Teaching for Understanding

1. Teaching for Understanding A Characteristic of Effective Instruction in the Iowa Core

2. Outline • Overview and set the stage • What is understanding? • How to teach for understanding. With videos, research, opportunities for reflecting and discussing, and optional extended examples from mathematics

3. Purpose To stimulate thinking and discussion, and develop understanding of Teaching for Understanding

4. Overviewandsetting the stage

5. Do students learn by … ? • Receiving knowledge OR • Constructing knowledge [Discuss briefly.]

6. Should teachers … ? • Teach by telling (transmitting knowledge) OR • Teach by asking(orchestrating students’ active involvement in constructing knowledge) [Discuss briefly.]

7. Teaching for Understandingin the Iowa Core … Strongly emphasizes an interactive, constructivist approach(but not exclusively). • “Teachers need to learn when the interactive, constructivist forms of teaching are called for and when other less demanding, conventional strategies are appropriate.” • “The vision of practice engendered by teaching for understanding does not assume transmission strategies are inappropriate for all tasks. Some learning objectives might be best achieved through lectures [telling, direct instruction].” [McLaughlin & Talbert 1993, p. 4] Example: The convention that + means add and  means multiply

8. Iowa Core Mathematics We must shift from a paradigm of “memorize and practice” to one of “understand and apply” • This doesn’t mean no more memorizing and practice. The driving focus, however, is to understand and apply. • For practice to be effective, it must be meaningful – “build on and extend understanding” (Kilpatrick, et al. 2001) • “Students who memorize facts or procedures without understanding often are not sure when and how to use what they know, and such learning is often quite fragile.” (Bransford, et al. 1999)

9. What is understanding? Does the following video clip from the popular media illustrate “understanding”?  Mathematics scene from “Little Big League” [Find this three-minute video using on online search.For example, try:http://www.youtube.com/watch?v=VnOlvFqmWEY] [Discuss briefly.]

10. Optional Math Discussion See the next 9 slides for an optional in-depth discussion of this video clip …

11. Now that you’ve seen the movie and the math problem …

12. Joe can paint a house in 3 hours, Sam can paint it in 5 hours. How long does it take for them to paint it working together?

13. Analyze, Discuss, Solve • Answer • Did they get the right answer? • 15? 8? 4? 1 7/8?? • Strategy • What solution strategies did they use? • What solution strategies could be used? • Knowledge • Understanding? Procedural? Conceptual? Deep?

14. Understanding and Strategies in the Movie(Ever exhibited by your students?!) • Not a clue • “Math never did make any sense to me.” • No understanding • Combine all numbers every which way, hope for the best and maybe partial credit. • “Math is about computational procedures, I’m not sure which one to use or why, but I’ll give it a shot.” • No deep understanding • Magic formula • “Math is about formulas. Must memorize and match to the right problem.” • Maybe some understanding, or could just be rote memory.

15. Some Meaningful & ProductiveSolution Strategies • Think about it • Make sense of it • Estimate • Guess, test, and refine • Draw a (useful) picture • Make a table, look for patterns • Draw and trace a graph • Write and solve an equation • Derive and use a formula

16. Is there any math in the task?(concepts and procedures) • Fractions • Proportional reasoning • Computation • Estimation • Solving equations • Multiple representations (equation, graph, table, diagram) • Linear functions Lots of good mathematics!

18. And the formula in the movie? Instead of 5 and 3, use a and b: x/5 + x/3 = 1 • x/a + x/b = 1 So, bx + ax = ab (multiply thru by ab) and x(a + b) = ab (factor) Thus, x = ab/(a+b). So the player remembered the correct magic formula!

