Over the past twenty-five years, due to improvements in statistical software capabilities, it has become increasingly easier to obtain output for complex statistical procedures. Unfortunately, this does not mean that statistical analysis has become easier. In a relatively short period, it is possible to train an employee or student to be competent at getting computers to produce output, but not so aware of what that output actually means, or what pitfalls need to be avoided on the way to producing and interpreting results. For those of you who fall into the “good SAS programmer, non-statistician” category, this presentation should provide useful information on errors to avoid. These are not the type of errors that are highlighted in capital letters in your SAS log, and that is what makes them dangerous. For those who have forgotten more statistics than most people will ever know, today’s presentation will serve as a reminder of why you learned that information in the first place, and plant the scary thought that, although you assume certain steps are being taken by the people who produce those printouts on which you base your decisions, that may not always be the case.
Common problems occur when analysts proceed without getting to intimately know the data. These include failing to accommodate for sampling methods, overlooking data entry errors, incorrect distributional assumptions and just plain poor measurement. To illustrate how these usual suspects occur and how to avoid them, I chose three common scenarios. These include use of large-scale databases, relatively modest surveys done by a business and the evaluation of the effectiveness of a program.

SCENARIO 1: MULTI-USER LARGE-SCALE DATABASES

Wonder of SAS: Public and proprietary datasets
There are an enormous number of datasets already in existence. These range from large-scale studies of hospitals, drugs or consumer purchasing behavior collected by ones employer to datasets collected by the federal government and free to the public. For anyone interested in research and without an infinite budget, publicly available datasets are everywhere from the Census Bureau to the Agency for Health Research Quality to the California Department of Public Education. Any faculty member, student or staff of the hundreds of universities and agencies that belong to the Interuniversity Consortium for Political and Social Research can download files from this site.
Often these files can be downloaded already as SAS datasets, or text files with accompanying SAS programs for reading in data, creating formats, labels and even re-coding missing values. This may come to over a thousand lines of code that you can just download and run. Incidentally, most of these files come with a link to related literature.
and download SAS datasets, or text files So, you begin your research program with data collected, data collection section written, data cleaned and a preliminary literature search conducted. I throw this in because every single time I have mentioned this resource graduate students, and even professors who teach statistics and research methods have groaned and said, “Why did no one mention this before now?”

Especially with so many working professionals taking graduate (and so many professors teaching those courses) the time saved is truly a wonder and lets you concentrate on the actual academic material.

http://www.icpsr.umich.edu/ICPSR/index.html

From the ICPSR site, I downloaded the American Time Use Survey and came up with a chart. Now, here is the first danger of SAS. I did many, many things wrong already, not the least of which is that I produced that chart using SPSS. The chart looks good. When I examined the log, there were no errors. It is based on a sample of over 13,000 using a well-designed survey conducted by the U.S. Department of Labor, the same organization that conducts the U.S. Census. You could draw conclusions from this chart, interpreting it fairly easily. For example, the average woman without children spends 400 minutes a day alone. There is only one problem. The results are incorrect.

Dangers of programming with SAS
This is the first danger of SAS. I know that I should always check the data. I know that a few outliers can really throw off your data. I know I should have read about the source of the data before I did any statistics and I know that any code I get from any source, no matter an expert within my own organization or an institution, the first thing I should do is read all of it. The Users’ Guide alone was 63 pages, though, not to mention the codebooks and data dictionary for each file, lengthy questionnaire, documentation on linking files. Reading all of that would take several times longer than writing a program.

When I did go back and read the documentation, I found the statement in Chapter Seven
“Users need to apply weights when computing estimates with the ATUS data because simple tabulations of unweighted ATUS data produce misleading results.“

Fortunately, as I learn through reading all 63 pages of documentation, the American Time Use Survey staff have helpfully provided a weight variable to multiply it by the response for each individual.

WHY?
In a representative random sample of 13,000 people, where everyone had an equal chance of being included, you would feel fairly confident in generalizing to the whole population. However, they did not do that. They took a stratified sample so that they would be able to have a large enough number from certain groups to make generalizations to the population.
Think of it this way — you have a sample of 10,000 people that is 25% from each of four ethnic groups, Caucasian, African-American, Latino and Asian-American. You can generate pretty good estimates for each group on that, but you cannot add the whole sample together and say that the country is 25% Asian-American.
Before we perform our calculations, we have to weight each response. Maybe our person in North Dakota might represent one-fourth of a person, because that state was over-sampled, while each Californian in the sample might represent three people. To get an accurate population sample, we multiply the weights by the number of minutes or hours a person report doing a particular activity. Then we divide by the sum of the weights. The equation provided by ATUS looks like this

Ti = ∑ fwgt Tij
————
∑fwgt

The average amount of time the population spends in activity j Tj is equal to
• The sum of the weight for each individual multiplied by the individual responses of how much time they spend on activity j
• Divided by the sum of the weights.

