2013 Conference Proceedings - University of Nevada, Las Vegas

For example, let us consider what happens to these students as they begin to study three digitmultiplication. Suppose the teacher begins the study of three-digit multiplication with theproblem 234 × 23. One possible genotype of the traditional two-digit multiplication algorithm issomething like “Multiply the bottom right number by the top right number, then the bottom rightnumber by the top left number, then the bottom left number by the top right number and finallythe bottom left number by the top left number”. This genotype will typically lead to confusion(or even despair) when confronted with a three digit multiplicand, for “left-right” may or maynot admit a “center”. (This depends upon how fully or deeply ingrained a learning allele is.)However, a student who has a more iterative gene (e.g., “multiply the right-most bottom columnby all the columns on the top, then proceed to multiply the next bottom column to the left by allthe columns on the top, and keep doing this until there are no more bottom columns”) is likely tobe bored by the new problem, for three digit multiplication is nothing new. Notice carefully thatthe possible reactions to the “new” material range from “no connection to the old material” to“nothing is new here”. Certainly, many other reactions are possible, but even this exampleindicates that the genotype, not the phenotype, is important for learning.A similar analysis could be used if this two-digit multiplication were used as a precursor tomultiplying algebraic binomials (e.g., as a “warm-up problem” or to “activate prior knowledge”).In this case, a “left-right” student is likely, because of what amounts to a lack of distributiveproperty ability, to write something like (x + 2)(x + 2) = x 2 + 4.Based on these hypothetical students, it is apparent that the ability to solve a supposed “prerequisite”problem is not sufficient to determine if the necessary pre-requisites are part of astudent’s cognitive make up; the ability to solve such a “pre-requisite” problem is not sufficientto indicate readiness for learning. Although this may not be surprising to many, two importantpoints arise from this consideration: First, phenotype assessments, while easy to administer, donot result in a complete picture of student understanding. Second, curriculum should be focusednot on what types of problems (phenotypes) should be solved by students but rather on whatadditional genes are needed to combine with a student’s existing genotype. Although formativeassessments are common, we claim that more meaningful formative assessment is not possiblewithout considering the observation of genotypes. Thelen (1997, 2005) argued that metaphorsboth constrain and enable what we can see; the genetic metaphor calls for a different type offormative assessment. Further, considering the genetic processes that drive the dynamics ofProceedings of the 40 th Annual Meeting of the Research Council on Mathematics Learning 2013 191