# College Math Teaching

## September 14, 2013

### Reality of modern college teaching: students with Asperger’s syndrome

Filed under: academia, mathematics education, student learning — Tags: — collegemathteaching @ 4:43 pm

One of the major changes I’ve encountered since I started college teaching (first as a teaching assistant in 1986; then as a new professor in 1991) is that students with Asperger’s syndrome have been showing up.

Most of the time, it isn’t a big deal; the worst I’ve had is one of these students became completely disoriented when he got to class and someone was sitting in “his” seat (no, I don’t make seat assignments; this is college).

This semester, I have a transfer student (not sure why he transferred); in spots he is “disruptive to a minor degree”: you have to remind him that there are 34 OTHER students in the class; this isn’t a one-on-one dialogue just for him.

Also, I sometimes make side remarks (to explain a point to another student) and use analogies; that just confuses the heck out of him. But I am not going to stop being effective with the other 34 students just for him; I just tell him “see me in office hours” or “don’t worry about this”.

On the other hand, he is relatively easy to work with in office hours; the one-on-one exchanges are usually reasonable and pleasant.

Hence, when I see he is getting confused, I tell him “for this point, see me for office hours.”

I’ve searched the internet to see what is out there; most of it is what I already know and much of it is a series of tired cliches, finger wagging, etc. I haven’t found much of the following: “I had these issues in my calculus class; here is how they were resolved” or “these issues COULDN’T be resolved.” Sometimes they aren’t up to the task of being in college.

But, overall, it seems to be this way: we are told to be “more productive” which means more students per semester (105 students in 2 sections of calculus and 1 of differential equations). So no, one cannot tailor lessons and work to the learning style of a specific student, especially if that student is an outlier. One has to teach to a type of average or to the class as a whole; one can adjust for a class full of, say biology students, or one full of engineers or one full of computer science majors.

These students require time, more attention and resources and these COST MONEY. This is where some of the increased educational expense is coming from (some from technology as well). At times, it appears as if colleges and universities are being tugged in different directions.

## February 8, 2013

### Issues in the News…

First of all, I’d like to make it clear that I am unqualified to talk about teaching mathematics at the junior high and high school level. I am qualified to make comments on what sorts of skills the students bring with them to college.

But I am interested in issues affecting mathematics education and so will mention a couple of them.

1. California is moving away from having all 8’th graders take “algebra 1”. Note: I was in 8’th grade from 1972-1973. Our school was undergoing an experiment to see if 8’th graders could learn algebra 1. Being new to the school, I was put into the regular math class, but was quickly switched into the lone section of algebra 1. The point: it wasn’t considered “standard for everyone.”

My “off the cuff” remarks: I know that students mature at different rates and wonder if most are ready for the challenge by the 8’th grade. I also wonder about “regression to the mean” effects of having everyone take algebra 1; does that force the teacher to water down the course?

By Drew Appleby

I read Epstein School head Stan Beiner’s guest column on what kids really need to know for college with great interest because one of the main goals of my 40-years as a college professor was to help my students make a successful transition from high school to college.

I taught thousands of freshmen in Introductory Psychology classes and Freshman Learning Communities, and I was constantly amazed by how many of them suffered from a severe case of “culture shock” when they moved from high school to college.

I used one of my assignments to identify these cultural differences by asking my students to create suggestions they would like to give their former high school teachers to help them better prepare their students for college. A content analysis of the results produced the following six suggestion summaries.

The underlying theme in all these suggestions is that my students firmly believed they would have been better prepared for college if their high school teachers had provided them with more opportunities to behave in the responsible ways that are required for success in higher education […]

You can surf to the article to read the suggestions. They are not surprising; they boil down to “be harder on us and hold us accountable.” (duh). But what is more interesting, to me, is some of the comments left by the high school teachers:

“I have tried to hold students accountable, give them an assignment with a due date and expect it turned in. When I gave them failing grades, I was told my teaching was flawed and needed professional development. The idea that the students were the problem is/was anathema to the administration.”

