## The Assessment Project

#### Figure 1. Sample Open-Ended Math Problems Used for Instruction and Assessment in Project Classrooms

Find the missing digits:

What if your class were playing a game and your teacher gave you these numbers: 4, 6, 7, 2. How would you put the numbers in the boxes to make the largest possible answer?

You have 24 square tiles. How many different rectangles can you make, using all of the tiles for each one? Draw each rectangle on graph paper, label each side, and write the multiplication fact it shows.

What would you tell someone about multiplying by 1?

Gina says that when she can't remember a multiplication fact like 9 x 3, she turns it around to 3 x 9, and often she remembers that one. Can she do that? What would you tell Gina?

Joe had 5 sheets of paper. Each sheet had 4 large circles on it. In each circle, there were 2 stars. How many stars were there in all? Draw a picture to show how you got your answer.

I have 17 wheels, and each one is on a bike or a trike. How many bikes and trikes do I have? Is there just one answer? Explain how you did the problem.

## The Struggles

## The Successes

For many teachers, these successes continued into the next year. In two schools, by mutual agreement, we continued to meet with teachers on a less demanding schedule. In the spring of the second year, teachers' comments at an inservice given by one team for other teachers in their school showed a thorough understanding of project issues and ownership of original project aims:Teacher 1:The CU [University of Colorado at Boulder] people felt that if you taught the strategy, that was just like teaching an algorithm. So we wanted the kids to pull their own resources from their head. Each day we gave them problems that would be different and use a different strategy. We didn't want them to think that yesterday we used a table and then look at a problem today and automatically make a table....During my debriefing time on the following day, I wrote on poster paper three different ways that three different children solved the problem to show them that there is more than one way to solve the problem.Teacher 2:Confession time here. I remember five years ago, three years ago, we lived with that textbook. Well, we haven't used our math book too much this year. Maybe a little bit as a resource, but it's not like, page 36 today, 38 tomorrow, and the next day is more on 40.... So what I'm going to show you are just some examples that two years ago I would have said, “No way. Third graders cannot handle this.” But it's amazing, they can when they're exposed to it.

*scoring for*(the intended

*construct*, in measurement terms). By working with expository text, teachers were able to refocus attention on the purpose of summarizing (Davinroy and Hiebert 1994). They also became clearer about the multiple dimensions buried in their scoring rubrics—a tension that some resolved by giving two scores; for example, one for reading comprehension and one for writing, or one for the explanation and one for getting the right answer. Rubrics in math also shifted to focus more on the reasonableness of the answer and explanation rather than the accuracy of the calculation.

#### Figure 2. Sample Student Response to One Step from a Maryland Math Assessment Task

In the baseline year, most low-scoring students could not fill out the table in Step 4 and could not write an explanation. After the project year, low-performing students in the participating schools most frequently gave wrong answers of either 15 or 60; nonetheless, they completed the table, and gave explanations with real mathematical content (for which they received partial credit). For example: I counted by fours which is 60, then I went in the ones which is 15.On the cups as you go along you count four more each time.

## Implications for Staff Development

- appropriate materials to try out and adapt;
- time to reflect and to develop new instructional approaches; and
- ongoing support from experts to learn (and challenge) the conceptual bases behind intended reforms.