19. Problem-Based Instructional Tasks • Help students develop a deep understanding of important mathematics • Are accessible yet challenging to all students • Emphasize connections, especially to the real world • Encourage student engagement and communication • Can be solved in several ways • Encourage the use of connected multiple representations • Encourage appropriate use of intellectual, physical, and technological tools

20. End Optional Math Discussion

21. Now take a closer look atunderstandingandteaching for understanding

22. The Nature of Understanding • Connections • The mental attempt to connect something to something other than itself (Bartlett 1931, from Newton 2000) • The connecting of facts … the weaving of bits of knowledge into an integrated and cohesive whole (Nickerson 1985, from Newton 2000)

23. The Nature of Understanding 2. Structures (models, schema) • Only mental structures that answer the question “why” deserve to be called understanding. (Piaget 1978, from Newton 2000)

24. The Nature of Understanding 3. Performances (especially application & transfer) • Understanding is the ability to think and act flexibly with what one knows. • Neither basics or skills are worthwhile unless they can be mobilized in significant performances of understanding. (A performance view of understanding, in contrast to representational views of understanding in terms of structure or mental models.) (Perkins 1998, Gardner 1998)

25. The Nature of Understanding 4. Sense Making • “I truly believe that (a) mathematics coheres–it really does make sense!–and (b) mathematics can be taught so that students come to see it so.” • “Move beyond acquisition of facts to sense making in a subject area.” - All disciplines! (Schoenfeld 2009, McLaughlin & Talbert 1993)

26. The Nature of Understanding 5. Reflection • The most stringent criterion of understanding is the availability of knowledge to consciousness and reflection.(Brown 1986, from Newton 2000)

27. The Nature of Understanding 6. Depth and Type of Understanding • Iowa Core focuses on: Deep understanding Both conceptual and procedural knowledge Deep Conceptual and Procedural Knowledge

28. Deep Conceptual and Procedural Knowledge Deep knowledge (Willingham 2002): • rote • inflexible • deep structure Conceptual knowledge • What is it? (e.g., What is a fraction?) Procedural knowledge • How do you do it? (e.g., Add fractions.) (See separate presentation on Deep Conceptual and Procedural Knowledge.)

29. Deep Conceptual and Procedural Knowledge Does the following video clip from the popular media illustrate deep conceptual and procedural knowledge?  Video of Ma and Pa Kettle doing long division … (Find this two-minute video by searching online. For example, try: http://video.google.com/videoplay?docid=7106559846794044495#) [Briefly discuss.]

30. Discussion • Conceptual knowledge? Knowledge of numbers? Knowledge of operations? Any number sense? (An intuitive sense about numbers, their magnitude, structure, relationships, and operations) • Procedural knowledge? Knowledge of how to compute? Using correct procedures? Which ones? Why or why not?] • Deep knowledge? No evidence here!

31. How to Teach for Understanding Content QuestionsFor teachers to ask themselves about any topic as they prepare to teach for understanding: • What is it? (conceptual knowledge) • How to do it, compute it, operate on or with it? (procedural knowledge) • What is it good for? (relevance, application) • What is it connected to? (connections, structure, coherence)

32. How to Teach for Understanding Teaching and Learning QuestionsFor teachers to ask themselves about any topic to help focus on teaching for understanding: • Common misconceptions – What are they and how will I address/resolve them? • Questions – What questions will I ask to probe and deepen student understanding? • Scaffolding – What is just the right amount of support and structure I should provide for student learning? • Reflection – How will I provide time and design opportunities for students’ critical reflection?

33. How to Teach for Understanding “Looking across domains, studies consistently find that highly effective teachers support the process of meaningful learning by:” • Creating ambitious and meaningful tasks • Engaging students in active learning • Connecting to students’ prior knowledge • Scaffolding the learning process • Assessing student learning continuously • Providing clear learning goals • Encouraging strategic and metacognitive thinking (Darling-Hammond 2008)

34. Why teach for understanding? This is a “sea change” in pedagogy, so why undertake it? “These new demands [of the 21st Century] cannot be met through passive, rote-oriented learning focused on basic skills and memorization of disconnected facts… [We need] learning that enables critical thinking, flexible problem solving, transfer of skills, and use of knowledge in new situations.” (McCarthey and Peterson 1993, Darling-Hammond 2008)

35. Teaching for Understanding:a characteristic of effective instruction in the Iowa Core • Overview • What is understanding? • How to teach for understanding. With videos, research, opportunities for reflecting and discussing, and optional extended examples from mathematics