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. Three interrelated skills of Professional Noticing of children’s mathematical thinking: • Attending to the children’s work • Interpretingchildren’s work in context of mathematics • Deciding the 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. STAGES OF EARLY ARITHMETIC LEARNING (SEAL) 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.

6. SEAL LEARNING PROGRESSION OF EARLY QUANTITATIVE UNDERSTANDING VIA EXAMINATION OF COUNTING SCHEMES 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

7. 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?

8. Video-Based Assessment Prompts 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)

9. ATTENDING BENCHMARKS

10. INTERPRETING BENCHMARKS

11. DECIDING BENCHMARKS

12. Preliminary Analysis Conducted at 3 Research Sites Descriptive statistics of professional noticing measures by university Results of independent t-test comparing pre and post tests of all universities

13. the student blew through my last two tasks. I had her come up with a set of numbers that added up to 13 and then again for 10. I asked her for three pairs of numbers each time, and she easily dictated three sets each time. Lastly, I had drawn colored dots on index cards and flashed them to the student. She correctly stated how many dots there were each time, even when I had used two different colors. She did a great job. Overall for this interview, I originally thought I was pitching it a little over the student’s proximal zone of development by using both my own tasks and those from the 2nd and 3rd grade questionnaire since she is only in 1st grade. However, I really think it was pitched at the right level. I got to see the student pushed to the edge of her ability. In the end, the student confidently succeeded in all components of this interview after doing things her own way. The tasks and questions I presented in this interview were mostly to help me gain a better understanding of the student’s addition and subtraction skills and her knowledge about place value and number sequence. I loved the student’s answers for the screened addition and subtraction tasks. She has some pretty advanced explanations for a 1st grader! Her reasoning demonstrates that she is able to chunk multiple numbers in the same problem. After looking through the SEAL descriptions, her answers for these two problems combined with her answers for some of the other tasks, the student was using thought processes that demonstrated knowledge from Stage 5- Facile Number Sequence. A child in the facile number sequence stage might use doubles to work out other facts which the student demonstrated when she thought of 9+9 to find out 9+6. Also, a child in this stage might use knowledge of the inverse relationship between addition and subtraction which the student did when figuring out 14-3. A child at this stage can also count forwards and backwards by 2s, 10s, 5s, 3s, and 4s which the student can do forwards. I am not sure if the student can count backwards by any of those intervals, but I presume that she cannot do it automatically. She might be able to prove me wrong, but based off of what I have observed, it would be a task that she could figure out if she had a long time, but it would not come naturally. I If I had to place the student in a stage, I don’t think she has quite mastered all aspects of the facile number sequence, but I do believe she is inching closer to it. The one task that makes me weary to put her into such a high stage was the bundling of the sticks since she did not use her knowledge of tens to solve the problem in the most efficient way; she counted by ones instead of using the bundles. In all, I feel confident to say that the student is a mixture of the characteristics from Preservice Elementary Teachers’ Professional Noticing in Clinical Contexts

14. Questions?