This is coded in SAS very simply, with a BY statement added to provide means by gender and presence/absence of children in the home.

Proc means data= mylib.AT25vars ;
VAR time_alone ;
BY tesex child ;
WEIGHT tufinlwgt ;

Done. Quick. Easy. Wrong.

The first clue was that there were five different possible values for gender. Referring back to the codebooks, it is discovered that unusable data were broken down into multiple categories, including “refused to say”, “not applicable” and so on and numbered these -1 , -2 . As a result, the above statements, with correct weighting variable, scientifically constructed sample and automated data analysis can produce output showing that the average American spending negative time on a given activity. The illogic of that result is obvious. The real danger is when 5- 10% of the data has those negative numbers. One may have used a very sophisticated procedure, the data looks pretty much correct but is actually wrong.

DATA mylib.AT25vars;
SET mylib.ATUS ;
ARRAY rec {*} _numeric_ ;
DO I = 1 TO dim(rec) ;
IF rec{i} < 0 THEN rec{i} = . ;
END ;
KEEP list the variables ;

A few simple tips are illustrated above to reduce the other type of error, the type that shows up as ERROR in your log.

Use the * to assign the dimension to the array to be however many variables there happen to be. This will prevent errors caused by miscounting the number of variables in the array, an error that could occur easily in a case such as this with over 500 variables.

Use _numeric_ to avoid typing in all 510 variables and ensure that none are inadvertently left out of the list.

The DO loop recodes all of the missing values to truly missing. Using the DIM function will do these statements from 1 to the dimension of the array.

Recoding the incorrect data and using a weight statement will give the correct means.

EXAMPLE 2: Stratified random sample with surveymeans

It is a common scenario to have a sample that is not simply random but stratified, either proportionally or not, cluster sampled or any combination. For example, some specific group, such as small rural hospitals or special education classrooms have been purposely over-sampled to give a large enough sample within groups. Subjects may have been selected within a cluster, such as students in classrooms at selected schools, or all of the subjects at specific car dealerships during October. In these situations, failing to account for the stratification will give the incorrect estimate for standard errors. Fortunately, the SURVEYMEANS procedure can be used for a range of sampling types including non-proportional stratified samples, stratified samples, cluster samples and more.

In this example, a sample was selected from the American Time Use Survey with a minimum of 40 per stratum, stratified by education and gender.

The code below uses the SURVEYMEANS procedure to provide the correct means and standard errors. The TOTAL = specifies a dataset which contains the population totals for each strata. The STRATA statement gives the two variables on which the data were stratified. The WEIGHT statement uses the samplingweight variable which, if PROC SURVEYSELECT was used for sample selection, has already been included as a variable by that name.

PROC SURVEYMEANS DATA= mylib.AT25vars TOTAL = strata_count ;
STRATA tesex educ ;
VAR list of vars ;
WEIGHT samplingweight ;

SAS also provides PROC SURVEYFREQ and PROC SURVEYREG that will produce frequency and regression statistics and standard errors for survey data.

Take-away message – as the calculation has gotten easier, each new release of SAS including increasingly sophisticated statistical techniques, the need for understanding the assumptions underlying those calculations has increased.

SCENARIO 2: CONSUMER SURVEY

The first common example used an existing large-scale database. The next example comes from a mid-sized company providing services to people with chronic illness. Their target market was Indian reservations in the upper Midwest and their objective was to determine how people obtained information related to disabilities and chronic illness.

Danger of Using SAS: Don’t be too smart
A cynic might caution consultants never to be smarter than their clients want them to be. The fact is these clients were very smart. They knew exactly what they wanted. What any client wants is not to see how brilliant you are but the answer to their question, which in this case is simply:

“How do people on the reservations get information? There is an assumption that these are remote, low-income communities and almost no one is using the Internet or email. This is the first large sample collected. What does it tell us about how to reach our customers?”

