### On teaching college students what they should already know

Rudbeckia Hirta succinctly explains that if you can’t do algebra, then you can’t take calculus:

Due to reasons beyond my understanding, high school math and college math are completely unaligned. The K-12 system sends us students whose knowledge is a mile wide and an inch deep: we get students who are shaky at algebra, frightened of fractions, and unsure of how to find the areas of basic plane figures (

and completely unable to accept the idea that it is a reasonable request to ask them to solve non-standard problems where the method of solution is not immediately obvious), but they have been exposed to matrix arithmetic, computations from polynomial calculus and other supposedly “advanced” procedures. You would think that the “college prep” track would prepare students for college, but it doesn’t. Recently I read somewhere (maybe in Focus?) that there is more calculus taught in high schools than there is at colleges.

See also: mathematics by pattern matching, word problems, et al. So much of this problem can be traced to the fact that students do not understand that an equation is a relationship among quantities. Each equation they see is a concept unto itself, to be memorized and applied to the word problem on the test that looks like the word problem that I did on the blackboard and that used a similar equation. If the word problem on the test doesn’t look like any of the problems I did in class, then that question is “totally unfair”. Most students will leave the question blank, or solve a completely different problem (one with a ready-made equation) in the space provided.

This New York Times op-ed is about teaching freshman English to illiterate college students, but I’m sure that anyone teaching freshman math to innumerate college students can find plenty to relate to. I’m not sure I’m willing to buy into, wholesale, the author’s belief that content should be ignored in favour of form, but I can’t argue with success:

On the first day of my freshman writing class I give the students this assignment: You will be divided into groups and by the end of the semester each group will be expected to have created its own language, complete with a syntax, a lexicon, a text, rules for translating the text and strategies for teaching your language to fellow students.

…14 weeks later - and this happens every time - each group has produced a language of incredible sophistication and precision.

How is this near miracle accomplished? The short answer is that over the semester the students come to understand a single proposition: A sentence is a structure of logical relationships. In its bare form, this proposition is hardly edifying, which is why I immediately supplement it with a simple exercise. “Here,” I say, “are five words randomly chosen; turn them into a sentence.” (The first time I did this the words were coffee, should, book, garbage and quickly.) In no time at all I am presented with 20 sentences, all perfectly coherent and all quite different. Then comes the hard part. “What is it,” I ask, “that you did? What did it take to turn a random list of words into a sentence?” A lot of fumbling and stumbling and false starts follow, but finally someone says, “I put the words into a relationship with one another.”

*An equation is a relationship among quantities*. How many times have I tried, and failed, to get this idea across? Imagine getting students to realize it for themselves! Unfortunately, many students lack the intuitive ideas about math needed to know, even subconsciously and with the help of leading questions, that equations are anything other than a jumble of letters, numbers, and symbols. (Even though I spent twenty minutes showing how we could use the definition of a circle and the Pythagorean Theorem to derive the equation of a circle in the Cartesian plane, nearly all of my students were angry that I wouldn’t provide the formula on the test. I mentioned, incorrectly, that they could derive the formula themselves, on the test, if they needed to; of course, I was wrong.) But I would gladly sacrifice 80% of the poorly-learned content in a first-year college math course if I could instead effect a solid understanding of what equations really meant - and how to get one from a sentence or two of information. I wonder if Fish’s lesson could be adapted to the first-year math classroom.

At Critical Mass, where I found the NYT piece, Erin O’Connor isn’t optimistic even about applying it to the English classroom:

[M]ost university composition courses are taught by graduate students who are a) not necessarily good writers themselves, and b) often more interested in using the composition classroom to practice teaching the content they hope to teach as non-composition teaching English professors, and you’ve got a situation in which the Fish vision, regardless of its merits, is pure pipedream.

Ditto. No one teaches precalculus if they can avoid it; at Island U, it got passed from temps to new faculty to the department head, who teaches the courses that no one else will teach. Everyone says that the precalculus course needs to be completely revamped, but given the option, they’d rather take a less thankless teaching assignment than put forth the effort to revamp it. And around and around we go.

In the fall I gave the pre-service teachers a somewhat similar assignment:

Of course I couldn’t spend 14 weeks on this because I had about a zillion things that needed to be covered.

Interesting. How did it go? What sorts of things did they come up with? What sort of guidance did you offer as they worked on it (based on how open-ended this activity is, I assume it was an in-class thing, rather than a homework assignment)? What sort of math did you use to follow-up on this? (I’m guessing place value, the decimal system and such.)

This definitely sounds better than the math-for-future-elementary-school-teachers course offered at my old school, which covered, among other things, “the three models of multiplication”. (For the curious, there’s the grid method, the chart method, and the lattice method. I am not making this up.) Mind you, I’ve never heard of such a course that was anything but completely depressing to teach - if your blog posts are even somewhat representative, I’m kind of afraid to read your response.

See “Equal Sign Considered Harmful'’ .

*sigh* http://www.livejournal.com/users/r6/11375.html

Why must this text entry box delete everything that I write between angle brackets?

