Conceptual Understanding
Let's start with an area in which there is a clear victory for constructivism: the importance of conceptual understanding. There is a large literature demonstrating the limits on generalization of math skills that can occur when instruction focuses exclusively on learning facts and procedures.
Consider decimal fractions. Learning how to add and subtract decimal fractions is often a struggle for children because of preconceptions they bring to the task from integer arithmetic. For instance, with integers, numbers with more digits represent larger quantities than numbers with fewer digits, e.g., 509 is more 7. However, this relationship breaks down for decimal fractions, where .7 is more than .509. Research has shown that children who receive sufficient practice doing arithmetic with decimal fractions may be able to add and subtract correctly while having a very narrow understanding of the meaning of the numbers they are computing. For example, they may be able to add .5 and .4, but may be unable to provide a number between .5 and .4, or to identify the pizza that has .5 of its slices remaining.
Studies have shown that providing children with practice on visual representations of decimal fractions can help children transfer their knowledge to problems on which they have not been trained. For example, one set of researchers had children practice locating decimal fractions on a visual number line. Zero and one were marked on the scale.. Given a fraction such as 0.509, children would try to locate it on the number line, and would receive feedback on whether they were correct.
Children who received practice on this visualization task were subsequently better able to solve other decimal tasks such as choosing the decimal fraction that is nearest to, greater than, or less than a target. Thus practice with the number line visualization gave children a way of representing the meaning of decimals that allowed them to transfer their learning to problems that were different in form but similar in the underlying mathematics to the problems on which they were trained.
Thus, conceptual understanding is a good thing because it can tie together mathematical tasks that might otherwise seem disconnected to a child.
After acknowledging the importance of conceptual understanding for mathematical proficiency, the story gets more complex. In the basic constructivist view, conceptual understanding is best arrived at through discovery learning with real-life problems, is sufficient in itself for mathematical proficiency, and can be generated through teaching methods that appeal to children's natural curiosity.
However, a number of studies have demonstrated that conceptual understanding can be produced through a variety of pedagogical techniques, including carefully sequenced direct instruction on the underlying principles, practice on a wide variety of problem types, and exposure to worked examples. In other words, the type of knowledge that allows a learner to solve many types problems doesn't need to be discovered by the learner to be effective. For example, children who practiced placing decimal fractions on a number line, in the study I've just described, still made mistakes after considerable practice. A frequent error was to ignore a zero in the tenths position, thus treating .03 as .30. A subsequent study showed that simply telling children to "notice the first digit" before they solved problems substantially enhanced their performance compared to basic discovery learning with the number line problems. In other words, providing direct instruction on what to attend to created conceptual understanding.
A second finding that complicates the basic constructivist view is that discovery activities may substantially compromise learning unless the child already has mastered the background knowledge that is relevant to the problem to be explored.
Cognitive psychologists have discovered that humans have fixed limits on the attention and memory that can be used to solve problems. One way around these limits is have certain components of a task become so routine and over-learned that they become automatic. For instance, children who are first learning to read have great difficulty paying attention to the meaning of the text because they are working so hard on sounding out the words. However, once decoding has become fluent, the child no longer needs to think about it, and the brain resources that the child has available to solve real-time problems can be directed towards understanding the meaning of print.
Although teaching children to understand the principles of multiplication by having them double a cookie recipe may seem like a good idea, if the child doesn't know the times table, the cognitive requirements of working with cookie dough and the cognitive requirements of multiplication will be too much to handle and will detract from learning. A clear instructional implication is that discovery activities should come later in a sequence of instruction, after children have acquired the requisite background knowledge to handle open-ended, real life problems. I am not saying that discovery activities should wait until graduate school. They can occur at any grade. However, the child should be prepared for the activity so that he can focus on what is important to the instructional goal. This is a basic principle of instructional design that is often ignored in approaches that rely on discovery activities.
A third finding that complicates the basic constructivist view concerns the inefficiency of discovery methods. Per the previous discussion of the number line, it may take a very long time for some children to discover that they have to pay attention to the first digit in solving decimal fractions. Why not tell them? As the famous psychologist Jerome Bruner said about discovery learning back in 1966, "it is the most inefficient technique possible for regaining what has been gathered over a long period of time." The algorithms, procedures, and facts of mathematics are powerful cultural inventions that have accumulated over thousands of years of human history. We simple cannot expect every child to discover the Pythagorean theorem. As with the previous point, this argues for discovery methods being used judiciously. There is a time and a place in an instructional sequence for children to apply their skills and to learn how to solve authentic open-ended problems. It is not everyday or everyway.
The final tenet of constructivism that requires a more nuanced view is that learning should be intrinsically motivated; it should be fun. Part of the rationale for discovery learning with real life problems is that drill and practice on math facts and procedures are not fun. If math isn't fun, children won't learn it, or so the thinking goes.
This slide includes data from 37 countries in which 8th grade students took the 1995 TIMSS assessment. The X axis represents how much the children said they liked math. The Y axis represents their math scores. The correlation is - .63 between math scores and how much kids like math. This is a very strong negative relationship. Children in countries that were at the top on math achievement, such as Japan, don't like math. Children in countries at the bottom of international achievement, such as South Africa, like math more than children from any other country in the world. Perhaps you find that as interesting as I do.
Don't get me wrong. I am not arguing for the pedagogical value of drudgery. Whatever we can do to make any academic subject more intrinsically interesting is worthwhile, as long as we don't compromise the content along the way. In mathematics, we can use computer-based animation, real-time feedback, appropriately placed real life problems, and social processes to make learning more fun. The younger the child, the more important these motivators will be. However, the type of practice that results in skills becoming automatic typically takes considerable repetition and time-on-task. This is true for hitting a tennis ball or playing the violin or decoding written text or doing mathematical calculations. Doing something over and over again until you don't have to think about it may rarely be great fun, particular in the context of other ways that children can spend their time. By failing to acknowledge that mathematical learning involves work, the United States may be placing a ceiling on the levels of proficiency that it can expect its students to achieve.