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Every time we teach calculus we discover that a significant portion of our students never understood arithmetic. I don't mean that they can't multiply numbers, but rather that they don't know intuitively that a car going 15 miles an hour goes 1 mile in 60/15=4 minutes (i.e. that division is the arithmetic operation corresponding to this problem).

It would be entirely inappropriate to teach them as if they were 10 years old, even if we had 3 months to teach them arithmetic.

Usually these are fairly intelligent individuals considering that they managed to get through high school mathematics well enough to get into a good college or university despite this handicap, and this deficiency in their background is not their fault in any way. They are likely to be able to pick up arithmetic quite quickly, and figure out from that why they have been a little befuddled through all of high school math.

I would hope this problem has been studied and ways to help these students have been proposed.

I am looking for references either to resources for these students or resources for instructors trying to help these students in the context of a calculus (or precalculus) class.

Alexander Woo
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    I don't think I agree with your premise. Students may very well have learned arithmetic at some point, but they have a lot going on in their lives. It doesn't take much time away from mathematics for that knowledge to vanish. I think it's difficult sometimes for working mathematicians to appreciate this dynamic. But most mathematicians I know forget huge chunks of mathematics they knew as undergraduates -- only they haven't used that branch of mathematics recently, so that part of their mathematical mindset is near inert. – Ryan Budney Nov 05 '10 at 19:19
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    The cynical side of me says that this is precisely why my PhD institution gave the junior instructors the option of requiring a qualifying exam for the intro calculus classes (those who don't pass can make up for it with a summer class). I am interested in any solutions: this may be a problem that I will run into in the future. (As an aside to the OP: I am somewhat uncomfortable with you speculation on why the students aren't clear with the concepts. Perhaps you can remove it without changing the focus of the question?) – Willie Wong Nov 05 '10 at 19:21
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    Continued... For example, my grandfather had essentially no formal education at all, but he understood arithmetic in his bones -- he was a farmer so basic issues of proportionality were things he had to deal with all the time. So he taught himself on his own. The problem is that students don't perceive a natural persistent need for quantitative reasoning in their lives. IMO this isn't something that can be readily addressed in one course. – Ryan Budney Nov 05 '10 at 19:24
  • And they don't have an intuitive understanding of why a zero in a denominator with a nonzero numerator implies a vertical asymptote. Early this semester I ran past a calculus class the question of what $2^x$ approaches as $x\to -\infty$ and gots lots of wild guesses. Then I went through the idea that if you repeatedly cut something in half it approaches 0. Then I told them they would need this later and asked who would remember it when that time came. They all said they would. And a month later they did! But then nearly two months later they didn't do as well until reminded. – Michael Hardy Nov 05 '10 at 20:04
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    +1 because I see a lot of calculus/precalculus students struggling with elementary school math and --- regardless of how they got that way --- I'd love to have some resources to point them towards. I'll second Willie's suggestion as I'd hate for this useful question to be derailed by a debate over the source of student misunderstandings. – Ross Churchley Nov 05 '10 at 20:10
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    Willie and Ross - I hope I have addressed your concerns? I want to make it clear this is a question about diligent intelligent students who unfortunately happen to have this deficiency in their backgrounds but have actually managed to cope with it quite well. – Alexander Woo Nov 05 '10 at 20:21
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    There's also this issue: If one has, e.g. $\dfrac{8}{3}\cdot 51$, a student divides $8$ by $3$, getting $2.66667$, then multiplies that by 51, getting $136.00017$, and the student of course thinks that's more accurate than the "rounded" answer $136$. But if you write $$ \dfrac{8}{3} \cdot 51 = \dfrac{8}{3} \cdot(3 \cdot 17) $$ and then cancel the 3s, you've got exactly 136. Tell a student that one person got $136.00017$ and another got $136$ and ask which one is rounded. Maybe this is a major part of the answer to my question that I asked earlier in theese comments. – Michael Hardy Nov 05 '10 at 23:27
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    +1. Many "pedagogy" questions on MO consist largely of bitching about lower-division students' errors and misconceptions. (Indeed, some comments here border in that direction.) But you are bringing up a question that I have absolutely faced, don't have a good answer for, and have seen many instructors fail spectacularly at addressing this. I know how to teach arithmetic one-on-one, and I suspect that many people do. But I don't know how to do it in an undergraduate-classroom setting. – Theo Johnson-Freyd Nov 06 '10 at 00:37
  • @Willy Wong: At UC Berkeley, the lowest-level course we teach is our "precalculus" class, and students are directed into it if they fail the some "qualifying" exam. I have taught the precalculus class, and count it as one of my favorite classes to teach. However, even there, the curriculum does not include basic arithmetic skills (I think it should). So although I agree that some instructors use "you should move to the precalculus class" as a way to get rid of the students who don't know arithmetic, and although maybe I know more than they about teaching arithmetic, it is a subpar solution. – Theo Johnson-Freyd Nov 06 '10 at 00:43
  • At SPC, they used to have a class called "Physics for Poets", effectively a basic introductory course on Physics for those who did not intend to become Physics majors but needed to fulfill a Basic Sciences curriculum requirement. At TTI, they didn't have anything like that. Even pre-meds who wanted to take Biology or Chemistry pretty much had to take the hard-core difficult classes intended as the introductory course for those who would major in that subject. However, in the context of a calculus class, the obvious question is why are they in this class without meeting the prerequisites? – sleepless in beantown Nov 06 '10 at 08:55
  • Translation guide for above: (done so that I don't step on any toes or anyone's feelings...): SPC=small private college, TTI=tiny technical institute, ESU=enormous state university (thanks, Tank McNamara, for the college name). There's an introductory "pre-test" for some physics classes taught by a colleague of mine, and she always finds that 10% of the students fail the basic concept of understanding what a variable is, and how to simplify equations. She tells these students that they do not qualify for the course. Students who need remedial arithmetic ought not be in calculus. – sleepless in beantown Nov 06 '10 at 09:02

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I have TA'ed a "Mathematics for Future Elementary School Teachers" course. The point of the course is to develop a deep understanding of elementary school math (read: An actual understanding, rather than a knowledge of how to do computations). The book we used was Sybilla Beckmann's "Mathematics for Elementary Teachers".

