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.
- Most real problems are messy. It is not immediately apparent which formula you should use.
- 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.
- In life, you have to ask your own questions most of the time. Someone else doesn’t give them to you.
- Questions that can be answered in 15 seconds aren’t the kind that really promote thinking.
” Find a function that expresses where a child sits on a seesaw in terms of her weight.”
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.
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.
- Give students fewer problems.
- Give them the time to solve those problems on their own.
- 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.”
A 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. 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.
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
Ten years of graduate school, two masters degrees, one doctorate and there are four statements that still stand out decades later.
Dr. Irv Balow, our research methods professor had just been asked by a frustrated student,
“It seems like every time we ask you a question, the answer is always, ‘The data don’t support that hypothesis’ or ‘It depends on other variables’. Isn’t there anything that you can say a straight yes or no?”
Dr. Balow pondered this seriously for moment and responded,
“After twenty-five years of research experience, the only thing I can say with great certainty about education and psychology is this – All the simple answers are wrong.”
I am having a great time working on our new course in Developmental Psychology. Many beginning students in psychology find it frustrating that they cannot get “a straight answer” to such questions as, “Why did my cousin, who has two parents who never drink, become an alcoholic?” or “Why do some children who seem to be developing so well before school suddenly have problems when they begin kindergarten or first grade?”
Such frustrations led Urie Bronfenbrenner to comment that we are left with “… a science that tells us precious little about the questions which beckon us to it.”
In fact, with all due respect to Dr. Bronfenbrenner, it seems to me a little naive that we should expect COMPLETE answers to any complex question about human behavior. Let’s take you as an example. (Yes, I am talking about you). You no doubt have relatives who did not attend college, despite the fact that they are about your age, perhaps from the same family, had some of the same teachers, lived in the same community. I challenge you to name a SINGLE reason. I bet you can name a great many reasons, but not point to the ONE and ONLY ONE reason why you attended college and they are not.
One reason life is not so simple is even when you things have the same number, they can actually be different experiences. For example, I just read a New York Times article about Marian Radke-Yarrow, who was doing research at Harvard for years, on maternal depression. It turns out that not all depression is the same. Some people are “irritably depressed”, other people are “silently depressed” and still others have bipolar disorder where they are alternately depressed and manic.
An article published over twenty years ago, also in the New York Times challenges the assumption that brothers and sisters really do grow up in the same environment. In other words, each person has different friends, different teachers, and even different experiences in the same family. One child may be closer to his father while the other is closer to his mother. A child who is academically talented may feel challenged to achieve by parental demands that he gets straight A’s, while another child who does not have the same ability may be depressed and discouraged, feeling he will never measure up.
So… this is how the course is beginning – with the question of “Why do people turn out the way they do?”
On our Julia Group Forum, Ronda asked if students only take courses like developmental psychology to try to understand themselves better. My answer was that some do, and so, for what I think is that large majority of students, this will be a good way to start.
Today I am on my soapbox. These are words to the wise for working people everywhere, but especially to a certain generation – mine – and to a certain gender – women. If “getting the job done” requires that you work 70 hours a week while other people work 40, then the solution is that you need to stop doing it. I’m not talking about the time crunch when everyone is working on a big project until the wee hours of the morning. I mean those situations where other people go out to play golf on Saturday while you are in the office because “No one else knows how to ___ .” Fill in the blank. Then stop doing it. If you died, they would find someone else and there is no reason for you to do the work of two people unless you are getting the pay of two people, and even then only if you want to do it.
So.. having learned this belatedly, here is what I am doing in my spare time instead. First, I signed up for Flickr. I don’t even own a camera, other than the one on my iPhone, however, people are continually sending me photos, particularly about judo. Also, several articles I read lately have mentioned Flickr as one of the few things my former favorite technology company, Yahoo had done right lately. A flickr account allowed me to add a slideshow to my personal blog, which is mostly about judo. (Among other reasons, before the whole statistical programming, founding companies gig, I was world judo champion.) Since I have a Yahoo account it took me about five seconds to set up my Flickr account. This is a very wise design because it allows them to sign up people like me who are only mildly interested in their product.
