I often teach a course with the enigmatic title “Fundamentals of Mathematics I”, intended for liberal arts majors. This is usually the last encounter with math for students in non-scientific disciplines. The syllabus contains a decent amount of optional topics so it is quite possible to tailor the material to one’s taste and professional interests. As many of us are well aware, a course like this poses its unique challenges. Unlike calculus, where the topics are fairly standard, a “fundamentals” course must, in my view, depend much more on mathematical ideas and much less on drudgery computations. Of course, students need to compute, but the result of their computational efforts should be exciting and fun, rather than a more or less meaningless answer that matches the solutions page at the end of the chapter.

These are the topics I usually teach: set theory, logic, group theory, and combinatorial methods. All lend themselves to a great deal of enjoyment, where students are confronted with deep ideas (e.g. what is truth? what is counting? does infinity come in different “sizes”?). Some students feel bewildered when they discover how difficult it can be to “count”.

It is a time-tested favorite of mine to teach them about groups. But unlike a formal course in abstract algebra, I tell them briefly what a group is, how abstract notions can be fun and useful, and after showing them the cyclic groups, both infinite and finite, I proceed fairly quickly to the dihedral groups D₃ and D₄ (of orders 6 and 8 respectively). I construct them as the groups of rigid symmetries (rotations and reflections) of the triangle and square. D₃ is such a revelation since it is the smallest group that happens to be non-commutative; and I usually spend several lectures drawing pictures and discussing composition of symmetries. It is really exciting to reveal to them how “multiplication” is not a universal idea and it can be defined as a non-commutative binary operation. After producing the multiplication tables of both D₃ and D₄ we set out to explore the orders of individual elements as well as the subgroup structures of each one of these groups. We emphasize the subgroups of rotations and reflections.

One of my favorite results in elementary group theory is Lagrange’s theorem: If G is a group of order n and H is a subgroup of order m, them m divides n. I take advantage of the sheer simplicity of this result and “test” it for D₃ and D₄. It is not magic, I tell them, it’s a theorem!… I also exploit Lagrange’s result by bringing the (finite) cyclic groups back and sharing the pleasant fact that the converse of Lagrange is true in that context: if C_n is the finite cyclic group of order n and m is any divisor of n, there exists a subgroup (necessarily cyclic) of C_n whose order is m. And yes, we “test” the truth of this and explore the consequences when n is prime.

Many topics presented in more advanced courses in combinatorics, abstract algebra, logic, etc. can certainly be made accessible to a liberal arts audience. The trick, of course, is to explain the ideas in layman’s terms, progress to some level of formalization, and work out many enlightening examples. Teaching this course has been very satisfactory indeed. My hope is and has always been to leave my students with some long-lasting interest in the ideas behind mathematics, as well as a taste of what mathematicians do.

In a recent New York Times op-ed piece, Angelina Jolie revealed that she had a double mastectomy to reduce her risk of breast cancer.  She had a family history of cancer and tested positive for flaws in the BRCA 1 gene.  She wrote: “My doctors estimated that I had an 87 percent risk of breast cancer …”  Her disclosure was big news, but according to literature on judgement under uncertainty, the statistics she mentioned are likely to be misinterpreted.

While never explicitly stated, the 87 percent risk is presumably the cumulative lifetime risk.  My assumption is based on a Times article published in the following day: “Women who carry BRCA mutations have, on average, about a 65 percent risk of eventually developing breast cancer, as opposed to a risk of about 12 percent for most women…  Ms. Jolie wrote that the estimate for her was 87 percent.”

Twelve percent (for most women) is consistent with the familiar statistic that “one in nine” women would develop breast cancer.  This figure has become a mantra in the popular press and breast cancer screening programs, and has terrified many people.  (See this New York Times Magazine cover article entitled “Our Feel-Good War on Breast Cancer.”)  But what does it mean?  According to an article published in the New England Journal of Medicine, many younger women view this 12-percent statistic as a short-term probability and grossly overestimate the risk of breast cancer in a 10-year period.

Source: K.-A. Phillips, G. Glendon, and J. A. Knight, “Putting the risk of breast cancer in perspective.” New England Journal of Medicine, 340, 141-144 (1999).

Based on the above table, we see that 1+3=4 out of 1,000 women developed breast cancer in their 30s, and 5+8=13 out of 1,000 women during their 40s.  In other words, a woman entering her 30s has a 0.4 percent chance of breast cancer in the next 10 years.  The risk of breast cancer increases with age; a woman entering her 40s has a 1.3 percent chance of the disease in the following decade.  These figures are far less than 12 percent that many people were led to believe.  Breast cancer is much more common at older ages, which makes sense.  Note that cancer incidence and cancer mortality are not the same: 1+2+3+3+3+4+5+6+6=33 out of 1,000, or 3.3 percent women will die of breast cancer by age 85.  But we also see that about six times as many women die of cardiovascular disease.  For a further discussion, Gerd Gigerenzer’s book Calculated Risks is an excellent reference.

The “one in nine” statistic is rarely explained properly, and misinterpretation fuels unnecessary levels of fear.  For this reason, the general public needs to approach Ms. Jolie’s 87 percent figure with caution.  Most importantly, her situation is not representative of what a typical 37-year-old woman would face.

