Prepared by David Kovara and Diana Wang (additions and comments by Frawley)
Over the past decade, cognitive scientists have become increasingly interested in how we learn mathematical concepts. Because mathematical concepts are notoriously difficult to acquire, cognitive researchers are attempting to discover more effective teaching methods, an attempt which relies primarily upon an understanding of how we process math concepts, and the type of instructions we are currently receiving. Issues in mathematical instruction basically fall into two categories: skill learning (knowing what do with an addition sign), and conceptual understanding (handling the big picture). Teaching methods have generally evolved through the current century from programs focused on skill learning (e.g., under the influence of individuals like Thorndike) to programs stressing instead conceptual understanding (e.g., under the influence of individuals like Bruner). Despite this pendulum-like trend, however, it is generally recognized that it is not a matter of becoming either skillful or conceptually aware, but, rather, that both are necessary for mathematical success. Ongoing work on how children learn mathematics would help us determine how better instructional programs can be designed since they can take into account how children learn math and maintain the relationship between skill and understanding.
Note: compare the difference between -- and interdependence of -- procedural and declarative knowledge, discussed much earlier in the course with respect to memory.
One of the major complaints about teaching methods, in fact, is that current educational systems have produced an artificial separation between skill learning and conceptual learning, a separation that may, in fact, be too difficult to reconcile. But scientists are trying. There is a big reform movement currently underway attempting to determine a set of standards applicable to successful math instruction. There has even been a "coming-out" of the back-to-basics movement focusing on skills and an emphasis on greater conceptual understanding. The search for standards essentially relies on the following theoretical arguments:
1. Understanding is the construction of a relationship between facts, ideas, actions, and so on. [That is, knowledge involves the constructed of interconnected representations of a domain.]
2. Understanding places learners in a good position to make sense of subsequent instruction. [That is, interconnected representations provide a good basis for future learning.]
3. Understanding allows students to invent new procedures and modify old ones for new problems. [That is, interconnected representations as a good base for future learning also provide the means for generalizing to new problems.]
Take, for example, the following problem: Mary has 28 stickers. She buys 35 more. How many stickers does she have now? An understanding of this problem requires at least the following:
1. quantities can be grouped and represented in various ways [is 28 four groups of 7, a series of individuals from 1 to 28, etc.?]
2. quantities/numbers can be decomposed and recombined without changing values
3. operations on quantities require that like units get combined
4. the skills to execute the above three
In trying to determine standards of understanding, however, scientists have discovered numerous problems. The definition of understanding itself is very controversial. Assessment of understanding relies on tasks, which vary in each scholastic environment. It is therefore extremely difficult to find reliable, consistent results.
More particularly, it is necessary to see that: (1) "understanding" and "skill" are not consistently defined in the reasearch on mathematics learning; (2) instruction itself influences the relationship between understanding and skill execution, and so there is no pure experimental situaiton; (3) the relationship between skill and understanding is dynamic and changes over time; (4) explanations of performance must go beyond correlations to causes. Suppose that we find that children make systematic errors in learning certain aspects of arithmetic (which they do!). We must be sensitive to all four of the previous factors to get the proper explanation. What is their understanding and what is their skill? How have they been instructed in this area in the past? How have their beliefs and skills changed? What predicts this behavior?
Professor Hiebert described a school-based experiment sought to create a stable, consistent environment within which relevant data could be gathered in relation to understanding. Hiebert and colleagues followed a group of children for three years (first grade through third) as they were taught math. These children were interviewed periodically to assess understandings and skills.
Half of the children learned in a conventional teaching environment. The curriculum followed the textbook, emphasized paper and pencil computational skills, and prescribed standard computational algorithms. The other half, however, learned math in an "alternative" environment, which situated problems in a story-telling context, encouraged the use of physical materials, allowed students to invent alternate solving procedures, and emphasized sharing and explaining procedures. Although the results showed advantages in both styles of teaching (conventional learners were better at addition, while not as successful at subtraction), the alternative teaching methods produced higher overall performance levels.
Despite the high levels of attention math learning has been receiving in the cognitive world, scientists are still far from determining the ideal curriculum. But as Pearla Nesha states in the end of her article assigned in preparation for Prof. Hiebert's lecture, "developments in research and instruction give hope that the more knowledge we gain about the cognitive depths underlying mathematical performance, the better and less failure-fraught our math instruction will become."