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A Case for Understanding

March 20, 2010

I’ve had several interesting conversations recently centered around decimal arithmetic and whether or not students should understand why the standard algorithms for decimal arithmetic produce correct answers.  For example, when multiplying .2 \times 1.39 we should first think:

2 \times 139 = 278

and then think:

In .2 the decimal is moved over one spot.  In 1.39 the decimal is moved over two spots.  One plus two is three.  So we should move the decimal in our answer over three spots.  Our final answer is:

.278

Most of the objections to teachings why these algorithms work fall under a few (interconnected) umbrella reasons:

Reason 1:  It is too hard for many students to understand the why.  It is easier for them to simply memorize all of the rules (with no understanding informing their actions).

Reason 2:  There is simply no time to teach reasons why.  There is so much in the curriculum that teachers need to rush through things and the time necessary to explain why these algorithms work simply does not exist.

Reason 3:  We want students to do as well as possible on the end of year test (for NCLB) and to maximize our scores on this test we must focus on drilling the procedure rather than any sort of explanation.

There are VERY important reasons why we should want students to understand but what I want to focus on here is why I think teaching for understanding would like be  1)  Easier 2)  Faster  3) Would result in higher test scores.

Let us first be clear about why the standard procedures for decimal arithmetic work, and how this could be taught to students:  Decimals are a particular type of fraction, one whose denominator is a power of ten.  When viewed in this light, the “lining up of the decimal point” in decimal addition and subtraction is exactly finding a common denominator.  The “counting” procedure for decimal multiplication comes from multiplying the corresponding fractions and knowing that (10^a) (10^b) = 10^{a+b}.  The simplest way to teach students how to do decimal arithmetic is to simply tell them that they are a special kind of fraction.  Then when the student is told to compute .2 \times 1.39 they will do:  \frac{2}{10} \times \frac{139}{100} = \frac{278}{1000} = .278.   If students do decimal arithmetic in this fashion it is very likely that they will discover the shortcut (the standard procedure) themselves.  If they do not, they can be informed of the shortcut after they are comfortable with the conversion method.

Let us now discuss why this way of teaching (referring back to fractions) could very well be:

1)  easier for students

2) take less time in the classroom and

3)  result in higher test scores.

First, consider that with the new approach dispenses with three procedures (addition and subtraction of decimals, multiplication of decimals, and division of decimals) that students would otherwise need to learn.  This of course makes things easier for the student and frees up class time.  The learning of these three procedures is replaced with the the very easy skill of converting a decimal to fraction.  This is an easy task and a very important connection for the students to understand.  The two conversions that happen in each problem involving decimal arithmetic are a great opportunity to solidify the important connection between decimals and fractions.

Second, the new approach allows students to make sense of what is being done.  In addition to promoting a healthy conception of mathematics (things ought to make sense!), this greatly increases the likelyhood of students paying attention and thinking.  This actually makes things much easier for them and has the potential to raise test scores (remember that my focus is on these “practical matters” for this post).

Finally, teaching in this way provides the perfect opportunity to build multiple layers that will cement important knowledge in students minds.  First, students will get much more practice with fraction arithmetic and changing decimals to fractions, and they do this inside of a larger problem (this is good).  Second, and more importantly, the information about fraction arithmetic and the connection between decimals and fractions are crucially used in the subsequent demonstration (or discovery) of the shortcuts for decimal arithmetic.  It is a well known fact that when we use information to build further conceptual knowledge that uses that information, we are MUCH more likely to remember and correctly use that information later.  I think that this would have significant effects on students test scores for test questions that deal with either fractions or decimals (this is most decidedly NOT an exclusive or).

Now, a couple of comments:

I recently had the opportunity to observe a 6th, 7th, and 8th grade classroom at a local middle school.  I looked in the 6th grade math book, the 7th grade math book, and the Pre-Algebra book.  In all three of the books, fraction arithmetic and decimal arithmetic were taught.  In ALL THREE OF THEM decimal arithmetic came FIRST.  I was astounded.  If one is following the textbook (and decide not to refer to what occurs in subsequent chapters of the book), one cannot teach decimal arithmetic in the way that I have outlined above (which seems to me the most mathematically natural way to teach it).