Now with an understanding that 87 percent is more likely to be the cumulative lifetime risk of breast cancer for a woman (with BRCA mutations) who lives past the age of 85, the next question is what action should someone take.  There is no easy answer, as it depends on an individual’s own personal tolerance for risk, and the cost and benefit of mastectomy.  According to a study that followed up 214 women at high risk of breast cancer who had undergone mastectomy at the Mayo Clinic in Minnesota, incidence of breast cancer was reportedly to be reduced by 92 percent.  Again, we need to understand how this number was obtained.  By comparing the 214 women with their sisters who had not undergone mastectomy, about 38 were expected to develop breast cancer, but actually only 3 were actually observed.  In terms of relative risk reduction, , which is very impressive.

However, we can present the exact information in a different way.  Mastectomy prevented breast cancer incidences among 214 women, which also means that women had no benefit from mastectomy.  For some medical professionals, they prefer expressing clinical implications of the findings in terms of “number needed to treat.”  In this case, the number of patients who would need to be treated to prevent a bad outcome is then .  It has been shown that results expressed as the relative risk reduction and those expressed as the number needed to treat have different influences on decisions about treatment.

From the Book of Common Prayer, “we have left undone those things which we ought to have done, and we have done those things which we ought not to have done.”  One is frequently faced with such choices because most actions in our life do not guarantee absolute certainty.  The Mayo Clinic study demonstrates that among the high-risk group, most women (5 in 6) would not develop breast cancer even if they kept their breasts, and a few (1 in 71) would develop breast cancer even if they had their breasts removed.  For educators, I think it is important to ensure that students appreciate the meaning of probability and are able to describe risk in a variety of ways, so as they are empowered to make informed decisions by themselves.

In 2010, two Harvard economists, Carmen Reinhart and Kenneth Rogoff, circulated a paper demonstrating that GDP growth is negatively correlated to public debt (debt-to-GDP ratio to be more precisely).  Their paper was highly influential and has been used to support the global austerity agenda.  For instance, it was cited in the “Paul Ryan Budget,” p. 80.  But the problem is that the paper is rife of basic math errors, as pointed out in a recent paper by Thomas Herndon, Michael Ash and Robert Pollin of the University of Massachusetts, Amherst.  On April 19, Paul Krugman wrote an op-ed piece entitled “The Excel Depression” for the New York Times publicizing an embarrassing Excel error in Reinhart and Rogoff’s spreadsheet.  (You can see the Excel screen capture here; instead of AVERAGE(L30:L49), Reinhart and Rogoff entered AVERAGE(L30:L44), excluding Australia, Austria, Belgium, Canada, and Denmark from their calculation.)  A week later, Reinhart and Rogoff contributed their own op-ed piece; they suggested that their mistake was inconsequential and continued their claim that “growth is about 1 percentage point lower when debt is 90 percent or more of gross domestic product.”  In their online appendix, the chart below was shown to make such a point.

Source: Reinhart and Rogoff’s New York Times online appendix.

You can find extensive discussions over the Reinhart-Rogoff controversy on the internet.  Here I just want to highlight some of the technical issues involving merely elementary arithmetic.  In fact, Thomas Herndon, a PhD candidate, discovered RR’s Excel error through a course assignment.  Students were supposed to replicate the findings of a famous paper in order to learn econometric techniques.  As RR’s work involves essentially calculating the means and medians, Herndon’s professor almost didn’t let him take on the project.  (See this New York Magazine article and this Colbert Report interview.)

A casual reader might get an impression of downward slide in GDP growth based on the above chart, but a careful reader might have noticed that the chart compares medians and arithmetic averages.  Students who have taken an introductory statistics course know that medians and means (often loosely referred to as averages) can be quite different if the distribution is skewed.  Furthermore, means and medians without accompanying measure of how much the data is spread out (such as standard deviation or interquartile range) provide incomplete information.

I wrote to Professor Ash and he kindly supplied additional information.  He and Professor Pollin published a response to Reinhart and Rogoff in the Times, with a comprehensive technical supplement.  They also made the data public to allow a close examination.  To illustrate the effect of Reinhart and Rogoff’s Excel error, one can make a comparison between the New York Times chart and a corrected one based on the inclusion of Australia, Austria, Belgium, Canada, and Denmark.  This inclusion alone increases median GDP growth by 0.3 percentage points in the 90 percent and above public debt/GDP category.

Median figures for blue line are from the NY Times; median figures for red line are from Table 1 of technical supplement by Ash and Pollin, “RR calculation method but with corrected spreadsheet.” The Excel error alone is responsible for a difference of 1.9-1.6=0.3 percentage points in RR’s above 90% debt/GDP category.

It is unclear why RR excluded available data of earlier years for Australia, Canada and New Zealand.  By including these three countries, it further increases median GDP growth in RR’s highest debt/GDP category to 2.5%, which is only 0.4 percentage points lower than that in the next highest debt/GDP category.

Median figures for blue line are from the NY Times; median figures for red line are from Table 1 of technical supplement by Ash and Pollin, “recalculation with both corrected spreadsheet calculations and inclusion of Australia, Canada and New Zealand early years.” The difference in median GDP growth is only 2.5-2.9=-0.4 percentage points between 60 to 90% category and above 90% category.

