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Learning with UnderstandingEntry Task

- Individually, take two minutes to write your own definition of Learning for Understanding.

Lesson ObjectivesEach student will

- Learn that eating candy in class is unacceptable
- Develop and refine a definition of Learning for Understanding
- Compare & contrast student definitions of Learning for Understanding against those of leading theorists
- Utilize Zoltan’sStages of Learning
- Apply theories of Learning for Understanding to mathematics instruction

William ‘Dry Reading’ Brownell (1947)What is Understanding?

- Understanding:

“Meaning is to be sought in the structure, the organization, the inner relationships of the subject itself.”

- Premise:

“The basic tenant in the proposed instructional reorganization is to make arithmetic less a challenge to the peoples’ memory and more a challenge to his intelligence.”

Brownell (1945)A Longstanding Debate

- “It is a mistake to suppose that meaningful arithmetic is something new, something cut out of the whole cloth, as it were, during the past twenty or twenty-five years.”

Brownell‘Meanings of’ vs. ‘Meanings for’

- Meanings of (pure math):

Mathematical understandings

- Meanings for (applied math):

Applications of mathematics to real life

BrownellDegrees of Meanings

- Meanings are relative, not absolute
- Degrees of exactness, depth, and growth depend on the audience
- Levels (degrees of meaning) build

BrownellMeanings of Arithmetic

- Basic concepts

(whole numbers, fractions, percents, etc.)

- Fundamental operations

(+, - , ×, ÷)

- Principles, relationships, & generalizations

(e.g. additive identity, commutative property)

- Decimal number system

BrownellInteractive Interlude

Consider Brownell’s definition of Learning for Understanding: “Meaning is to be sought in the structure, the organization, the inner relationships of the subject itself.”

Compare and contrast your definition of Learning for Understanding (from the entry task) with Brownell’s.

Zoltan ‘Maverick’ Dienes (2004)Father of the Base 10 Block“What is Understanding”

- Understanding:

Art (an excerpt)

“This is where the artistic creation begins,

Creatures might have no legs or the fish have no fins,

What the brush puts on canvas has been through the fire,

Let us get to the meaning we shall all inquire.”

From the crucible of such creations’ wild act

Comes a language that does not describe any fact !

Mental states are thus formed recipients’ minds

So that each one a meaning quite readily finds.”

ZoltanTeacher as Source of Knowledge

- Teacher delivery of content is not an issue
- Technology incorporation may be necessary, but it is not sufficient (i.e. not a ‘silver bullet’)
- Student may be unable to ‘receive’ knowledge
- Student may be unwilling to ‘receive’ knowledge—in which case ‘artificial’ motivation may be employed (Behaviorism?)

ZoltanStages of Learning

- Initial Interaction
- Discovery of regularities in situations and consequent play with sets of rules or restraints
- Comparison of several games possessing the same structure (search for isomorphisms)
- Representation of isomorphic situations in one, all-embracing, usually graphical form.
- Study of the representation by the description of its properties (symbolizing)
- Formalization (proof)

ZoltanGames as Stages of Learning

- Developed games in response to stages of learning
- Preliminary games
- Structured games
- Practice games

ZoltanInteractive Interlude

Circular Villages

Organize yourselves into groups of 3 – 4. Your teacher will distribute the Circular Villages handout. Use Zoltan’s Stages of Learning to complete steps 1 – 3. Raise your hand if you need assistance and follow all directions carefully or there will be no break.

Jerome ‘Renegade’ Bruner (1996)What is Understanding?

- Understanding:

“Is the outcome of organizing and contextualizing essentially contestable, incompletely verifiable propositions in a disciplined way.”

BrunerBruner’s Conceptual Framework

- A person categorizes new objects and events that occur in his environment according to the properties they are seen to have in common with other objects and events previously categorized.

Warp to Forms of Representation

BrunerBruner in Historical Context

- 1957: USSR launches Sputnik
- 1959: Woods Hole Conference on curriculum reform
- 1960: The Process of Education

Bruner (1960)The Process of Education

- “Any subject can be taught effectively in some intellectually honest form to any child at any stage of development.”

BrunerForms of Representation

- Enactive mode

Involves human motor capacities

- Iconic mode

depends on visual or sensory organization

- Symbolic mode

involves reasoning, words, and language

- New forms are added and become dominant, but old forms remain

Skip to Bruner Interlude

Bruner (1966)Toward a Theory of Instruction

- “A theory of instruction … is in effect a theory of how growth and development are assisted by diverse means.”

BrunerSpiral Curriculum

- Teach a meaningful topic at a level appropriate for the child and revisit later to create a more explicit and mature understanding.

BrunerInteractive Interlude

In groups of 3 – 4, compare and contrast each other’s definitions of Learning with Understanding. Discuss each other’s definitions in light of those of Bruner, Zoltan, and Brownell.

Richard ‘Buzzword’ Skemp (1971)What is Understanding?

- Understanding:

“To understand something means to assimilate it into an appropriate schema.” (schema: a conceptual structure)

- Premise:

The widespread negative attitude towards mathematics is caused by the widespread failure to teach relational mathematics.

Skemp (1977)Instrumental & Relational Understanding

- Instrumental Understanding:

Possessing a rule and the ability use it—rules without reasons

- Relational Understanding:

Knowing both what to do and why

a

b

a

a2

ab

b2

b

ab

SkempExample of Instrumental & Relational Understanding- Instrumental Understanding: Justification by FOIL

(a + b)2 = (a + b)(a + b) = a2 +2ab + b2

- Relational Understanding: Picture ‘proof’

SkempInteractive Interlude Part I

In groups of 2, use Skemp’s definition of Instrumental Understanding (possessing a rule and the ability use it—rules without reasons) to identify, list, and discuss mathematics examples of textbook and classroom instrumental explanations.

