I had a very difficult homework assignment the other day. I had an exam the following day, and besides spending my evening studying, I stayed up late into the night trying to complete this mentally taxing project to turn in with the exam. It was an assignment for my math methods class. I had to construct 150 base-ten blocks. I chose to build them with popsicle sticks.
My friends have given me a tough time about this class. As they sit in the living room frantically typing papers, I spend my late-night hours cutting out laminated choo-choo trains to use as counters in an arithmetic lesson. This class is the quintessential example of why elementary ed majors are looked down upon for the simplicity of their major: "You're doing elementary school math. That's not a real major!" I have to challenge this stereotype. I can acknowledge that elementary education may be a simple major at its surface, but if pre-service teachers decide to delve into their courses, they can explore questions of content and pedagogy that are critical to the foundation of the intellectual conditioning of subsequent generations.
In my math methods course, every unit is framed along the same dichotomy: conceptual v. procedural understanding. In teaching children math, it is essential that they first develop a concept, and then develop procedures for effectively using the concepts quickly and in useful ways. For example, if I say the word triangle, what do you think of? What image comes to mind? A polygon with three sides and three vertexes? Is it white? Is it filled in? Is it equilateral? Obtuse? Right? Acute? No matter what your initial conceptualization is, you are able to call to mind the idea of the form of a triangle. This conception is critically important in you being able to properly apply procedures to the triangle. Procedures such as the Pythagorean Theorem or an Area equation can be applied in a rote manner if a child does not understand the concept, he/she will be able to use the procedure, but will not understand when to apply the procedure in a dynamic concept. If you say "solve for x" he/she will be able to. If you give him/her a word problem, he/she will hand you back a blank paper with a puzzled look.
In thinking about children's conception of basic mathematical concepts, I began to wonder about why a breakdown seemed to occur in middle and high school for many students. At one point, students begin to start saying "I can't understand this. I just don't get math." No matter how much time a tutor/teacher/parent/friend might spend with the struggling student, it sometimes seems like the student will never understand the concept. I wondered where our conception of mathematics failed, but perhaps, more importantly, how that influenced out lives.
Fortunately, at the same time as I was having these questions, I purchased a book that a friend had recommended to me over the summer: "Innumeracy: Mathematical Illiteracy and Its Consequences" by John Allen Paulos. As can be infered by the title, Paulos argues that most of our society is innumerate: we do not know how to read and conceptualize numbers and mathematical processes. The effects of this poor mental grasping lead to personalization of events over accurate mental constructions, pseudosciences, and major misunderstandings of chance and meaningful coincidences. The implications of this book are staggering. Paulos uses basic rules of probability to explain that the meaning humans attach to seemingly meaningful events is terribly misplaced. Such "meaningful" events are really just boung to happen based on the probability of such events. To illustrate, he explores the theory of "seven degrees of separation." We are amazed when we sit down on an airplane only to find that the person sitting next to us once lived in the house that our best friend's cousin's father-in law once owned. This seemingly chance occurance, Paulos explains, really isn't that rare or meaningful. According to a few of his estimates, if each adult in the U.S. knows about 1,500, then there is a 99 in 100 chance that any two adults would be linked by no more than two intermediates. This is but one example he fabricates to tear apart common perceptions of reality and piece them back together using probability and mathematical inferences.
Paulos casts his message in a dry, condascending tone. He seems to really look down upon the many who do not view life through his mathematical perspective. He constantly takes jabs at the social sciences. Despite this thinly veiled negativity however, he offers a few solutions that enable society as a whole to become more nummerate. The most exciting of his propositions for me was his suggestion of the playing of mind games. He suggests that when we teach or learn math, we should not focus on the pure mathematical operations, but rather place the algorithms in a context that forces us to correctly determine which operation should be applied. His favorite example involves the simple equation 1+1=2. Easy right? He explains that although math might seem very straightforward, if this tool is not applied correctly in life, it can be very misleading and lead to false information rather than truth. 1 bowl of oatmeal plus 1 bowl of hot water does not equal two bowls of hot cereal. To train the mind not to committ such mistakes, he advocates the use of games. He proposed questions just for the sake of the creative mathematical excercise they encourage. How many paper clips would it take to fill up the hands of every five year old in the United States? How many children have been born in the past thirty seconds? By answering such questions, the mind is forced to determine which information and operations are necessary. Math is a creative process.