While most economists acknowledged some correlation between high debt and low GDP growth, the above 2 charts illustrate that the claim “growth is about 1 percentage point lower when debt is 90 percent or more of gross domestic product” by Reinhart and Rogoff is based on sloppy math and is unsubstantiated.

In the Times online appendix, Reinhart and Rogoff stated that they “gave significant weight to the median estimates, precisely because they reduce the problem posed by data outliers.”  Yet two paragraphs later, when reporting their findings in a paper published in 2012 (joined by Vincent Reinhart), they gave the means only.  They compared the mean from 1800 to 2011, 2.3 percent, to the mean of Herndon et al. from 1945 to 2009, 2.2 percent.  If we just cherry pick numbers, let us compare the tainted result that Reinhart and Rogoff published in the Times with means from 2000 to 2009 only.  Contrary to Reinhart and Rogoff’s central claim, GDP growth in the over 90 percent public debt/GDP category has actually outperformed GDP growth in the 60 to 90% public debt/GDP category.

In this chart, the median figures for blue line are again from the NY Times, and the mean figures are from Table 4 of technical supplement by Ash and Pollin. Data from 2000 to 2009 contradict Reinhart and Rogoff’s claim.

But the above chart is not particularly meaningful, because the standard deviation of each GDP growth figure (as shown in Ash and Pollin’s supplement) is at least 0.3.  A more responsible way to present the means is the following graph, taken from their supplement.

Source: technical supplement by Ash and Pollin, Figure 2.

People who have working experience with data know that mean or median conceals a lot of information.  In this case, it is evident that there is a wide range of economic performance outcomes in a given category.  Using the file containing data from 1946 to 2009 (the basis of the New York Times line chart above), one can make a scatterplot.  It is apparent that the relationship between public debt and GDP growth varies significantly.

Source: The file RR-processed.dta posted in U Mass website. Working spreadsheet was provided by RR; the U Mass group corrected errors and cleaned up the data.

By now, you should have realized that the same data can be graphed in different ways, which can lead to a vastly different impression.  The Reinhart-Rogoff affair offers educators a great opportunity to introduce many other quantitative reasoning topics, such as which measures of central tendency to use, correlation does not imply causation (does high debt cause low growth, or the other way around?), and so on.  It is quite feasible to ask students to perform their own analyses and make their own inferences, and I strongly encourage you to do so.

As walking around inside the newly opened National Museum of Mathematics (MoMath), Professor Frank Wang (LaGuardia Community College) and I saw many, many kids, playing, chattering, taking pictures, laughing. In the small two-story museum located right in front of Madison Square Park (the home of Buckyball), everything was colorful, interactive, and looking fun. I caught a group of 8 to 10 children working on (or goofing around with) some sort of a computerized painting activity.

Me: “Are you guys from New York?”

Kids: “(Altogether) Brooklyyyyynnn!”

M: “What grade are you in?”

K: “Fourth and siiiixth!!”

M: “(Hmm. Where are the fifth graders?) How do you like this place?”

K: “Greeaaaaaaat!”

M: “Do you like math?”

K: “YEEEEEEEEEEEESSS!!” 

Wow. Love the enthusiasm. But when did they rehearse these responses? Anyway, I was happy that they said they liked math, and that they seemed to be genuinely enjoying the museum.

It was more like a playground or amusement park than a museum. I had to fight several kids to have my turn to ride a tricycle with square wheels. All these photos can probably tell we were highly engaged there–making tesselations with monkey magnets, becoming a tree, walking and hopping on the discoteque-ish “Math Square,” rolling objects of different shapes to see how they move forward, etc. In between the fun activities, Professor Wang pointed at some of the displays (attractions? rides?), saying, “I can use this for my Calculus 3!” In fact, he reported last week that he had used the picture of a pyramid (below) for one of the course’s exams. The question went:

“The pyramid shown in the picture was built on a square base with sides measuring about 230 m. The height of the pyramid is 146 m. What is the angle indicated in the picture?”

Great! MoMath can be useful for college math teaching!

After the 2 hours of our visit, however, I had to say, “I would give it a C-,” to which Professor Wang responded, “I would give it an INC.” Here is why. First, although it had almost been a month since its grand opening, more than a few attractions were not ready. Each of them had a “Problem Solving in Progress” sign, which was cute at first. But after 5 or 6 or them, I felt increasingly bitter and mistreated, and had to mumble, “Man, I paid $16 to see these cute signs?”

Second, true, it was fun, but it was so unclear how exactly these fun displays are math. We could not find sufficient explanations for any of the exhibits on the interactive touch-screen monitors located nearby, or on the MoMath website. If we cannot connect these activities and the math taught at school, how can we send the “Math is fun” message to children? I could go on and on, but then realized this New York Times article expresses what I wanted to say more beautifully.

Having said all that, I still consider myself a big supporter of MoMath. It is a great idea and the museum has infinite potential. When someone asked me, “How was MoMath?” I would answer, “Don’t go yet. It’s not ready (Obviously I was still bitter about my 16 bucks).” But then, I thought: No. Everyone, Go! Experience the exhibits and give the creators tons of feedback. Tell them to make their website more informative. Tell them to give us deeper mathematical insights. Tell them to show us math can be really fun! Personally, I would love a math-themed cafe on the premises to serve cakes and pies of all sorts of shapes, and mysterious drinks named after mathematicians.