One last comment.  The state of Illinois has a very nice web tool called the Illinois Interactive Report Card.  In looking at several middle schools and the high schools that they feed into I noticed the following (disturbing) trend:  Since the year 2002 (the first available data), middle schools show a steady increase in the percentage of students that meet or exceed standards.  Over the same period of time, the high schools that these middle schools feed into show an unsteady decrease in the percentage of students that meet or exceed standards.  I propose the following explanation (this may not account for all of it, but I suspect it does account for some):  In middle school there are few enough things that simply memorizing the procedures for everything can be an effective strategy (if the end goal is to do well on the state test).  In high school things there are too many topics discussed (and they are complicated enough), that this strategy simply will not hold up as well for the general population.  Of course this strategy does hold up for those who deem it important enough to spend lots of time and effort on “studying” (unfortunately I have usually found that studying focuses much more on recall than on reason).

I suspect that if students in middle school were more accustomed to thinking (and less accustomed to recalling procedures), there would be a much greater chance of them thinking and understanding what was taught to them in their high school classes, and those test scores would also increase.

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Teaching Philosophy

March 1, 2010

I wish to put down in writing certain of my postulates on teaching.  I may slowly enlarge the list.

Postulate:  There is no hope for a teacher to get students seriously engaged in the material if they themselves are not seriously engaged in the material.  Imagine an Trigonometry teacher who would be bored to discuss the mathematics of their course with colleagues outside of class.  I contend that such a teacher has no chance of seriously engaging the students in their classroom.

Some Books I like

October 13, 2009

Mathematics Education:

Arithmetic for Parents (Ron Aharoni)

Knowing and Teaching Elementary Mathematics (Liping Ma)

Algebra (Israel Gelfand)

Geometry (Harold Jacobs)

Differential and Integral Calculus (Richard Courant)

Calculus Made Easy (Silvanus Thompson)

General Education:

Montaigne’s Essays (particularly on the education of children)

Rousseau’s Emile

Spencer’s Education:  Intellectual, Moral and Physical

Autobiographies:

Benjamin Franklin’s Autobiography

John Stuart Mill’s Autobiography

Bertrand Russell’s Autobiography

Tolstoy’s Confessions

Frederick Douglas’ Autobiography

Automathographies:

Hardy’s A Mathematicians Apology

Halmos’ I Want to be a Mathematician

On Limits in Calculus Courses

October 7, 2009

How should limits be taught in a freshman (or high school) calculus class?  In particular, what role should two well-known, yet generally disliked, Greek letters play in the beginning weeks/months of the course?

It is surely the case that if one is to do things properly (i.e. rigorously) one must deal with \epsilon - \delta arguments.  However, it is also surely the case that if one is to do things properly, one must prove, for example, the extreme value theorem.  I think this is a viable way to teach calculus, though I think it unlikely something like this would actually be taught outside of a place like Harvard (I surely don’t think something like this is likely to happen any time soon at the University of Illinois).

The question that remains is what should be done in calculus courses that do not attempt to have a completely rigorous treatment of the subject.  I think that what currently is done is very poor.  As I have looked at books and tried to see what is taught, it seems that the usual treatment goes something like this:

First:  Give an intuitive notion of what a limit is (this might be precisely stated, but it is rarely, if ever, precisely referred to).  Work with this a bit to evaluate or “conjecture the value” of some limits.

Second:  Formally state the definition of limit.  Work with the definition to prove trivial things like \lim_{x \rightarrow 4} \sqrt {x} = 2.

Third:  Work  with nontrivial limits, essentially disregarding the definition of limit for the intuitive notion worked with initially.

It seems that by treating limits as such, we do a great disservice to students and give them, in fact, the very opposite impression of mathematics than what we should give them.  What they see is that mathematics (notation and arguments) turn something intuitively obvious (e.g. \lim_{x \rightarrow 4} \sqrt {x} = 2) into something very difficult (perhaps totally incomprehensible).  In fact what mathematics does (or should do) is take very difficult problems and make them comprehensible (mathematical language allows us to precisely talk about something that is very difficult and mathematical arguments allow us to come to correct conclusions about those difficult things).  Instead of viewing mathematical notation and arguments as valuable tools, students may very well get the impression that they are something that instead muddy the waters.