Wonder of SAS: Enterprise Guide
This question could have been answered with Graph N Go and descriptive statistics procedures such as PROC FREQ, which are part of Base SAS. Enterprise Guide was chosen as a solution because it was faster to produce visually appealing output and was more easily understood by clients with no programming background.

Here were the steps in Enterprise Guide (illustrated in the PowerPoint accompanying this paper).

1. Write three lines of code, including the run statement, because to use a format created to label the charts.
2. Point and click to create two charts that are saved as jpg files.
3. Point and click again to get frequencies (Describe > One-Way Frequencies)
4. Point and click for correlations. (Analyze > Correlations)

Photoshop was used to put the two charts on one page with a title.
graphs2.jpg
Conclusions:
Residents in the target population are as likely to use the Internet daily for information as to read a daily newspaper.
There is no correlation between Internet use and traditional media use.

SCENARIO 3: PROGRAM EVALUATION
In the previous scenarios, the variables to be used were already identified and it was assumed that whatever measure being used was reliable and valid. In evaluations of specific programs or products, this assumption cannot always be made.

In this example, a test comprised of multiple choice questions and short case studies was developed by the client and used for the first time in this evaluation. No assumptions can be made and we need to begin by testing everything from whether there are gross data entry errors to internal consistency reliability. This is before we get to answer the client’s question, which was whether the two-day staff training program was effective.

Just for fun, I decided to complete the whole project using only two procedures.

Wonder of SAS: One step produces multiple steps in psychometric analysis
Not coincidentally, the WHERE statement below illustrates a point that should be familiar by now – know your data! An assumption of most statistical tests is that the data are independent, random observations. If pre- and the post-tests of the same people are included, this assumption is grossly violated.

PROC CORR DATA = tests ALPHA ;
WHERE test_type = “pre” ;
VAR q1 – – q40 ;

Descriptive statistics, which the CORR procedure produces by default, include the
minimum and maximum for every variable, which are checked to make sure that nothing is out of range. Nothing is less than zero or greater than the maximum allowable points for that item. All of the items are at least within range.

Next step, as any good psychometrician knows is to examine the item means and variances. If there is any question that everyone got wrong (mean = 0) , or that everyone got right, then it will have a variance of zero. These items would bear further examination, but there aren’t any on this test.

Coefficient alpha is a measure of the internal consistency reliability of the test. The .67 value is undesirably low. This analysis shows several items that have a negative correlation with the total, which is inconsistent with the assumption that all items measure the same thing, i.e. knowledge of regulations, specific concepts and best practices applying to the jobs these individuals do. Answering an item correctly should not be negatively related to the total score.

Possibly two different factors are being measured. Maybe these items that are negatively correlated with the total are all measuring some other trait – compliance or social appropriateness – that is perhaps negatively related to staff effectiveness.

The correlation matrix, also produced in the same step, is scrutinized to determine whether these items negatively correlated with the total score perhaps correlate with each other. It seems that these are just bad items. As we examine them, they ask questions like “Which of these is a website where you could look up information on …..” which may be a measure of how much time you spend at work goofing off searching the Internet.

Having verified that no data are out of range, gone back to a DATA step , deleted bad items to produce a scale with acceptable reliability and used a SUM function to create a total score for each person, pre- training and post-training, it is time for the next step.

Wonder of SAS: The General Linear Model Really is General
A valid measure of employee skills should be related to job-specific education, but not to age. To determine how much the test score initially was determined by education and age one could use a PROC REG but I decided to use Proc GLM because it is the General Linear Model and, after all, regression is a special case of the general model.

TITLE “Regression” ;
PROC GLM DATA=in.test2 ;
MODEL score = age years_of_ed ;
WHERE test_type = “pre” ;

The focus of the training was on care for people with chronic illness. Perhaps if a person had a chronic illness him/herself or had previous experience in a job providing such care it would be related to their pre-existing knowledge.

This is an obvious 2 x 2 Analysis of Variance, which can be done with a Proc GLM, because, of course, Analysis of Variance is another special case of the General Linear Model. So, I did it.

TITLE “2 x 2 Analysis of Variance ” ;
PROC GLM DATA=in.test2 ;
CLASS disability job;
MODEL score = disability job disability*job ;
WHERE test_type = “pre” ;

Whether or not a subject had disability was unrelated to the score but experience on the job was. The test has adequate reliability, some evidence for validity.