Sorry. I am calm now.

I will say that just about any linguist in the world would *love* to teach Stanley Fish’s linguistics class (which he, for reasons unknown to me, calls English). I admit that I laughed when people talked about changing fonts for plurals, though.

Well, the assignment didn’t go as well as planned — I naively overestimate the abilities of my students. Had I had more class time to devote to it, it might have gone better. It was placed after I had taught base n stuff (only one group came up with base 5 as their system — and that was the group that included the pre-vet student taking the class to raise his math/science gpa) but before I had taught why the various standard algorithms work.

Btw, lattice multiplication rocks. When I taught the various methods on multiplication (and, fool that I am, I also taught “Russian peasant” multiplication), my challenge to my students was for them to prove that the algorithm always provided the right answer. (Were they successful? Guess.) And I totally think that everyone should be shown the geometric version of the distributive law.

I get the same kinds of comments from the (just as innumerate) students in my (core clas - required of all business majors)Intro Finance class, since it’s equation rich. Even though it’s all pretty low level algebra - linear equations, with some exponents in the time value of money section.

One of my previous teachers (obviously much smarter than me) say that “Math is just a shorthand way of describing the real world”. Understand the world it’s describing, and the Math is easy. DOn’t understand it, and it’s just a bunch of squiggles.

r6 - Because it thinks you’re trying to use HTML. (And only limited HTML is legit in this place.) Interesting post, though, about the equals sign as binary relation versus equals sign as placeholder - that’s a subtle point, but possibly an important one. I know that a lot of my students don’t know what the equals sign means. (Though I suspect that their problems are even deeper, as many of them are so weak in algebra that they don’t know that 2(x+3y)

means the same thingas 2x+6y.)RH - geometric version of the distributive law? I’m curious. A few years ago, I started explaining the distributive law with animals and cages. You have three cages, each with two dogs and five cats. How many of each type of animal do you have? Now, what’s 3(2d+5c)? And from there it’s a quick jump to (a+3b)(2d+5c) - replace (a+3b) with x, and repeat the procedure twice…

If you can get past the fact that the focus is on how teaching speakers of Black English if different from teaching speakers of standard English, Eleanor Wilson Orr’s _Twice as Less_ has some interesting examples of how to teach students to understand that equations are relationships among quantities. If my copy hadn’t wandered off, I would give an example, but I remember being impressed by the difficulty of the relationships (equations) she asked students to analyze, including questions about whether a given equation expressed the relationship in a word problem and why.

It’s the sort of understanding that would be incredibly useful for middle school teachers. Because if you understand what the equation means, the solving part comes much more easily.

The geometric version of the distributive law (assuming that Becky’s talking about what I think she’s talking about) is fairly straightforward. Treat multiplication as dealing with areas. So 2x3 means that we have a rectangle with one side of length 2 and the other side of length 3.

Now, what happens if your rectangle has side lengths 2 and x+3y? Well we can break down the side that has length x+3y so that we have one part with length x and another part with length 3y and our rectangle is now composed of two smaller rectangles. One has area 2x and the other has area 6y. Add those together and we have 2x+6y.

And of course from there it’s a short hop skip and a jump to multiplying not just binomials but any polynomials.

Algebra tiles can be helpful for students who like to move things around and really see what’s going on. I suppose legos or any sort of rectangular blocks could also work in a pinch.

RH - link to example of lattice multiplication? I’ve never heard of it.

r6 - I used to use arrow when I was doing derivations, as opposed to solving equations. This was so I could follow the flow of my work when I wanted to double-check my steps.

Anyway, remembering that rant by V.I. Arnold I linked to yesterday at livejournal, I’ll have to keep that in mind when teaching the kids math. I think I’ll go the ole Euclid route, and teach number theory via geometry.

I realise this may be a blindingly obvious question, but just how much/often have you repeated “an equation is a relationship among quantities”? If the students want something by rote, maybe make it that. I remember my high-school history teacher who would say “all history is an interpretation” at least three times a week, and put that on every quiz. It was on the final twice, as I recall. You could even have a follow-up question about “What does that mean?”.

Here’s the article Meep mentioned; worth reading in full.

Lisa - it’s not an obvious question at all - at least, it wasn’t to me the first time I taught precalculus (ah, the joys of having everything documented and archived). This past term, I took to repeating that mantra every single time I presented a word problem on the blackboard - and I’d explain what quantities we were dealing with, and how they were related. The problem, though, is that I can’t explain that sort of thing without using a specific word problem as an example, and when I do

that, they focus only on what is necessary to solve the exact problem that is on the blackboard.After one particularly miserably-done test, I allowed students to submit corrections for the especially-poorly-done word problems. They had to follow a specific template for their solutions: they had to state what quantities they were looking for, which of those quantities were known, which were unknown, and how they were related. Then they had to give an equation based on that. This activity went ok-but-not-great; I fear that the undergraduates in the tutorial centre are a little overzealous with the help they provide. In any case, that’s an activity that I’d like to emphasize more next time (simply having them parrot that an equation is a relationship among quantities is not good enough; it’s the

applicationof that idea that they just don’t get), but that’s just one of around a dozen things I’d like to do in a course that already requires the instructor to cover more content than is practical with such weak students.It seems as if the problem you’re struggling with is that your students have no mathematical intuition. They don’t think in math.