“hahahaha!! Hold the kids responsible and you will get into trouble! I worked at one school where we had to submit a written “game plan” of what WE were going to do to help failing students. Most teachers just passed them…it was easier. See what SGA teacher wrote earlier….that is the reality of most high school teachers.”

“Pressure on taechers from parents and administrators to “cut the kid a break” is intense! Go along to get along. That’s the philosophy of public education in Georgia.”

“It was the same when I was in college during the 80’s. Hindsight makes you wished you would have pushed yourself harder. Students and parents need to look at themselves for making excuses while in high school. One thing you forget. College is a choice, high school is not. the College mindset is do what is asked or find yourself another career path. High school, do it or not, there is a seat in the class for you tomorrow. It is harder to commit to anything, student or adult, if the rewards or consequences are superficial. Making you attend school has it advantages for society and it disadvantages.”

My two cents: it appears to me that too many of the high schools are adopting “the customer is always right” attitude with the student and their parents being “the customer”. I think that is the wrong approach. The “customer” is society, as a whole. After all, public schools are funded by everyone’s tax dollars, and not just the tax dollars of those who have kids attending the school. Sometimes, educating the student means telling them things that they don’t want to hear, making them do things that they don’t want to do, and standing up to the helicopter parents. But, who will stand up for the teachers when they do this?

Note: if you google “education then and now” (search for images) you’ll find the above cartoons translated into different languages. Evidently, the US isn’t alone.

Statistics Education
Attaining statistical literacy can be hard work. But this is work that has a large pay off.
Here is an editorial by David Brooks about how statistics can help you “unlearn” the stuff that “you know is true”, but isn’t.

This New England Journal of Medicine article takes a look at well known “factoids” about obesity, and how many of them don’t stand up to statistical scrutiny. (note: the article is behind a paywall, but if you are university faculty, you probably have access to the article via your library.

And of course, there was the 2012 general election. The pundits just “knew” that the election was going to be close; those who were statistically literate knew otherwise.

## December 13, 2012

### Domains and Anti Derivatives (Indefinite Integration)

Grading student exams sometimes inspires me to revisit elementary topics. For example, I recently spoke about some unusual (but mostly correct) integration techniques used by students on a final exam.

I’ll recap (and adjust the example slightly): on a recent exam, a student encountered $\int \frac{2}{1-x^2} dx$. I had expected the student to use the usual partial fractions expansion to obtain $\int \frac{1}{1+x} dx + \int \frac{1}{1-x} dx = ln|1+x| - ln|1-x| + C$ which is valid when $x \ne \pm 1$. I admit to being a bad professor and not being picky about domains.

But one student noticed the $1 - x^2$ in the denominator of the fraction and so used the trig substitution $x = sin(\theta), dx = cos(\theta) d\theta$ which leads to the following integral: $\int \frac{2}{cos(\theta)} d\theta = 2ln|sec(\theta) + tan(\theta)| + C$ which leads to $2ln|\frac{1}{\sqrt{1-x^2}} + \frac{x}{\sqrt{1-x^2}}| + C = 2ln|\frac{1+x}{\sqrt{1-x^2}}| = ln|1+x| - ln|1-x| + C$ for $x \in (-1,1)$. Note that, strictly speaking, the “final answer” is really defined for all $x \ne \pm 1$ though the equalities do not hold outside of the domain for $x$ used in the original trig substitution.

And yes, I was a bad professor; I gave full credit to this answer even though we “lost domain” during the string of equalities.

But that got me to wondering: is there a trig substitution that works for $|x| > 1$? Answer: of course:

$\int \frac{2}{1-x^2} dx = -\int \frac{2}{x^2 -1} dx$. Now use $x = sec(\theta), dx = sec(\theta) tan(\theta) d\theta$ which leads to $-2\int csc(\theta) d\theta = 2ln|csc(\theta) + cot(\theta)| + C = 2ln|\frac{x}{\sqrt{x^2-1}} + \frac{1}{\sqrt{x^2 -1}}| + C$ which leads us to our ultimate solution for $|x| > 1$

So, if one REALLY wanted to use trig substituions for this problem, one could and do it in a way to cover the entire domain.

