assessment workshop: may 16-18, 2004, washington, dc 1 classroom assessment in secondary mathematics...

Assessment Workshop: May 16-18, 2004, Washington, DC 1

Classroom Assessment in Secondary Mathematics

George W. BrightUniversity of North Carolina - Greensboro

[email protected]

Jeane M. JoynerMeredith College, [email protected]


Dynamic Classroom Assessment

• Professional development materials for middle grades and high school mathematics teachers; published by ETA/Cuisenaire.

• Product of a project which is supported in part by a grant from the National Science Foundation (Grant #9819914).

• All conclusions and opinions expressed are those of the authors and do not necessarily reflect the position of the Foundation or any other government agency.


Classroom Assessment

• Classroom assessment is the process by which teachers gather information about what students know and can do and then use that information to make more effective instructional decisions.

• Classroom assessment involves planning effective mathematics instruction based on understanding how students think about key ideas.


Why Classroom Assessment?

• Classroom assessment links teaching with learning.

• When classroom assessment is used appropriately, students learn more.

• Teachers need to understand more about how students think about mathematical ideas.


Need

Instruction in how students learn and how learning can be assessed should be a major component of teacher preservice and professional development programs. This training should be linked to actual experience in classrooms in assessing and interpreting the development of student competence.

(Pellegrino, Chudowsky, & Glaser, 2001, p. 14)


Overview of Professional Development

Core module: 30 hours (ten 3-hour sessions)• Model of classroom assessment and assessment methods• What students “know” and what they “show” they know• Questions to clarify and probe students’ thinking

Extension modules (three at 10 hours each)• Revisiting Students’ Understanding• Revisiting Student Self-Assessment• Revisiting Substance and Presentation

Total: 60 hours of professional development over time


Direct Effects

Black and Wiliam (1998) conclude from an examination of 250 research studies on classroom assessment that “formative assessment does improve learning” -- and that the achievement gains are “among the largest ever reported for educational interventions.” The effect size of 0.7, on average, illustrates just how large these gains are…. If mathematics teachers were to focus their efforts on classroom assessment that is primarily formative in nature, students’ learning gains would be impressive. These efforts would include gathering data through classroom questioning and discourse, using a variety of assessment tasks, and attending primarily to what students know and understand.

(Wilson & Kenney, 2003, p. 55)


Residual Effects

A recent review (Black and Wiliam, 1998) revealed that classroom-based formative assessment ... can positively affect learning… [S]tudents learn more when they receive feedback about particular qualities of their work, along with advice on what they can do to improve. They also benefit from training in self-assessment, which helps them understand the main goals of the instruction and determine what they need to do to achieve. But these practices are rare, and classroom assessment is often weak… Teachers must have tools and other supports if they are to implement high-quality assessments efficiently and use the resulting information effectively.



Fundamental Belief

When teachers understand what students know and can do, and when teachers use that knowledge to make more effective instructional decisions, the net result will be greater learning for students and a greater sense of satisfaction for teachers.


Foundation

Every assessment… rests on three pillars: a model of how students represent knowledge and develop competence in the subject domain, tasks, or situations that allow one to observe students’ performance, and an interpretation method for drawing inferences from the performance evidence thus obtained.


DCA provides opportunities to learn about each of these three elements.


What Teachers Bring to the Task

What one believes about the nature of learning will affect the kinds of assessment data sought and the chain of inferences drawn.


While most teachers are given a great deal of guidance on what to teach, they have considerable latitude on how to teach…. Teachers report designing their instruction using resources and strategies grounded in their background knowledge, experiences, and beliefs.

(Weiss, Pasley, Smith, Banilower, & Heck, 2003, p. 104)


Model for Learning

In cognitive theory, knowing means … being able to integrate knowledge, skills, and procedures in ways that are useful for interpreting situations and solving problems….

The situative perspective proposed that every assessment is at least in part a measure of the degree to which one can participate in a form of practice….

Some students, by virtue of their histories, inclinations, or interests, may be better prepared than others to participate effectively in this practice.

(Pellegrino, Chudowsky, & Glaser, 2001, p. 62-65)


Usefulness of the Model

The cognitive perspective can help teachers diagnose an individual students’ level of conceptual understanding, while the situative perspective can orient them toward patterns of participation that are important to knowing in a domain….

The cognitive perspective can help teachers focus on the conceptual structures and modes of reasoning a student still needs to develop, while the situative perspective can aid them in organizing fruitful participatory activities and classroom discourse to support that learning.

