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Carnegie Learning | www.carnegielearning.com Solving for x’s and why’s: Bringing Cognitive Science into the Math Classroom Kevin Judd Western Regional Vice President Carnegie Learning, Inc.® South Dakota Laptop Institute 2007

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Carnegie Learning | www.carnegielearning.com Welcome

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Carnegie Learning | www.carnegielearning.com Learning by Doing “We learn by doing – that is the thing. For though you think you know it, you have no certainty until you try.” - Sophocles, 270 B.C.

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Carnegie Learning | www.carnegielearning.com Sources How People Learn and How Students Learn from the National Research Council – John Bransford Larry Lowery from Lawrence Lab at Berkeley and FOSS Al Corbett, John Anderson and Steve Ritter at Carnegie Mellon University The Teaching Gap by James Stigler and James Weisert.

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Carnegie Learning | www.carnegielearning.com “How People Learn” research Learners use what they already know to construct new understandings. How People Learn: Brain, Mind, Experience and School Bransford, J.D., Brown, A.L.& Cooking, R.R. (Eds) 1999, National Academy Press.

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Carnegie Learning | www.carnegielearning.com An example of prior knowledge: HOUSE “How People Learn” research

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Carnegie Learning | www.carnegielearning.com “How People Learn” research Learning with understanding is enhanced when new knowledge is built on the major concepts of a discipline. How People Learn: Brain, Mind, Experience and School Bransford, J.D., Brown, A.L.& Cooking, R.R. (Eds) 1999, National Academy Press.

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Carnegie Learning | www.carnegielearning.com Learners take control of their learning with metacognitive strategies. Write a sentence describing how you found these answers. You will be expected to share your answers with the class. Explain to your partner how you got your answer “How People Learn” research

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Carnegie Learning | www.carnegielearning.com Learners must be actively engaged. “How People Learn” research

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Carnegie Learning | www.carnegielearning.com Learners’ motivation and sense of self affect what and how much they will learn. “How People Learn” research

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Carnegie Learning | www.carnegielearning.com “How People Learn” research Learners utilize different approaches, strategies and styles based upon individual learning opportunities and prior experiences. How People Learn: Brain, Mind, Experience and School Bransford, J.D., Brown, A.L.& Cooking, R.R. (Eds) 1999, National Academy Press.

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Carnegie Learning | www.carnegielearning.com “How People Learn” research

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Carnegie Learning | www.carnegielearning.com “How People Learn” research Learners cement learning through the use of distributive practice (vs. mass practice). How People Learn: Brain, Mind, Experience and School Bransford, J.D., Brown, A.L.& Cooking, R.R. (Eds) 1999, National Academy Press.

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Carnegie Learning | www.carnegielearning.com Social interactions enhance learning. “How People Learn” research

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Carnegie Learning | www.carnegielearning.com “How People Learn” research Learning is situated in activity and is shaped by the context and culture in which it occurs. How People Learn: Brain, Mind, Experience and School Bransford, J.D., Brown, A.L.& Cooking, R.R. (Eds) 1999, National Academy Press.

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Carnegie Learning | www.carnegielearning.com Context is important

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Carnegie Learning | www.carnegielearning.com Context is important

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Carnegie Learning | www.carnegielearning.com Context is important

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Carnegie Learning | www.carnegielearning.com Result-unknown Problems 1. Story When Ted got home from his waiter job, he took the $81.90 he earned that day and subtracted the $66 he received in tips. Then he divided the remaining money by the 6 hours he worked and found his hourly wage. How much per hour does Ted make? 2. Word-equation Starting with 81.9, if I subtract 66 and then divide by 6, I get a number. What is it? 3. Equation (81.90 - 66) / 6 = x Result-unknown Problems 1. Story When Ted got home from his waiter job, he took the $81.90 he earned that day and subtracted the $66 he received in tips. Then he divided the remaining money by the 6 hours he worked and found his hourly wage. How much per hour does Ted make? 2. Word-equation Starting with 81.9, if I subtract 66 and then divide by 6, I get a number. What is it? 3. Equation (81.90 - 66) / 6 = x Start-unknown Problems 4.Story When Ted got home from his waiter job, he multiplied his hourly wage by the 6 hours he worked that day. Then he added the $66 he made in tips and found he earned $81.90. How much per hour does Ted make? 5. Word-equation Starting with some number, if I multiply it by 6 and then add 66, I get 81.9. What number did I start with? 6. Equation 6x + 66 = 81.90 Start-unknown Problems 4.Story When Ted got home from his waiter job, he multiplied his hourly wage by the 6 hours he worked that day. Then he added the $66 he made in tips and found he earned $81.90. How much per hour does Ted make? 5. Word-equation Starting with some number, if I multiply it by 6 and then add 66, I get 81.9. What number did I start with? 6. Equation 6x + 66 = 81.90 Cognitive Analysis

