My reason for trying to find that book (both Google and Amazon have not helped me) is that I was going to use it as support material for today's blog on learning mathematics. For you see, the courts just ruled that the Seattle Public Schools must go back and redefine their mathematics curriculum, that their present one does not meet the criteria set forth in policy by the Seattle School board---which ironically had accepted the present plan of study.
I am on an ethics kick today. I don't have the court order to the Seattle Schools in front of me--just a new article. I also don't have that book on How Women Learn, a research item I believe from Harvard. I don't have the textbooks as selected by the Seattle School Board for the Mathematics curriculum. As I might say when teaching my sailing class, we in a fog bank with little references. For those who want to argue with me, you have the advantage, point to your side.
It appears that the Seattle school system had a committee of teachers, administrators and parents study the mathematics curriculum for about a year and decided to go in a certain direction. Other parents and teachers did not like the decision and challenged the boards approval. And that is where we sit today. But I suspect there are other cities in the United States agonizing over the same decision. How should we teach mathematics?
The interesting thing to me--no, make that fascinating is that much of the mathematics dilemma can be found in our philosophy of education. Plato started all this. Blame him. But blaming anyone does not solve our present problem.
I've done this once before with early readers of this blog. I have an in-class assignment for you. Take a sheet of paper and with a quarter draw three circles on your paper. Location is not important but give yourself some space for notes. Now take a dime and circle it inside the first three circles. Should look like you have three wheels--okay so far?
Put the letter "I" in each of the smaller circles. This represents the individual or the student in a learning situation. You can title your first pair of circles Idealism. Now make a few arrows going from the outer circle toward the inner circle. The idea behind idealism is that it is a top down type of philosophy. I suppose a good example of this type of thinking can be found in our military. The president says this will be and then the Generals and Admirals pass the word on to lower officers until it reaches the privates and seaman at the lowest ranks.
An example of a curriculum within Idealism would be the Great Books Selection. Experts in literature would select what they thought was the best and presented it to the rest of us. How about mathematics--how would that get taught? Numbers would be presented to the student. Probably addition next then subtraction. Once that is managed, multiplication tables would be memorized. Rote learning. Then we face division. The teacher would probably show the student on paper or a white board and then have them duplicate the problem. Making sure the student understands would be the solving of the mathematics problems and presenting it to the teacher. At higher levels of understanding a student would be taught algebra and geometry in the same manner. An example of a problem, the solution done by the teacher and then the students does the problem.
Now please understand that this whole example is highly simplistic. And idealistic as well. I'm telling you something except you don't have to respond by giving me an example of what I just explained. You can teach most anything by this method, English, Social Studies, Health, Reading--the list goes on. You need a teacher and an understanding of what needs to be taught.
There are three subjects that are crucial to becoming educated; reading, writing and numbers, i.e., arithmetic. Why so? Because they deal with abstractions. You are presently reading this missive because you understand that a collection of abstracts means a "word." You and I hopefully agree with what that "word" means to each of us. The Letter "T" has evolved from ancient Greek lettering into something that you and I and others all agree on. It can be a seven foot high letter "T" as on a roadside advertising sigh or a minute one in a stock market report. But both "T"s we recognize as an abstraction that we agree on. Think of all the different types of "T"s that we see daily. Amazing isn't i"t"?
The same holds for writing. Getting children to write a word means that they not only have an understanding of the word, they need to form those letters that constitute the word. And then they have to form their letters with a pencil so that someone else can recognize it.....or they can punch it in on their Blackberry. (I can't believe I just wrote that--oh my, life is changing)
My point here is that reading, writing and numbers are abstractions and each needs to have a meaning connected to that item. That is why I think elementary teaching is difficult. How do you attach meaning to an abstraction? I have often pondered how the Chinese teach their abstractions. I would have like to visit their elementary schools.
Idealism is probably learning effective in teaching abstractions. You tell the student, they use the abstraction and then give you an example of it. If correct you go on. This is how much mathematics is being taught today in the public schools. Idealism.
Now let's take your second set of circles. Label it Realism. This time make arrows going from the inner circle to the outer circle. Four or five will do. It is symbolic that the individual "wants to learn" and asks the environment or adults. It is not uncommon in preschools to have letters or different sizes and shapes for the children to play with. When a child holds up a green letter "T" and waves it at a teacher, she responds by saying "T" Later on when the child hold up another perhaps smaller letter "T" the teacher agrees that it is a "T" as well. In this manner the child "learns" the alphabet on their own. Seeing a letter "T" in a book makes that child understand that it is something important and begins the understanding of the different letters. We can do the same with numbers. Combining numbers and letters tell us something different but the child has to discover these combinations.
Again I have simplified this example. In the Idealism mode, a student memories the learning. In the Realistic mode the student discovers or owns the learning. Which one is the better method? I really don't know. I suspect that sometime in the future we will do a report on the genes of each child and know by those genes which mode of learning is optimum. At present we can only use trial and error. Maybe our form of education is correct. A student has this teacher who is idealistic for one year or a subject and then another teacher for a year or subject who is realistic.
What? What? What should we do with the last set of circles? I'm glad you asked. Label it Pragmatism. The draw arrows going to and from each circle. An arrow with points on either end. In this case we have an individual or student exploring the world but getting advice, instruction and guidance from outside; i.e., the adults. Developed by John Dewey (the philosopher not the librarian although I honor both of them) he insisted that it was a different philosophy of education. Dewey said that the only constant was change itself. Therefore the curriculum was always in flux. Given today's technology advances I suspect he would say "I told you so." Learning has to be adjusted for the time and the individual. Tough job.
So what is Seattle school board going to do? They selected the realistic mode but the courts were not accepting of their choice. I suspect they will go back to the idealistic mode. It will be interesting to see what happens.
You've read all this abstraction and you understand it all. You need to be very concrete and go thank a teacher.