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


Slide 2:*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.


Slide 3:*disclaimer There are at least five models for cognitive theory. I will talk about two of them in an instructional context.


Slide 4:theory practice questions introduction


Assessment of Student Learning :Assessment of Student Learning definition? ?


Slide 6:1. Provides information you need on your student’s learning. A participatory, iterative process that: (from Higher Learning Commission) ?


Slide 7:2. 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) ?


Slide 8:Suppose we center this assessment process on the students. ?


Slide 9:2A. Engages you in analyzing and using this information to confirm and improve teaching. A participatory, iterative process that: (modified from Higher Learning Commission) ?


Slide 10:2B. Engages students in analyzing and using this information to confirm and improve learning. A participatory, iterative process that: (modified from Higher Learning Commission) ?


Slide 11:3. Produces evidence that students are learning the outcomes you intended. A participatory, iterative process that: (from Higher Learning Commission) ?


Slide 12:3. 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. ?


Slide 21:Possible student scripts for factoring ?


Slide 22:what happens? ?


Slide 23:the problem here is that the student ran the wrong script ?


Slide 24:Faulty Student script: Problem says factor Squared term in front? Three terms? Run trinomial factoring script. ?


Slide 25:my hint ?


Slide 26:what just happened? ?


Slide 27:student “ran” trinomial factoring script ?


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Slide 29:now what happens? ?


Slide 30:Student script: Problem says factor Four terms? Run factor by grouping script. ?


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Slide 32:Successful problem solvers have a large variety of flexible schemata. ?


Information Processing Theory :Information Processing Theory Knowledge Declarative Procedural ?


Slide 34:Knowledge Declarative Procedural ?


Slide 35:How do we form procedural knowledge? ?


Slide 36:Procedural knowledge is activated when the if of an if-then relationship is encountered. ?


Slide 37:Students need to practice recognizing and categorizing problems into schemata. ?


Slide 38:Students need to encounter problems that refine and revise their schemata. ?


Slide 39:What makes schemata? abstraction gist-extraction interpretation ?


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Slide 41:Encountering objects in new ways revises and refines schemata. ?


Slide 42:remember this? How can we change this in a way that helps students revise and refine their schemata? ?


Slide 43:Why does this approach help to revise schemata? ?


Slide 44:Initial script: If the problems are different, then the answers are different. ?


Slide 45:Learners need to have multiple encounters with objects or events in different ways ?


Slide 46:Learners need to have multiple encounters with objects or events in different ways ?


Slide 47:that’s not repetition per se ?


Slide 48:Repetition ?


Slide 49:Abstraction ?


Slide 50:Gist extraction ?


Slide 51:The gist of what you saw: Cool poster Sing Uhr 06 Must stand back to read. ?


Slide 52:the gist-extraction process revises your actual experience in order to store the memory ?


Slide 53: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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Slide 57:After the student leaves class, what is retained in their memory? ?


Slide 58:gist of today’s class: If you have to multiply polynomials, use FOIL (first, outer, inner, last). ?


Slide 59:Even worse, an interpretation has been made: All multiplication involves FOIL. ?


Slide 60:Let’s say (instead) I teach my students this way. ?


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Slide 64:After the student leaves class, what is retained in their memory? ?


Slide 65: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. ?


Slide 66:interpretation: filling in things that were not said or seen ?


Slide 67:? One common interpretation:


Slide 68:? One common interpretation:


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


Slide 70:Diagnostic Tools ?


Slide 71: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. ?


Slide 73: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. ?


Slide 74: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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Slide 76:But now you have one more thing to grade, right? ?


Slide 77:nope ?


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Slide 82: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. ?


Slide 83:Revise and Refine ?


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Slide 91: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. ?


Slide 92:Assess Understanding Categorize this Picture ?


Slide 93:Does this help? ?


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Slide 99: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. ?


Slide 100:Metacognitive Skills ?


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Slide 103:5 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. ?


Slide 104: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. ?


Slide 105: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 ?