Thursday, December 29, 2005

My thoughts on "Math is hard" - the myth

Before I begin, my blog mama has posted her thoughts on this here. If you haven't read them, go take a look.

I don't think that math is hard even though there are some maths which totally befuddle me (probability being the best example). However math, like most disciplines, requires building a foundation before erecting the house. I would like to write a proof for one of those interesting little factoids that arise in math. I don't know if it's feasible and I don't care. This is a fun mental exercise for me. But before I can even begin, I have to learn some additional math - my foundation isn't complete yet. It's a very good thing that I enjoy math.

Now getting to the title of this post. Earlier this month I had a short email discussion about the myth of mathematics. Actually, it wasn't that short and I'm going to copy the bulk of what I had to say here. But rest assured, there is no actual math in these posts - just my ideas and opinions.

The email exchange started with a request that I explain what it was that I called the myth of mathematics. This was my somewhat long winded reply.

I started thinking about this when I took the Praxis and some education classes. (I wasn't sure if I wanted to teach.) On average non-mathematics majors who were planning on teaching took the math section of the Praxis at least twice before passing. Many took it three or more times. I think the most retests I heard about was five. I wasn't overly concerned about the students who were planning on teaching English or History or anything that wasn't math or science, but virtually all of the students preparing to teach on the elementary level needed several tries before they passed. At the same time I kept on hearing "I can't do math, I'm bad at math, math is so hard." Well maybe I'd buy that differential equations are hard, but the Praxis math section is almost all arithmetic! So here are all these future elementary teachers who will have to teach arithmetic telling me that they can't do it, hate it, think it's hard, and so forth. And they're proud of it and see nothing wrong with repeating a basic test multiple times. At the same time, students who needed to retake the reading and writing sections would barely admit to that. Not passing the reading and writing section was socially unacceptable even for the math majors.

This is the mythology of mathematics - the belief that math is difficult and can only be done by "special" people. Math people are the priests in some esoteric religion that the ordinary man (or woman) is barred from. This is crap.

Before you can reveal the falsity of this myth, you need to understand that mathematics is not arithmetic. Just as you need to learn to count before you learn to add and subtract, you need to learn arithmetic before basic mathematics are accessible. Counting is a universal skill - baby's have a sense of number and basic counting skills before they are verbal; many animals also exhibit an awareness of numerosity and number sense. Baby's understand that if you put two stuffed toys behind a screen, when the screen is removed there should be two toys not one or three. This means that children are primed by biology to count, to add and to subtract before they ever enter a classroom.

Then school begins. The teacher honestly believes that arithmetic is mathematics and that math is hard. The teacher probably also dislikes and maybe fears arithmetic/math. Even if the teacher never says that math is hard and gives no outward indication of dislike or fear, children are almost eerily aware of the underlying feelings. So with no intention of perpetuating the myth, the teacher does. And when the student goes home, they will more likely than not see these same feelings in their parents.

Interestingly enough, through the fourth or fifth grade students say that they like math. There seems to be a delayed reaction. This may be a response to the mathematics standards that teachers are expected to meet. As the standards demand a more technical background, the ability to meet the standards becomes harder to reach. State standards are usually based on the standards set my the National Council of Teachers of Mathematics (NCTM). See this web page for an overview of the standards - The problem? The teachers have no idea what the NCTM is talking about since the standards are written by and for mathematics educators. NCTM does state that mathematics should be taught by math professionals, but how many school systems can actually afford (and justify) the additional staff for this one subject?

To add insult to injury, the state of New Hampshire standards expect elementary teachers to cover the following math topics in the elementary grades - Problem Solving and Reasoning, Communication and Connections, Numbers, Numeration, Operations, and Number Theory, Geometry, Measurement, and Trigonometry, Data Analysis, Statistics, and Probability, Functions, Relations, and Algebra, Mathematics of Change, Discrete Mathematics (see for information).

So the little aliens get out of elementary school and have, usually for the first time, a math teacher. The math teacher has a vested interest in maintaining the idea that math is special and not every student is going to be able to do math. Again, this is crap. Every student has the same evolutionary history and so has the same cognitive skills. (I'm ignoring special ed students, I know.) Some students will be better at accessing these skills, but this is no different than any other skill. Some people can draw or write or play baseball (and so forth) better than others. Math is the same.

Once we learn that the cognitive skills used in math have been in play since early man began using tools, educators can start teaching to these innate skills. Math is no longer a secret religion, but a tool and the myth that "math is hard" should evaporate.

