### Thursday, July 20, 2017

## Algebra or Stats?

Apparently, the California community college system is considering allowing students in non-STEM majors to fulfill a math requirement by taking statistics, rather than algebra.

The idea behind the proposal is twofold. First, algebra generates more student failure and attrition than almost anything else. (One of the guest speakers at Aspen said that his one piece of advice to any college president looking to improve graduation rates would be to fire the math department. We laughed, but he didn’t seem to be kidding.) Second, in many fields, algebra is less useful than statistics.

The objections are obvious. Most basically, it looks like watering-down. If the solution to increased college completion is to get rid of anything difficult, then college completion itself becomes meaningless. There’s an “exposure” argument, too, that says that many students don’t know they like math or STEM until they’ve found themselves wrestling with it; deprive them of that exposure, even “for their own good,” and the downstream effects are predictable. Pragmatically, there’s an argument from transfer; many four-year colleges won’t take math courses that don’t have an algebra prerequisite. And from within, there’s a valid argument to the effect that if you don’t have basic algebra, you won’t be able to generate most statistics; at most, you might be able to consume them.

For example, when I took stats, I was captivated by the idea of controlling for a variable. (“You can DO that?”) But the idea of a variable came from algebra. If you don’t have some level of algebra, I’m not sure how much sense the concept of controlling for one would make. Correlations and standard deviations also rely on some knowledge of algebra. “Base” and “rate” make sense algebraically. I’m not sure how the course would work.

It’s true that drop/fail rates for algebra courses tend to be higher than for stats courses. It’s also true that in my own scholarly discipline, and in my line of work, I use stats far more than I use algebra. I find the “median” of a distribution on a regular basis, but I don’t remember the last time I used the quadratic formula. It just doesn’t come up.

But the argument from usefulness is stronger against many other fields, and it doesn’t get deployed against those. We have a history requirement for A.A. degrees, for example. From a pure ‘usefulness’ standpoint, that’s hard to justify. But there’s a general consensus that the skills students develop through the study of history are valuable, even if it can be hard to demonstrate in as linear a way. (Knowing the future would be far more useful, but it’s hard to find good materials.) We have a humanities requirement that’s entirely independent of usefulness. Honestly, if we want to argue usefulness, I could imagine a compelling argument that history and literature majors shouldn’t have to take lab sciences. The usefulness argument is a slippery slope.

Stats courses tend to lend themselves to “general education” kinds of applications very well. I’m a fan of questions based on epistemology: “what statistical evidence would prove this claim?” Political journalism offers no shortage of “how to lie with statistics,” which can be excellent fodder for sharpening students’ critical thinking. Just being able to distinguish between correlation and causation is valuable.

I’d be curious to hear from folks who’ve taught a stats class that didn’t assume any previous knowledge of algebra. Can you sneak the relevant algebra in through the stats? Are the students able to grasp concepts like “control for a variable” without knowing what a variable is? If the students are able to get the critical thinking and quantitative reasoning skills from a stats class without an algebra prereq, I’m on board. I just don’t know if they can.

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Full disclosure - I work at the on the Carnegie Math Pathways at the Carnegie Foundation. I'm a sociologist by training and have had lots of qualitative and statistics coursework in grad school which I loved. One of the things I learned working here is that every single one of the math scholarly societies believes that there are several good alternatives to providing students with intellectually rigorous and conceptually rich quantitative reasoning learning opportunities. They also pretty much agree that the traditional algebra sequence is not one of them. Take a look at TPSEMath and their Math Advisory Group work (https://goo.gl/wjusMH) for a recent overview of the space. It has been a real eyeopener for me. The recent interview of Eloy Ortiz Oakley on NPR was a great summary - although the headline is completely wrong. The last thing Eloy wants to get rid of algebra. What he wants is for every student to get a rich and challenge math learning opportunity that aligns with his or her college and career goals.

My view matches what was said above. Statistics is usually thick with word problems, so it contains more analytic thinking than algebra. They got rid of word problems ages ago, including most of the pseudo-problems of the good old days. (Who ever asked you to figure out what coins you have in your pocket? Who still has coins in their pocket!) Algebra will still be there (leading to Business Calculus and STEM classes), and it needs serious work so students will have had to think with math before they get to a physics or chemistry class.

That said, the solution in Florida was to create classes called Math for Liberal Arts that have almost no algebra but lots of applications. (They have been around for many decades, replacing College Algebra for, say, History majors. Students don't have to memorize formulas, but they do have to know how to apply them.) These classes are forbidden to STEM and Business majors, but they do a great job with mortgage rates and basic statistics, things that are really useful for people in fields that don't even require a knowledge of "real" statistics.

By the way, the assumption is that they have all taken and "passed" algebra in high school so it isn't like they have never seen algebra. They just hate it.

