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Multi-stage Arithmetic Direction

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  1. Multi-stage Mathematics Instruction Jeff Knisley East Tennessee State University MAA-Southeastern Section, Spring 2005

  2. I enjoy teaching… • Teaching is its own reward. • The satisfaction of seeing students progress • The joy of studying and sharing mathematics for a living • The mutual benefit of the student-teacher relationship • Favorite Quote: They won’t care what you know until they know that you care.

  3. But are they learning anything… • Performance doesn’t always imply learning • In the 70’s, a student was taught MACSYMA but not Calculus • Student “Aced” a series of MIT Calculus Tests • I have often wondered… • Are they learning, or are they just good at taking my tests? • How close are they to understanding a concept well enough to put it to good use? • How can I make mathematics and problem solving less frustrating for them?

  4. How do students learn math? • Early on, I felt I had to have an answer to this question in order to teach at all. • Reviewed Math Education Research • Explored Cognitive and Applied Psychology • Had some experience in Artificial Intelligence • I combined the research, some observations, and some simple experiments into a “macro-model” of how a student learns math

  5. Outline • Mathematics Education, Applied Psychology • What the Experts say about learning • Some observations and simple experiments • A “Macro-model” for mathematical learning • As a guide for implementing new curricula • Not an exact description of mathematics students • Using the model to identify “Best Practices” • As an indicator of what works and what doesn’t • As a guide for using Technology in mathematics

  6. What the Experts Say…Math. Ed. • Individual Learning Styles • Some students are visual learners, some learn by synthesizing ideas, some learn by imitation • Each student has a preferred learning style • Preferred style used to construct concepts • Kolb Learning Model • Those who learn by building on previous experience • Those who learn by trial and error • Those who learn from detailed explanations • Those who learn by implementing new ideas • Much of this from R. Felder, Engineering Education

  7. What the Experts Say…Psychology • Learning Models • Different people associate new ideas to old ones at different rates • Different people memorize information at different rates • Individual Differences in Skill Acquisition • Give subjects simple “air traffic control” game • First they discover simple heuristics—how to land planes, how to create holding patterns • Their game abilities improve in “jumps” as they develop strategies that allow sophisticated actions

  8. What the Experts say…AI • Heuristic reasoning • Associates a pattern with an action • Closest Pattern determines method used • Criteria for “closest” often yields incorrect result • Is knowledge without understanding • Heuristics = Rote Learning • Reduces learning to a set of rules to memorize • Replaces comprehension with association

  9. Pattern Action Heuristic yields Incorrect Result: Example: Heuristic Reasoning Problem: Simplify

  10. Observation: Math Learning Types • Kolb Learning styles translated into math • Allegorizers: They prefer form over function, and often ignore details • Integrators: They want to “compare and contrast” the known with the unknown. • Analyzers: They desire logical explanation and detailed descriptions • Synthesizers: They use known concepts like building blocks to construct new ideas. • Other models yield similar Math Styles Allegory (noun): figurative treatment of one subject under the guise of another (Webster)

  11. 3 c b a Experiment: Pythagorean Theorem • Procedure: • Present an example right triangle with sides given and hypotenuse unknown • Prove the Pythagorean theorem and use it to determine the hypotenuse • Measure the three sides of the triangle and show it satisfies Pythagorean theorem • Distribute paper with right triangle with unknown hypotenuse ( and rulers) ?

  12. Expected Observations • Allegorizers: (@ 15 % of sampled students) • They reduce learning to a set of “Case Studies” • They look in the text for a worked example • Integrators: (@ 60 % of sampled students) • Ruler is known, Hypotenuse is unknown • They use the ruler to measure the hypotenuse • Analyzers: (@ 20 % of sampled students) • They use the Pythagorean theorem • They seem to want a logical explanation • Synthesizers: (A handful, at best) • Use theorem and explore to find a 3-4-5 triangle

  13. Observation: Topic implies Style • Style is a function of student and topic • A student may be an analyzer in Linear Algebra • Same student may be an allegorizer in Statistics • We resort to Heuristics when all else fails • Math Ed research shows that even the best students fail to understand limits • Students pass tests on limits by resorting to heuristics—memorization and pattern-based association

  14. Definition of the Macro-Model • Students acquire new concepts by progressing through 4 stages of understanding • Allegorization: A new concept is described in terms of existing knowledge (i.e., intuitively) • Integration: Comparative analysis is used to distinguish new concept from known concepts • Analysis: New concept becomes part of existing knowledge. Connections and explanations follow. • Synthesis: New concept is used as a “building block” to establish new theories, new strategies, and new allegories

  15. The Importance of Allegories • Learning begins with allegory development • New concept stated in a familiar context • Allegory is description within the given context • Insufficient allegorization prevents learning • Failure to Allegorize forces a Heuristic Approach • Some “good students” have sophisticated heuristics

  16. B1 C1 D1 E1 F1 D1 C1 B1 A1 A1 A1 A1 A1 A1 A1 A1 immobilize (capture is figurative language) A2 A2 A2 A2 A2 A2 A2 A2 B2 C2 D2 E2 F2 D2 C2 B2 Example: Chess without Allegories Valid moves for a given token are determined by token’s type. Each player attempts to capture the other’s F token. Each player receives 8 “A” tokens, 2 each of “B,” “C,” and “D” tokens, and 1 each of “E” and “F” tokens