SkempInteractive Interlude Part II

In groups of 2, use Skemp’s definition of Relational Understanding (knowing both what to do and why) toattempt to provide relational explanations for the traditional algorithms for dividing two fractions or for finding the area of a circle (your choice).

Warp to Hiebert & Carpenter

SkempInstrumental/Relational Mismatches

- Instrumental Students + Relational Teacher = Frustrated Teacher
- Relational Students + Instrumental Teacher = Frustrated Student (negative, far-reaching consequences)
- Instrumental Teacher + Relational Textbook = Frustrated Author

James ‘Abbot’ Hiebert & Thomas ‘Costello’ Carpenter (2000) What is Understanding?

- Understanding:

“We understand something if we see how it is related or connected to other things we know.”

(Hiebert cites Brownell, Carpenter & Hiebert)

- Premise:

“Understanding is the most fundamental goal of mathematics instruction” (emphases added)

James & ThomasImportance of Understanding

- To really know math requires that it is understood; utility is not enough
- Confidence
- Engagement
- Internally rewarding/satisfying (compare to Behaviorism)

James & ThomasThe Path to Understanding

- Reflecting (origins in cognitive psychology, emphasis on internal mental operations)
- Communicating (origins in social cognition with emphasis on the context of learning and social interaction)

Warp to Pros & Cons OR Warp to Exit Task

James & ThomasClassroom Framework

Reflecting & communicating are crucial. We need a framework within which these can happen.

- Classroom tasks
- Teacher’s Role
- Classroom social culture
- Mathematical tools
- Equity and accessibility

James & ThomasFramework 1—Classroom Tasks

- Make math problematic
- Connect with students where they are at (zone of proximal development)
- Leave behind something of mathematical value (usefulness is important [applied math] but pure mathematics must also result)

James & Thomas Framework 2—Teacher’s Role

- Ideas & methods must be valued
- Share essential information (not too hot, not too cold, jusssst right)
- Establish classroom culture

James & ThomasFramework 3—Classroom Culture

- Ideas & methods must be valued
- Students choose & share methods
- Mistakes are learning opportunities
- Correctness resides in mathematical argument

James & ThomasFramework 4—Mathematics Tools

- Meaning of tools must be constructed by each student
- Used to solve problems
- Used to record, communicate, & think

James & ThomasFramework 5—Equity & Accessibility

- Tasks accessible to all students
- Each student has a ‘voice’ in the classroom
- Every student contributes

Warp to exit task

Rote

Within its own context, instrumental mathematics is usually easier to understand.

Rewards are more immediate & more apparent

One can get the correct answer quickly and reliably

Harder to retain (in absence of repetition)

Conceptual

Transferable

Easier to remember/recover

Harder to learn

Relational knowledge can be effective as a goal in itself

Relational schemas are organic in quality

Reduces repetitive practice

Safeguards against ‘silly’ answers

Versatility of attack

‘Rote’ vs. ‘Conceptual’ Pros & ConsTeaching for UnderstandingGeneralImplementation Issues

- Standardized/Required Testing
- Breadth of coverage
- Assessment difficulties
- Teacher acceptance

Teaching for UnderstandingRural Implementation Issues

- Good ole boys
- Good enough for me good enough for my kids
- Change is bad
- Lack of resources (e.g. technology)
- Homework support
- Projects that require out-of-class research

Final InterludeExit Task

Individually, refine and explain your original definition of Learning with Understanding. If you believe that your original definition does not need to be refined, please explain why.

When you are done, you may be excused if you can demonstrate, via possession of at least 5 packages of Smarties, that you participated in today’s class.

References

- Bart, W. (1970). Mathematics Education: The Views of Zoltan Dienes. The School Review, Vol. 78, No. 3.
- Brownell, W. (1945). The Natural Sciences and Mathematics. Review of Education Research, Vol. 15, No. 4.
- Brownell, W. (1947). The Place of Meaning in the Teaching of Arithmetic. The Elementary School Journal, Vol. 47, No. 5.
- Bruner, J. (1966). Toward a Theory of Instruction. W.W. Norton & Company, New York.
- Bruner, J. (1966). The Culture of Education. Harvard University Press, Cambridge, MA.
- Bruner, J. (1966). The Process of Education. Harvard University Press, Cambridge, MA.
- Dienes, Z. (1960). Building Up Mathematics. Hutchinson Educational LTD, London.
- Hiebert, J, Carpenter, T, & others. (1996). Problem Solving as a Basis for Reform in Curriculum and Instruction: The Case of Mathematics. Educational Researcher, Vol. 25, No. 4.
- Hiebert, J, Carpenter, T, & others. (2000). Making Sense: teaching and learning mathematics with understanding.
- Kilpatrick, J., Wearver, J.F. (1977). The Place of William A. Brownell in Mathematics Education. Journal for Research in Mathematics Education, Vol. 8, No. 5
- Noddings, N. (1994). William Brownell and The Search for Meaning. Journal for Research in Mathematics Education, Vol. 24, No. 5.
- Schoenfeld, A. (1992). Learning to Think Mathematically: Problem Solving, Metacognition, and Sense-Making in Mathematics. Handbook on Research for Mathematics Teaching and Learning.
- Schoenfeld, A. (2005). Mathematics Teaching and Learning. University of California, Berkeley.
- Skemp, R. (1971). Psychology of Learning Math. Penguin Books Ltd, Harmondsworth.
- Skemp, R. (1976). Relational Understanding and Instrumental Understanding. Mathematics Teaching, 77.
- Skemp, R. (1987). The Psychology of Knowing Math. Lawrence Erlbaum Associates. Hillsdale, NJ.

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