That said, the solution in Florida was to create classes called Math for Liberal Arts that have almost no algebra but lots of applications. (They have been around for many decades, replacing College Algebra for, say, History majors. Students don't have to memorize formulas, but they do have to know how to apply them.) These classes are forbidden to STEM and Business majors, but they do a great job with mortgage rates and basic statistics, things that are really useful for people in fields that don't even require a knowledge of "real" statistics.

By the way, the assumption is that they have all taken and "passed" algebra in high school so it isn't like they have never seen algebra. They just hate it.

I'm somewhat biased here, as statistics is central to most of bioinformatics (which has been my field for the past 22 years). I see statistics as far more essential to high school education than geometry or calculus, but I don't see how to teach even rudimentary statistics without at least algebra 1.

The statistics that bioinformatics and bioengineering students are required to take at my university requires fairly heavy-duty integral calculus. (They refer to the algebra-based statistics course that the biology students take as "baby stats" or "high-school stats".) So what I regard as bare minimum statistics may be far beyond what people are envisioning in a "stats for all" course.

If people wanted to remove a mostly useless math course from the high school curriculum, I'd argue that dropping geometry would be of more use than dropping algebra. Not that geometry isn't useful and even fun for some people, but it has very little transference to other subjects. Statistics would be a better core for teaching reasoning than geometry is, and teaching logic directly would be better for the cultural role of mathematical proofs. The main reason for enshrining geometry is the medieval quadrivium (arithmetic, geometry, music, and astronomy) and most schools have dropped astronomy and made music an extra option. I think that we'd be better off nowadays concentrating on the trivium (logic, rhetoric, and grammar), which seem to be getting short shrift in the schools.

The statistics that bioinformatics and bioengineering students are required to take at my university requires fairly heavy-duty integral calculus. (They refer to the algebra-based statistics course that the biology students take as "baby stats" or "high-school stats".) So what I regard as bare minimum statistics may be far beyond what people are envisioning in a "stats for all" course.

If people wanted to remove a mostly useless math course from the high school curriculum, I'd argue that dropping geometry would be of more use than dropping algebra. Not that geometry isn't useful and even fun for some people, but it has very little transference to other subjects. Statistics would be a better core for teaching reasoning than geometry is, and teaching logic directly would be better for the cultural role of mathematical proofs. The main reason for enshrining geometry is the medieval quadrivium (arithmetic, geometry, music, and astronomy) and most schools have dropped astronomy and made music an extra option. I think that we'd be better off nowadays concentrating on the trivium (logic, rhetoric, and grammar), which seem to be getting short shrift in the schools.

Are you talking about high school here? You typically need to pass an algebra course in high school to graduate. But plenty of college majors don't require any math courses at all.

Stats was the only math requirements for my human services bachelor's degree. With my Math SAT scores I had to placement test and somehow placed into calculus (spare me). Thankfully, stats didn't have a pre-req. It was, without a doubt, the best I've ever done in math. Years later the faculty told my mom, (the registrar) that I probably should've gotten an A. Would've been my only A in math...ever. I took Algebra in 9th grade. So, clearly not a ton of algebra is needed to get through stats because I remembered about a smidgen of it.

Why can we have a course that's some combo of algebra lite and real world stats. There would be transfer issues, of course. But aside from STEM fields, I can see that being the exact amount of math most college students need.

Why can we have a course that's some combo of algebra lite and real world stats. There would be transfer issues, of course. But aside from STEM fields, I can see that being the exact amount of math most college students need.

Having read the article !!! I would like to clarify a few things for Johan and GSwoPumps. But first, some humor.

The author of the NPR story described "intermediate algebra" (which is functionally equivalent to high school algebra 1, where using the quadratic formula is its highest-level skill) as "abstract algebra". I had all I could do to stay in my chair.

Yes, they are talking about dropping "intermediate algebra", which is something that colleges and universities in Florida did in response to a legislative mandate to assume that all PUBLIC HS graduates were qualified to take college level reading, composition, and math classes. Someone needs to look at the data, but this move can only be positive. Many students despise algebra but find formal logic interesting and fun. It can also help in the STEM areas, because it makes intermediate algebra (which only earns college credit at a CC, but does not count as "college level") into a class that is required for only two classes: College Algebra and Real Statistics. (Our stats class does not require College Algebra, for the few majors that require statistics but don't mandate algebra, but Business majors usually take College Algebra and Business Calculus before Stats. You don't use logarithms and factor cubics in a soft-ware driven statistics class.) Intermediate Algebra is no longer required for the two Liberal Arts math classes, so its rigor can be increased to help students get ready for a pre-calculus sequence.

GSwoPumps: the stats they are talking about are quite basic, what might be in a HS statistics class, if it is what is in one of our Liberal Arts math classes. Just enough to be helpful to a crimial justice or social work major.

