N OTICING N UMERACY N OW!

1. N3: NOTICING NUMERACYNOW! RESEARCH FUNDED BY THE NATIONAL SCIENCE FOUNDATION: Transforming Undergraduate Education in STEM (TUES) Award # 1043667, 1043656, 1043831

2. About Us Preservice Teacher Preparation Collaborative * Comparison Implementers

3. Instructional Module

4. Professional Noticing Attending to the children’s work Interpretingchildren’s work in context of mathematics Deciding appropriate next steps 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, 169-202.

5. 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, 2836-2843.

6. 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.

7. 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

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

9. 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?”

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

11. Assessment Score Levels Interpreting Deciding Attending Level 4Elaborate 3Salient Accurate Appropriate & Connected 2Limited Limited Adequate, Disconnected 1Inaccurate Inaccurate Inappropriate, No Rationale

12. 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

13. 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

14. 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

15. Preliminary Analysis of Three Research Sites Descriptive statistics of professional noticing measures by university Results of ANOVAcomparing pre and post assessments of all universities

16. Professional Noticing Measure Descriptive Statistics – All Sites

17. ANOVA comparing pre and post

18. Questions? tinyurl.com/noticingnumeracynow

19. 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.” ELABORATE SALIENT “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.” INACCURATE LIMITED

20. 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.” 4 3 “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.” 2 1

21. 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 “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).” “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.” INACCURATE LIMITED

22. 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.” 3 “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).” 1 2 “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.”

23. Deciding Benchmarks Appropriate Decision with… “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. 20-6. 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.” Connected Rationale Adequate Decision with… Inappropriate Decision with… Littleor No Rationale Disconnected Rationale

24. Deciding Benchmarks “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. 20-6. 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.” Appropriate & Connected Adequate, Disconnected Inappropriate, Disconnected

25. Deciding Benchmarks “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. 20-6. 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.” 3 2 1

26. PSET Professional Noticing in Clinical Context

27. Instructional Module