(Pellegrino, Chudowsky, & Glaser, 2001, p. 62-65)


Overview

There are three main ideas in the core module:

• Exploring a model of classroom assessment and a variety of assessment methods (Sessions 1–3)

• Revealing what students “know” and what they “show” us about what they know (Sessions 4–6)

• Asking questions to clarify students’ thinking (Sessions 7–9)


Model for Classroom Assessment

LearningTargets

AssessmentMethods

Inferences About

Thinking

Instructional Decisions

Purposes

Communication

Documentation


National Standards

The DCA materials respond to the recognition in national mathematics standards that classroom assessment is a critical part of effective teaching and learning and that improving the quality of learning occurs in classrooms, as individual teachers work closely with groups of students.


Division of Fractions

1. 6 ÷ 2/3 =

A. 9 B. 4 C. 1 D. 1/9

2. Draw a picture to represent 6 ÷ 2/3 and write the answer illustrated by your picture.

3. 2/3 of what number equals 6?


Discussion: Division of Fractions

• How are these three problems alike?

• How are the problems different?

• What inferences might we make based on the different student responses?


Another Solution

• Draw a picture to represent 6 ÷ 2/3, and write the answer illustrated by your picture.

• What do you know about this student’s reasoning?

• How sure are you about your inferences?

Student’s Response:

Answer: 9


Perimeter

• Here is a task:

Find the perimeter of the figure on the right. Show your work.

• Here is a variation:

Describe the shape shown on the right. Without actually computing the perimeter, list everything you know that would help you describe the perimeter.

111


Overview






Teachers’ Generalizations

At times, teachers’ knowledge about what students know and can do is unfocused; that is, teachers’ generalizations about students’ thinking may be highly influenced by memorable instances of students’ thinking rather than being based on carefully constructed inferences supported by clear frameworks for mathematics and students’ development.

Teachers’ instructional decisions flow from their generalizations about students’ thinking, so it is important that generalizations be as accurate as possible.


Conceptual Knowledge and Procedural Knowledge

We can think of conceptual knowledge as the “big ideas” in mathematics.

We can think of procedural knowledge as “ways to do things.”


Substance and Presentation of Mathematical Ideas

We can think of the substance of an idea as the “mental construct” that is in someone’s head.

We can think of the presentation of an idea as the way that mental images get communicated.


Combinations of Two Pairs of Ideas

Substance Presentation

Conceptual

Procedural


A Simple Problem

What is 2 more than 3 times 4?

1. How you would solve the problem?

2. How might students solve the problem?


Student Solutions to a Simple Problem

What is 2 more than 3 times 4?

Student A Student B

(3 + 2) x 4 = 20 3 x 4 = 12

+ 2

14

Student C Student D

3 x 4 = 12 + 2 = 14 (3 x 4) + 2 = 14


General Chart

Substance Presentation

Concept Describing a concept and creating examples and non-examples

Recognizing a concept in different representations

Procedure Explaining the reasons for steps

Carrying out the steps efficiently


Multiplying Mixed Numbers

This is one of the test items for the chapter on fraction operations:

3 2/3 x 4 1/2 =

Student A3 2/3 x 4 1/2 = (3 + 2/3) (4 + 1/2) = 12 + 2/6 = 12 1/3

If you were to write Student A a note, what would you say?


Multiplying Mixed Numbers

This is one of the test items for the chapter on fraction operations:

3 2/3 x 4 1/2 =

Student B3 2/3 x 4 1/2 = 32/3 x 41/2 = 1312/6 = 218 2/3

If you were to write Student B a note, what would you say?


Helpful Feedback

Feedback is most helpful when it

• is clearly focused on an important idea

• is specific to the task

• takes into account what students seem to understand

• supports conceptual development


A Key Distinction for Errors:

Miscommunicated Understanding

versus

Communicated Misunderstanding


Overview






Questioning as an Element of Instruction

Questioning is among the weakest elements of mathematics and science instruction, with only 16 percent of lessons nationally incorporating questioning that is likely to move student understanding forward.

Lessons that are otherwise well-designed and well-implemented often fall down in this area.

(Weiss, Pasley, Smith, Banilower, & Heck, 2003, p. 65-67)


Questioning Patterns in Instruction

By far, the most prevalent pattern in mathematics and science lessons is one of low-level ‘fill-in-the-blank’ questions, asked in rapid-fire, staccato fashion, with an emphasis on getting the right answer and moving on, rather than helping the students make sense of the mathematics/science concepts.