The grades we have given are part of formative assessments. We are so looking forwrad to our next visit.

On a side note: Later that day, I visited the Metropolitan Museum of Art to see a tiny show for European lace. The lace from a couple of centuries ago was so quiet and delicate that it made a great contrast with the MoMath’s vibrantly colored high-tech exhibits; however, it was undeniably math. The show ended last month, but Selected Highlights can be seen online.

A professional mathematician has, through exposure to many trials, become something like a desert creature, capable of supplying “meaning” metabolically by an internal gland rather than imbibing it from without. It can be difficult to diagnose “meaning deprivation” in our more delicate students, who can begin to wilt in what we perceive to be a perfectly cool and saturated environment.

In this post I want to experiment with what I think is a helpful beginning in a characterization of meaning for the purposes of mathematical pedagogy. The definition in question is that “The meaning of a proposition is the means of its verification,” and it occurs in Wittgenstein’s Lectures on the Foundations of Mathematics.

It seems that anything W says captures only clumsily his actual view. I take the above slogan only as a starting point for a discussion.  The views of Wittgenstein relevant to this post are explained better and at greater length in this summary of the Philosophical Investigations.

The Lectures on the Foundations of Mathematics (LFM) was transcribed from classroom discussions that occurred in Wittgenstein’s later period (c. 1939), during his “second coming” at Cambridge.  (In the early lectures, Alan Turing appears as a participant.) In this period, the philosopher was elucidating his ideas on language games and forms of life. A recurring theme in this line of thought is that the meaning of any word (or phrase) is best understood by careful examination of the way in which the word is used. To recruit another slogan: meaning is use.

A corollary of this view is that “essentialist” descriptions of activities and words can lead to confusion if taken too seriously.  The notion of a “game” encompasses many different types of activities:  soccer, chess, fencing, horseback riding, etc.  For any formal definition of what a game should be, it seems possible to find an instance that either shouldn’t be included in the extension of the definition, or else should be included, but isn’t.  Any dictionary definition is fairly easy to “break;”  because of the attention W called to this issue, people have gone to great lengths to perfectly specify what is meant by game. The conclusion we are intended to draw is that the things we call games bear a “family resemblance” to one another, but are not all skewered by any expressible criteria (to a mathematician, this brings to mind the notion of an ultrafilter).  What the word “game” means is not something that can be cashed out in terms of other English words–it is part of our common form of life that we understand the role of the word game in discourse.

Wittgenstein adheres to his anti-essentialist philosophy, emphasizing the role of words in everyday life, even in the context of mathematics.  This naturally creates an interesting tension, because, superficially at least, mathematics is teeming with formal definitions, words with “objective” meaning, and essentialist characterizations.

When we are not in a reflective mood (when we are “just working”) we are happy to say that a theorem is true just if the condition it describes really obtains:  All isosceles triangles have two angles the same, etc.  In this naive mode, which is the mode in which we teach, the meaning of comes from the fact that it describes an objective state of affairs.  We need to contrast this with the “meaning is use” view.  In the “meaning is use” mode, I will try to cash out the meaning of a proposition:

The assertion that my keyboard is 24 inches wide means that I can find other objects, such as measuring tapes, that others also believe are 24 inches, and put them against my keyboard in such a way that myself and others will be satisfied that my claim is true. In contrast to this view of the meaning of the phrase “my keyboard is 24 inches wide,” there is the more metaphysical and more absolute understanding that in some sense my keyboard really is 24 inches, and that my utterance merely calls attention (correctly) to that state of affairs.  From the first point of view, the meaning of the length of my keyboard has a normative component, and arises in a complex way from the role of length, and other notions, in my form of life.    From the second point of view, the length of my keyboard is an empirical question, and empirical data alone determines the truth value of the claim. This way of viewing questions of length might be characterized as being positivist.

To give a more abstract example of “meaning as use”, and an application to mathematics, consider the following statements.

In some sense these claims are all of the same form, namely that of a cardinality claim. But they are all of an extremely different character from one another. I will summarize for each (except the last one!) a corresponding method of verification.

1. This is “immediately clear.”
2.  We count the members of the set one by one until we utter (perhaps silently) the word “ten.”
3. By definition of — this is a tautology.
4. We construct an abstract map (there are many issues buried here).
5. This statement has no known method of verification.

The above indicates that each of the cardinality claims has a different subjective import from the others, but we want to say that there is more involved than a difference of accompanying sensation. The first two claims have “naive” or “empirical” meanings which feel to be in rough agreement with the meaning that derives from their role in language games.  In particular the symbols “two” and “ten” are given more or less their ordinary meaning.

For the latter three statements, this correspondence seems to come apart. There is on the one hand the “naive” understanding that there are two sets that stand in the stated relation to one another. But now (in the previous sentence) my use of the words “are”, “sets” and “stand” have taken on nonstandard (specialized) meanings. They are part of a very different language game.  Verifying that there are even integers is a different sort of practice than counting out five apples.  Some of the ordinary sense of cardinality is preserved in the new game, and some of the sense has changed.  It is not the same “cardinality game” as in the finite case, though it bears a clear resemblance.  With a certain amount of technical training, which emphasizes the use of a “one-to-one” correspondence, these cardinality games can be seen as “the same.”  When we believe that they really are the same, however, W would argue that we have been bewitched by language.