But…as our existence and uniqueness theorems imply, once we get a candidate for an anti-derivative that “works” or the domain, it really doesn’t matter if we did “illegal” steps to get it; we need only show that it is an anti derivative and is valid for the entire domain for the integrand.

Now if one wants a more detailed discussion on domain issues for anti-derivatives, I can recommend the article The Importance of Being Continuous by D. J. JEFFREY which appeared in Mathematics Magazine, Vol 67, pp 294 – 300. (reprint can be found here, scroll down a bit; this mathematician has written quite a bit!). Note: I can recommend this little paper as it talks about the domains of the anti derivatives themselves and not just the domains assumed in doing the calculations along the way or the domains of validity of the substitutions. Note: integral tables and computer algebra systems don’t always give the anti derivative with the “largest” possible domain. One has to watch for that.

## December 4, 2012

### Teaching Linear Regression and ANOVA: using “cooked” data with Excel

During the linear regression section of our statistics course, we do examples with spreadsheets. Many spreadsheets have data processing packages that will do linear regression and provide output which includes things such as confidence intervals for the regression coefficients, the $r, r^2$ values, and an ANOVA table. I sometimes use this output as motivation to plunge into the study of ANOVA (analysis of variance) and have found that “cooked” linear regression examples to be effective teaching tools.

The purpose of this note is NOT to provide an introduction to the type of ANOVA that is used in linear regression (one can find a brief introduction here or, of course, in most statistics textbooks) but to show a simple example using the “random number generation” features in the Excel (with the data analysis pack loaded into it).

I’ll provide some screen shots to show what I did.

If you are familiar with Excel (or spread sheets in general), this note will be too slow-paced for you.

Brief Background (informal)

I’ll start the “ANOVA for regression” example with a brief discussion of what we are looking for: suppose we have some data which can be thought of as a set of $n$ points in the plane $(x_i, y_i).$ Of course the set of $y$ values has a variance which is calculated as $\frac{1}{n-1} \sum^n_{i=1}(y_i - \bar{y})^2 = \frac{1}{n-1}SS$

It turns out that the “sum of squares” $SS = \sum^n_{i=1} (y_i - \hat{y_i})^2 + \sum^n_{i=1}(\hat{y_i} - \bar{y})^2$ where the first term is called “sum of squares error” and the second term is called “sum of squares regression”; or: SS = SSE + SSR. Here is an informal way of thinking about this: SS is what you use to calculate the “sample variation” of the y values (one divides this term by “n-1” ). This “grand total” can be broken into two parts: the first part is the difference between the actual y values and the y values predicted by the regression line. The second is the difference between the predicted y values (from the regression) and the average y value. Now imagine if the regression slope term $\beta_1$ was equal to zero; then the SSE term would be, in effect, the SS term and the second term SSR would be, in effect, zero ($\bar{y} - \bar{y}$). If we denote the standard deviation of the y’s by $\sigma$ then $\frac{SSR/\sigma}{SSE/((n-2)\sigma}$ is a ratio of chi-square distributions and is therefore $F$ with 1 numerator and $n-2$ denominator degrees of freedom. If $\beta_1 = 0$ or was not statistically significant, we’d expect the ratio to be small.

For example: if the regression line fit the data perfectly, the SSE terms would be zero and the SSR term would equal the SS term as the predicted y values would be the y values. Hence the ratio of (SSR/constant) over (SSE/constant) would be infinite.

That is, the ratio that we use roughly measures the percentage of variation of the y values that comes from the regression line verses the percentage that comes from the error from the regression line. Note that it is customary to denote SSE/(n-2) by MSE and SSR/1 by MSR. (Mean Square Error, Mean Square Regression).

The smaller the numerator relative to the denominator the less that the regression explains.

The following examples using Excel spread sheets are designed to demonstrate these concepts.

The examples are as follows:

Example one: a perfect regression line with “perfect” normally distributed residuals (remember that the usual hypothesis test on the regression coefficients depend on the residuals being normally distributed).

Example two: a regression line in which the y-values have a uniform distribution (and are not really related to the x-values at all).

Examples three and four: show what happens when the regression line is “perfect” and the residuals are normally distributed, but have greater standard deviations than they do in Example One.