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Carnegie Learning | www.carnegielearning.com Result-unknownStart-unknown 20 30 40 50 60 70 80 90 Equation Word-equation Story problem Number Correct Koedinger, K.R., & Tabachneck, H.J.M. (1995). Verbal reasoning as a critical component in early algebra. Paper presented at the annual meeting of the American Educational Research Association, San Francisco, CA. Results

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Carnegie Learning | www.carnegielearning.com Example of basic cognitive research Logical Reasoning Fact: All cards have letter on one side and number on the other Rule: If there’s a vowel on one side, there’s an odd number on the other side. Question: Which card(s) do you turn over to verify the rule?

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Carnegie Learning | www.carnegielearning.com Example of basic cognitive research Fact: All cards represent people - the person’s drink is on one side, age is on the other. Rule: If the person is drinking alcohol, he or she must be over 21 Question: Which card(s) do you turn over to verify the rule?

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Carnegie Learning | www.carnegielearning.com Example of basic cognitive research Rule: If X then Y To verify, check X and not Y

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Carnegie Learning | www.carnegielearning.com 1. Multisensory – 1st hand experience 2. Pictorial – Video, photo, diagram, etc… 3. Symbolic – Reading, solving equations, etc… “How People Learn” research

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Carnegie Learning | www.carnegielearning.com “How People Learn” research Skills or Subjects??? Skills: Examples - piano, multiplication tables, reading Generally learned by practice. Subjects: Examples – social studies, art, mathematics Generally learned by rehearsals.

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Carnegie Learning | www.carnegielearning.com Expert – Novice studies Experts recognize relationships better. Experts generalize better. Experts notice details better. Experts have a tremendous amount of prior knowledge in the area of their expertise. Experts are not more intelligent. All people can become experts. Experts often can not explain their expertise to novices (Expert Blind Spot).

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Carnegie Learning | www.carnegielearning.com What’s important to learn?

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Carnegie Learning | www.carnegielearning.com Teaching Math– the current state Typical class Warm-up, check homework Demonstrate a procedure with definitions Students practice the procedure Homework 80% of the mathematics concepts are only stated not developed 95% of seatwork engaged in practicing routine procedures

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Carnegie Learning | www.carnegielearning.com Unskilled 60% Skilled 20% Professional 20% Skilled 65% Unskilled 15% Professional 20% 19502000 National Summit on 21 st Century Skills for 21 st Century Jobs Ramifications – skill level changes

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Carnegie Learning | www.carnegielearning.com Ramifications in student performance Typical urban school system 15,346 9 th graders, 9,788 10 th graders, 8,031 11 th graders, and 6,461 12 th graders 67% of African-American students and 77% of Hispanic students failed the mathematics state assessment

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Carnegie Learning | www.carnegielearning.com Another Urban System Grade Number of Students % of 9 th Graders 968,599 100% 1052,317 76% 1138,908 57% 1228,132 41%

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Carnegie Learning | www.carnegielearning.com Potential Earnings by Educational Attainment Education Attained Unemployment RateMedian Earnings in 2001 Less than HS8.5%$22,350 HS grad5.3%$29,187 Some college4.8%$34,340 Associate4.0%$36,399 Bachelors3.1%$46,969 Masters2.8%$56,589 Doctorate1.6%$75,182 Professional1.6%$82,421 Bureau of Labor Statistics

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Carnegie Learning | www.carnegielearning.com Back to the Future in Mathematics Education – Lynn Steen, Ed Week 4/7/04 Recent reports dealing with the mathematical expectations of higher education and the world of work show that little has changed in the last 20 years. (Since A Nation at Risk) Only two out of three students graduate from HS with three years of HS math Only two out of three students graduate from HS which means least than 50% of HS graduates meet the standard set by A Nation at Risk.