This is an interesting topic and one could go on much further, but to come back to pedagogy: It can arise naturally in a mathematics class that a superficial similarity, perhaps of notation, conceals a deep difference. One can see this in very basic examples such as and

We might produce either of these “off the cuff” as a generic example of a quadratic equation, but the physical act of finding a solution to one or the other is unique in each case. Naturally there is a sense in which they are the same (both are solved by the quadratic formula) but

1. we would not consider a student educated who could not articulate the important differences between the two equations
2. one cannot see the sameness of the two equations without training

History illustrates the second point well: Al-Khwarizmi gives six types of quadratic equation rather than the modern general form of . Expressed in post-cartesian notation, with coefficients assumed to be positive, Al-Khwarizmi’s equations are:

1. Squares are equal to roots ()
2. Squares are equal to numbers ()
3. Roots are equal to numbers ()
4. Squares and roots are equal to numbers ()
5. Squares and numbers are equal to roots ()
6. Roots and numbers are equal to squares ()

Where we (after intense training) see one type of equation, Al-Khwarizmi sees six.

Note that the equation is not given a category, because its solutions are imaginary.

This is a recurring difficulty in pedagogy: From a certain point of view, two things are “the same.” But the meaning of sameness is the technical mathematical notion of equivalence, not the everyday idea of similarity. The meaning of sameness must be illustrated by its use. Seeing things as the same is an achievement that comes from mastery of a certain kind of language. This is a form of language which we are constantly illustrating to our students (while perhaps consciously teaching them something entirely different).

Consider the difficulty of teaching students the meaning (or use) of a formula. I have in the past been surprised that in calculus many students do not understand the meaning of an equation with several terms, such as

They see the rule (if they are trained to perceive the statement as a rule) but they do not know its use.  I find it very challenging to see this difficulty from the students’ point of view.  What doesn’t make sense?  Why are they not satisfied that they understand, if they see the symbols, grasp the corresponding meanings, understand arithmetic and the notion of equality?  What is left to know?

It is easy to brush off these questions–students will get used to the equation.  Indeed they will.  But what changes?  Is their confidence in their own understanding of the concept slowly increased simply by long exposure?

I think this is a situation where Wittgenstein’s theory of meaning can play a useful role.  What the students are missing, in my view, is exactly the use of the formula in our mathematical form of life (from which the meaning of the formula derives).  How is the formula used?  It is used to solve problems about gravitational attraction between two masses.  And what does this consist in?  Well, suppose there are two particles and you know that masses of both, but not the force between them, then…  

The use of the formula is to play a certain kind of computational game called “algebra”, with which we are deeply familiar.  Even if the student knows the algebra game, she will not “understand” the new formula until the symbols in the formula are connected with algebra–connected with what the formula “actually does”.

I have had very good professors in my life who have actually said to the class:  “Why do you not understand?  It is only addition, subtraction, multiplication and division!”

And they were right in a way.  It is very puzzling that students understand arithmetic, but do not understand easily something like a rational function.   Wittgenstein points to a solution to this paradox which I find insightful:  When we (mathematicians) see a rule expressed as an equation, we not only see the rule, but, through our training, know its use.  This knowledge of use (or its absence) in mathematics is what confused students rightly intuit that they are missing.

Wittgenstein is very sensitive to the problem of learning to use a rule. Here is an extended excerpt from LFM:
(Wittgenstein speaking)

Suppose that I write down a row of numbers

and say, “What series is this?” Lewy suddenly answers, “Now I know!”–It came to him in a flash what series it is.

Now what happened when he suddenly understood what series it was? Well, all sorts of things might have happened. For instance, the formula might have come into his mind–or he might have pictured the next number. Or he might have said, “Now I know!” and gone on correctly.

But suppose that the formula had struck him. Does that guarantee that he will go on and continue the series in the right way? Well, in an overwhelming number of cases, yes; he will go on correctly. But now suppose that he goes on all right until 100, and then he writes “20,000″. I should say, “But that is not right. Look, you have not done to 100 the same as you did to 99 and all the previous numbers.” But suppose he stuck to it and said that he had done the same thing with 100 as he had done with 99.

Now what is doing the same with 100?–One might put the point I want to make here by saying, “99 is different from 100 in any case; so how can we tell whether something we do to 99 is the same as something we do with 100?”

…

Yet suppose I say, “You don’t know what it is to do the same to 100 as 99. Well, I’ll show you. This is the same as this.” But he might reply, “Very well, but how am I to apply that definition to this case?”

This passage illustrates that the problem of teaching a rule is bound up in the teaching of the use of the rule. It seems to me that use is often precisely what is missing in a student’s understanding in many practical situations. In the past I have been trying to prove a limit to the class and have given evidence with a graph of the function that the limit is such and such. But frequently this explanation is met with the silence of incomprehension. Obviously the students have not connected the graph of a function with its use. Perhaps they have not even connected the idea of a limit with its use.

To come back to the argument begun above, to understand the statement it is necessary to understand how the statement is verified, be it with a graph or a calculation. The formal definition of a limit is of no use to a student who is unable to ground any of the terms used in the definition, and who moreover has no experience in the *use* of the definition.