At the end of the course, most students could really explain why 2/3 of 4/5 of a cup of milk was 8/15 of a cup of milk, and could draw a picture which showed why it was true. Ditto for the addition of fractions, and the algorithms for addition, multiplication, and division. I had many students who were flabbergasted that no one had ever shown them why these things were true before. Of course, I didn't actually show them: Sybilla's book is geared toward activities which help students to discover why these things work on their own or in small groups. The role of the teacher is to direct and clarify.

The reason that this course works, though, is because the students (at least initially) think that they are only learning how to explain these things to elementary school students. You never come right out and say "You do not understand addition, and I am going to show you". So it is a unique circumstance. Even then there are many students who resist the course because they feel like they don't have to put in any work to understand such "basic concepts". A lot of these students turn around when they realize that they do not really understand, and see that they are doing poorly on examinations and homework. Some of them do not ever feel comfortable enough to face their ignorance, and these people generally do not do so well in the course. A teacher must be humble enough to realize when they do not understand something, so it is a good thing that this course is a requirement for future teachers.

If you are serious about starting a course focused on elementary school math at the college level, which I think is a GREAT idea, I would use Beckmann's book. It is really fantastic. If you want more info, like an actual plan for a quarter's worth of work, I could email one to you.

Steven Gubkin
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  • BTW, how do you convince students to cancel before multiplying? They see $20 \cdot 6/15$ and change it to $120/15$ and then divide. – Michael Hardy Nov 05 '10 at 22:29
  • I do not think this is a big conceptual error - it is not the most efficient way to do things, but I do not think that is a big deal. So I would mention it, but if it didn't stick I would let it go. – Steven Gubkin Nov 05 '10 at 22:40
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    So you're computing a binomial coefficient like $\binom{20}{8}$ and you've got $\dfrac{20\cdot 19 \cdot 18 \cdots 13}{8\cdot 7 \cdot 6 \cdots 1}$ and the student reads the dots between numbers as a verb in the imperative mood commanding the reader to multiply. This seems like a case where (1) efficiency is worth it, and (2) maybe knowing that the denominator must cancel out completely helps conceptual understanding. – Michael Hardy Nov 05 '10 at 22:48
  • BTW, did you do Euclid's algorithm in that course? That's something where the computationally efficient way to do it is not the one that makes it easiest to understand. Knowing that $\gcd(74451,21827) = \gcd(74451-21827,21827)$ is the essential idea. – Michael Hardy Nov 05 '10 at 22:54
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    I do not have any specific recommendations about encouraging people to cancel first. – Steven Gubkin Nov 05 '10 at 23:39
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    +1 for this answer. @Steven-Gubkin, excellent answer for the specific situation of teaching soon-to-be-teachers, and possibly for others also. I'll have to take a look at this book and remember to recommend it to those who are willing and able and interested. Also, another +1 if I could give it to you for recognizing what an exceptional circumstance that class is, and for your well-presented and apropos response, particularly your commentary on the role of teachers to "direct and clarify" and let the students learn and master the material. – sleepless in beantown Nov 06 '10 at 09:30
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    I also taught this course and can testify that it works. One way to tell it works is how stressful it can be to teach (at least without Steven's gift of diplomacy). Afterwards I mentioned I received some of the harshest evaluations of my career. A colleague, a master teacher who had also taught it, gleefully asked to see them. After perusing them he said somewhat less cheerfuly: "these aren't so bad," apparently compared to his own. (Some positive reviews I got were also my best ever.) Beckmann spent 10 years on the book, which has been recognized as the best such text available. – roy smith Feb 01 '11 at 19:02
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    One remark on a popular geometry illustration repeated in this book relates to the "proof" of SSS congruence using a rigid model triangle of straws joined along a thread. This actually proves "rigidity", i.e. lack of infinitesimal deformations. This implies the moduli space is discrete, not necessarily a singleton. Indeed the same model shows SSA implies rigidity, but of course not congruence; i.e. this moduli space is in general 2 point. Thinking about these questions motivated me to read about Cauchy's rigidity theorem for convex polyhedra, and Connelly's non -convex counterexamples. – roy smith Feb 01 '11 at 19:14
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    @roy Actually I had a very strange experience. I taught two recitations of the same class. For some reason one class was always really full of energy, fun, laughter, and insight, whereas in the first class everyone was very quite, reserved, and obviously didn't care about the material. Evaluations followed suit: One class gave the best reviews I have ever received, and in the other they were just terrible. I still really don't know what happened, even after pondering it for over a year now. – Steven Gubkin Feb 01 '11 at 23:19
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    @Steven: Evaluations are problematic. As a nervy young teacher with two identical sections of precalculus, I tried an experiment. I handed out the evaluations in one class attached to the back of the final exam on exam day. In the other class I passed out champagne and glasses, followed by the evaluations, on the last day of class. Guess which class liked my teaching infinitely more than the other? (I eventually lost that position.) – roy smith Feb 03 '11 at 00:38
  • @Steven Gubkin: I've also had that experience you mentioned in your last comment above: http://www.angrymath.com/2015/07/the-difference-teacher-makes.html – Daniel R. Collins Nov 14 '15 at 15:31