ASSESSING STUDENTS UNDERSTANDING THROUGH A PROBLEM-BASED TASK
Finally, assessment is what guides our instruction. There are many formative and sum- mative assessment strategies that teachers employ in their mathematics classroom. We used the five strands of mathematics proficiency to assess student understanding. In the condensed version of Adding it Up (NRC, 2001), the authors used Understanding, Computing, Applying, Reasoning, and Engaging (UCARE) as an acronym to stand for the five strands.   
- 4. Reasoning: (Adaptive Reasoning) Using logic to explain and justify a solution to a problem or to extend from something known to something not yet known.
- 5. Engaging: (Productive Disposition) Seeing mathematics as sensible, useful, and doable—if you work at it—and being willing to do the work (NRC, 2001, p. 9).
We used the five strands as assessment criteria in our rubric for assessing students’ mathematics proficiency. In this UCARE Rubric, we provided a place for comments because we believe that this assessment is more important as a way to gauge how the students are showing these strands of proficiency and rating scale would not be accurate. The comments can help gauge the areas that a teacher may want to focus on for the development of the child’s mathematical proficiency (see MMI Toolkit 2).
In one of the research lessons, we presented the students with the problem below called Measuring Cups (see Text Box 2.2). It was a problem that was rich in doing mathematics and provided opportunities for teachers to assess the different dimensions on the UCARE Rubric.
Text Box 2.2 A Math Happening 2: Measuring Cups
Lucy has measuring cups of cups of sizes 1 cup, У cup, 1/3 cup, and У cup. She is trying to measure out 1/6 of a cup of water and says, “if I fill up the У cup and then pour that into the 1/3 cup until it is full, there will be 1/6 cup of water left.”
- 1. Is Lucy’s method to measure 1/6 of a cup of water correct? Explain
- 2. Lucy wonders what other amounts she can measure. Is it possible for her to measure out 1/12 of a cup? Explain.
- 3. What other amounts of water can Lucy measure?
- —Problem from the Illustrative Math Website
One of the students excitedly worked on this problem at the problem center. He demonstrated all aspects of the UCARE criteria. He showed engagement with the task using the measuring cups experimenting and making conjectures. Soon he and his partner were explaining to each other, and he used what he knew to reason through Lucy’s method. He applied what he knew about fraction equivalence to demonstrate
Figure 2.4 Student engaged in mathematizing the problem. Source: Authors.
that he could formulate and carry out a plan and solve the problem using appropriate math strategies by drawing a picture and connecting it to the procedure of renaming the fraction to equivalent fractions. In this way, he demonstrated mathematical proficiency in terms of understanding, computation, application, reasoning, and engagement.
Think about it!
How can you use the UCARE rubric to assess your students’ mathematics proficiency? What are some other ways you are able to assess these important strands of Conceptual Understanding, Procedural Fluency, Strategic Competence, Adaptive Reasoning and Productive Disposition?
-  Understanding: (Conceptual Understanding) Comprehending mathematicalconcepts, operations, and relations—knowing what mathematical symbols, diagrams, and procedures mean.
-  Computing: (Procedural Fluency) Carrying out mathematical procedures, suchas adding, subtracting, multiplying, and dividing numbers flexibly, accurately,efficiently, and appropriately.
-  Applying: (Strategic Competence) Being able to formulate problems mathematically and to devise strategies for solving them using concepts and proceduresappropriately.