So what do we do?  It seems to me there are three natural options.  The first option has already been mentioned:  treat things rigorously.  Not only do you state the definition of limits, but you use the definition of limit in explicit nontrivial ways (of course if calculus is taught like this, students should also be required to rigorously do mathematics on their homework, exams, etc.)  The option on the other end of the spectrum is to never mention the definition of limit, to work with it as a purely intuitive notion, and to not worry about establishing a solid logical foundation on which calculus may securely rest.

The middle option (which to me seems very plausible at a school like the University of Illinois) does just the opposite of what seems currently prevalent.  Whereas now the difficulty of abstraction is fully present and the other difficulties are not present at all, this option would allow any natural difficulties and only deal with a portion of the abstract difficulty.  What I propose is this:  for any limit that shows up in the course (say \lim_{x \rightarrow c} f(x) = r) you should expect students to be able to answer a question like:  Find positive real number \delta so that:  If  |x -c| < \delta then |f(x) - r|< \frac{1}{100}.  Of course the analogous thing should be required for limits as x goes to infinity.

You may say that it only a small step from this to the full definition, but I think otherwise.  I think we should not underestimate the difficulty of successive quantifiers for beginning calculus students.  Also, it seems clear that if a student is unable to do a problem like this, being able carefully prove that \lim_{x \rightarrow 4} \sqrt {x} = 2 is rather meaningless.

It seems to me that this would be a good step in the right direction for students.  We are clearly moving in the direction of rigor,  yet we retain some degree of tangibility.  Also, the difficulties that the students will face are meaningful difficulties (they will deal with more than abstractions of trivialities).

On the Product and Quotient Rules

September 27, 2009

I was recently reading in Courant’s Differential and Integral Calculus and I found a much better formulation of the product and quotient rules than what typically appears in modern textbooks (the English edition of Courant’s book, by the way, is now 75 years old.  Does anyone know if this is its last year of copyright in the US?).  The usual formulation of the product and quotient rules is as follows:  Let r(x) = f(x) g(x) and s(x) = \frac{f(x)}{g(x)} where f(x) and g(x) are differentiable functions.  Then:

r^\prime (x) = f(x) g^\prime (x) + f^\prime (x) g(x)

and

s^\prime (x) = \frac{g(x)f^\prime (x) - f(x) g^\prime (x)}{ [g(x)]^2}

Compare this with Courant’s formulation:

\frac {r^\prime (x)}{r(x)} = \frac{f^\prime (x)}{f(x)} + \frac{g^\prime (x)}{ g (x)}

and

\frac {s^\prime (x)}{s(x)} = \frac{f^\prime (x)}{f(x)} - \frac{g^\prime (x)}{ g (x)}

Courant’s formulation of these rules is much simpler (especially in the case of the quotient rule) and show that the product and quotient rules are very closely related (of course they have to be very closely related since division is the inverse of multiplication; what is great about Courant’s formulation is that this relationship is very apparent).

Another point in favor of Courant’s formulation is that is extremely easy to generalize.  It is one easy step from Courant’s formulation to:

\frac{(f_1 f_2 ... f_n)^\prime}{(f_1 f_2 ... f_n)} = \frac{f_1^\prime}{f_1} + \frac{f_2^\prime}{f_2} + ... + \frac{f_n^\prime}{f_n}

This gives a simple proof of the power rule for natural numbers (Courant proves the power rule this way in his book).  It is also just one easy step to full product/quotient rule:   If q = \frac{(f_1 f_2 ... f_n)}{(g_1 g_2 ... g_m)}, then:

\frac{q^\prime}{q} = (\frac{f_1^\prime}{f_1} + \frac{f_2^\prime}{f_2} + ... + \frac{f_n^\prime}{f_n}) - (\frac{g_1^\prime}{g_1} + \frac{g_2^\prime}{g_2} + ... + \frac{g_m^\prime}{g_m})

I should mention that this generalized formula is a simple application of logarithmic differentiation.  This gives credence to the notion that the truly important differentiation rule is the chain rule.  Everything else easily follows from that.