What about the related equation that determines the kids we want in math? To move students toward successful math learners, we must first understand where they are mathematically. e = evaluate. Diagnosing math understanding, though, is complicated. A general math achievement test, like a state test or typical progress monitoring assessment, only provides an indicator of general health. Is the kid about where he or she is supposed to be? If not, you don’t know why. Since a general assessment covers a broad range of material, you typically only have one or two items related to a specific area of content. How much can you tell from that little bit of information?
Let’s look at some items related to fractions to dig into this question. Here’s one from the grade 8 NAEP test in 2007. Only 49% of students answered it correctly. The most common incorrect response was E, which, oddly is the only sequence in which both the numerator and denominator go from greatest to least, the opposite of what the question requires. However clever our error analysis, we will never know if the selection of E represented a common misconception (that smaller numbers, in this case numerators, means bigger quantities when it comes to fractions) or just a futile guess with the last available choice.
Even individual correct responses can be deceiving. We did some field research for a fraction program Tom Snyder Productions is releasing this fall. We found students who consistently exhibited their understanding of adding fractions with problems like this one. Note how the numerators are added while the denominator remains constant. The student even shows the answer in simplified form. You might conclude from this single item that the student “understands” adding fractions. However, the same student suddenly forgot this understanding when confronted with the task of adding fractions of unlike denominators. Where did the fifteenths come from? Analyzing this second item in isolation might lead one to conclude that the student is applying whole number concepts to fractions. But that wasn’t the case with the previous problem. To diagnose what’s really going on with this student we need to dig deeper.
What we really need, I think, are layered assessments. The high level ones -- periodic achievement tests -- act as triage. They let us know which students need more in-depth diagnosis. But what troubles should we diagnose for? I suggest targeted key concepts, like those identified by the NCTM Curriculum Focal Points or the National Math Advisory Panel’s algebra foundations. Focus on the foundations, like fluency with whole numbers and fractions. Employ targeted, deep assessments to get a true window into what’s happening inside our struggling students’ heads.