The real question of interest is whether scores increased from pre- to post-test for the experimental group but not the control group. For this I am going to use, of course, the General Linear Model for a repeated measures ANOVA

TITLE “Repeated Measures Analysis of Variance ” ;
PROC GLM DATA = in.mrgfiles ;
CLASS test_group ;
MODEL score score2 = test_group ;
REPEATED test 2 ;
LSMEANS test_group ;

It can be seen that the model is highly significant, that there is a significant interaction of test * test_group, exactly as hoped, and the output from the LSMEANS statement provides the welcome information that the trained group increased its score from 45% to 68% while the comparison group score only showed an increase of less than 2% – from 57.4% to 59%.

CONCLUSION
SAS has made it possible for thousands of people to obtain output of statistical procedures without ever needing to understand the assumptions of independently sampled random data, what an F distribution is, the impact of measurement error on correlation obtained or even what error variance is. This is a mixed blessing as ERROR-free logs in no way guarantee error-free results or interpretation.
On the other hand, SAS, particularly with the Enterprise Guide, has made great progress in making statistics accessible to a wider audience and perhaps moved more statisticians to understand that the best statistical analysis is not the one that the fewest people can understand but that the most people can understand. So far, I can’t find anything bad about that.

Contact information:
AnnMaria DeMars
University of Southern California
Information Technology Services, Customer Support Center
Los Angeles, CA
ademars@usc.edu
(213) 740-2840

Every now and then I post a mistake I made using either statistical software or statistics. Students often get discouraged feeling they make so many mistakes and they will never get it all right. No one gets it all right all the time.

Obvious mistake of the day ….

I was making minor cosmetic changes on a production job on a test computer and none of the data for the current week were being selected. This is bad. It kept reading only 20 observations. I reviewed the code for a subsetting IF statement or other logic that would delete all but 20 observations. Nope, looked fine. It was supposed to read the records in the last week:

If rec_day > today() – 8 then do :

Using the SAS function today() and then doing a bunch of stuff.

I created a null dataset and put the value of today() to my SAS log

Data _null_ ;

set odd_data ;

put “Record Day = ” rec_day ;

put “Today = ” today () ;

Only 20 records were printed and sure enough, none of them were within the last week. The value of today() was exactly what it should be.

what_the_hell.jpgI was testing the new SAS 9.2 — could that be it ?

Then, it dawned on me in one of those moments where you slap your head and can’t believe you missed it. Earlier in the day, I had been running another job trying to format some output to be exactly right. Since I didn’t want pages of output, I had set

options obs = 20 ;

I had just closed the old program, opened the new one and kept on going.

My code was fine, the options were still in effect. I re-set

options obs = max ;

and life was good again.

jenn_small1.jpgI have been asked several times by students in my classes if I would consider writing a blog on statistics and statistical programming. Apparently, blogs are “in” with people younger than me, as witness daughter #2 at left who has just informed me that people do not use the term “in” any more. Whatever.

Giving it some thought, I could see three advantages to a blog on statistics, statistical software and common errors.

  1. Even after twenty-five years of experience, I am still making mistakes every day, so there will be no shortage of topics.
  2. It may help to dispel the myth that some people cling to that math, statistics and computer programming is something that you are either good at or not, the sole domain of those whos brains work differently than the normal people. On the contrary, I would say that these are both areas where I am very good and where I learn every day. These two are related.
  3. Speaking of the whole learning thing, it is possible that one could learn from reading a blog on other people’s mistakes. In fact, I have now added to my infinite to-do list, “Find blogs of other people’s mistakes.

SAS on UNIX – a lesson about how there is no place like home

A little background – the university where I work as a consultant has one of the top high performance computing centers in the world. Way cool. Everyone who has an account has a home directory where their personal information is stored – login files, etc. You can run small programs there and save small files. (I define small as anything under 20,000 records or so.) If you are working on a major project you may have something like the entire Medicaid records database which would take up a huge amount of disk space. Rarely would you work on a project like that by yourself and it doesn’t make sense for everyone to have a copy of some enormous file in their home directory, so you would have a project directory where your data are stored and shared with other people.

Yesterday, I kept trying to log in to my HPCC account and I got a message saying “explicit kill or server shutdown”. I thought perhaps there was something wrong with my Windows machine. Let’s face it, there’s always something wrong with Windows. I did all of the usual things – closed the XWin program, restarted the computer. I tried logging on to my account using two other computers and I still could not log on.

