Cognitive Assessments for Math

Developing Cognitive Math Assessments for Daily Classroom Use : 

Developing Cognitive Math Assessments for Daily Classroom Use Maria H. Andersen Muskegon Community College AMATYC 2008, Washington DC

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*disclaimer I am not a cognitive psychologist (yet). I have only read some books. If you want more detailed information, you should probably do the same.

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*disclaimer There are at least five models for cognitive theory. I will talk about two of them in an instructional context.

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theory practice questions introduction

Assessment of Student Learning : 

Assessment of Student Learning definition? ?

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1. Provides information you need on your student’s learning. A participatory, iterative process that: (from Higher Learning Commission) ?

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2. Engages you and others in analyzing and using this information to confirm and improve teaching and learning. A participatory, iterative process that: (from Higher Learning Commission) ?

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Suppose we center this assessment process on the students. ?

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2A. Engages you in analyzing and using this information to confirm and improve teaching. A participatory, iterative process that: (modified from Higher Learning Commission) ?

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2B. Engages students in analyzing and using this information to confirm and improve learning. A participatory, iterative process that: (modified from Higher Learning Commission) ?

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3. Produces evidence that students are learning the outcomes you intended. A participatory, iterative process that: (from Higher Learning Commission) ?

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3. Produces feedback to the student that they are learning the outcomes you intended. A participatory, iterative process that: (modified from Higher Learning Commission) ?

We use information to confirm and improve learning goals. : 

We use information to confirm and improve learning goals. (from assessment) ?

Students must learn cognitive monitoring to achieve learning goals. : 

Students must learn cognitive monitoring to achieve learning goals. (from cognitive theory) ?

Cognitive Theory : 

Cognitive Theory Schema Theory Information Processing Theory What makes schemata? ?

Schema (pl. schemata) : 

Schema (pl. schemata) a mental representation of what all instances of something have in common ?

Schemata… : 

Schemata… 1. Categorize your experiences ?

Schemata… : 

Schemata… 2. Help you remember what you are experiencing. ?

Schemata… : 

Schemata… 3. Help you comprehend what you are experiencing. ?

Schemata… : 

Schemata… 4. Are important in developing the ability to problem solve. ?

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Possible student scripts for factoring ?

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what happens? ?

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the problem here is that the student ran the wrong script ?

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Faulty Student script: Problem says factor Squared term in front? Three terms? Run trinomial factoring script. ?

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my hint ?

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what just happened? ?

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student “ran” trinomial factoring script ?

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?

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now what happens? ?

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Student script: Problem says factor Four terms? Run factor by grouping script. ?

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Successful problem solvers have a large variety of flexible schemata. ?

Information Processing Theory : 

Information Processing Theory Knowledge Declarative Procedural ?

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Knowledge Declarative Procedural ?

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How do we form procedural knowledge? ?

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Procedural knowledge is activated when the if of an if-then relationship is encountered. ?

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Students need to practice recognizing and categorizing problems into schemata. ?

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Students need to encounter problems that refine and revise their schemata. ?

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What makes schemata? abstraction gist-extraction interpretation ?

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Encountering objects in new ways revises and refines schemata. ?

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remember this? How can we change this in a way that helps students revise and refine their schemata? ?

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Why does this approach help to revise schemata? ?

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Initial script: If the problems are different, then the answers are different. ?

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Learners need to have multiple encounters with objects or events in different ways ?

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Learners need to have multiple encounters with objects or events in different ways ?

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that’s not repetition per se ?

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Repetition ?

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Abstraction ?

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Gist extraction ?

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The gist of what you saw: Cool poster Sing Uhr 06 Must stand back to read. ?

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the gist-extraction process revises your actual experience in order to store the memory ?

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Let’s say I teach my students to multiply polynomials in the following way.* *I don’t do it this way, but let’s just say I do. ?

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After the student leaves class, what is retained in their memory? ?

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gist of today’s class: If you have to multiply polynomials, use FOIL (first, outer, inner, last). ?

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Even worse, an interpretation has been made: All multiplication involves FOIL. ?

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Let’s say (instead) I teach my students this way. ?

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After the student leaves class, what is retained in their memory? ?

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gist of today’s class: If you have to multiply polynomials, multiply each term in the first expression by each term in the second expression. Cues: Draw arcs to help, line up like terms. ?

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interpretation: filling in things that were not said or seen ?

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? One common interpretation:

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? One common interpretation:

Put it inPractice : 

Put it inPractice Revise and Refine Metacognitive Skills Diagnostic Tools Assess Understanding

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Diagnostic Tools ?

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Students undergo imperfect gist-extraction, and the flawed schemata they form must be identified. We need to be able to identify where the false schemata are. ?

Building a Diagnostic Tool : 

Building a Diagnostic Tool Think back to the last time you gave a test on lines. Write down some examples of mistakes that students made. ?

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Misapply the negative on a fraction. Difficulty evaluating expressions with x and y. Don’t know the difference between vertical and horizontal. Mistakes in subtracting signed numbers in the slope expression. Incorrectly solve equations with negative coefficients. Simplifying point-slope form when it involves fractions. Can’t plot points involving fractions. ?

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Can’t plot points involving fractions. Incorrectly solve equations with negative coefficients. Difficulty evaluating expressions with x and y. Misapply the negative on a fraction. Mistakes in subtracting signed numbers in the slope expression. Simplifying point-slope form when it involves fractions. Don’t know the difference between vertical and horizontal. ?

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But now you have one more thing to grade, right? ?

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nope ?

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Summary: Diagnostic Tool Identify false schemata Purpose: How to construct: Think about past student behavior, write short questions that target common errors at end of chapter. How to assess: Tally sheet. Students self-assess. ?

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Revise and Refine ?

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Summary: Revise & Refine Revise & Refine existing schemata to improve learning. Purpose: How to construct: Think about what gets muddled when a new concept is added. How to assess: You monitor progress and students self-assess. Simple answer-checking is key. ?

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Assess Understanding Categorize this Picture ?

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Does this help? ?

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Summary: Assess Understanding Improve Information Processing and gist-extraction. Purpose: How to construct: Work with “first-steps” and categorization of problem types. How to assess: Student or group makes first attempt, then go over assessment as a class. ?

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Metacognitive Skills ?

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5 Bring it on!!! 4 I know I will pass, grade? Not sure. 3 Well, we’ll see how it goes. 2 I think I should have attended more class. 1 Not a clue. ?

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Summary: Metacognitive Skills Improve metacognitive skills. Purpose: How to construct: Look at learning objectives and look where objectives overlap. How to assess: Student makes initial assessment prior to studying for exam. Second assessment after studying. Reflection after exam. ?

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Recommended Reading: Cognitive Development and Learning in instructional Contexts, by James P. Brynes Presentation at: www.tcmtechnologyblog .blogspot.com (right-hand side) Workbooks of Assessments and Activities: Cengage Learning, author: Maria H. Andersen ?