First, I created some x values and then came up with the line $y = 4 + 5x$. I then used the formula bar as shown to create that “perfect line” of data in the column called “fake” as shown. Excel allows one to copy and paste formulas such as these.

This is the result after copying:

Now we need to add some residuals to give us a non-zero SSE. This is where the “random number generation” feature comes in handy. One goes to the data tag and then to “data analysis”

and clicks on “random number generation”:

This gives you a dialogue box. I selected “normal distribution”; then I selected “0” of the mean and “1” for the standard deviation. Note: the assumption underlying the confidence interval calculation for the regression parameter confidence intervals is that the residuals are normally distributed and have an expected value of zero.

I selected a column for output (as many rows as x-values) which yields a column:

Now we add the random numbers to the column “fake” to get a simulated set of y values:

That yields the column Y as shown in this next screenshot. Also, I used the random number generator to generate random numbers in another column; this time I used the uniform distribution on [0,54]; I wanted the “random set of potential y values” to have roughly the same range as the “fake data” y-values.

Y holds the “non-random” fake data and YR holds the data for the “Y’s really are randomly distributed” example.

I then decided to generate two more “linear” sets of data; in these cases I used the random number generator to generate normal residuals of larger standard deviation and then create Y data to use as a data set; the columns or residuals are labeled “mres” and “lres” and the columns of new data are labeled YN and YVN.

Note: in the “linear trend data” I added the random numbers to the exact linear model y’s labeled “fake” to get the y’s to represent data; in the “random-no-linear-trend” data column I used the random number generator to generate the y values themselves.

Now it is time to run the regression package itself. In Excel, simple linear regression is easy. Just go to the data analysis tab and click, then click “regression”:

This gives a dialogue box. Be sure to tell the routine that you have “headers” to your columns of numbers (non-numeric descriptions of the columns) and note that you can select confidence intervals for your regression parameters. There are other things you can do as well.

You can select where the output goes. I selected a new data sheet.

Note the output: the $r$ value is very close to 1, the p-values for the regression coefficients are small and the calculated regression line (to generate the $\hat{y_i}'s$ is:
$y = 3.70 + 5.01x$. Also note the ANOVA table: the SSR (sum squares regression) is very, very large compared to the SSE (sum squares residuals), as expected. The variance in y values is almost completely explained by the variance in the y values from the regression line. Hence we obtain an obscenely large F value; we easily reject the null hypothesis (that $\beta_1 = 0$).

This is what a plot of the calculated regression line with the “fake data” looks like:

Yes, this is unrealistic, but this is designed to demonstrate a concept. Now let’s look at the regression output for the “uniform y values” (y values generated at random from a uniform distribution of roughly the same range as the “regression” y-values):

Note: $r^2$ is nearly zero, we fail to reject the null hypothesis that $\beta_1 = 0$ and note how the SSE is roughly equal to the SS; the reason, of course, is that the regression line is close to $y = \bar{y}$. The calculated $F$ value is well inside the “fail to reject” range, as expected.

A plot looks like:

The next two examples show what happens when one “cooks” up a regression line with residuals that are normally distributed, have mean equal to zero, but have larger standard deviations. Watch how the $r$ values change, as well as how the SSR and SSE values change. Note how the routine fails to come up with a statistically significant estimate for the “constant” part of the regression line but the slope coefficient is handled easily. This demonstrates the effect of residuals with larger standard deviations.

## December 1, 2012

### One challenge of teaching “brief calculus” (“business calculus”, “applied calculus”, etc.)

Today’s exam covered elementary integrals and partial derivatives; in our course we usually mention two variable functions and show how to calculate some “easy” partial derivatives.

So today’s exam saw a D/F student show up late (as usual); keep in mind this is an 8 am class (no class prior to it). He, as usual, got little or nothing correct. Of course we had the usual $\int \frac{1}{x^2} dx = ln(x^x) + C, \int^1_0 3e^{5x}dx = (15e^5 -15) + C$, etc.