You might be being too hard on them.

Personal Experience:

I’m an engineer. When I was in undergrad the process that you described above was required format for statics, Dynamics and Mechanics of materials. It’s a nice method of a complicated problem (multiple variables, several equations unclear problem statement)

Problem Format was worth 1/5 of your homework grade.

GIVEN: State problem and give information

FIND: What are you looking for

DIAGRAM: (first rule of mechanical engineering: draw a picture. ChemE’s and EE’s didn’t use this as much)

EQUATIONS: What formulas will be used.

Knowns: What variables do you know from the given

Unknowns: what variable will need to be solved for to find the answer.

Solution: This is where the work goes.

Answer: State the answer clearly, with words. Comment as needed. If the correct answer was silly say so. I had a professor that would give information that intentionally lead to silly answers to teach us to pay attention. If you didn’t point it out than you lost partial credit.

Right answer, correct work (use of equations, math, FBD etc) improper format was worth 80%. (70% if I couldn’t find your answer quickly) I used to grade statics as a work study and was told to stress this. If they didn’t fill in all of the ‘busy work’, there was no way to tell if they understood the problem or were just plugging numbers into a formula and hoping.

The idea is that after you solve problems though this method long enough you’ll eventually start to think that way naturally. What’s going on? What am I trying to find? What are the forces involved? What physics governs this type of behavior? What equations can I use? what are my knowns and unknowns? Show how i used it all and than tell you what I found.

Once you got to 300+ level classes the professors would stress this less and less.

I’m not saying that’s this would work for you but it really sounds like your students have no idea how to solve a quantitative problem. It might be useful to teach them that. I know you have a lot to cover, but the format was a handout and about 20 minutes in class. It sucked for the easy short problems, because it took longer to fill in the format than to solve the problem but it was very helpful later on. I should point out that we’d usually only have 3-5 problems per day in class. Hope it helps.

here’s how i handled the prerequisite problem:

on day one, hand them a short take-home test containing very basic problems covering concepts and skills that you very explicitly say you will take for granted or only review very briefly in class. tell them that this test is not for a grade. rather, they must come into your office to have you check it, or else you will not give them their first exam. (this gets them into your office at least once, and you can show them when they get there that you’re there to help.) tell them that if they are unable to do more than a couple of these problems without a calculator or an open math book, or if the test takes more than x minutes, then you predict that they will probably fail this course. remind them that you are being kind by letting them know this early. then tell them that if they have trouble, and take this course anyway, you’ll help them to the best of your ability, but “don’t come whining to me if you still can’t do the work, because I gave you fair warning.”

maybe some variation of that would work for you.

Here’s a link for

Lattice Multiplication.

I found it with a google search on “Lattice Multiplication”

Incidentally, there’s a wealth of information on assorted bi-ways of high school mathematics in Alfred Posamentier et al’s book on Methods of Teaching Secondary Mathematics. It’s a great book for any math instructor. Especially those of us who are teaching secondary mathematics (or lower) to college students.

Tim - That’s pretty much what I did.

As for other suggestions - this is what I’m working with. The problem is that students learn little useful math (”how to graph stuff on the TI-83+” doesn’t count) in high school, and remember nothing. I pretty much have three months to teach five years’ worth of background. I’m a pretty good teacher, but a miracle worker I am not.

Does it seem like the students that don’t get it are trying and failing or just not really that interested? I know you talk a lot about grade grubbers and complainers, (and I like those stories!) but are the majority clueless or lazy? One of the reasons it worked in engineering schools is that everyone knew they needed to master the material and worked at it. My comment assumed that they were trying, but just had /no/ idea how to solve an open ended problem quantitatively

A little of each, and plenty of both. There are plenty of students who are both clueless

andlazy, and do not understand that their grade is a reflection of how much of the math they should know, they do know, rather than a consequence of their divine right to the grade they want. And for many of them, it’s not just that they have no idea how to solve an open-ended problem quantitatively - they also do not know basic algebra. As I’ve written elsewhere, my precalculus students were, on average, doing math at a grade seven level.Attitude, though, is the biggest problem. Even many of those who have the intelligence and ability to learn to solve open-ended quantitative problems resist my efforts to teach them. In high school, they saw examples of every single type of problem they’d see on a test. I don’t do that, therefore, I am a bad teacher.

Rudbeckia…

My thought was, if you used all the letters 0, A, B, C, D on numbers… how would you have any room left for the arithmetical operations?

Consequently… I figured you would need one symbol for each operation (A = addition, B = subtraction, C = multiplication, D = division), and then use a unary system based on 0

(’0′ = 1, ‘00′ = 2, etc)…

so we would have, for example, 0 A 0 for ‘1 + 1′

or am I missing something?