I was then asked where the myth originated and answered as follows:

Origins of the math myth are rooted in the historical development of mathematics within western culture and society. The earliest artifacts of western mathematics appear in the mideast in Mesopotamia and then later in Egypt. In both cases, mathematics was the purview of the priests and scribes. Although the math itself was not simple, its needs were - tax collection and calculating the movement of the heavens for predictions. In the cultures and society of that time, the people learning and doing math were largely from the priestly classes and male. Calculating for commerce was a matter of keeping a tally - simple counting and arithmetic. As mathematics emigrated into Greece, the mathematicians were still primarily male and upper class.

It is also of note that mathematics was defined as geometry. Algebra arrived in the west around 500 AD. Muslim scholars had developed algebra for use in trade and tax collection. By this time western culture was pretty thoroughly male dominated and the educational institutions and educators were to a large extent tied to the church. Although there were female mathematicians, the prevailing societal rules did not encourage women to learn or practice mathematics. So math was developed and used first by priests who were male. Later use was pretty much restricted to males who were members of the wealthy ruling classes. Since the practicioners of math were deemed special in some way, mathemathics acquired a reputation which grew into myth.

Reading, by the way, does not suffer from this elitism. I think because learning to read the Bible would have been considered "good" in some sense.

Notice that I've said western mathematics.

Since we all engage in symbolic reasoning via language everyday, that should not make mathematics unaccessible. Although I question the theory that mathematics is entirely an adaptation of language based brain function, a good portion of its outward character is.

The final question was on my definition of cognition and cognitive evolution. So for your edification, here's the last of my weird ramblings.

The point of the research I want to do in graduate school is to understand the cognitive processes underlying math and then find ways to use this information to develop a better way of teaching math. At this point, I'm just a tad light on specifics partially because (with one exception) there are no math cog theories that cover actual mathematics rather than numbers and arithmetic.

The 19th century did see a radical change in mathematics. Virtually all mathematics prior to that time, including calculus, had been developed largely to fulfill practical needs (I know that I'm glossing over the few people like Euclid). All math was essentially applied math. By the middle of the 19th century a radical expansion of mathematics was in full swing. Topology, set theory, non-euclidean geometry, multi-dimensional maths, ring and field theories in algebra, etc. were created by the mathematicians within a very short period of time. This was to me the birth of "pure" mathematics - math for math's sake with no need for application to real world problems. The fact that these new maths found applications later on is nice, but wasn't the point behind their development.

I have a theory - one that I'd like to look at further later on. I believe that human cognition went through an evolutionary change. This change was the result not of biologic pressures but cultural pressures. Cognition has evolved over time. When early hominids started using stone tools, there was a cognitive change. Another occurred when tools changed from one all purpose tool to many specific tools. Yet another when early humans began to bury their dead. The appearance of body decoration and the later appearance of cave art and 3-dimensional art are also indicative of evolving cognitive processes. Each of these changes took thousands of years to develop and, I think, were driven by a Darwinian style evolutionary process.

The emergence of new mathematics in the 19th century was too sudden to be the result of this type of evolutionary process so culture must have been the driving force. (Language, by the way, evolves through cultural/societal pressures and does so at a fairly rapid pace.) Understanding this change may be one of the clues to understanding how we "do" math. I haven't decided yet, but it's fun to think about.

Wednesday, December 28, 2005

Math is hard?

The latest post at Ars Mathematica is very simple - "Math is hard. Discuss." The comments from mathematicians and math educators while few so far are very interesting. If you have a burning desire to see what mathematicians think of math and math education, go read the comments.

After you've read the comments, come back here and tell me what you think. Is math hard? Why?

Sunday, December 25, 2005

Greetings and Solicitations

A while back I worked at a company that I referred to as the UN of companies. Each year the department I worked in had a holiday lunch with the employees bringing in food from their respective countries. The last year I was there the only day when most of the people could get together happened to be the first day of Hanukkah. One of the employees wrote "Happy Hanukkah" on the white board after much discussion on the correct spelling. Another added "Merry Christmas." Then everyone started adding greetings for their particular holidays. So there we were eating really great food and adding holiday greetings until the board was covered with as many different religious greetings as there were different people. This always struck me as the perfect Christmas meal - Christians, Jews, Muslims, Buddhists, Taoist, Hindus, and probably others I've forgotten celebrating the holiday season and breaking bread together.

So in that spirit, I wish you all greetings and solicitations this holiday season.

Tuesday, December 20, 2005

Apparently disconnected ramblings

Can you find the connecting thread?

This morning I got to arrange payments for yesterday's heating oil delivery. Whoopee. So why have the gas prices come down since their spike after the hurricanes, but heating oil hasn't?