Let us consider what implications a Wittgensteinian mathematical philosophy should have for the teaching of mathematics. If we agree with Wittgenstein about meaning as use, the problematics of learning a rule and (something not discussed here) the nature of essential properties, is there a practical way in which pedagogy should change? We will ignore issues of whether W himself would view the content of certain mathematics classes as subject to criticism (as he might if the material uncritically employed some aspects of set theory).

An individual committed to the above view of meaning in mathematics sees the teaching of mathematics in any fashion, even the most formalist, as the teaching of a form of language.  The instructor initiates his or her students into a form of life, and a collection of language games, through some process that we will not attempt to explain, but which certainly involves many demonstrations of the use of technical words and concepts.  (Though we consider the details of the “acculturation” question as beyond the scope of this essay, it is concerned with Wittgenstein’s notion of training, about which much has been written.)

Since “training” is what occurs in math instruction in any case, it is not necessarily helpful to try to do it deliberately.  Any teaching style acculturates the students into ways of doing mathematics, and by facilitating the absorption of these norms, the work of the teacher is done.  This is true if the class is taught in the style of Lakatos or in the laconic tradition of Bourbaki.

But perhaps there is room for a few fine points.  I think one consequence of Wittgenstein’s view is an appreciation of the great deal of time needed to understand mathematics. One can be convinced (wrongly) that he or she is teaching only rules and facts, and that learning facts and rules is easy; facts and rules are comprehended and left behind in short order. But Wittgenstein shows us the same process through the lens of language learning.  We all know that absorbing a language is hard, time consuming, and fraught with a thousand exceptions, idioms and complexities. To learn a language one has to be immersed in it for a long period of time.

If one views mathematics as a language, or even as a culture, or a form of life, then it is clear that very little can be absorbed in a single semester course.  The students in these courses are likely to be bewildered by an onslaught of meaningless symbols and techniques, for which they never learn to associate any use or value.

The necessity for immersion indicated by W’s mathematical theory is one I think we can make some use of at our university.  At CUNY students are frequently challenged in their educational efforts exactly where the issue of immersion is concerned.  Namely, many of our students commute long distances, are unacculturated to mathematics previous to matriculating, and frequently have extra-curricular responsibilities that restrict the time available for academic work.

Beyond immersion, the metaphor of mathematics learning as language learning suggests that a family of techniques found to be useful in the teaching of language may benefit mathematics students as well.  Some things that come to mind are the following:

1. Long interactive days of mathematics instruction
2. Friendly departments, which can serve as hubs for the dissemination of mathematical culture
3. Engagement with those who are already fluent :  mathematics clubs, tutoring, and recitation sections

Exposure in early childhood is another possibly fruitful linkage, but not one that we can make use of at the post-secondary level.

Of course all of these practices are already in place at the best departments and the most math friendly schools.  Perhaps, on campuses where some of the above could be done better, we should make efforts to do so.

Since its birth on February 1, 2012, CUNYMath Blog has now reached the 50^th post!

My 2012 was the Year of Math. I had the pleasure of working with many CUNY math professors and other math-related people, who are so passionate about math and math education. Thank you, Math Blog authors, for your wonderful contributions. Thank you, Math Blog readers, for your direct and indirect comments and thoughts. Thank you, all the professors I have and haven’t met, for being so brilliant and nice (and at times funny and quirky). Thank you, great math-popularizing authors including Steven Strogatz. “The Joy of x” is the first e-book I ever purchased, and I love it. And thank you, MoMath, which I haven’t visited yet. My next post will probably be a MoMath visit report!

So, New Year’s. This is my 13^th in the States but I’m still not used to the idea or practice of having to start to work on January 2^nd. The ever-hardworking Japanese are allowed to be lazy (or so we feel) only during the first three days of the year. And my seasonally lazy Japanese self says: Hey, it’s New Year’s. Stop working and celebrate. Celebrate others’ work.

I surrendered and decided to introduce the following quotes regarding (math) education that I encountered fairly recently. This one is by David McCullough.

“We need to revamp, seriously revamp the teaching of the teachers. I don’t feel that any professional teacher should major in education. They should major in a subject. Know something. The best teachers are those who have the gift, and the energy, and the enthusiasm, to convey their love for science or history or Shakespeare or whatever it is. Show them what you love is the old attitude. We’ve all had them, where they can change your life, they can electrify the morning when they come into the classroom.” (60 Minutes aired on November 11, 2012)

Video:David McCullough

He starts talking about education at the 8:45 mark. Whether or not you agree on his view on teachers, you may find the entire segment delightful.

The following are from “Adventures in teaching: A professor goes to high school to learn about teaching math” by Darryl Yong (Notices of the American Mathematical Society, 2012, Volume 59, Number 10).