But there was this too: note that we had barely discussed partial derivatives and how to calculate them “by the formula”. But I did give the following bonus question: “is it possible to have a function $f(x,y)$ where $f_x = x^3 + y^3$ and $f_y = 3xy$? Yes, this is a common question in multivariable calculus (e. g., “is this vector field conservative?”) but remember this is a “brief calculus” course.

A few students took the challenge; some computed $\int(x^3 + y^3)dx = \frac{x^4}{4}+ xy^3 + C, \int (3xy^2)dy = \frac{3}{2}xy^2+C$ and noted that the two functions cannot be made to match (I didn’t expect them to recognize that functions of one variable alone represents constants of integration). Some took the second partials and noted $f_{xy} = 3y^2, f_{yx} = 3y$ and that these don’t match. Again, this was NOT a problem that we practiced.

Another instance: given the ideal gas law $PV = nRT$ I challenged them to show $\frac{\partial P}{\partial V}\frac{\partial V}{\partial T}\frac{\partial T}{\partial P} = -1$ and someone got it!

Bottom line: in one course, we have some bright, interested students who enjoy thinking and we have some who either don’t or can’t. This makes teaching difficult; if one tries to “teach to the mean” one is teaching to the empty set. It is almost: either bore half the class, or blow away half the class.

## September 11, 2012

### Two Media Articles: topology and vector fields, and political polls

Topology, vector fields and indexes

This first article appeared in the New York Times. It talks about vector fields and topology, and uses finger prints as an example of a foliation derived from the flow of a vector field on a smooth surface.

Here is a figure from the article in which Steven Strogatz discusses the index of a vector field singularity:

Note: the author of the quoted article made a welcome correction:

small point that I finessed in the article, and maybe shouldn’t have: it’s about orientation fields (sometimes called line fields or director fields), not vector fields. Think of the elements as undirected vectors (ie., the ridges don’t have arrows on them). The singularities for orientation fields are different from those for vector fields. You can’t have a triradius in a continuous vector field, for example.

Comment by Steven Strogatz

Our local paper had a nice piece by Brian Gaines on political polls. Of interest to statistics students is the following:

1. Pay little attention to “point estimates.”

Suppose a poll finds that Candidate X leads Y, 52 percent to 48 percent. Those estimates come with a margin of error, usually reported as plus or minus three or four percentage points. It is tempting to ignore this complication, and read 52 to 48 as a small lead, but the appropriate conclusion is “too close to call.”

2. Even taking the margins of error into account does not guarantee accurate estimates.

For example, 52 percent +/- 4 percent represents an interval of 48 to 56 percent. Are we positive that the true percentage planning to vote for X is in that range? No. When we measure the attitudes of millions by contacting only hundreds, there is no escaping uncertainty. Usually, we compute intervals that will be wrong five times out of 100, simply by chance.

Note: a consistent lead of 4 points is significant, but doesn’t mean much for an isolated poll.

## February 16, 2012

### The “equals” sign: identities, equations to be solved and all that…

Here is the sort of thing that got me thinking about this topic: a colleague had a student complain about how one of her quiz problems was scored. The problem stated: “show that $\sqrt{2} + \sqrt{3} \neq \sqrt{5}$“. She was offended that her saying “$\sqrt{x} + \sqrt{y} \neq \sqrt{x+y}$” wasn’t enough to receive credit and would NOT take his word for it. In fact, she took this to the student ombudsman!!!

But that raised the question: “what do we mean when we tell our students “$\sqrt{x} + \sqrt{y} \neq \sqrt{x+y}$“?

Of course, there are some central issues here. The first issues is that our “sure of herself” student thought that “$\sqrt{x} + \sqrt{y} \neq \sqrt{x+y}$” meant that this relation is NEVER true for any choice of $x, y$, which of course, is false (e. g. let $y = 0$ and $x \ge 0$.) In fact, $\sqrt{x} + \sqrt{y} \neq \sqrt{x+y}$ is the logical negation of the statement $\sqrt{x} + \sqrt{y} = \sqrt{x+y}$; the latter means that “this statement is true for ALL $x, y$ and its negation means “there is at least one choice of $x, y$ for which the statement is not true. “Equal” and “not equal” are not symmetric states when it comes identities, which can be thought of as elements in the vector space of functions.

