While there can be overlap between some types of formative and summative assessments, and while there are both informal and formal means to assess students, in this article, I will primarily offer suggestions for informal, formative assessment for the mathematics classroom, particularly in the first of the three categories suggested by Clarke & Wilson:
- The student’s mathematical content knowledge.
- The student’s mathematical processes, such as reasoning, communicating, problem solving, and making connections.
- The student’s mathematical disposition, such as attitudes, persistence, confidence, and cooperative skills.
If you agree with the notion that words are labels for concepts, then you will want to use the 1, 2, 3, 4, 5 idea shown below:
Indicate your knowledge of each word by writing a 1, 2, 3, 4, or 5 in front of the word. The numbers signify the following five statements:
- I’ve never even seen the word/phrase.
- I’ve seen the word/phrase, but I don’t know what it means.
- I know the word/phrase has something to do with…
- I think I know what it means in math
- I know the word/phrase in one or several of its meanings, including the meaning for mathematics.
———— Unit 2: Using Measures and Equations ————-
- length of a segment
- central angle of a circle
- complementary angles
- vertical angles
- right triangle
- solving an equation
- rational number
- perfect square
- scientific notation
- right angle
- supplementary angles
- acute triangle
- equivalent equations
- irrational number
- perfect cube
- absolute value
- congruent segments
- vertex of an angle
- straight angle
- congruent angles
- obtuse triangle
- square root
- real number
- cube root
I prefer to use this as both an informal pre- and post-assessment. At the beginning of a new unit or chapter (and again at the end), I give students a sheet similar to the one shown above, with vocabulary terms for the unit listed. [The first time you use this idea, it is necessary to go over the five different levels of word knowledge, but students easily understand the idea that there are words they have never heard of and words that they know in several ways (and everything in between these two).] It is important to pronounce the words as the students read them and rate their own level of knowledge of the word because there are words that students recognize when they hear them but don’t recognize when they see them. Then, to assess content knowledge, for all words that the students rated as 4’s or 5’s, ask them to write their best understanding of what that word means in mathematics. This is not used for a grade but rather, as formative assessment to give an idea of students’ understandings of the concepts before and after the unit of instruction.
A second way of assessing students’ content knowledge, is giving students a sheet with 5 rows and 4 columns at the beginning of the week. Then, each day, either as students enter class, or as the closing activity for the day, four problems from a previous day’s lesson or homework are given, and students enter each problem (and solution) in the four spaces for the day. The teacher can check these quickly or have a row grader check them. These may be collected each day or at the end of the week, depending on the teacher’s plan for using the assessment information.
The third suggestion for formative assessment of content knowledge is performance assessment. Entire articles (and books) have been written on the next suggestion for formative assessment of mathematical content knowledge, but even though I cannot fully explain it in the context of this article, I would be remiss not to mention the idea of performance assessment. Performance assessments are assessments “in which students demonstrate in a variety of ways their understanding of a topic or topics. These assessments are judged on predetermined criteria” (ASCD, 1996, p. 59). Baron (1990a, 1990b, and 1991) in Marzano & Kendall (1996) identifies a number of characteristics of performance tasks, including the following:
- are grounded in real-world contexts
- involve sustained work and often take several days of combined in-class and out-of-class time
- deal with big ideas and major concepts within a discipline
- present non-routine, open-ended, and loosely structured problems that require students both to define the problem and to construct a strategy for solving it
- require students to determine what data are needed, collect the data, report and portray them, and analyze them to discuss sources of error
- necessitate that students use a variety of skills for acquiring information and for communicating their strategies, data, and conclusions (p. 93)
Begin exploring various formative assessment tools with your students to determine their content knowledge in mathematics. You will learn a great deal – and then be able to help your students learn even more!