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Prerequisites for college math

Are there any high school math teachers – or, better yet, developers of high school math curricula – who read this blog? I’ve decided to take a break from my usual undirected griping about how woefully unprepared students are to do college math, and divert those energies into something more productive – a working list of what students should be learning in high school math classes in order to prepare them for college (or even, for what they are supposed to be learning later on in high school).

Some months ago Rudbeckia Hirta drily observed that contrary to what one might assume, the college prep track in the high school math system does not, in fact, prepare students for college math. As far as I can tell, high school math in general doesn’t prepare them for damned near anything. In fact, I think that students often learn negative amounts of math in high school: in grade school, they learn how to perform basic mathematical operations and such, and by the time they’ve received their diplomas, they’ve been trained to leave those tasks to their calculators. (See: inability to add fractions, talldarkandmysterious.ca, 2004-present.) The claim that foisting heavy-duty calculators upon students frees them to do more complex and creative tasks is wishful thinking that, in my experience, has no grounding in reality.

When I started teaching, I anticipated that students would be weak in much of the material that they were expected to have learned a year or two earlier. The actual state of affairs was far more dire: many couldn’t do the math that they should have learned a full decade earlier. Most weren’t just weak in math; they didn’t even know what math was.

They didn’t know what an equation symbolized, or even that it was supposed to symbolize anything at all: to them an equation was just a jumble of symbols. They looked at me blankly when I asked them to think not only about what steps they needed to perform in order to solve a problem, but also about why they needed to perform those steps.

They had no experience, nor understanding, of how to reason logically when presented with quantitative problems. Students threw fits when asked to combine simple techniques in…basic ways” (link via Chris Correa) – no one had ever required them to do that before. In his book Innumeracy, John Allen Paulos’ lamented:

Elementary schools by and large do manage to teach the basic algorithms for multiplication and division, addition and subtraction, as well as methods for handling fractions, decimals and percentages. Unfortunately, they don’t do as effective a job in teaching when to add or subtract, when to multiply or divide, or how to convert from fractions to decimals or percentages.

I disagree somewhat with Paulos, who wrote Innumeracy before calculators were ubiquitous: elementary schools no longer do manage to teach the basic algorithms very well. Other than that, he’s correct. Students’ depth of mathematical knowledge is so shallow that they can’t even figure out when perform basic mathematical operations.

The necessary groundwork for doing mathematics at the college level – or even at the high-school-courses-in-college level – is more basic than anything that students supposedly learning in high school in the most superficial and fleeting manner. And they routinely leave high school without it.

Based on my experiences, students graduating from high school should, in order to succeed in even the most basic college math classes:

  1. Be able to add, subtract, multiply, and divide fractions. Moreover, they should understand that the horizontal bar in a fraction denotes division. (Seem obvious? I thought so, too, until I had a student tell me that she couldn’t give me a decimal approximation of (3/5)^8, because “my calculator doesn’t have a fraction button”.)
  2. Have the times tables (single digit numbers) memorized. At minimum, they should understand what the basic operations mean. For instance, know that “times” means “groups of”, which will enable them to multiply, for instance, any number by 1 or 0 without a calculator, and without putting much thought into the matter. This would also enable those students who have not memorized their times tables to figure out what 3 times 8 was if they didn’t know it by heart.
  3. Understand how to solve a linear (or reduces-to-linear) equation in a single variable. Recognize that the goal is to isolate the unknown quantity, and that doing so requires “undoing” the equation by reversing the order of operations. Know that that the equals sign means that both sides of the equation are the same, and that one can’t change the value of one side without changing the value of the other. (Aside: shortcuts such as “cross-multiplication” should be stricken from the high school algebra curriculum entirely – or at least until students understand where they come from. If I had a dollar for every student I ever tutored who was familiar with that phantom operation, and if I had to pay ten bucks for every student who actually got that cross-multiplication was just shorthand for multiplying both sides of an equation by the two denominators – I’d still be in the black.)
  4. Be able to set up an equation, or set of equations, from a few sentences of text. (For instance, students should be able to translate simple geometric statements about perimeter and area into equations. ) Students should understand that (all together now!) an equation is a relationship among quantities, and that the goal in solving a word problem is to find the numerical value for one or more unknown quantities; and that the method for doing so involves analyzing how the given quantities are related. In order to measure whether students understand this, students must be presented, in a test setting, with word problems that differ more than superficially from the ones presented in class or in the textbook; requiring them only to parrot solutions to questions they have encountered exactly before, measures only their memorization skills.
  5. Be able to interpret graphs, and to make transitions between algebraic and geometric presentations of data. For instance, students should know what an x- [y-]intercept means both geometrically (”the place where the graph crosses the x- [y-]axis”) and algebraically (”the value of x [y] when y [x] is set to zero in the function”).
  6. Understand basic logic, such as the meaning of the “if…then” syllogism. They should know that if given a definition or rule of the form “if A, then B”, they need to check that the conditions of A are satisfied before they apply B. (Sound like a no-brainer? It should be. This is one of those things I completely took for granted when I started teaching at the college level. My illusions were shattered when I found that a simple statement such as “if A and B are disjoint sets, then the number of elements in (A union B) equals the number of elements in A plus the number of elements in B” caused confusion of epic proportions among a majority of my students. Many wouldn’t even check if A and B were disjoint before finding the cardinality of their union; others seemed to understand that they needed to see if A and B were disjoint, and they needed to find their cardinality – but they didn’t know how those things fit together. (They’d see that A and B were not disjoint, claim as much, and then apply the formula anyway.) It is a testament to the ridiculous extent to which mathematics is divorced from reality in students’ minds that three year olds can understand the implications of “If it’s raining, then you need an umbrella”, but that students graduating from high school are bewildered when the most elementary of mathematical concepts are juxtaposed in such a manner.)
  7. More generally: students should know the basics of what it means to justify something mathematically. They should know that it is not enough to plug in a few values for x; you need to show that an identity, for instance, is true for all x. Conversely, they should understand that a single counterexample suffices to show that a claim is false. (Despite the affinity on the part of the high school text I am working for true/false questions, the students I am working with do not understand this.) Among the educational devices to be expunged from the classroom: textbooks that suggest that eyeballing the output of a graphing calculator is a legitimate method of showing, for instance, that a function has three zeroes or two asymptotes or what have you.

What else? I invite comments from everyone with a dog in this fight – college students who’ve taken math classes, college math instructors, curriculum developers, textbook authors. What unwritten (or written!) prerequisites are students missing? What should every student know in order to succeed in the class you took, taught, or wrote the text for?

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