are really mathematics, not numeracy: precalculus, abstraction, set theory, calculus. These are topics that are more akin to teaching literature, not literacy. If we make the distinction clear in our minds and in our standards and programs, I think that would be a big step forward. Yes, it would be nice if everyone comes out of High School knowing calculus and being able to use it when and as needed; just as it would be nice if everyone came out of High School knowing the Shakespeare plays, the major works of American literature, etc. It would probably make for a better society over all. But the need is for a certain level of literacy and numeracy, in order that society has citizens that can live productively.

As a non-math head like those above, I wonder then if we should have less depth in math and science subjects as we move up through middle school into high school. All kids will have to then take a sort of algebra lite, geometry lite, chemistry lite, biology lite, and physics lite in middle school into 9th grade perhaps. Once there, kids can then choose which of the following subjects most appeal to them. Referring to Pajares, we might be more apt to build self-efficacy as we encouragingly compel students to become proficient at the basics of these subjects and allow those seeking more advanced learning to delve into subjects they find interesting.

It seems to me that kids forced to sit through a course for a whole academic year studying something they have only a passing interest in ultimately impacts negatively student motivation, and teacher efficacy.

If people are selected for college on the basis of a high math score this will correlate pretty well with ability to study at a college level. Although the resulting sample will probably be biased to a higher quantitative visual-spatial IQ as opposed to verbal IQ, the math and verbal componenets of IQ are highly correlated. So using math ability as a selection criterion will work out OK.

It doesn’t matter that much even if students selected for math ability study no math at all in college. Because of the high correlation between verbal and quantitative IQ a sample selected for high math ability will likely also be well above average in verbal ability.

This is not to say that the use solely of scores on math tests is the optimal way to select people for college study. Just that such a method will be reasonably successful in producing students capable of college level study.

I posted a comment a few days ago and according to the blog site it is still waiting moderation. What is the normal wait time for approval?

Regards,

Allan

Allan Jones

President, Emaginos Inc.

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“The convention opened with a prayer, a very fervent prayer. If the Lord can see His way clear to bless the Republican Party, the way it’s been carrying on, then the rest of us ought to get by without even asking for it.” — Another timeless Will Rogers quote He could as easily have cited the Democrats!