Chapter 03

3 | 2 Copyright © Cengage Learning. All rights reserved. Informing Our Decisions: Assessment and Single-Digit Addition

3 | 3 Copyright © Cengage Learning. All rights reserved. Mathematical Routine: How many squares are not shaded?

3 | 4 Copyright © Cengage Learning. All rights reserved. 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.

3 | 5 Copyright © Cengage Learning. All rights reserved. Assessment for Instruction Pedagogy

3 | 6 Copyright © Cengage Learning. All rights reserved. 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

3 | 7 Copyright © Cengage Learning. All rights reserved. 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

3 | 8 Copyright © Cengage Learning. All rights reserved. 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

3 | 9 Copyright © Cengage Learning. All rights reserved. Traditional Assessment Short Answer Multiple Choice Matching Fill-in-the-blank, True-False Raw Scores, Percentages, Checklists, Rubric Scores

3 | 10 Copyright © Cengage Learning. All rights reserved. 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.

3 | 11 Copyright © Cengage Learning. All rights reserved. Alternative Assessment Open-ended questions Communication Observations Interviews Journals Performance Assessments Portfolios

3 | 12 Copyright © Cengage Learning. All rights reserved. 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

3 | 13 Copyright © Cengage Learning. All rights reserved. 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

3 | 14 Copyright © Cengage Learning. All rights reserved. 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)

3 | 15 Copyright © Cengage Learning. All rights reserved. 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.

3 | 16 Copyright © Cengage Learning. All rights reserved. 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

3 | 17 Copyright © Cengage Learning. All rights reserved. Journals Through journal writing, we can: Assess children's reflections of their own capabilities, attitudes & dispositions, Evaluate their ability to communicate mathematically, through writing

3 | 18 Copyright © Cengage Learning. All rights reserved. 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

3 | 19 Copyright © Cengage Learning. All rights reserved. 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

3 | 20 Copyright © Cengage Learning. All rights reserved. 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

3 | 21 Copyright © Cengage Learning. All rights reserved. Recording Assessment Data for Alternative Assessments Rubric Scores Checklists Anecdotal Notes

3 | 22 Copyright © Cengage Learning. All rights reserved. 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)

3 | 23 Copyright © Cengage Learning. All rights reserved. Comprehensive Rubric

3 | 24 Copyright © Cengage Learning. All rights reserved. Video Analysis

3 | 25 Copyright © Cengage Learning. All rights reserved. Single-Digit Addition and Subtraction Content

3 | 26 Copyright © Cengage Learning. All rights reserved. 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

3 | 27 Copyright © Cengage Learning. All rights reserved. How do Children Develop? Problem Types Join Separate Part-part-whole Compare Problem-solving Strategies Direct Modeling Counting Known Facts Derived Facts

3 | 28 Copyright © Cengage Learning. All rights reserved. Analyzing Problem Types Semantic versus Computational

3 | 29 Copyright © Cengage Learning. All rights reserved. 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

3 | 30 Copyright © Cengage Learning. All rights reserved. Video analysis

3 | 31 Copyright © Cengage Learning. All rights reserved. Generalizations One More/One Less Ten More/Ten Less Combinations of numbers to ten Commutativity Doubles and near doubles Making a ten

3 | 32 Copyright © Cengage Learning. All rights reserved. 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

3 | 33 Copyright © Cengage Learning. All rights reserved. Practice for Quick Recall Meaningful practice Games Music Timed tests?