An illustration: Mathematical Task Analysis

We spend a short time in-class going over the four categories. Each group of students (students sit in groups of three to four) opens a copy of the textbook to a problem set and identifies the task level for three problems. I then introduce the concepts of ``funneling'' and ``scaffolding.'' Funneling occurs when a task's level is reduced through hints, suggestions, or partial completion of the problem on the part of the instructor, tutor, or peer. Scaffolding is support offered to the student, usually through probing questions, that does not reduce the cognitive level of the task. The students all recognize the ``I don't know how to get started'' plea and the easing of cognitive difficulty level possible through funneling.

This 10 minute activity reaps great rewards later in class on several levels. First, the activity provides language for students to use in communicating with each other. I have heard students ask each other ``do you really need funneling?'' Sometimes the answer is ``yes'' and the group offers support for building understanding. Sometimes the answer is a sheepish ``no'' and the result is a student who redoubles efforts to engage with the mathematical task. A second benefit of the activity is that students can identify what they do well. Some students report analyzing their homework and exam problems to ``help decide which ones I should really worry about.''

In Fall 2005, student familiarity with the Task Analysis Guide allowed them to complain to me quite effectively. I was absent from school, attending a conference, and asked a graduate TA to ``cover for me'' in Liberal Arts Mathematics^{3} while I was gone. When I returned from the conference, during the usual *Comments and concerns* time at the beginning of class, six of the 34 students raised their hands. The story that emerged was that the person who had covered my class was ``nice but she treated every problem like a procedures - without- connections question'' and ``she funneled so much all over us there was nothing for us to do.'' Several students bemoaned the fact that ``we had to watch her do math, we didn't get to do any!'' The graduate teaching assistant who had taught the class reported being disconcerted by the questions my students asked during class. The topic for the day was conversion factors between standard and metric units. Some students had asked if scientific notation was connected to the idea of converting measures. The TA had said, ``No, that's in Chapter 3, this is Chapter 2.'' The student's response had been ``I know they are in different chapters, but they are both math so they must be connected somehow. What's the connection?'' The students reported that they decided scientific notation was a very convenient tool for moving within the metric system and wondered why the TA did not note it.