So, $\sqrt{x} + \sqrt{y} \neq \sqrt{x+y}$ means that $\sqrt{x} + \sqrt{y}$ and $\sqrt{x+y}$ are not equal in function space, though they might evaluate to the same number for certain choices in the domain.

So, what is the big deal?

Well, what about equations such as $x^2 + 3x + 2 = 0$ or $y^{\prime \prime} + y = 0$?
These are NOT equalities in the space of functions; the first means “what values in the domain does $f^{-1}(0)$ take given $f(x)=x^2 + 3x + 2$ and the second asks one to find the inverse image of 0 for the operator $D^2+1$ where the domain is the set of all, say, twice differentiable functions.

But, but…would the average undergraduate student understand ANY of this? My experience tells me “no”; hence I intentionally allow for this vagueness and only address it as I need to.

## May 16, 2011

### Teaching Mathematics in this era: The Helicopter Parent

Filed under: calculus, mathematical ability, mathematics education, student learning — collegemathteaching @ 12:45 pm

Mathematics Education
I was somewhat taken aback at this Daily Kos diary:

I was asked how my daughter is doing. She has 16 credits to go to her bachelor’s degree but it might as well be 2 more years. She’s up in Michigan licking her math wounds. She’s trying to gear up for what she has to do to complete her degree. She had her math ass kicked this last term – again. My daughter, the high school mathlete can’t hack collegiate Calculus II, Differential Equations or Multivariate Analysis (aka Calculus III on steroids) well enough to get a 73.5% or higher which is a requirement for her Physics B.A. degree. She can get 70% and 71%, but not the 73.5% deemed necessary. I mentioned that our current strategy is to find an instructor who’s actually teaching these classes in another school in town (we have 6 major colleges and universities in town to choose from) and transfer in the credits so she can get her degree.

Well, that statement set the mouse amongst the pooties.

How could I say that! Was I saying the instructors weren’t teaching! (That would be sacrilegious in this group.) What’s strange is that I wasn’t criticizing Chibi’s elementary, middle or high school math experience. Her high school math classes were the last classes where actual math instruction occurred. We complain about education and math education in particular, but Chibi got decent math education through to her Senior year in high school. She did well there. It’s university math outside the physics department that’s giving her trouble. She aces all her Physics & Calculus classes that require her to calculate vortexes, how moisture flows through tail pipes, predicting the sizes of hail stones or whatever; these aren’t difficult for her. It’s the math for math’s sake classes, where there’s a bare equation with no real world application or context for solving it that kicks her butt. The lack of practical application simply stymies her mathlete abilities. […]

One thing is for certain, as long as we continue to teach collegiate math using 100 year old methods, the U.S. math competencies will remain where they are.

So do you get this: his daughter did well in high school mathematics but didn’t do well in college mathematics, so it must be the fault of the mathematics professors….ALL of them. Then he blames our position with respect to the world in mathematics education on the university mathematics faculty.

Let’s examine this for a bit:

The disappointing performance of U.S. teenagers in math and science on an international exam, in scores released yesterday, has sparked calls for improvement in public schools to help the country keep pace in the global economy.

The scores from the 2006 Program for International Student Assessment showed that U.S. 15-year-olds trailed their peers from many industrialized countries. The average science score of U.S. students lagged behind those in 16 of 30 countries in the Organization for Economic Cooperation and Development, a Paris-based group that represents the world’s richest countries. The U.S. students were further behind in math, trailing counterparts in 23 countries.

The gap is already there by the time the kids are 15 years old….and exactly how is this the fault of university mathematics faculty?
Remember, his complaint was that his daughter got good grades in high school mathematics but didn’t get them in college mathematics…in several classes. What was the constant there? It was his daughter, of course.

In short, this person makes a sweeping claim based on the lack of success of HIS KID. THIS is part of the “helicopter parent” era.