“I have won teaching awards at the institutions where I’ve worked, but I intentionally held low expectations for my effectiveness as a high school teacher. Even so, I felt depressingly ineffective as a teacher most of that year. While it’s not wise to generalize from a single case, my experience shows that having strong content knowledge in one’s field is a necessary but insufficient condition for student learning to take place.” (p. 1410)

“I learned that, regardless of how “tough” some students are or how weak their math skills are, teenagers still love feeling successful when they become good at something or when they figure something out… I found that 95 percent of the cases when one of my students was disruptive or seemed disinterested in learning were the result of the student not understanding what to do or how to do something.” (p. 1411-1412)

“I initially spent a great deal of time thinking of fun or creative lessons that would get students excited. These lessons rarely worked because they were often too complicated or inappropriate for my students’ mathematics development. Instead, I began to design my lessons and accompanying student work so that (1) all of my students could successfully complete the first problem or task independently, and in which (2) the sequence of problems/tasks matched my students’ tolerance for challenge and self-concept. This strategy not only increased student learning but also eliminated most of the discipline issues in my class and relieved the pressure of having to develop whiz-band “fun” lessons every day.” (p. 1412)

This great article was introduced to me by Professor Hunter Johnson, who is also a Math Blog author. This program of Visiting Faculty Permit in California sounds just so valuable. Does anyone know whether New York has something similar?

A meeting prolonged, trains delayed, and I ended up being 30 minutes late to arrive at the meeting spot in Brooklyn Heights. Waiting for me was Professor Janet Liou-Mark, waving and all smiley… outside in the rain! Professor of Mathematics and the Honors Scholars Program Director at New York City College of Technology, she has been awarded a number of research grants and scholarships. Professor Liou-Mark is described as having “such an inspirational attitude towards teaching and learning mathematics,” by one of her colleagues. Let me add to that description, “… and so, very, utterly kind.” Right. Brilliance and kindness are not necessarily incompatible.

As customary, I’d like to start by asking about your childhood. Were you born in New York?

No, I was born in Manitoba, Canada. It is where my father completed his doctorate degree. Then we moved to Brooklyn, where I spent my childhood. My father has been a professor of physics at Brooklyn College for 42 years, still teaching and very active in his research. My mother was a master tutor in mathematics at Kingsborough Community College for 15 years. She decided to retire to take care of my children when I joined City Tech. I liked mathematics and was expected to do well in it when I was a child, and had the full support from my parents necessary to succeed. 

You’re a CUNY kid in the true sense! So, choosing a career as a mathematics professor was pretty straightforward for you?

It was not straightforward. When I went to New York University, I was unsure what I wanted to major in.  My parents encouraged me to continue taking mathematics courses even when I told them I wanted to be pre-med.  After an incident, I discovered that I could not withstand the sight of blood. This happened during my sophomore year and it ended my pursuit to be a medical doctor.  At that point in time I had already completed many of the mathematics requirements, so naturally I decided to declare mathematics as my major. During the summer before my junior year, I was fortunate to get an internship position—a job for graduate students enrolled in MBA programs. My supervisor had asked me to continue during the academic year.  Since it was expected for me to pursue a graduate degree, I wanted to see if the business world was for me. After working there for a year, it became clear that the business industry was not the environment I wanted to be in.  When the prospects of getting a doctorate in mathematics education at NYU were brought to my attention, I knew immediately that this degree is where my passion for mathematics and teaching can be combined.    

Some parallels between your story and Professor Kathleen Offenholley’s are so intriguing… Please tell us your teaching philosophy.

My teaching methodology and philosophy are mainly shaped by my parent’s influences.  The passion for teaching mathematics came from my mother who received her Master’s degree in mathematics from Brooklyn College.  She taught me that students need to be independent thinkers, the importance of practice to gain mastery, and the necessity to show your passion in teaching mathematics so others can be influenced by it. She is truly a remarkable woman, and her patience can be compared to a saint.  She can be explaining the same mathematics problem over and over again, and her soft-spoken voice would not have even a trace of impatience.  My father taught me how to be a great teacher.  His meticulous and detailed notes depicted a teaching philosophy which is to never assume that the students have the sufficient background knowledge – spending a few minutes reviewing concepts will increase retention and persistence especially in mathematics and science courses. 

How you describe your parents reminds me of “grit,” one of the hot topics in education nowadays.  More and more people believe that educators and parents should nurture children’s persistence and resilience to set them up for their long-term success, rather than focusing solely on the children’s current academic performance and test scores. Through modeling, your parents seem to have taught you how to be gritty, and how to teach students to be gritty. Now, you have a 16-year-old son and a 14-year-old daughter. How would you describe yourself as a parent?

I try not to become the typical Asian parent, or one of those “tiger moms”! I believe that children need to be problem solvers and critical thinkers. They should be encouraged to explore, to make mistakes, to find solutions, and to use their creativity.  I just tell them they have to try their best and with the knowledge that they gain, they need to pass it on to others, helping and assisting others as much as to their ability. My children are very talented in science and mathematics, but I will get upset when they do not do homework, though. When they get lazy, that is when I am on their case!

Speaking of tiger moms, I heard that your doctoral dissertation was trying to disprove the Model Minority Myth regarding the stereotype, “All Asians are good in math.” I re-read Chapter 8 of Malcolm Gladwell’s Outliers last night (which talks about Asians’ mathematics achievement), and as an Asian, I’m VERY interested to learn what you found.