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Carnegie Learning | www.carnegielearning.com NAEP 2005 Math Assessment 4 th Graders Scoring At or Above Proficient 36 percent of all students 19 percent of Hispanic students 13 percent of African-American students 19 percent of low-income students Ramifications

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Carnegie Learning | www.carnegielearning.com NAEP 2005 Math Assessment 8 th Graders Scoring At or Above Proficient 30 percent of all students 13 percent of Hispanic students 9 percent of African-American students 13 percent of low-income students Ramifications – who’s successful?

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Carnegie Learning | www.carnegielearning.com The Solution?

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Carnegie Learning | www.carnegielearning.com The Real Solution Aligning teaching to learning Vs. Making learning align with teaching

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Carnegie Learning | www.carnegielearning.com Solving problems to learn mathematics Vs. Learning procedures to solve problems The Real Solution

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Carnegie Learning | www.carnegielearning.com Carnegie’s Curriculum Problem Solving Procedures Students LEARN math by DOING math Students KNOW more than they SHOW Students should never ask THE question Reverse the traditional approach Real World Problem Solving

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Carnegie Learning | www.carnegielearning.com Carnegie Learning’s Solutions Bridge To Algebra Algebra I Geometry Algebra II Integrated Mathematics I Integrated Mathematics II Integrated Mathematics III Professional Development Pedagogical Mathematical Content

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Carnegie Learning | www.carnegielearning.com Classroom Curriculum Classroom – 3 days per week (60%) Collaborative learning Conversations about mathematics Writing about mathematics Student presentations Real World Problem Solving

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Carnegie Learning | www.carnegielearning.com Task 2: For this problem we will need to clearly identify the problem situation, produce a table of values, construct a graph, and find an algebraic rule. Your teacher will be giving your group overlays for you to put your answers on to share with the class. share table of values with the class. algebraic rule. graph, problem situation, Classroom Curriculum Task 1: Answer the following questions using complete sentences. 1.How much would you charge for an order of 3 shirts? 10 shirts? 100 shirts? Write a sentence describing how you found these answers. 2.How many shirts could a customer purchase with $50? $100? $1000? $52.50? Write a sentence describing how you found these answers. 1.How much would you charge for an order of 3 shirts? 10 shirts? 100 shirts? Write a sentence describing how you found these answers. 2.How many shirts could a customer purchase with $50? $100? $1000? $52.50? Write a sentence describing how you found these answers. Your job at U.S. Shirts is to calculate costs for various t- shirt orders. For each order, U. S. Shirts charges $8 per shirt plus a one time set up charge of $15. complete sentences.

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Carnegie Learning | www.carnegielearning.com Bridge to Algebra

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Carnegie Learning | www.carnegielearning.com Computer Curriculum Computer Lab – 2 days per week (40%) Students discovering mathematics Coaching by intelligent Tutor One-on-one student-Tutor relationship Teacher spends time with students who need it most Real World Problem Solving

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Carnegie Learning | www.carnegielearning.com Computer Curriculum

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Carnegie Learning | www.carnegielearning.com Five “Dos” to help students learn Do make students “do the math” Do let students struggle Do use word walls Do create student work walls Do provide students with ample classroom time to do presentations of their solutions

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Carnegie Learning | www.carnegielearning.com Five “Don’ts” to help students learn Don’t give students answers – always answer a question with a question Don’t make assumptions about levels of difficulty or levels of ease Don’t help students “set up” problem solving tasks Don’t keep students from working together - allow students to assist each other Don’t let students give short answers - require students to write all answers in complete sentences

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Carnegie Learning | www.carnegielearning.com Thank you! Kevin Judd kjudd@carnegielearning.com 1-888-851-7094 x477

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