Education ∪ Math ∪ Technology

# Month: March 2015(page 1 of 1)

Almost every school district across the United States is thinking about how they use data to inform instruction. Not all of them are doing so in ways that I think are likely to lead to useful change.

Below is an example of the kind of data that has no useful impact on instruction ever. The data content in the picture below is high but the information content is low. How exactly is this information supposed to help a teacher make sense of what she should do with her students?

Here’s a slightly more useful variation on the same data where it has been organized to potentially be suggest some next steps. However, there’s a critical piece missing in this data – the actual task students did! Without knowing what the questions were that were asked and without knowing what the responses below refer to, this is meaningless information. The only take-away that I have from the information below is that it is unlikely for these multiple-choice questions that students were guessing.

This leads me to believe that one of the issues we have with this data is that it has compressed the information about what students did to such a great degree that it is impossible to use the information meaningfully. All of the rich work students have done and what they have thought about has been compressed into a few numbers and consequently making decisions based on those numbers alone is incredibly difficult.

What can we do differently? One option is to consider hypotheses about why students may have actually chosen those multiple choice responses like the following. Again, the question itself is critical but at least this leads to some potential things to give feedback to students about.

But these suggestions for what students thought about if they chose a response are just hypotheses. They are thought through from an adult’s perspective on the mathematics and these kind of analyses are rarely informed by detailed and thoughtful research into children’s mathematical thinking.

While multiple-choice questions are an inexpensive and relatively time-efficient way to gather evidence of student performance, they make it difficult to really capture the richness of student thinking. What, for example, do you think the student below was thinking about when they worked on this problem?

While it is clear that this sample of student work provides potentially powerful information, the challenge with looking at individual student work is it can be extremely time-consuming and challenging to study a variety of student work and come up with systematic responses to that group of students. We have to somehow combine the richness of information provided by what students actually did with a systematic approach so that we can approach teaching an entire classroom full of students.

One approach we have used in our project to diagnostic assessment is to systematically look at student work and decide what approaches students took and attempt to group their work with other students who appear to have thought similarly. A teacher I know literally uses the desks in his classroom and spreads out all of his student work across the room trying to make sense of the different mathematical ideas students used. Other teachers record their interpretations of the approaches the students used in spreadsheets, like the picture below suggests.

But still, no matter how we organize the information, the question remains, what do we do with it? A further challenge is, how do we use information on what we uncover about student thinking in a more timely fashion rather than after our class is over?

I’m going to suggest some approaches for both of these questions in a follow-up post, but I’d love to know how you respond to student data, both after the fact and during a class.

Last Saturday, I tried a new instructional activity called Choral Counting. This activity was recommended to me by Magdalene Lampert and comes out of the work she and others have done to support high-quality ambitious teaching.

I’m experimenting with these instructional routines in part because I hope to support teachers in the project I support in using them in their instruction, which I can hardly be expected to do without experiencing them from the inside myself. It’s clear that this activity is much richer mathematically than one might expect from just hearing about it. One finding I have already is that I have to do a much better job of planning how I will use my space when recording the numbers!

The basic idea of choral counting is easy — students count in unison and you write down the numbers as they chant, and then pause students to ask them questions about the numbers. What I learned Saturday is that there is a lot of potential in this activity to bring out rich mathematical ideas for discussion as a group.

I chose to start at sixteen and count by fives. While the counting was going on, I noticed my students paused a bit at 76. So I stopped and asked them why they think they paused. They also paused at 101, which during my lesson planning I had anticipated they would do. They also had different responses at 111 (121 was next, which I expected).

I don’t know why they found 76 more challenging. Maybe because it was the first time they had to use a number in either position higher than 6? 101 is clear – Many students thought “tendy one” and self-corrected before they spoke, which slowed them down. I thought this myself when I first counted through this routine! Many students said 121 instead of 111 and I remember my own son doing something similar when he was learning how to count.  I also paused a couple of times to ask students what pattern they noticed and at one point I asked them to predict what the number would be if we counted four more times. One student proved her answer by counting up by fives, another student said it would be twenty more because four times five is twenty.

Here’s the data collected by my assistant teacher (one of the children’s older sister comes to the class and she has happily volunteered to record information for me and to walk around the class generally supporting students).

Once we got to 131, I asked students what would be the next number that starts with 2. Two students came up with different mathematical arguments that the next number would be 201. One student said that the first digit was clearly a 2 and that the next digit must be a zero, because we are counting by fives. The last digit must be a 1 because our pattern always alternates between 1 and 6 and 1 is smaller than 6. The other student said that if we counted 19 fives or 95 higher from 101, that would be 196 which does not start with 2, but that if we counted 20 fives or 100 higher from 101, that would be 201 which must be the next number starting with 2 since we won’t count a number between 196 and 201.

Although the activity went longer than I expected, it was incredibly rich and worth doing.