I tried using Fetch on my Mac to upload a file. The little Fetch dog kept running and running but nothing happened. No error message, just a continually running little Fetch dog.

Since it wasn’t my computer – I had tried logging on with three computers using three different operating systems – it must be my account. I logged in with another account no problem. Hmmm … definitely my account.

I logged in using PuTTY on my Windows machine thinking perhaps something had gone wrong with the XWin settings. I tried editing a file using Pico and it would not let me save it, saying “Disk quota exceeded”. This really made no sense since the quota for my project directory had been doubled to 200 GB due to another anomaly, a day or so ago. I had tried to copy a file and received a message “disk quota exceeded”.

I had looked at the files in my project directory and everything seemed fine. However, I asked the kind folks at HPCC to increase my quota to 200 GB and they did. Still, I did not think I had over 50 GB in my project directory.

Having managed to log in with PuTTY, I searched around my home directory, did an ls and found that I had inadvertently at some point saved a very large file to my project directory instead of my home director by having my home directory in the LIBNAME statement instead of my project directory. This had greatly exceeded my personal disk space quota of 100 MB. So … I moved the file from my home directory to my project directory giving myself gonzo amounts of space again and life was good.

LIBNAME libref “~/somename” ;

saves things to your home directory. I know that. I must have just been in a hurry.

Since I had just enough space in my home directory for the dataset to completely fill it up, my SAS job ran fine without errors and I was locked out of my account and over quota for a day.

LIBNAME libref “/projectdirectory/subdirectory” ;

saves it in your project directory.

Beware the ~ !!!!

Today, I had an epiphany. As I was reading a book on statistical modeling, I read a statement I had read dozens of times before about how beta-weights are partial derivatives and the light went off as I thought,

“Duh, of course! I can’t believe I have never realized this. It is so obvious. OF COURSE, the beta-weight has to be a partial derivative. A derivative is the rate of change and the partial derivative is the rate of change with other variables held constant and so of course a beta-weight in a multiple regression is the same thing.”

In my defense I should point out that my last Calculus course was probably ten years before my first inferential statistics course, so it is not completely stupid that I did not put two and two together at once. (My niece would comment that this is actually far beyond two plus two. Humph!)  Still, it is embarrassing to relate that for twenty years I have been using matrix algebra to solve these equations without really thinking very deeply about the underlying mathematics.

Of all of the people I know, Dennis is the only one who, instead of,

“What the hell are you talking about?”

would say,

“That is so obvious. I can’t believe you never thought of that before,”

In fact, that is exactly what he said. Well, he didn’t add that he couldn’t believe I hadn’t thought of it, because he wants peace and marital bliss in the house.

Why did it take me so long to put these obvious facts together, even if they were learned ten years apart? The embarrassing truth is, like everyone else, I often did not learn more than I had to. There were courses to pass, children to raise, articles to write. Today, for the first time, I had the leisure of sitting down with no intent other than to completely understand the mathematical underpinnings of the statistical techniques that I use, even if it took reading the same page ten times or reading a hundred different books until I got it completely. It didn’t take a hundred books or even reading one book ten times. All it took was giving my complete, undivided attention to understanding it instead of ‘getting through it’.

What a concept.

The General Linear Model is general (also linear and a model, but that is another topic). What is general about it? What does it do? One way to try to understand a statistic is by the underlying mathematics. I read a paper today where most of it was written in Greek. Seriously. There were a lot of equations where the natural logarithm ofamhead1.jpg the probability of event A was multiplied by the probability of event B from which was subtracted the log of the probability of events other than A.

If you had included all of the Greek letters and formula it was even less comprehensible than it sounds. I did finally understand what they were doing after I read it three times. At the end, I looked like this.

I think a more useful way for understanding statistics, for most people, is to look at the types of questions you are able to answer.

Almost all questions that can be stated:

Is there a relationship between this thing and this other thing?

.. can be answered using the General Linear Model. Another way this type of question can be put,

Is the difference in scores of variable X, between this group and some other group, greater than one would expect to find purely at random?