(Weiss, Pasley, Smith, Banilower, & Heck, 2003, p. 65-67)


Typical Questions

Unfortunately, … teachers’ questioning is dominated by recall questions….

If the teacher limits questions to a narrow band of procedural questions, the answers given may not be sufficient for the teacher to make informed inferences about the breadth or depth of students’ understanding.

That is, the teacher may take a series of correct answers by a student as evidence of understanding, when in fact it is very limited evidence merely of the student’s ability to give the correct answers.

(Wilson & Kenney, 2003, pp. 55-56)


Rigor

Fewer than 1 in 5 mathematics and science lessons are strong in intellectual rigor; include teacher questioning that is likely to enhance student conceptual understanding; and provide sense-making appropriate for the needs of the students and the purposes of the lesson.

Overall, 59 percent of mathematics/science lessons nationally are judged to be low in quality, 27 percent medium in quality, and only 15 percent high in quality.



Questions and Equity

Differences in assessment outcomes are often explained by using coarse categories for students, such as gender, race, or social class. This explanation tends to treat all those within a category as homogeneous.

(Wilson & Kenney, 2003, p. 56)

Classroom assessment, when appropriately carried out, opens a teacher’s eyes to the individual differences among students in a class.


Engaging Questions

• Engaging questions invite students into a discussion, keep them engaged in conversation, invite them to share their work, or get answers “on the table.”

• These are “open” questions that invite students into a conversation.

• How can we decide what value the question mark stands for?

Two similar rectangles

6

3

?

4


Refocusing Questions

• Refocusing questions help students get back on track or move away from a dead-end strategy.

• These questions respond to thinking that seems dead end.

• What are the corresponding parts of these rectangles?


6

3

?

4


Clarifying Questions

• Clarifying questions help students explain their thinking or help you understand their thinking.

• In response to a student who says that the answer is 5, the teacher asks, How did you get 5?


6

3

?

4


Clarifying Questions to Reveal Students’ Thinking

The purpose of clarifying (probing) questions is

• getting more information out of a student’s head,

• not putting more information into a student’s head.


Water in a Bucket Problem

A 1-kilogram bucket contains 99 kilograms of water.

1. What percent of the total mass of the water and bucket is water?

2. If 5 kilograms of water are removed, what percent of the combined mass of the remaining water and the bucket is water?

3. How many kilograms of water must be removed from the bucket so that the remaining water is 50% of the combined mass of the water and the bucket?

4. Suppose W is the amount of water removed and R is the percent of water in the combined mass of the remaining water and the bucket. Write an equation that shows the relationship between W and R.


Goals: Water in a Bucket Problem

• What learning targets might you be teaching or assessing with this task?

• What parts of the relevant mathematics are conceptual? What parts are procedural?

• Write questions that you could use to probe students’ conceptual understanding.

• Write questions that you could use to probe students’ procedural understanding.

• Which type of question is easier to write?


Surprises from Conversations

When teachers talk to students, they are frequently surprised by the ways that students reason mathematically.

Sometimes the surprise is in the high level of sophistication students exhibit, and sometimes it is the unusual ways that students interpret what they are asked.


Talk Less.

Listen More!


Teacher Knowledge

Teacher content knowledge is clearly not sufficient preparation for high quality instruction….

Teachers also need expertise in helping students develop an understanding of that content, including knowing how students typically think about particular concepts; how to determine what a particular student or group of students is thinking about those ideas; and how the available instructional materials … can be used to help students deepen their understanding.



Teacher’s Knowledge of Students’ Thinking

Teachers need to be able to hear and see expressions of students’ mathematical ideas and to design appropriate ways to respond.

A teacher must interpret students’ written work, analyze their reasoning, and respond to the different methods they might use in solving a problem

(Kilpatrick, J., Swafford, J., & Findell, B., 2001, p. 369-370)


In Conclusion

The real importance of classroom assessment is the way it helps teachers make instructional decisions so that they can align instruction more closely to the needs of students.

Information from classroom assessment has no impact until actions are taken based on our inferences about thinking and understanding.

assessment workshop: may 16-18, 2004, washington, dc 1 classroom assessment in secondary mathematics...

Documents

formative assessment

assessment methods

learning students

classroom questioning

revisiting students

variety of assessment

weak teachers

achievement gains