Saturday, March 24, 2007

Learning about Fractions

Today's review comes from a chapter in a book in which a series of studies about fraction-learning is used as an argument in favor of the author's view of learning. The focus of this post will be on the fraction studies themselves.

Epigenetic Foundations of Knowledge Structures: Initial and Transcendent Constructions
by Rochel Gelman, UCLA

In S. Carey and R. Gelman, (Eds.). The epigenesis of mind: Essays on biology and cognition Hillsdale, NJ: Erlbaum Associates. (pp. 293-322). 1991

Internationally, fractions and numbers with decimal points are a big sticking point with students, even into high school. Kids don't really grasp deep down what these sorts of numbers are, so they just memorize a bunch of methods to cope with them, that end up not working all the time.

The group of studies related in this chapter all involved testing children who had been learning about fractions and non-whole numbers with a standard textbook/workbook series that first introduced the most basic concepts in kindergarten, and added more lessons each year thereafter. Gelman was an author on all of them.

The first study tested children in kindergarten through second grade. They failed miserably on the researchers' tests. While they had mostly memorized facts such as that "1/2" was pronounced "one-half", the understanding went no deeper.

The second study tested kids in grades 2 through 8. The children's scores were divided between the regular students and the ones enrolled in the gifted classes. All of the children did poorly in all of the tests in the 2nd and 3rd grades. The first difference between the groups appeared in the 4th and 5th grade scores. There were 5 types of questions asked. For each question type, about half of the regular kids got the answer right, while at least three-quarters of the gifted kids got the answer right. The 7th grade gifted students all got perfect scores, while the regular students still did not reach that point by 8th grade. The children were also asked to explain their reasoning behind their answers for certain questions. All of the gifted children gave better explanations (even though the youngest kids still got lots of answers wrong), which indicates that they were all somewhere along the path of developing a sense about what these strange numbers really mean.

The third study looked at a new way of introducing the concepts of fractions. The researchers worked with kindergarteners and first graders. They worked with some of the children on some tasks using a number line and shapes (circles or squares that were whole, in halves, thirds, etc.). They also related the shapes to the fraction names. All of the children were then tested using a bigger number line as well as with other tests. In general, the children with the lessons did better than those who did not work first with the researchers, but they did best when the names of the fractions were not used. In other words, they could deal with the shapes (and the ideas that they represented) but not with the names that had already been drilled into them at school!

The conclusion to this chapter was that kids come into school convinced that numbers are only things that can be counted. This is enough for a young child to know, but not enough for a true understanding of the larger world of mathematical reasoning. Something needs to happen in the child's understanding before the wide world of math can be pried open by the child. Unfortunately, the way that fractions are generally introduced can prevent the child from including it as a part of the number stream. The author suggests immersing children in number experiences, and allowing them to practice using part-whole concepts, before going on to teach the mechanics of using fractions.

How Waldorf Schools Teach Fractions
There is a big to-do among Waldorf teachers about the "9-year-old change", which my oldest daughter is now entering into. The children become more grounded in their bodies and more aware of themselves as individuals. They begin to understand part-whole relations because they can finally feel, deep down, what being whole is. This is why fractions are not introduced until 4th grade, a point at which the entire class has finished the change. Then, the children are deliberately led to the fractional concepts by manipulating blocks, drawing items and partial items, woodworking, and other hands-on activities. Only after they have spent several months working with the unspoken concepts are they given names and written labels. And the whole class knows what these numbers are and how they fit into the counting numbers by the end of the year.

By waiting until the kids are ready, and then immersing them in appropriate learning situations, all before even thinking about introducing labels (both written and spoken), the Waldorf curriculum actually goes well beyond the recommendations of Gelman and her colleagues on the ideal way to teach fractions.

Friday, March 23, 2007


Why I am starting this blog and what are my initial plans for the site? Enquiring minds want to know!

I send my kids to a Waldorf school, and I am going to be reviewing journal articles from a variety of fields (mostly neuroscience) that seem relevant to this educational model. Am I biased? I hope not! However, you need to understand this about me: I chose a Waldorf education for my children because I already knew that the vast majority of the brain research supports it. If I do manage to find a study that questions Waldorf methods, I will still review it.

We recently had an all-school parent meeting, focused on satisfaction and retention. The point that I brought up was that it can be very alienating to keep hearing the curriculum explained in terms of "because Steiner said so". I happen to think that the guy was a genius, and way ahead of his time, but the fact remains that when a parent asks "why?" and the answer is that some guy who has been dead for 80 years said so, it actually gets interpretted this way: "Steiner said so and no one has come up with a valid justification ever since." Which isn't true! After this meeting, a number of people urged me to write an article (or a series) for the Waldorf parent magazine, Renewal. The thing is, with 4 kids aged 8 and under, I don't have 40 hours to sit undisturbed and hammer something out.

Hence, the blog format. I am hoping to review 1 or 2 articles per week; after 6 months or so, I ought to have something of sufficient depth and breadth to work with. In the meantime, if you all could tell me how clear my writing is, ask me questions that are most pertinent to you, and suggest articles to review, I would greatly appreciate it! All of this will help make my eventual article(s) of the highest possible quality.

At first, I will be reviewing papers that are lying around my house. I graduated from MIT nearly 10 years ago, and then spent 2 years in grad school, so the freshest papers that I have around here are at least 8 years old. But this is not a problem! Cutting-edge research may play a small part in this enterprise, but bodies of work that span decades will probably end up being more useful, so the age of the materials is nothing to worry about. And then, after exhausting my in-house collection, I will go seeking other papers; I suspect that a lot of them will be recent.

So, that is my modus operandi. I look forward to working with you to make this site as useful as possible!