This is really just another way of saying the first question, that is, “Is there a relationship between group membership and X?” Some examples of where the General Linear Model can be used:

  • Testing for the significance of differences between the mean scores of two different groups. For example, if one wanted to test to see whether the difference in average salaries of men and women is greater than one would expect by chance. If you are familiar with statistics, you may realize right away that this could be done with an independent t-test. The t-test is a specific case of the general linear model.
  • Testing the difference between the mean scores of the same group taken at two different times. For example, we might want to determine whether the amount of time devoted to leisure activities declines after a child is born, or is this just a myth. We could survey people in the year before a child was born and the year after. You may recognize this as a dependent t-test. This is another specific case of the general linear model.
  • Predicting scores on one variable from another variable can be done using the general linear model. For example, I might want to know whether it is possible to predict marital happiness a year after marriage from the number of months a couple dated before marriage.

Not every question can be answered with the General Linear Model. If you have two categorical variables, such as, gender and being a jerk and you want to answer the question, does one gender have a higher proportion of jerks than the other, you would not use the General Linear Model. You would use a chi-square for this question. Or, you could just ask me. (Short answer – yes. )

Here’s a clever idea —

Let’s say you want to predict who dies within the next year (I bet you are a lot of fun at parties). Moreover, you have a hypothesis to test that married people are less likely to die than single people. There are a number of factors that relate to both marriage and death. Married people see the doctor more often, they report less depression and are more likely to be employed. People who receive less health care, are unemployed and depressed are all more likely to die. Sucks to be you.

You could do an analysis where you control for doctor visits, depression and employment. OR, you could create a propensity score, using all three of those variables and predict how likely a person is to get married, also known as a propensity score. Then you could match people on propensity scores, which effectively controls for all of those other variables related to marriage. In very simple terms, this would be kind of like having equal numbers of unmarried, depressed, employed people and married, depressed, employed people all of whom haven’t seen a doctor in a decade.

Since you have controlled for these other related factors, you could then see if marriage is really related to lower mortality. I threatened my late husband that if he died and left me with all of these young children that I would spend all of our money on gigolos in the Bahamas.

beach-palm-trees.jpgI thought that perhaps threats like these would be related to fewer deaths among married people. In my case, it did not work, he died anyway and I did go to the Bahamas eventually, with my new husband who is a software engineer trained as a physicist. No gigolos were involved. I am not certain if that is a good thing or a bad thing.

If you are interested in learning about computing and using propensity scores with SAS, you can find more information here.

When I read textbooks, whether in mathematics or other fields, these are usually as boring as watching a light bulb flicker. Searching the Internet for Algebra problems can get to be pretty depressing. (Whether someone who spends her spare time looking for Algebra problems might already have mental health issues is a separate question not to be discussed at this time.)

Seriously, though, I don’t believe math is inherently boring. Today, I am doing a repeated measures Analysis of Variance. The question I want to answer is how far you can go from the original plan for a training program before it ceases to be effective. No one would imagine that if, instead of teaching Algebra on-line for an entire semester, you walked  up to a group of students with a flat piece of slate and a rock, scratched out the Associative Property:

(a +bX) +cY = a + (bX + cY)

then went out for beer for the rest of the semester, that the students would learn an equivalent amount as in our full-semester, state-of-the-art course. Where is the dividing line, though? How many days could you skip? COULD you replace the computers with sharp rocks and flat pieces of slate and learn just as much? One way to test for this would be to check the significance of the interaction effect between type of class and the improvement on test scores.

I could go into great detail about what we are actually doing, and I probably will next time, but for now I am going to lament the sad state of Algebra. Here are a few examples of Algebra problems

The DeVry University page has questions about how much things cost if apples are fifteen cents and oranges are thirty-five cents or what the area of a circle is when r is increased by three.

The Broome Community College page asks you to factor 16x – 8.

This GRE practice site is a little better. It asks questions to problems that are mildly interesting, such as calculating total income from investments with different rates of return.

There are thousands of sites like those above, and these reflect nearly every Algebra textbook in America. One thing these all have in common is that I don’t much like them. We are asking students to apply a formula to a neat little problem. There are several reasons these are not the way I think we should teach Algebra.

  1. Most real problems are messy. It is not immediately apparent which formula you should use.
  2. Students are learning procedures rather than understanding mathematics. When a problem looks like this, apply the first formula. When it looks like that, apply the second formula. But why? I think there is a big difference between learning rules and thinking. A really big difference.
  3. In life, you have to ask your own questions most of the time. Someone else doesn’t give them to you.
  4. Questions that can be answered in 15 seconds aren’t the kind that really promote thinking.

