1. Developing Mathematics Lessons “The Big Ideas”

2. The Verbs of Doing Mathematics Explaining Investigating Exploring Interpreting Analyzing Describing Deriving Reasoning Justifying Proving Conjecturing Calculating Examining Constructing Grappling Engaging

3. Instrumental vs. Relational Understanding What are some key ideas from the article? What quotes did you make note of? What questions do you have? What ideas about teaching did this article generate?

4. Instrumental vs. Relational Understanding – Key Ideas Instrumental is easy way out for teachers – easier to give rules than to explain why Rules don’t necessarily lead to ability to apply Relational understanding can be reconstructed There could be a place for both types of understanding

5. Instrumental vs. Relational Understanding - Quotes The analogy comparing learning to navigate a town The kind of learning which leads to instrumental mathematics consists of the learning of an increasing number of fixed plans, by which pupils find their way from particular starting points (the data) to required finishing points (the answers to the questions). In contrast, learning relational mathematics consists of building up a conceptual structure (schema) from which its possessor can (in theory) produce an unlimited number of plans for getting from any starting point within his schema to any finishing point.

6. Instrumental vs. Relational Understanding - Questions When is it okay to give instrumental explanations? Is it ever okay? What type of understanding do I value? What is my definition of mathematics?

7. Instrumental vs. Relational Understanding – Ideas about teaching Our role as mathematics teacher needs to change from providing students with instrumental knowledge to facilitating opportunities for students to construct relational knowledge

8. What I do in my Classroom What are some key ideas from the article? What quotes did you make note of? What questions do you have? What ideas about teaching did this article generate?

9. What I do in my Classroom – key ideas Finding the balance – freedom to think and engage with mandated curricular requirements Multiple points of entry so students can work from where they are Allowing all students the opportunity to engage with the topic Being creative Asking good questions that help students to further their own thinking

10. What I do in my Classroom – Quotes I have no time for the argument that there are some pupils who will never be able to ‘do’ trigonometry. I know this is probably true but I do not consider it part of my job to decide in advance who those people are. P.57 ‘every time you put your hand up you shut someone else’s brain off!’ p.61

11. What I do in my Classroom – questions How do I envision my own classroom? How will I ensure that all students have an opportunity to engage with the mathematics? What strategies can I use to ensure that certain students do not dominate the discussion and deny others the opportunity to think for themselves?

12. What I do in my Classroom – ideas about teaching