Mainly here, I have my notes – points they made that I found interesting to think about. I am leaving many of these in “note” form because I had written them as points to consider.
Gary spoke first, asking us to consider a problem that asks us to factor a trinomial. He asked us to look at two questions –
What mathematical thinking are you likely to get from your students? Recall, either they know what to do or not.
What kind of mathematical thinking would we like to get? The reasoning and sense making – the thinking.
In the High School Focus on Reasoning and Sense Making book, there are four reasoning habits given:
- Analyzing a problem
- Implementing a strategy
- Seeking and using connections
- Reflecting on a solution to a problem
It’s more about the nature of what students are doing in the classroom – are they just mimicking or are they thinking?
What makes a task use reasoning and sense making?
- multiple points of entry
- trial and error – why is something right, why is something wrong
- more than one answer
- discussion was helpful
- simple to get into – not that elaborate
-The math might be useful in solving the problem.
-It makes sense to do this math to answer an interesting question.
The mathematics you are doing makes sense.-You understand how the mathematics works.
-You understand how the particular mathematical content and mathematical reasoning relate to the problem at hand (or contributes to a deeper understanding of the mathematics itself)
You can make (mathematical sense) sense of your solutions to others.
The teacher of mathematics should pose tasks that are based on –-sound and significant mathematics
-knowledge of students’ understandings, interests, and experiences
-knowledge of the range of ways that diverse students learn mathematics
-Vary the task
-Provide a task with multiple entry levels and appropriate teacher support.
You can also minimize the reasoning and sense making that students might experience in a supposedly good task by:-Discussing your answer before the students have had a chance to think about the problem for themselves.
-Only calling on students who you think know the answer
-Body language (similar to what Dan had talked about in his keynote)
Eric Robinson cited Questioning Our Pattern of Questioning (Herbell-Eisenmann & Breyfogle, 2006) –
Many teachers do either Imitation-Response-Feedback type questioning or funneling (keep narrowing your questions to get where you want to be). They proposed an alternative pattern – Focusing, where the teacher listens to the students’ responses and guides them based on what the students are thinking rather than what the teacher thinks is the best solution.
It allows the teacher to understand what the students are thinking
- It provides students an opportunity to clarify their thinking well enough to communicate to the class
- It provides members of the class opportunities to encounter different approaches to the problem
NSF-funded currciula – http://www.imp.org/, www.wmich.edu/comp/, www.montana.edu/wwwsimms, www.comap.com/mmow/, http://www.cmeproject.edc.org/ - I have to check the slide once it's posted to make sure I have the un-linked ones correct.
Illustrative Mathematics mentioned this morning
Heinemann – search tasks **this was audience mentioned, but I'm not finding it.
Mathematics Assessment Project
They also suggested that you can adapt an existing task to better meet the needs of your students. This can be done by changing the content, adjusting the mathematical level (without adjusting the necessity of reasoning and sense making), or even going to the point of creating a new task.
Gary closed by going back to the original problem – factor x^2 – 7x + 12. Suggestions to change it to involve more reasoning and sense making:
- Maybe put it in the context of the area of a rectangle.
- The easy fix he proposed was to change it to: Find coefficients for x so that you can factor the trinomial x^2 - ?x + 12