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RESEARCH FUNDED BY THE NATIONAL SCIENCE FOUNDATION: Transforming Undergraduate Education in STEM (TUES) Award # , ,

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About Us Preservice Teacher Preparation Collaborative Jonathan ThomasNorthern Kentucky University KY Center for Mathematics Edna O. SchackMorehead State Sara EisenhardtNorthern Kentucky Molly H. FisherUniversity of Margaret YoderEastern Kentucky Janet TassellWestern Kentucky Cindy Jong*University of Todd Brown*University of Greg Gierhart*Murray State * Comparison Implementers

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Instructional Module

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Attending to the children’s work Interpreting children’s work in context of mathematics Deciding appropriate next steps Professional Noticing Jacobs, V. A., Lamb, L. L. C., & Philipp, R. A. (2010). Professional Noticing of Children’s Mathematical Thinking. Journal for Research in Mathematics Education, 41,

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Pedagogies of Practice Decomposition of professional noticing Representations video of early number sense diagnostic events Approximations PSETs conduct diagnostic interview with child Grossman, P. (2011). Framework for teaching practice: A brief history of an idea. Teachers College Record. 113, 12,

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Early Numeracy Stages of Early Arithmetic Learning Learning Progression Early Quantitative Understanding Examination of Counting Schemes Olive, J. (2001). Children's number sequences: An explanation of Steffe's constructs and an extrapolation to rational numbers of arithmetic. The Mathematics Educator, 11, 4-9. Steffe, L. (1992). Learning stages in the construction of the number sequence. In J. Bideaud, C. Meljac, & J. Fischer (Eds.), Pathways to number: Children’s developing numerical abilities (pp. 83–88). Hillsdale: Lawrence Erlbaum. Wright, R. J., Martland, J., & Stafford, A. (2000). Early numeracy: Assessment for teaching and intervention. London: Paul Chapman Publications/Sage.

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Early Numeracy Stages of Early Arithmetic Learning Stage 0:Emergent Counting Scheme Stage 1:Perceptual Counting Scheme Stage 2:Figurative Counting Scheme Stage 3:Initial Number Sequence Stage 4:Intermediate Number Sequence Stage 5:Facile Number Sequence

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To what extent can teacher educators facilitate the development of Preservice Elementary Teacher (PSET) professional noticing (attending, interpreting, and deciding) of children’s mathematics? PRIMARY RESEARCH QUESTION

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Professional Noticing Assessment “I have seven little bears... But now I have too many shells. I have eleven shells. (Jon shows the eleven shells then covers them with his hand.) How many shells am I going to have left over?”

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1.Please describe in detail what this child did in response to this problem. (Attending) 2.Please explain what you learned about this child’s understanding of mathematics. (Interpreting) 3.Pretend that you are the teacher of this child. What problem or problems might you pose next? Provide a rationale for your choice. (Deciding) Professional Noticing Prompts Jacobs, V. A., Lamb, L. L. C., & Philipp, R. A. (2010)

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Assessment Score Levels Attending InterpretingDecidingLevel 4 Elaborate 3 SalientAccurateAppropriate & Connected 2 LimitedLimitedAdequate, Disconnected 1 InaccurateInaccurateInappropriate, No Rationale

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Growth of PSET PN: Attending “He knew that since the teacher said he had too many shells he had to do subtraction. He also knew that because the teacher said left over he had to do subtraction, or see what the difference was. The child understood key words and phrases and understood how to take away to get the right answer. He used the bigger number and took away using the smaller number and realized that 11-7=4.” “In response to this problem this child first counted the bears and found that there were seven. From there he used his fingers and counted up from seven until he got to the number eleven. He had four fingers up so he said that that was his answer.” POST PRE

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Growth of PSET PN: Interpreting “It seemed that instead of subtracting seven from eleven he used the problem 7+?=11, and came up with four by counting from seven to eleven instead of from eleven to seven.” “This child does not count on; he needed to count the bears from one in order to count the remainder of the shells. He uses his fingers to count when materials are unavailable to him. He understands associating one object with a number and adding a value with each corresponding object added.” POST PRE

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Growth of PSET PN: Deciding “I would pose more bears than shells. Or only have shells exposed, so he couldn't count the bears. How many shells must I take away to get 7 bears? Other ways of getting answer and using subtraction.” “I would screen both of the counters. This requires the student to use a different type of counters (fingers) but he might run into trouble because he will be counting past 10. I[t] would be interesting to see how he got the answer.” POST PRE

