Learning Algebra, In Steps

Today's study breaks down the different skills needed by a student to be able to complete an algebra problem. It turns out that there are four separate types of knowledge needed, and a person must master the easiest before progressing to the next, etc. So, without further ado, here is:

"A Developmental Model for Algebra Symbolization: The Results of a Difficulty Factors Assessment", by Neil T. Heffernan and Kenneth R. Koedinger. Published in Proceedings of the Twentieth Annual Conference of the Cognitive Science Society, 1998, edited by Morton Ann Gernsbacher and Sharon J. Derry, pp 484-489.

The Experiment
During the first month of school, 74 urban ninth graders enrolled in a regular-level algebra class were given a test made by the experimenters that included word problems of different levels of complexity.

The first type asked for each student to make a single-operation formula out of numbers. Example: "Sue made 72 dollars by washing cars to buy holiday presents. She decided to spend 32 dollars on a present for her mom and then use the remainder to buy presents for her sisters. How much can she spend on her sisters?" Answer: 72-32.

The second type asked for each student to make a single-operation formula out of a combination of numbers and variables. Example: "Sue made 72 dollars by washing cars to buy holiday presents. She decided to spend 'm' dollars on a present for her mom and then use the remainder to buy presents for her sisters. How much can she spend on her sisters?" Answer: 72-m.

The third type of question asked each student to make a double-operation formula out of only numbers. Example: "Sue made 72 dollars by washing cars to buy holiday presents. She decided to spend 32 dollars on a present for her mom and then use the remainder to buy presents for each of her 4 sisters. She will spend the same amount on each sister. How much can she spend on each sister?" Answer: (72-32)/4.

The fourth type was the most challenging. Each student was asked to make a double-operation formula using both numbers and variables. Example: "Sue made 72 dollars by washing cars to buy holiday presents. She decided to spend 'm' dollars on a present for her mom and the remainder to buy presents for each of her 4 sisters. She will spend the same amount on each sister. How much can she spend on each sister?" Answer: (72-m)/4.

The actual questions were varied, such that any given test paper only had one question about Sue washing cars, but the test questions were all equivalent to the examples listed above and in the paper.

The Results
The lowest-performing students were able to do the simple arithmetic formulas (Type 1). The next step was to to the more complicated arithmetic expressions (Type 3). Once they are good at that, they start to understand the single-operator variable problems (Type 2). All students in the study who could correctly answer the Type 2 questions also correctly answered the arithmetic ones. The last step to being able to master algebra is the double-operator variable problems (Type 4).

Most people assume that it is the language used in word problems that makes algebra so hard; given the design of this experiment, the researchers have very conclusively shown that language comprehension is not the issue. After all, the language used in the Type 2 and 4 problems was exactly the same, and yet one was significantly more difficult than the other.

The researchers ran a separate experiment to see what is needed by students to make that final leap into fully grasping algebra. Since it seemed like students have trouble combining multiple steps together when there are variables involved, they trained some students for an hour on the following types of problems, dubbed symbolic substitution problems: "Let X=72-m. Let B=X/4. Write a new expression for B that composes these two steps." Answer: (72-m)/4. Student performance on algebra word problems increased after just one hour of practicing these sorts of problems.

What this shows is that it is not only the language of word problems that needs to be understood, but the language of abstract mathematical concepts needs to be understood, as well. Not only do students have to understand what the problem says, but how to create the right abstract representation, which are two very different things!

Here's a way of thinking about it: Lots of people say that if you can read, you can cook. My husband says that this baloney, because he can read, but he can't cook--cookbooks are written in a foreign language, as far as he in concerned! All of those verbs mean something very specific, and involve skills that he generally does not have; everything is also written in shorthand. He says it's like this: If you can read a cookbook and perform all of the required actions, then you can cook!

How Waldorf Education Teaches Algebra
Arithmetic is introduced in the first grade, and its form takes on both of the first two types of algebra skills, both inside and outside of oral word problems. For instance, they try to figure out what can be added to 7 to equal 12 (a type 2 problem). This early work with numbers and variables skips over the work with complex number problems; however, this is gradually introduced the next couple of years with number adventures, where the children do a lot of operations to a given starting number in their heads. While they may not be writing down these concepts, they are busy internalizing them.

The final step is introduced slowly over the 6th-8th grade years (particularly in 8th grade), as the children work with formulae that they have themselves discovered, the manipulative rules of algebra (many of which they discovered but did not name back in first grade, like 1+3=4 and 3+1=4 (the commutative property)), polynomials (which can be designed into symbolic substitution lessons), and finally, algebra word problems.

The early, deeply-ingrained numeracy lessons of the Waldorf curriculum lay an excellent foundation for introducing the abstract concepts of algebra; based on the study and my inquiries with a Waldorf teacher, the hurdles that the students need to jump over are slowly laid out in a logical way, making 8th grade algebra a fun, playful challenge, instead of drudgery.

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.