THIS I like, from Drexel University, the Algebra problem of the week. For example,

” Find a function that expresses where a child sits on a seesaw in terms of her weight.”

This I like, from the Julia Forum,

If you woke up in the morning and everything was twice as big, how could you know?

Part of learning Algebra, I think, should be requiring students to come up with questions as well as answers. Questions could be either useful ones, such as about the effectiveness of changing course design, or simply interesting, like how you could know if the whole world doubled in size. You see, I absolutely believe that Algebra can be both interesting and useful. Unfortunately, the way it is generally taught, it is neither.

I was reading a book this week, Mathematics for the Intelligent Non-mathematician. If it was a person, this book would be your grandmother, not terribly exciting but pleasant to spend time with and if you paid attention you were likely to learn something.

Since I use mathematics for my living, you might reasonably wonder why I would be reading this book. The answer is that I believe in considering different perspectives. I’ve never really quite “got” the whole humanities thing. When I took history in school, I was secretly thinking, “They’re all dead. Get over it.” In English class, I was the kind that made teachers throw up their hands in despair. They wanted me to discuss, “The deep meaning of Moby Dick, what do you think it is really about?”

What did I think it was really about. I thought it was about a big white whale, for crying out loud, because it said that on the first page and about seven hundred more times throughout the book.  The title? That’s the name of the whale, hello? Apparently, that was not the correct answer and you are supposed to say that it is a metaphor for the universal struggle of man against the sea, or man against himself or for man’s domination of marmots.

As you might guess, the second I had the opportunity for classes in college like Accounting, Calculus and Statistics where the questions had actual answers, like 42, I jumped at the chance. This isn’t to say that I made A’s in all of those classes initially, as that would have interfered with my plan of going to parties at night and sleeping through the morning. This plan was ended through a talk with the dean and some threatening words about losing my scholarship and having to find $20,000 under a mattress. Heck, I didn’t even own a mattress, much less $20,000 to find under it.

So, here I am thirty years after graduation looking at mathematics from a more naive point of view, which brought out a couple of points I had never really given much thought.

The first is that mathematics is the most general thing in the world. You cannot apply psychology to rocks or biology to building a space shuttle or oceanography to orthopedic surgery. However, as the author said, you can count devils or angels, whales or stars. In fact, when I went from being an industrial engineer to studying for my Ph.D. in Educational Psychology I used the exact same equations I had applied to predict which cruise missile would fail testing before launch to predict which child with a disability would die within the next five years. (Yeah, I wasn’t a lot of fun at parties back then.)

The second interesting point was one that is obvious after someone else states it, i.e., some ideas in mathematics are more important than others. For example, it is a fact that the digits in multiples of nine always add up to nine, e.g., 2x 9 = 18  and 1+8 = 9. This is not a key fact on which a lot of mathematics is based.  So, this led me to thinking about the ideas in mathematics that I think are crucial and wondering about what other people think.

I always thought that the basic properties of real numbers, such as the distributive property –

A x B = B x A    or A+ B = B +A was one of the most fundamental ideas in mathematics.

A second really important idea was the associative property,  –

A(B+ C) = AB + AC

and the commutative property is a third

(4A + 2B) + 11C = 4A + (2B + 11C)

Once a student understands these properties, it opens up an enormous number of problems that he or she can now solve.

And that is why I like teaching Algebra.

reading_julia.jpg
Silence is one of the most under-used teaching techniques. As Julia learns mathematics, I notice major differences in the way my husband and I respond to her. After I ask her a question, I wait for an answer. The period at the end of that sentence is deliberate. I don’t do anything else. I don’t give her any prompts or hints. If she whines that she can’t get it, I tell her to keep thinking about it. If she comes up with the wrong answer, I tell her that it’s wrong and she should try again. Almost always, she can find the mistake she made.

Dennis, like most people, will try to help her if she doesn’t answer right away, by giving her a hint. Often that makes it more difficult to solve the problem because she now has the original problem to solve plus trying to figure out how the hint relates to it, not an easy task for a fourth-grader. Alternatively, he will give her the answer and then tell her to try the next problem, which is always just like the previous problem, that being the way math textbooks in America are structured. Since she could not figure out the previous problem, she is not going to get this one, either.

Dennis has degrees in Mathematics and Physics from UCLA. He was an excellent student in math and he acts the way his teachers acted in school. Paradoxically, this is not the way he learned mathematics. He had taught himself Calculus by the eighth grade from books he checked out of the public library.