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Results of ANOVA comparing pre and post assessments of all universities AttendingInterpretingDeciding NMSDM M University 0Pre-Test Post-Test University 1Pre-Test Post-Test University 2Pre-Test Post-Test All Participants Pre-Test Post-Test Descriptive statistics of professional noticing measures by university ScaleNF (3,91)P Partial Eta Squared Attending < Interpreting < Deciding < Preliminary Analysis of Three Research Sites

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Professional Noticing Measure Descriptive Statistics – All Sites AttendingInterpretingDeciding NMSDM M Scale Pre- assessment Post- assessment

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ANOVA comparing pre and post All UniversitiesScaleNF(3,91)P Partial Eta Squared Attending < Interpreting < Deciding <

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Questions? tinyurl.com/noticingnumeracynow

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Attending Benchmarks “He counted from one up when counting all of the bears. He then counted the remaining shells on his fingers to get the answer 4.” “Counted the bears individually then used his fingers to count up to 11.” “Instead of subtracting 11-7, he counted to seven and then used his fingers to see how many more it took to get to 11.” “The child subtracted in response to this question using his fingers as a manipulative. Starting with 11 & working backwards.” ELABORATE SALIENT LIMITED INACCURATE

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Attending Benchmarks “He counted from one up when counting all of the bears. He then counted the remaining shells on his fingers to get the answer 4.” “Counted the bears individually then used his fingers to count up to 11.” “Instead of subtracting 11-7, he counted to seven and then used his fingers to see how many more it took to get to 11.” “The child subtracted in response to this question using his fingers as a manipulative. Starting with 11 & working backwards.”

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“I learned that this child can add easier than subtract because instead of 7-11 he did 7+__=11. I also learned that he needs a representation of the numbers to solve the problem (the bears, his fingers, and shells).” Interpreting Benchmarks “This child understands a one-to-one correspondence with objects, he needs to touch the objects and he still uses his fingers to count on.” ACCURATE LIMITED INACCURATE “I learned that the child is able to count on from a given number. He didn't have to go back and start at 1.”

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“I learned that this child can add easier than subtract because instead of 7-11 he did 7+__=11. I also learned that he needs a representation of the numbers to solve the problem (the bears, his fingers, and shells).” Interpreting Benchmarks “This child understands a one-to-one correspondence with objects, he needs to touch the objects and he still uses his fingers to count on.” “I learned that the child is able to count on from a given number. He didn't have to go back and start at 1.”

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“I would ask the child to tell me why there were four shells leftover. This would tell us whether or not the child had an understanding of remainders. This will tell us if he has the concept of sharing equally, rather than giving the four shells to select bears.” “I might say "How did you get this answer" to see how they explained their logic.” “I believe that the next task should be a really small number subtracted by a very large number. Ex This problem would be harder to count on your hands and you could get a better understanding of his conceptual knowledge of the problem and addition itself.” Deciding Benchmarks CONNECTED RATIONALE DISCONNECTED RATIONALE LITTLE OR NO RATIONALE Appropriate Decision with… Adequate Decision with…Inappropriate Decision with…

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“I would ask the child to tell me why there were four shells leftover. This would tell us whether or not the child had an understanding of remainders. This will tell us if he has the concept of sharing equally, rather than giving the four shells to select bears.” “I might say "How did you get this answer" to see how they explained their logic.” “I believe that the next task should be a really small number subtracted by a very large number. Ex This problem would be harder to count on your hands and you could get a better understanding of his conceptual knowledge of the problem and addition itself.” Deciding Benchmarks APPROPRIATE & CONNECTED ADEQUATE, DISCONNECTED INAPPROPRIATE, DISCONNECTED

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“I would ask the child to tell me why there were four shells leftover. This would tell us whether or not the child had an understanding of remainders. This will tell us if he has the concept of sharing equally, rather than giving the four shells to select bears.” “I might say "How did you get this answer" to see how they explained their logic.” “I believe that the next task should be a really small number subtracted by a very large number. Ex This problem would be harder to count on your hands and you could get a better understanding of his conceptual knowledge of the problem and addition itself.” Deciding Benchmarks

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PSET Professional Noticing in Clinical Context

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Instructional Module

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