First, each of the panelists were given time to present on the topic. Then Ed posed two questions to the panel, which he had emailed to them in advance. Finally, there was some time for audience questions.
First up was Gail Burrell. Some of the main points I got from her presentation:
- What are standards? Are they the curriculum? Or are they?
- Kids can think and reason, but we don’t let them do that. It’s pretty hard to think and reason with a typical list of problems in a textbook.
- Students must learn mathematics with understanding, actively building new knowledge from experience and previous knowledge.
- How students should work – Standards for Mathematical Practices. These are the kinds of ways we want kids to engage in mathematics.
- Thinking and reasoning depends on the types of questions we ask. We should be asking questions like “what is the meaning,” “what are the characteristics,” “how are they different,” etc.
- Ask the types of questions that get at what kids think.
- You need to have opportunities for discussion in the mathematics classroom.
- Think about the math talk in my classroom. Who talks more – me, one or two students, or most of the class? How much do I talk? How much do the students talk? Do my statements of questions encourage thinking and reasoning? Do I spend the time necessary to let students think and reason?
Next we heard from Jeremy Kilpatrick. Points he made that caught my attention:
- The Content Standards – Process Standards Dilemma: Content Standards are the ones we usually talk about. Test items group nicely by content but not by process standards. Also content standards are the “coin of the realm” in discussions of curriculum and standards.
- How do we tie the process standards to specific content is the pressing problem for teachers with Common Core State Standards. (e.g. Reason about and solve one-variable equations and inequalities – grade 6)
- It’s easy to do the content thing. It’s hard to do the process thing.
- How do we do this? Teaching mathematics through problem solving.
- George Polya – Read Mathematical Discovery (out of print) and The Stanford Mathematics Problem Book (back in print now)
- Polya’s important maxims about problem solving: 1) Learn problem solving by imitation and practice and 2) Reflect on one’s practice (we have to think about what we have done)
- Sarah Donaldson dissertation – Teaching through problem solving: Practices of four high school mathematics teachers. She watched and interviewed 4 teachers who use problem solving in their classroom on a regular basis. Common practices of the 4 teachers:
They all taught problem-solving strategies
Modeling problem solving (pretend you haven’t seen the problem before, model the act of problem solving)
Limiting teaching input
Promoting metacognition (solving the problem and then talking about yourself solving the problem – think about what you are doing, “watch yourself”)
Highlighting multiple solutions (push for other ways to solve the problem)
Last was William McCallum. If you are not aware, he was part of the committee who wrote the Common Core State Standards, so many of his points had to deal with the CCSS themselves.Important points to me:
- Teachers often don’t support reasoning and sense making
- He starts with a Grecian Urn – has form and shape and many fine details. Write standards for them, you end up with a pile of rubble.
- Does the text capture the structure of the subject? All the pieces are there, but you can’t see the structure.
- The structure is the standards. Process comes in navigating the standards.
- How to navigate structure, how to proceed structure (Standards for Mathematical Practice)
- The focus and coherence of the Standards for Mathematical Content are part of the standards.
- • The individual standards are part of the standards
- • They fit together into clusters, with descriptive headings that are part of the standards.
- • Clusters fit together into domains, whose names and arrangement are part of the standards.
- • In high school, domains are arranged into conceptual categories, which are part of the standards.
- Students need to see expressions as numbers.
- Think about the standards as not just the individual standards, but also the whole structure.
- http://www.illustrativemathematics.org/ – allows you to look at the Common Core State Standards from grade to grade.
- When you look at the Grecian urn – you look at the whole picture, then look at the details. We should be doing the same idea with the standards.
Then Ed Dickey asked the following questions for the panel:1) As teachers of mathematics, how do you assess reasoning and sense making?
A MC Question:
In pyramids ABCD and EFGHI shown above, all faces except base FHGI are equilateral triangles of equal size. If face ABC were placed on face EFG so that the vertices of the triangles coincide, how many exposed faces would the resulting solid have?
5 – 6 – 7 – 8 – 9 (7 – testmakers thought was correct)
This is #44 on the 1981 PSAT.
The correct answer is 5 (found by a 17 year old student at the time). Tests had to be rescored and credit given for 5 or 7.
If only 5 had been keyed correct, the item would have lost almost all of its worth for measurement.
2) Will the adoption of the Common Core State Standards we have a broad national models of what to teach. What about Professional Development?
Park City Math Institute is the most effective model she knows.
We expect teachers to continue to do mathematics. Teachers are thinking about the practice of teaching. We have to look at what works as well as what doesn’t work. Teachers have to work together to become resources for their colleagues.
Finally we conlcuded with audience questions –
What are the advantages and disadvantages of the two models in Common Core State Standards?
Gail – we looked at the textbooks to see what they are delivering for kids. Would not pick a side. Wants to know what the materials provide.
Jeremy – Integrated is in general a better curriculum. If you are going to make a switch, you have to prepare teachers for the switch.
William agreeed with Gail.
We want a greater and deeper understanding of mathematics in elementary and middle school years. With the way the current system is, we have a mile wide and inch deep curriculum. Is there some thought of instead of age progression to do content progression?
Jeremy – some of the best Professional Development out there is taking a look at the progression of the standards. You have to decide what to emphasize.
Gail – HS teachers need to look at what is happening in the Middle Grades. (a lot of what we do in 9th grade now is there) Assumption is that we are not going to back and reteach. Have to find a way to build support when students fall behind in the early elementary grades. District wide intervention needed (like in reading).
Paul Foerester – we are not here to cover material, we are here to uncover the material. When we come to something students should know, he would say “You recall… (whatever it is).” and would expect students to recall whatever it was after that without him telling them.
Resources to look at:
Professional Teaching Standards NCTM 1991
PCMI Website at Math Forum – materials for those workshops.
Overall, I thought there were some interesting points in this session. No real "Wow!" moments for me, but I think the session was good in making clear why the Standards of Mathematical Practice is important.