The applications of such theories to my math class are staggering. When I focus on teaching children, I need to ensure that they fully understand mathematical concepts. What use is it to them to graph an equation if they do not comprehend that the line they create represents the set of solutions for the equation? As a teacher, I need to focus on creating lessons and activities where children are required to do more than compute problems on a worksheet. Their mathematical learning should be embedded in the real contexts of other classes. What's the point of learning arithmetic for a test when they don't know how to apply it to their lives?
So as I sit cutting out my choo-choo's I remind myself that not only are the fun and colorful, but they are also invaluable tools in helping children to understand the application of mathematical concepts.
I wonder how much of your instruction is influenced by a teaching philosophy foisted upon my generation back in the day called "New Math"?
ReplyDelete(for convenience, I include below some of what wikipedia has on the subject):
New Math emphasized mathematical structure through abstract concepts like set theory and number bases other than 10. Beginning in the early 1960s the new educational doctrine was installed, not only in the USA, but all over the developed world. Much of the publicity centered on the focus of this program on set theory (influenced ultimately by the Bourbaki group and their work), functions, and diagram drawings. It was stressed that these subjects should be introduced early. Some of this focus was seen as exaggerated, even dogmatic. For example, in some cases pupils were taught axiomatic set theory at an early age. The idea behind this was that if the axiomatic foundations of mathematics were introduced to children, they could easily cope with the theorems of the mathematical system later.
Adopted in the US as a response to the challenge of early Soviet technical superiority in space, it began to be challenged in the late 1960s. One of the most influential critiques of the New Math was Morris Kline's 1973 book Why Johnny Can't Add. Kline says certain advocates of the new topics "ignored completely the fact that mathematics is a cumulative development and that it is practically impossible to learn the newer creations if one does not know the older ones" (p. 17). Furthermore, noting the trend to abstraction in New Math, Kline says "abstraction is not the first stage but the last stage in a mathematical development" (p. 98).
The New Math method was the topic of one of Tom Lehrer's most popular parody songs, with his introductory remarks to the song: "...in the new approach, as you know, the important thing is to understand what you're doing, rather than to get the right answer."
So let's all sing together:
Hooray for new math,
New-hoo-hoo-math,
It won't do you a bit of good to review math.
It's so simple,
So very simple,
That only a child can do it!
cf. the following link at http://www.csun.edu/~vcmth00m/AHistory.html
ReplyDeleteAt the end of the 20th century, mathematics education policies in U.S. public schools were in a state of flux. Disagreements between parents and mathematicians, on the one hand, and professional educators, on the other, continued without clear resolution. Wilfried Schmid described the disagreements at the end of the 1990s succinctly:
The disagreement extends over the entire mathematics curriculum, kindergarten through high school. It runs right through the National Council of Teachers of Mathematics (NCTM), the professional organization of mathematics teachers. The new NCTM curriculum guidelines, presented with great fanfare on April 12 [2000], represent an earnest effort at finding common ground, but barely manage to paper-over the differences.
Among teachers and mathematics educators, the avant-garde reformers are the most energetic, and their voices drown out those skeptical of extreme reforms. On the other side, among academic mathematicians and scientists who have reflected on these questions, a clear majority oppose the new trends in math education. The academics, mostly unfamiliar with education issues, have been reluctant to join the debate. But finally, some of them are speaking up.
Parents, for the most part, have also been silent, trusting the experts--the teachers' organizations and math educators. Several reform curricula do not provide textbooks in the usual sense, and this deprives parents of one important source of information. Yet, also among parents, attitudes may be changing...
The stakes are high in this argument. State curriculum frameworks need to be written, and these serve as basis for assessment tests; some of the reformers receive substantial educational research grants, consulting fees or textbook royalties. For now, the reformers have lost the battle in California. They are redoubling their efforts in Massachusetts, where the curriculum framework is being revised. The struggle is fierce, by academic standards.
The stakes are high not only for mathematics education in the public schools, but also for the nation's colleges and universities. Through a domino effect that begins in the elementary school grades and works its way up the educational ladder, the so-called reforms promoted by the NCTM, and other education organizations, are sure to affect university level mathematics education. Without adequate foundations in arithmetic skills and concepts from elementary school, entering middle school students will be unable to progress to algebra. Without strong foundations in algebraic skills and ideas, the doors to subsequent meaningful mathematics courses will be closed.