1. Toward the identification of mathematical horizon knowledge Chantel C. Blackburn University of Arizona This material is based upon work supported by the National Science Foundation under Grant No. DRL- 1019860. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

2. Overview Horizon Content Knowledge Mathematical knowledge for teaching Current thinking and potential problems Identifying Horizon Content Knowledge Preservice teacher study Continued work with inservice teachers My Current Thinking on Horizon Content Knowledge

3. Mathematical Knowledge for Teaching Specialized content knowledge (SCK) Horizon content knowledge (HCK) Common content knowledge (CCK) Knowledge of content and students (KCS) Knowledge of content and teaching (KCT) Knowledge of content and curriculum Pedagogical Content KnowledgeSubject Matter Knowledge Horizon knowledge is an awareness of how mathematical topics are related over the span of mathematics included in the curriculum. (Ball, Thames, & Phelps, 2008)

4. HCK: What is it? A sense of the mathematical environment surrounding the current “location” in instruction Major disciplinary ideas and structures Key mathematical practices Core mathematical values and sensibilities (Ball & Bass, 2009)

5. HCK: What is it? A sense of the mathematical environment surrounding the current “location” in instruction Major disciplinary ideas and structures Key mathematical practices Core mathematical values and sensibilities (Ball & Bass, 2009)

6. HCK: What is it? If HCK is mathematical knowledge at the horizon then how/where should we define the horizon ? Horizon knowledge as content at the horizon Subject to change Is it pedagogically useful? Is it tenable for the majority of school teachers?

7. HCK: What is it? If HCK is mathematical knowledge at the horizon then how/where should we define the horizon ? Horizon knowledge as content at the horizon Subject to change Is the tenable for the majority of school teachers? Horizon knowledge as a way of looking at content with a mathematical eye

8. HCK: What is it good for? Making judgments about mathematical importance Hearing mathematical significance in what students are saying Highlighting and underscoring key points Anticipating and making connections Noticing and evaluating mathematical opportunities Catching mathematical distortions or possible precursors to later mathematical confusion or misrepresentation (Ball & Bass, 2009)

9. Preservice Teacher Study Participants: 15 preservice teachers PSTs conducted 3 one-on-one problem solving interviews with elementary school children addition/subtraction multiplication/division base ten/place value Data: Three written reflections Summary of student understanding Implications for instruction: Description of problem solving lesson or 3-4 problem solving tasks to address needs of students.

10. “Doing” for the sake of “Doing” TaskReason for Task

11. Preservice Teachers Orientation toward treating interview tasks as a curriculum with content to be mastered. Elly: I think that when she is surrounded by the topic of multiplication and division then P will be able to be successful in solving story problems similar to the ones in the interview. Barbara: These problems would not only help the students better understand place value, but it would also help the teacher evaluate the students that understand material and those who don’t. This would be a great precursor to a possible lesson…[on] either place value or extending their thinking on place value.

12. Inservice Teacher Reflection Georgia: We use this book and for some of us, myself included, sometimes it’s the end all be all. And I stick to it for several reasons. One, we’re told to stick to it ; you know, fidelity to the curriculum. The other thing is, I wouldn’t venture out on my own because I don’t have a more global picture, which I think tends to happen the longer we teach; the more isolated we become where in terms of: I know what my part in the building blocks is and so I can do that one pretty well, maybe not even great, but don’t bother me about anything that goes beyond that boundary …I never thought of myself as a person who lives in the box but I’m livin’ in the box and then I have to admit to myself that I’m not as creative about thinking outside of the box in math…because I don’t feel like I have that much experience.

13. HCK and “Resolution”

15. Preservice Teachers Kelly: Challenging him with problems that evoke the concept of grouping will provide him with strategies to solve various word problems that caused him difficulty in the second interview. Additionally, it will help his transition into skip counting and start an understanding of place value.

16. “Doing” to head in a “Direction” TaskReason for Task

17. Pedagogically Useful HCK Knowledge that supports a mathematical way of looking at content that is: Sensitive to the role of curriculum in teaching Defines what should be taught/learned (and this can change) Defines how students (and consequently teachers) are evaluated Sensitive to student understanding Students come with prior knowledge Students are headed somewhere Knowledge of the existence of a mathematical horizon Finding mathematical connections in a “greater context” at any given moment Lesson Unit Grade level