My grades are in. A little self-congratulation going on here and bragging
Course Section/TitleFinal GradeCredits
1 PSYC-495-01 Sem: Nat & Orig ConsciousA3.00
2 PSYC-340-01 Psychological TestingA3.00
3 PSYC-254-03 Research Methods LabA1.00
4 PSYC-252-02 Research Methods PsycA3.00

The Pennsylvania courts have thrown ID out of the science classroom. My faith in the intelligence of at least some people is restored. Part of the judge's 139 page opinion includes the statement that "We have concluded that it is not [science]" (from CNN) .

I learned a new word today. While on what already feels like the world's longest break from school, I'm going over the journal articles I pulled with information on previous studies relating to cognitive metaphor and simile. The basic idea is to chart out the methodologies used to find evidence for cognitive metaphors and determine how to use these methods when I design a test for my research project. I need a test that will show evidence of unconscious use of a containment metaphor. Supposedly we all have this innate metaphor (I have a study that showed its existence in children between 9 and 19 months). According to one theory of mathematical cognition, this is a grounding metaphor that we use when doing set theory.

And the new word is "telic" - (adj.) 1. gram. expressing end or purpose; 2. tending to a definite end. From Gk "telikos," pertaining to an end or cause. (from Random House Webster's Unabridged Dictionary, 2nd Edition. 2001)

Friday, December 16, 2005

School's out

New classes begin on January 17th. I already miss going to classes, but I have a small stack of books to read - "Red Lily" by Nora Roberts (3rd book in the Garden Trilogy), "Fermat's Enigma" by Simon Singh (from my cousin), "The City of Fallen Angels" by John Berendt, "Guns, Germs and Steel" by Jared Diamond (a friend sent me this one and the Berendt book), "Handbook of Mathematical Cognition" ed. by Jamie I.D. Campbell, and more chapters in "Analysis With an Introduction to Proof" by Steven R. Lay. Doesn't this sound like fun?

Sunday, December 11, 2005

What ever possessed me to...

choose a book that is heavy on philosophy for one of two critiques/analyses? I so don't care for philosophy. I've been trying to finish the ###### paper for a week and a half. At this point I'd like to finish the paper by saying that the rest of the book is crap. And I'm procrastinating.

Friday, December 09, 2005

Five Weird Habits

Teresa has tagged me with this meme although she asks if it should be the "5 habits of Weird People?" Perhaps it should be the 5 Hobits of the weirdly ordinary world. I don't know, but here you go.

1) the number one weirdest habit according to my friends is reading math text books for fun.

2) I was going to say that backing up my research files in 3 different places was weird, but that just may be anal.

3) after I eat a Dove dark chocolate I always fold the wrapper in thirds, then in thirds again, and then roll it up. Well sometimes I save them because I like the message.

4) I add up the numbers on license plates, but I have to find them in consecutive order (this semester I've gotten 10 thru 39).

5) I don't think I'll admit to any more, this is beginning to sound like I have a little personality problem.

Now the problem of tagging 5 people. I emailed Teresa and asked if I could arbitrarily pick five people from The Bad Example Family since I don't actually know 5 bloggers. Since she said it was okay, here they are:

_Jon of We Swear - cause the blog momma said you were just dying to answer this.

Sarah of That's Not Very Nice! because she tagged me before. - Well shoot, Sarah's already been tagged with this one. If anybody wanders by and wants to take her place, feel free.

VW Bug at One Happy Dog Speaks because my dog is one happy dog and he's speaking right. now.

Well that's three.

I guess for the fourth I'll see if I can tag Jenna (even thought she's not in the family) at Sorgin because she really needs more to do as the semester winds down.

Hmmmm. Who for a fifth?
Okay, how about Jeff at Ponytailed Conservative because I adore people who aren't at all PC.

Thursday, December 08, 2005

What do you want to be when you grow up?

One of the women I work with asked me this morning what I was going to be when I grow up. While checking my email tonight, the answer appeared. Now, back to the world's most boring paper. *grin*

I can't think of a title!

There used to be a bulletin board where I made lots of friends like Teresa my blog momma. One of the people there was a young woman who is now in her first year of college. She is awesomely (is that a word?) smart! In between completing various papers and getting ready for final exams, I've been reading her blog. And learning. Woohoo! Go check it out.

Sunday, December 04, 2005

Thursday, December 01, 2005

Things I've never wondered about

The latest post on Ars Mathematica starts with "If you’ve ever wondered what a spin glass was..." Now this is something I've never wondered about (if you have, follow the link to the Wikipedia entry) and, in my usual totally out of touch with reality fashion, I kept on thinking "spin the bottle" instead of "spin glass." The post was filed under "physics" so great confusion ensued - why would physicists be writing about spin the bottle? This meant that of course I had to follow the link to the referenced article - Spin Glasses for Pedestrians. After reading the abstract I've decided that it may be a very convoluted and wordy explanation of "spin the bottle" after all.