My three recommendations for anyone who wants to be a better math teacher.

  1. Give students fewer problems.
  2. Give them the time to solve those problems on their own.
  3. Be quiet and let them do it.

Sites I liked today on teaching Algebra
Purple Math – I especially liked their “how do I really do this stuff” lessons. Readable and easy to understand. Also recommended for adults who knew they once knew, e.g. what a negative exponent was. Those of you who have not had a math class in years can peruse this site for lots of those moments when you smack your forehead and say, “Oh,yeah, THAT’S what that is.”

Teaching College Math Technology Blog – offers thoughts on demonstrations, learning activities and the use of technology.

The Wolfram Demonstrations Project is way cool – I say this being full aware of the fact that if there is such a thing as  a visual learner, I am not it. You can download the Mathematica player for free and run anyone of their demonstrations. Be aware that even with high-speed access the player takes a long time to download. Be patient.

When I look at the wealth of resources, from the straightforward, readable pages on Purple Math to the high-tech demonstrations of the Wolfram Project, it is hard to believe that every math class in this country is not an amazing place to learn. One reason why is that after teachers have finished teaching, tutoring students after school, grading papers and preparing for the next day’s lesson, they just don’t have time. In the summer, far too many are painting houses, teaching summer school or other second jobs just to make ends meet.

I really do think one solution for teachers, just like for Julia, is providing time and silence. If  we paid our teachers for those two months in the summer to come in and work on making their mathematics classes better, I wonder how our schools would change for the better.

Out of all the thousands of pages of all of the textbooks you ever read in your life, how many sentences can you remember? One that has remained with me for over twenty years was in the required book for my inferential statistics course,

“If something exists, it must exist in some quantity and that quantity can be measured.”

neanderthalA lot of people disagree with that idea and there are whole volumes written about how people like me are backward neanderthals. In addition to the redundancy of “backward neanderthal” (is there a “progressive neanderthal”), this is factually incorrect as one can see by the pictures I have helpfully provided of a neanderthal and me. Me in sunglasses The one wearing sunglasses is not the neanderthal.

To me, the whole idea of measurement is fascinating. The belief that everything from how good of a mother you are to the love you have for your spouse to intelligence can be somehow reduced to a number strikes some people like a science fiction story.

Let’s think about how this can really be done. Let’s take “a good family”, how could you possibly measure if someone comes from a good family?

You’d start with asking questions, and there really is a limited set of questions that most people would agree upon. Few people outside of mental institutions would ask such questions as:

Do you own a kazoo?

How many grapefruits are in your refrigerator?

What color is your llama?

Many more people would ask questions that sound like good measures, such as:

How often do you read to your child?

Do you sing songs to your child that teach patterns, like Ten Bears in a Bed?

Have you taken your child out somewhere in the past month (such as relative’s house, museum, church)?

Even if these questions sound good, they may not be good measures. Let’s assume we agree that not all families are equally good. Then a question that everyone answers the same is not a good question for our measure. So, we start with item analysis. First, we do a frequency distribution. If everyone gives the same answer, there must be something wrong with this question and we throw it out.

Second, we get the mean (average), standard deviation (the average difference from the mean) and graph the frequency distribution. Below is a graph of some data I just happened to have laying around. This is one of the things I love about my life, that I just happen to have data laying around. Also the fact that I don’t know whether the correct word is ‘lying’ or ‘laying’. If this was The Phantom Tollbooth, I would SO be living in Digitopolis.

Data from random test

As you can see, there are not very many people with really low scores and not very many people with really high scores. Most people fall in the middle range and that is what we would expect. As I tell my children all the time, “No one has a perfect family, so shut up.” If there are perfect families, there are few of them. There are also very few truly horrible families where children are kept in cages and force-fed mud. Even in the middle, we expect some variation. Some families are a little better than average, some are a little worse.

When we have a few hundred scores like this, it is useful to just stare at the data. It is also helpful to take a look at the numbers – what is the mean, what is the standard deviation, the minimum and the maximum? If the average was 42 and the maximum was 300,000,000 I think I would wonder whether this was a valid measure of quality of family life. Instead, I think it might be a measure of how much the family owns in stock, for example, which I don’t think is the same as how good your family is, no matter what some Republicans might think.

This is just the beginning. Once we have computed item level statistics and examined the mean, variance and distribution of the total score – wait! there’s more. That will have to wait  until another day

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