Second, I started checking out iGoogle gadgets. Turns out that, since I have a blog on blogger I somehow have a Google account. As with Yahoo, a very smart move. Sometimes designers assume that making it almost effortless to sign up and/or use their product will only attract the clueless, and, as I have been known to say myself, “Stupid people with money” is probably not a very lucrative target market. In truth, very few people will be as interested in your product as you are, especially, as in the case of gadgets, the “product” is four quotes of the day or a National Geographic photo of the day. Making it very, very easy to use is smart marketing. Didn’t I just say I was tired of working all the time?
Twitter is another mini-application I am checking out. I included the link to the Twitter blog here because I found that more interesting than the Twitter home page itself. I am not entirely convinced that this is anything that will interest me, but I did notice Twitter updates on someone’s blog, and Jennifer Laycock had some interesting and positive comments on her blog about it, so I figured, what the heck.
I am taking a month off after next week, come hell or high water, and I have a whole lot of things planned for “wasting” my time. That includes messing around more with all of the above, making real use of my O’Reilly account (the media company, not the idiot on TV), writing a couple of articles for scientific journals, checking out a few more new applications or mini-apps, reading my favorite technology blogs and about a dozen books. And you know what? I will feel no guilt! I learned SAS programming, statistics, FORTRAN, BASIC and HTML all because I thought it would be cool stuff to know. Turns out you can make money with that computer stuff – would would have guessed it back in the 1970s when I was in college? The truth is, you can’t predict what will be useful.
Even if it all turns out to be as useful as programming in MUMPS, I don’t care. I’ll be 50 years old this year, I’ve been working for 35 years and if I want to be useless for a month or so, I earned it!
It is has become increasingly evident that there is not an IQ cut-off for being allowed to post on the Internet. I wrote about this a while back in my personal blog, suggesting that if you would not tell 30,000 people individually how drunk you were last night and what you tried to do to your pet ferret as a result, it is probably no wiser to post it on a web page 30,000 people access each month.
Fortunately, there are some sites that are both intelligent and interesting. Here are a few I read lately:
Always On – has a good article on intelligent and not-so-intelligent ways to think about web marketing. I sigh every time I hear people use phrases like “social networking” or “viral marketing”. Their basic plan seems to be that they will put up a site and everyone will go there, create web pages, post on their forum and the site owner will become rich. The question they fail to ask is, “Why would people want to go to your site with nothing on it?”
Wired magazine - is one of the few I actually read in print. The cover stories more often than not fall into the ‘why-the-hell-would-anyone-over-15-read-this’ category of the latest computer games, Japanese animation and who is downloading photos of what celebrity who just got naked/busted/married. Once you skip over those, though, there is usually pretty good information on the tech industry, new technology coming down the pipeline. They have had some interesting articles on Yahoo lately, one of my favorite spots on the Internet that lately just doesn’t seem to get it. Yahoo was one of the first places to allow you to create a personalized page, they had a calendar with email and text reminders for people like me who are always over-scheduled, but somewhere they just lost their edge. I went to their misnamed Yahoo tech page, that should have been called instead, “The random electronic junk you can buy page”. It was very sad.
Giga OmniMedia is less than two years old and they already meet my cut-off for intelligent technology, selected based on a random sample of one – me. It is a daily on-line news reader on technology. Some of their articles I could care less about. For example, I have zero interest in gaming and I am both president of the company and a big supporter of telecommuting so I don’t need an article on how to convince the boss to let you telecommute. Still, I have never once gone to their site where I did not find at least a couple of articles that interested me. Today, I thought the article on the four themes in emerging technology companies was great. I am still thinking about it (so much for multi-tasking ruining your short-term memory!) Tomorrow, I am sure I’ll find something else good.
I found newsgator after searching forever to find news sites that had more intelligent coverage than which celebrity was in rehab and what company was being bought/laid off employees/losing money and how the employees felt about it (summary:bad. They lost their jobs, their company stock is now worth 22 cents a share and they are working for a 13-year-old boss. They feel bad). Newsgator has some overlap with Giga Omni, but enough features of their own to make it worth checking out.
Finally, of course, there is slashdot. which has been around so long I believe it predates the Internet and was originally chiseled on the walls of caves in Tibet.
So, there you go, recommended reading for the day. If you can’t find something to stimulate your brain in any of those sites, you’re probably dead.