chapter 03. 3 | 2 copyright © cengage learning. all rights reserved. informing our decisions:...

Chapter 03

3 | 2Copyright © Cengage Learning. All rights reserved.

Informing Our Decisions:Assessment and Single-Digit Addition


Mathematical Routine: How many squares are not shaded?


Conversation in Mathematics

• Discuss the method of assessment the teacher was using and what she was able to learn about the student’s problem solving abilities.


Assessment for Instruction

• Pedagogy


Why Alternative Assessment?

• Three components promoting systemic change: professional development, curriculum materials, & assessment. (Smith & O’Day, 1991)

• Assessment - least attention (Firestone and Schorr, 2004)

• Internationally - broad view of mathematical literacy (AAMT, 2002; NCTM, 2000) that includes a balanced acquisition of procedural proficiency and conceptual understanding


Notions from Principles and Standards for School Mathematics

• Assessment of instruction vs. Assessment for instruction

• Validity• Summative and formative• Accountability, stewardship • Traditional and alternative• Backwards Design


Backward Design

• Set general learning goal• Design and administer a pre-instruction

assessment• Determine your specific learning targets. • Determine acceptable evidence of learning• Design an instructional plan• Conduct interactive instruction/ongoing

assessment


Traditional Assessment

• Short Answer• Multiple Choice• Matching• Fill-in-the-blank, True-False• Raw Scores, Percentages, Checklists, Rubric

Scores


Item Writing Rules for Multiple Choice Questions

• Write a clear stem that does not require a reading of the options in order to be understood.

• Place most of the wording in the stem. This prevents having to select between lengthy answer options.

• Make sure the intended answer is clearly the best option.

• List options vertically.


Alternative Assessment

• Open-ended questions• Communication• Observations• Interviews• Journals• Performance Assessments• Portfolios


Open-ended Questions

• Answers to closed ended are predetermined and specific - # of primes between 10 & 20

• Open-ended allow for a variety of correct responses and elicit different thinking

• Both are appropriate for assessing students' mathematical thinking

• Open-ended take longer to score• Closed ended useful for covering broad range of

topics, but . . .• Don’t allow for the revealing of student thinking

like open-ended


Sam’s truck has a 20-gallon gasoline tank. Sam looked at his gauge and saw the reading below. What would be a reasonable estimate for how many gallons of gas Sam had used since he last filled the tank? Explain how you determined your estimate.

Example of an Open-ended Question


Communication

Communicate with and about math (NCTM, 1989) through:• Oral discourse (conversations, discussion, debates), • writing (essays, journals), • modeling and representing (manipulatives, pictures,

constructions), • performance (acting out, modeling)


Observations

• Observe with a specific goal in mind• Each child does not need be observed every

day• Assume role of a participant-observer; be part of

learning community, but also external to the environment.


Interviews

By conducting 1-1 interviews, we can assess:• Cognitive and affective development• How children model and communicate mathematical

concepts and skills

We conduct these interviews by:• Asking probing questions that guide them toward more

complex ideas• Asking prompting questions to help children attend to

misunderstandings and to scaffold success to the degree required


Journals

Through journal writing, we can:• Assess children's reflections of their own capabilities,

attitudes & dispositions,• Evaluate their ability to communicate mathematically,

through writing


Performance Assessments

• Students perform, create, construct, or produce• Assess deep understanding/ reasoning• Involve sustained work• Call on students to explain, justify, & defend• Performance is directly observable• Involve engaging ideas of importance & substance

(worthwhile math task)• Reliance on trained assessor’s judgments• Multiple criteria and standards are pre-specified and public

(rubrics)• There is no single correct answer (or solution strategy)• Performance is grounded in real-world contexts and

constraints


Portfolios

Portfolios are a collection of children’s work in which:• Children should be given the opportunity to provide input

regarding the portfolio contents• The type of items selected for the portfolio can be varied,

to reflect a real sense of the "whole" child


• Its contents are developed over time, allowing teachers to obtain information about children's learning patterns

• Items chosen by children - insight into their interpretation of their work, their dispositions toward mathematics, and their mathematical understanding


Recording Assessment Data for Alternative Assessments

• Rubric Scores• Checklists• Anecdotal Notes


Quick and Dirty Rubric

• 5 - Child really gets it, no errors• 4 - Child gets it, minimal errors• 3 - Child sort of gets it, inconsistent error pattern• 2 - Child doesn’t get it, consistent errors• 1 - Child is lost (sorry)


Comprehensive RubricRubric level Problem

Solving Communicating Reasoning Representing Connecting Procedural Conceptual

4 - Independent Understanding

Can solve the problem in two ways independ-ently

Can clearly explain the problem solving strategies

Can clearly justify the problem solving strategies

Can represent the problem in at least two ways independently

Can independently connect representations or strategies

Can solve the problem using a procedure independently

Can show thorough understanding of the problem and of the associated mathematics independently

3 Understanding with minimal help

Can solve the problem in two way with minimal help or one way independ-ently

Can clearly explain all but one part of the problem solving strategies

Can justify all but one part of the problem solving strategies

Can represent the problem in two ways with minimal help or one way independently

Can connect representations or strategies with minimal help

Can solve the problem procedurally with minimal help

Can show some understanding with minimal help

2 Understanding with substantial help

Can solve the problem at least one way with help

Can explain portions of the problem solving strategies

Can justify portions of the problem solving strategies

Can represent some of the problem with help

Can connect representations or strategies only with substantial help

Can solve the problem procedurally with substantial help

Can show some understanding with substantial help

1 Little understanding

Cannot solve the problem even with help

Cannot explain the strategies even with help

Cannot justify the strategies even with help

Cannot represent the problem even with help

Cannot connect representations or strategies even with help

Cannot solve the problem procedurally even with help

Cannot show understanding even with help


Video Analysis


Single-Digit Addition and Subtraction

• Content


Operation Sense

• Developing meanings for operations• Gaining a sense for the relationships among

operations• Determining which operation to use in a given

situation• Recognizing that the same operation can be applied

in problem situations that seem quite different• Developing a sense for the operations’ effects on

numbers• Realizing that operation effects depend upon the

types of numbers involved


How do Children Develop?

Problem Types• Join• Separate• Part-part-whole• Compare

Problem-solving Strategies• Direct Modeling• Counting• Known Facts• Derived Facts


Analyzing Problem Types

• Semantic versus Computational


Analyzing Solution Strategies

Direct Modeling• Joining all / counting all• Joining to• Matching

Counting• Counting on from first• Counting on from larger• Separating from• Counting down• Counting on to

Trial and error


Video analysis


Generalizations

• One More/One Less• Ten More/Ten Less• Combinations of numbers to ten• Commutativity• Doubles and near doubles• Making a ten


Steps in Which Generalizations are Developed

• Concrete, hands-on experiences• Using a model as a visual• Using symbols as a visual• Making mental calculations with the model in the

head• Making mental calculations using a generalized

rule, or known fact


Practice for Quick Recall

• Meaningful practice• Games• Music• Timed tests?

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