Johan: I was puzzled by your statement about "no math", because our state requires TWO "college level" math classes for all state universities and colleges, but the NPR article confirms that only 60% of college students are required to take any math in college. That might reflect high admission standardsat some places or football-driven universities that don't have an option like Liberal Arts math to keep athletes eligible for competition.

The author of the NPR story described "intermediate algebra" (which is functionally equivalent to high school algebra 1, where using the quadratic formula is its highest-level skill) as "abstract algebra". I had all I could do to stay in my chair.

Yes, they are talking about dropping "intermediate algebra", which is something that colleges and universities in Florida did in response to a legislative mandate to assume that all PUBLIC HS graduates were qualified to take college level reading, composition, and math classes. Someone needs to look at the data, but this move can only be positive. Many students despise algebra but find formal logic interesting and fun. It can also help in the STEM areas, because it makes intermediate algebra (which only earns college credit at a CC, but does not count as "college level") into a class that is required for only two classes: College Algebra and Real Statistics. (Our stats class does not require College Algebra, for the few majors that require statistics but don't mandate algebra, but Business majors usually take College Algebra and Business Calculus before Stats. You don't use logarithms and factor cubics in a soft-ware driven statistics class.) Intermediate Algebra is no longer required for the two Liberal Arts math classes, so its rigor can be increased to help students get ready for a pre-calculus sequence.

GSwoPumps: the stats they are talking about are quite basic, what might be in a HS statistics class, if it is what is in one of our Liberal Arts math classes. Just enough to be helpful to a crimial justice or social work major.

Johan: I was puzzled by your statement about "no math", because our state requires TWO "college level" math classes for all state universities and colleges, but the NPR article confirms that only 60% of college students are required to take any math in college. That might reflect high admission standardsat some places or football-driven universities that don't have an option like Liberal Arts math to keep athletes eligible for competition.

Like some of the other posters above, I also had grad training in stats, and I'm a strong advocate of requiring college-level intro stats rather than algebra for non-STEM majors. For the general population, it's far more helpful to understand what margin of error means in a political poll than how to factor a polynomial equation! In response to DD's comment:

Correlations and standard deviations also rely on some knowledge of algebra.

I'm not sure what DD's getting at here. The formula for the standard deviation requires no algebra, just basic arithmetic functions (e.g., square root, addition). You don't need to know a lick of algebra to calculate and meaningfully interpret standard deviations. I'll go as far as to say that I can't think of any of the essential concepts in an intro stats class that require any algebra skills. I know it probably makes the mathematicians cringe that students in a stats class are not able to derive the formula for the least squares regression line using differential calculus, but the calculation and interpretation of a regression line only requires arithmetic skills and some logical reasoning.

Correlations and standard deviations also rely on some knowledge of algebra.

I'm not sure what DD's getting at here. The formula for the standard deviation requires no algebra, just basic arithmetic functions (e.g., square root, addition). You don't need to know a lick of algebra to calculate and meaningfully interpret standard deviations. I'll go as far as to say that I can't think of any of the essential concepts in an intro stats class that require any algebra skills. I know it probably makes the mathematicians cringe that students in a stats class are not able to derive the formula for the least squares regression line using differential calculus, but the calculation and interpretation of a regression line only requires arithmetic skills and some logical reasoning.

SamChevre says

I have a solid undergrad stats background (4 or 5 stats-related classes with calculus as a pre-req), and work with math people daily. I tend to think that college-level math begins with Calculus (although I had never taken even an algebra class when I started college.)

I think that a class in "Statistical Reasoning" could be done well without algebra; it wouldn't prepare someone to generate statistics, but would prepare someone to read them. I think that it would be more valuable for most non-math majors than algebra, or even calculus.

I have a solid undergrad stats background (4 or 5 stats-related classes with calculus as a pre-req), and work with math people daily. I tend to think that college-level math begins with Calculus (although I had never taken even an algebra class when I started college.)

I think that a class in "Statistical Reasoning" could be done well without algebra; it wouldn't prepare someone to generate statistics, but would prepare someone to read them. I think that it would be more valuable for most non-math majors than algebra, or even calculus.

It's much easier to find adjuncts who know enough algebra to teach it acceptably well than adjuncts who know enough statistics to teach it acceptably well. A lot of decisions about lower division math classes are made based on the availability of teaching and grading power.

As an undergrad at a state university in Florida back in the late 1980s, I took the following four classes to fulfill my math requirements as a History major - Social Science Statistics, Introduction to Computer Programming, Introduction to Formal Logic, and Finite Math. I had tested into Pre-Calculus at orientation, but had no intention of going into a field that required higher math. Of course, I also made it through Algebra II in high school, so I had plenty of experience with variables by the time I hit college.

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