Assessments identify the mathematics content that is most valued for student learning. Just as important as mathematics standards and instructional practices, assessing learning is a hallmark of effective instruction. Summative assessments at particular grades measure students' mathematics learning for purposes of student, teacher, or school accountability. Formative assessments given throughout the school year provide feedback on student mastery of recently taught content and are useful for teachers in guiding instruction.
APEC Mathematics Assessment Database
The APEC mathematics assessment database provides sample assessment items. This database is part of a newly started United States project to collect sample items from summative assessments of APEC member economies. Each assessment item is reported by mathematics strand and by percentage of students answering the item correctly, which serves as an indicator of item difficulty. The assessment database includes assessment items from the following sources:
- Hong Kong, China: Territory-Wide System Assessments for grades 3 and 9, 2007
- Japan: Assessments for grades 6 and 9, 2007
- New Zealand: Secondary School Assessments, 2007
- United States: Massachusetts Comprehensive Assessment System, 2007
The Massachusetts assessment was selected for the United States because students from that state consistently perform better than students from any other state on the National Assessment of Educational Progress (NAEP). However, a comparison of the Massachusetts and Hong Kong Assessment shows there is work still to be done in the U.S. An analysis of the assessments revealed that items on the Japanese assessment are more mathematically challenging than those on Massachusetts assessment. The most difficult problem on the Massachusetts assessment was a number line problem. The Full Assessment Materials are available on a separate Wiki page, as are Overviews of Assessments found in the Database.
For the same grade (6th), the Japanese assessment questions appear to include items that are more mathematically challenging than Massachusetts assessment questions when analyzed by percent of test takers answering the question correctly.
To illustrate, in the example below, the most difficult Massachusetts problem with only about 25 percent answering correctly is a number line problem that challenges the student to recognize that the number line is marked off in thirds and then to perform a straightforward computation of 4 1/3 -1 2/3 = 2 2/3. Only 26 percent of Massachusetts sixth graders taking the assessment in 2007 gave the correct response to this item.
The Japanese example below is a much more conceptually demanding multi-step geometric math problem. A student must use map-reading skills, know the formulas for the areas of rectangles and parallelograms, use rules about parallel lines, and perform multi-digit multiplication. The problem also includes distracters, that is numbers not required to solve the problem. Correctly answering this problem requires a much greater application of mathematical thinking, conceptual understanding, and procedural knowledge than the single-step problem of Massachusetts' assessment above. Eighteen percent of Japanese sixth-grade students answered this item correctly.
- Massachusetts Grade 6 (2007) most difficult problem (26% correct)
Question: Using the number line below, what is the distance between point A and point B?
Answer: 2 2/3 fraction or equivalent
- Japan Grade 6 most difficult problem (18% correct)
Question: There are two parks close to Hiroshi’s house, as shown in the map below. Which park has the larger area, Central Park or East Park? Write your answer and the reason for your answer using words and mathematical expressions.
Answer: Central Park: A = 70 m x 150 m = 10,500 m2
East Park: A = 110 m x 100 m = 11,000 m2
East Park has the larger area.