That chapter of Outliers describes a study that used the Trends in International Mathematics and Science Study (TIMSS), and I used a similar data set from the National Education Longitudinal Study of 1988 for my dissertation on exploring the mathematics achievement patterns of Asian-Americans. Growing up, I noticedthat not all Asians are gifted in mathematics, but they were expected to do well, and they did with help from their teachers and parents. I was compelled to disaggregate the Asian-Americans in the study because many research studies would aggregate them as one group, generalizing their findings.  Important factors explaining mathematics achievement among the different Asian groups would be overlooked.  From my study I found not all Asian-American students perform alike.  When taking into account gender, generational status, home language, perceived parental involvement, reading achievement, self-attribution, and socioeconomic status, the mathematics performance from each group varied with each variable. Chinese-Americans and Korean-Americans performed better than Filipino-Americans, Japanese-Americans, and Southeast Asian-Americans. One explanation for this achievement was how students perceived their parent’s involvement in their success.  This research reinforces the fact that is important for parents of all ethnicity to have vested interest and to be actively involved in their children’s education, especially in mathematics.

Children’s academic successes, regardless of their race or ethnicity, require emotional and tangible support from adults. Now, let’s talk about one of your current projects. The Peer Assisted Learning Project won you the 2011 Chancellor’s Award for Excellence in Undergraduate Mathematics Instruction.

The grave results of pass rates in mathematics motivated me to create a program to assist our students from two fronts: supporting students in their mathematics courses and providing leadership development for STEM students (notably minority or at-risk). Many of these students enter college academically underprepared and lacking the necessary skills required for success. The Peer Assisted Learning Project is a modified version of the Peer-Led Team Learning (PLTL) instructional model where peers facilitate a group to work collaboratively.  I was introduced to this model by Dr. Dreyfuss who was at that time the project manager for the PLTL Dissemination grant at the City College of New York, and she now is leading this project with me at City Tech. By providing them with a culture of collaboration and by prompting them to integrate and transfer their knowledge, these students are better positioned to succeed in their academic life, especially in the STEM disciplines.

I understand you have many other projects going on as well… What are your favorite non-math-related activities? What do you do in your free time?

I do not really have any free time.  Aside from my full-time job, being a mother is also a full-time commitment.  I do love to scrapbook, and I have not done so in the past five years since I became involved with grants at the local and national level. 

A recurring theme in our conversation was “fulfilling others’ expectations.” If her parents’ high expectation was one of the driving forces for Professor Liou-Mark to do well academically, that is consistent with the outcomes of many psychological research studies. Phenomena discovered in those studies including stereotype threat (one of the most recent publications on this is Claude M. Steele’s Whistling Vivaldi) show that (perceived) others’ expectations could affect individuals’ behavior and performance; specifically, high expectations derive high performance and low expectations derive low performance. But then, just last week, I read an article about a professional football player. When he was in college, he told one of his professors that he wants to play in the NFL. The quarterback recalls that the professor laughed condescendingly and told him he would “never make it,” and he used her remarks as motivation to achieve his goal. The article seems to hint why the professor’s low expectation led to his high performance, but should this case be viewed as an exception? Haven’t all of us worked hard to defy others’ low expectations and to show them “Who’s laughing now”? Obviously, my interview with Professor Liou-Mark was so stimulating, and now I’m surrounded by my old social psychology textbooks.      

(This interview took place on October 2, 2012)

There is a concept I find very hopeful in contemporary mathematics education, the concept of Learning Trajectory (LT), whose simplest definition is derived from (Clements and Sarama, 2009): LT has a mathematical goal, a developmental progression, or a learning path to reach it, and the instruction that helps to move along that path.(p.17). The very idea of an LT in mathematics recalls the best, in my opinion, American pedagogical approach of the Texan Discovery Method created by Prof. R.L.Moore from Chicago School of mathematics of early nineteen hundreds. Each course of calculus, topology or any other mathematical subject designed through that method was constructed along a learning trajectory specified by the collection of problems and axioms, e.g Calculus 1 in 142 problems and axioms.

The main strength and virtue of learning trajectories lies in the possibility to design instruction both in agreement with the mathematical development of concepts and with student learning pathways.  They offer to the instructor a long range view upon mathematics involved and upon student learning, which can be utilized for cyclical 9from semester to semester) design of instruction. The art of design learning trajectories lies in creation of appropriate cognitive distance between two consecutive problems along the trajectory – they must be challenging but possible for students to deal with. The previous post On Understanding at https://cunymathblog.commons.gc.cuny.edu/2012/09/18/on-understanding-in-mathematics/  includes an excellent performance task from DOE NYC Mathematics Task Force, The Fir Tree designed along the learning trajectory with those characteristics. Its LT design made it an excellent measurement of understanding for the concept of a Variable.

At the same time teaching along a hypothetical learning trajectory gives a teacher sense of direction and increases awareness of learning process. So, with the introduction of Common Core Mathematics standards which employ, in its design, the learning trajectory approach in the service of understanding, there is hope and chance for success in Math Ed.

However, the teaching-research in the community colleges of the Bronx has suggested that student success, depends not only on its cognitive dimension, but in equal measure on affective and self-regulatory dimensions. Our colleagues in the Bronx CC pursue the path of mathematical creativity as the source of motivation for to enjoyment and to learning mathematics (described on the previous post On Creativity @ https://cunymathblog.commons.gc.cuny.edu/2012/09/27/on-creativity-in-mathematics/ ). The question of facilitation of the “self-regulatory practices” that is of “ how to study” for our college freshman, in for example, remedial algebra is still a mystery to me.