I’ll add a few thoughts from my experience: in high school, I made A’s in foreign language but had to work to earn a C in my junior level class in college. The professor was excellent; the material was just difficult for me. Even in mathematics: I did well in analysis, algebra, topology and ok in complex variables. I struggled badly in numerical analysis. Yes, my numerical analysis professor…was quite good and I said so on the student evaluations. I just didn’t do well IN THAT CLASS. It just took a long time for that material to make sense to me.

I am not saying that there aren’t some horrible mathematics instructors: there are. That is unfortunate, but there are there. But that is not an excuse for repeated poor performance in classes.

Upshot: there are some parents who will never admit that their student really isn’t that good; the fault will always lie elsewhere.

Note: I am NOT saying that not doing well in mathematics makes anyone a failure. For example: my step son took the multi-year path to get through the required calculus sequence required by his computer science program. He got his degree and is now earning (at least) triple my salary in the database industry.

## May 10, 2011

### Non-portability of mathematical skill

On my calculus final exam, I gave two questions about a metal plate of uniform density. The plate was easy to describe: it’s boundary was the $x$ axis and the parabola $y = 1-x^2$. In the first question, I asked for $M_x$ (the moment about the $x$ axis and in the second question, I asked for the center of mass (they could use symmetry to deduce $\overline x = 0$). So to find $\overline y$, they needed to find the area (mass) and $M_x$.

What astonished me is that a number of students missed the question “find $M_x$ ” completely but then went on to solve for $\overline y$ correctly!

This says something about the intellectually immature mind, but I am not sure what is says.

## March 10, 2011

### Students: do some problems without your book and notes!

Filed under: how to learn calculus, mathematics education, student learning — collegemathteaching @ 11:22 pm

I’ve found that some students make the mistake by always doing practice problems with their book and notes open. It is oh so easy to convince yourself that you know the material better than you actually do.

You don’t have to look far for instances of people lying to themselves. Whether it’s a drug-addled actor or an almost-toppled dictator, some people seem to have an endless capacity for rationalising what they did, no matter how questionable. We might imagine that these people really know that they’re deceiving themselves, and that their words are mere bravado. But Zoe Chance from Harvard Business School thinks otherwise.

Using experiments where people could cheat on a test, Chance has found that cheaters not only deceive themselves, but are largely oblivious to their own lies. Their ruse is so potent that they’ll continue to overestimate their abilities in the future, even if they suffer for it. Cheaters continue to prosper in their own heads, even if they fail in reality.

Chance asked 76 students to take a maths test, half of whom could see an answer key at the bottom of their sheets. Afterwards, they had to predict their scores on a second longer test. Even though they knew that they wouldn’t be able to see the answers this time round, they imagined higher scores for themselves (81%) if they had the answers on the first test than if they hadn’t (72%). They might have deliberately cheated, or they might have told themselves that they were only looking to “check” the answers they knew all along. Either way, they had fooled themselves into thinking that their strong performance reflected their own intellect, rather than the presence of the answers.

And they were wrong – when Chance asked her recruits to actually take the hypothetical second test, neither group outperformed the other. Those who had used the answers the first-time round were labouring under an inflated view of their abilities.

Chance also found that the students weren’t aware that they were deceiving themselves. She asked 36 fresh recruits to run through the same hypothetical scenario in their heads. Those who imagined having the answers predicted that they’d get a higher score, but not that they would also expect a better score in the second test. They knew that they would cheat the test, but not that they would cheat themselves.

Some people are more prone to this than others. Before the second test, Chance gave the students a questionnaire designed to measure their capacity for deceiving themselves. The “high self-deceivers” not only predicted that they would get better scores in the second test, but they were especially prone to “taking credit for their answers-aided performance”.

Bottom line: frequently quiz yourself by seeing if you can do problems without your notes or seeing if you can write out the proofs without references!

The research, published online Thursday in the journal Science, found that students who read a passage, then took a test asking them to recall what they had read, retained about 50 percent more of the information a week later than students who used two other methods.

One of those methods — repeatedly studying the material — is familiar to legions of students who cram before exams. The other — having students draw detailed diagrams documenting what they are learning — is prized by many teachers because it forces students to make connections among facts.

These other methods not only are popular, the researchers reported; they also seem to give students the illusion that they know material better than they do.

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