Beyond Behaviorism Learning for Understanding. Learning with Understanding Entry Task. Individually, take two minutes to write your own definition of Learning for Understanding . Lesson Objectives Each student will. Learn that eating candy in class is unacceptable
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“Meaning is to be sought in the structure, the organization, the inner relationships of the subject itself.”
“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.”
Applications of mathematics to real life
(whole numbers, fractions, percents, etc.)
(+, - , ×, ÷)
(e.g. additive identity, commutative property)
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.
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.”
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.
“Is the outcome of organizing and contextualizing essentially contestable, incompletely verifiable propositions in a disciplined way.”
Warp to Forms of Representation
Involves human motor capacities
depends on visual or sensory organization
involves reasoning, words, and language
Skip to Bruner 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.
“To understand something means to assimilate it into an appropriate schema.” (schema: a conceptual structure)
The widespread negative attitude towards mathematics is caused by the widespread failure to teach relational mathematics.
Possessing a rule and the ability use it—rules without reasons
Knowing both what to do and why
abSkempExample of Instrumental & Relational Understanding
(a + b)2 = (a + b)(a + b) = a2 +2ab + b2
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.
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
“We understand something if we see how it is related or connected to other things we know.”
(Hiebert cites Brownell, Carpenter & Hiebert)
“Understanding is the most fundamental goal of mathematics instruction” (emphases added)
Warp to Pros & Cons OR Warp to Exit Task
Reflecting & communicating are crucial. We need a framework within which these can happen.
Warp to exit task
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)
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 & Cons
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.