# Teachers – Formative Assessment – Informal Assessment of Students’ Content Knowledge in Mathematics

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:

1. The student’s mathematical content knowledge.
2. The student’s mathematical processes, such as reasoning, communicating, problem solving, and making connections.
3. 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:

1. I’ve never even seen the word/phrase.
2. I’ve seen the word/phrase, but I don’t know what it means.
3. I know the word/phrase has something to do with…
4. I think I know what it means in math
5. I know the word/phrase in one or several of its meanings, including the meaning for mathematics.

———— Unit 2:  Using Measures and Equations ————-

• continuous
• opposites
• line
• length of a segment
• ray
• central angle of a circle
• complementary angles
• vertical angles
• right triangle
• solving an equation
• rational number
• perfect square
• discrete
• scientific notation
• endpoint
• midpoint
• angle
• right angle
• supplementary angles
• acute triangle
• equation
• equivalent equations
• irrational number
• perfect cube
• absolute value
• segment
• congruent segments
• vertex of an angle
• straight angle
• congruent angles
• obtuse triangle
• solution
• 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!

# Teachers – Formative Assessment – Informal Assessment of Students’ Mathematical Dispositions

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 third of the three categories suggested by Clarke & Wilson:

1. The student’s mathematical content knowledge.
2. The student’s mathematical processes, such as reasoning, communicating, problem solving, and making connections.
3. The student’s mathematical disposition, such as attitudes, persistence, confidence, and cooperative skills.

Disposition is defined as “one’s usual mood; temperament, a habitual inclination, tendency.” With teenagers, it is not always easy to determine their disposition or temperament regarding anything, including mathematics. Too often, if their reference group has decided that “school is not cool,” then it is mandatory that they use all their body language and facial expressions (and sometimes words) to indicate their disdain for our beloved subject. We mustn’t accept this at face value (no pun intended). The ideas in this article will allow you to determine your students’ mathematical dispositions (sometimes without their knowing it!).

The first idea I want to offer came about almost by a fluke. Another teacher and I were creating an assessment instrument for some research we were conducting. We had about half a page left on this eight-page instrument – and didn’t want to waste the paper – so we decided to pose the prompt shown below. It turned out to be the best question of the whole assessment! Consider using this prompt at the beginning of the year – and then several other times throughout the school year to get a sense of the changes taking place.

As a math student in this class, I rate myself on the following scale (put an X on the scale where you rate yourself). 1 = Probably the worst in the class; 5/6 = Not too bad; not too good; 10 = Totally awesome! Maybe the best in the school

The reason I rated myself as a/an ____ on the scale above is because:

I have found that adolescent students are willing and able to be more truthful when asked to write than when asked to share their thoughts publicly. For this reason, I use learning logs as often as possible to learn about students’ dispositions toward mathematics. The term “learning log” is not one that I originated, but it is one that fits my philosophy of how writing looks in the mathematics classroom. The first part of the phrase “learning log” states the purpose of the writing: learning. The second part of the phrase “learning log” connotes a particular format, that is, running commentary. A log is not meant to be a polished piece of writing, taken through draft after draft. Commander and Smith (1996) define the purpose of learning logs as “reflections on specific cognitive aspects of learning…. (emphasizing) the connection or personal engagement with academic skills and techniques” (p. 447).

Using learning logs provides you with a variety of ways to assess students’ attitudes, beliefs, and stereotypes about mathematics. The following are some writing prompts I’ve found useful:

• What does a mathematician look like?
• My ability to do math is…
• When I am in math class, I feel…
• Mathematics has good points and bad points. Here’s what I mean…
• I study, I pay attention, I take notes, I read my math book, but I still don’t get math. True or False? Explain your answer.

Students’ answers to questions such as these provide insight to the teacher as s/he plans instruction. Ignoring students’ dispositions towards mathematics is done at teachers’ – and students’ – peril.

It’s not in anyone’s best interest to think that formative assessment is something that is ‘added on’ to our already full curriculum. Formative assessment is part of good teaching. There should be a seamlessness between instruction and assessment. Keep in mind that the word “formative” comes from the Latin word meaning “shape or form.” Formative assessment has as its purpose to shape upcoming instruction. Use and/or modify the ideas offered in this article. You will find that your instruction is more targeted and more effective. Then, design more ideas of your own – and share them with as many other teachers as possible.

# Teachers – Summative and Formative Assessment in Mathematics – What Are the Differences?

I’m a big fan of using definitions as a starting point for thinking about a topic…so let’s look at a definition of assessment from the National Council of Teachers of Mathematics (1995):

Assessment is…the process of gathering evidence about a student’s knowledge of, ability to use, and disposition toward, mathematics and of making inferences from that evidence for a variety of purposes (p. 3).

Depending on your age, this definition may describe the experience you had with assessment in mathematics during your school career, but for most readers, “testing” was really the only kind of “assessment” we knew. Like clockwork, at the end of every few sections of the math book, there would be a quiz (for a GRADE) and at the end of every chapter, there would be a TEST (for a MAJOR GRADE). Then, no matter what grades any of us received, we would go off to the next chapter, where the cycle began again.This type of testing (of which there are many varieties) is known in today’s parlance as “summative assessment,” defined as

“a culminating assessment, which gives information on student’s mastery of content” (Association for Supervision and Curriculum Development, 1996, p. 60).

The principal characteristics of summative assessment are that it:

1) occurs at the conclusion of a learning activity,

2) is to make a final judgment,

3) may compare students to other students, and

4) often results in a grade or some other ‘mark.’

In contrast, the principal characteristics of formative assessment include that it

• occurs during learning activities/experiences,
• is for the purpose of improving the learning, and
• will inform the teacher so that s/he can make adjustments if needed.

A useful definition of formative assessment is

“assessment which provides feedback to the teacher for the purpose of improving instruction” (ASCD, 1996, p. 59).

This concept of assessment meshes nicely with the NCTM definition shown above (i.e., “the process of gathering evidence about a student’s knowledge of, ability to use, and disposition toward, mathematics and of making inferences from that evidence for a variety of purposes”). Formative assessment – with or without that name – has always been around – depending on individual teacher’s attitudes towards this.  For the teacher who believes, as Grant Wiggins does, that “Good teaching is inseparable from good assessing,” there has always been an ongoing cycle of teaching, assessment, of the teaching, reteaching (as necessary), assessment, teaching, and so on. “Assessment should serve as the essential link among curriculum, teaching, and learning” (Wilcox & Zielinkski, 1997, p. 223).

So, the next time you hear others talking about assessment, ask if they are referring to formative or summative assessment. That will help you know what questions to ask next.