  17. Discussion: Learning Chess • Context is Medieval Military Figures • Game pieces themselves are allegories • Pawns are numerous but have limited abilities • Knights can “Leap over objects” • Queens have unlimited power • “Capture the King” is the allegory for winning • Colors are allegories • White Versus Black • “Battles” take place when 2 pieces occupy the same square on the checkerboard

  18. Components of Integration • Student now understands allegorically that there is a new concept to be acquired • Places a label on the new idea – i.e., a definition • Definition places concept into a mathematical setting • Compare and Contrast • How is new concept like known concepts? • How does new concept differ from known concepts? • We often neglect this stage • Visual Comparisons are the most powerful • Technology can be used to produce comparisons

  19. Tangent Planes Which plane, A and B, is tangent to the surface

  20. Analysis of a New Concept • The new concept takes on its own character • Explanations and origins are developed • Techniques for use of new concept are developed • The new concept becomes one of many characters • Connections to existing ideas are established • Sphere of influence becomes well-defined • What known concepts are related to new concept? • How are known concepts modified by the new concept? • Analysis desires that a great deal of relevant information be delivered quickly(i.e., lectures)

  21. Synthesis and Problem Solving • New idea becomes a tool • To create allegories for new ideas • To create new versions of existing knowledge • To solve problems and prove theorems • Strategy development • New concept and known concept are combined into sophisticated constructions • New concept is used to solve problems • Applications are desired and explored in depth

  22. Identifying Best Practices… • Too often, instruction is directed at analyzers • Lectures and techniques • Integrators and Allegorizers are lost/confused • Synthesizers may get bored and fall behind, making them allegorizers for later material • Most Students forced to use heuristics • Integrators and Allegorizers memorize rules • Analyzers often apply heuristics anyway • Example: Studies have proven this for Limits

  23. Example: Uncertainty Principle • Physics student asked me to explain Heisenberg Uncertainty to him from a Mathematical perspective • Mathematical: If A and B are self-adjoint and AB – BA = I, where I is the identity operator, and if f is in dom(A) ∩ dom(B) vector with ||f ||=1, then ||Af || ||Bf || > ½ • The amazing thing is that non-commutivity of A and B implies the lower bound

  24. Example: Heisenberg Uncertainty • Proof: Define Q(t) = || (A+itB)f ||2 Expand to show that Q(t) = ||Af ||2 + <i(AB-BA)f, f >t + || Bf ||2 t2 Q(t) > 0 implies that 4 ||Af ||2 ||Bf ||2 > |<i(AB-BA)f, f >|2 = 1 • Main Example: (Af)(x) = xf(x), (Bf)(x) = f ′(x) over L2(R)

  25. Heisenberg Uncertainty • Mathematically, Uncertainty is tricky f is differentiable a.e., but cannot allow f in dom(B) because || Bf || = 0 • So how to explain the mathematics of uncertainty to a physics student?

  26. Adjustable points Slope depends on ||Bf|| shorter steeper ||Af|| Area under curve = 1 ||Af|| = ||Bf|| = Using the Model • Allegory: relating derivative to multiplication operator • Integration: Interactive applet where they attempt to construct a function that minimizes uncertainty Area under the curve = 1

  27. Best Practices… Defining Roles • Role of the Teacher • Allegorization: Teacher is a story-teller • Integration: Teacher is a guide* • Analysis: Teacher is an expert • Synthesis: Teacher is a coach • Role of Technology • Integration = hands on student exploration • Technology for integration • After Introducing a concept • Before extensive lecture on the concept

  28. Example: Exponential Growth • How to introduce the exponential (and later, logarithmic growth) to a biology student? • Standard: so there is 2<e<3 such that • The fact that y=et satisfies y′=y does not mean all that much to a biology student

  29. Using the Model… • Allegory: Birth/Death processes • A population growing at a rate of k% per hour does not reproduce all at once. • Instead, reproduction takes place many times per hour • Exponential growth is a birth process with a constant % rate in which reproduction takes place arbitrarily many times per hour

  30. Introductory Systems Ecology… • Integration: • Divide time interval [0,t] into n short periods, where n is a very large integer • Having n generations of reproduction means n periods where each period has length h = t / n • “Probability” of reproduction in each time period is kh, which is % rate scaled over period • Simulation: Start with P individuals and let each reproduce over 1st period with probability kh. Repeat for all n time periods (

  31. Introductory Systems Ecology… • Start with P0individuals • After 1st period: P1 = P0 + kh P0 = P0(1+kh) individuals • After 2nd period: P2 = P1 + kh P1 = P0(1+kh)2 individuals • After nth period: Pn = Pn-1 + kh Pn-1 = P0(1+kh)n individuals • n arbitrarily large means n approaching ∞ • Definition: The exponential function is defined and from this we can derive all properties of the exponential.

  32. Best Practices…Technology • Technology as intermediate assessment • Multivariable Calculus = All quizzes are Maple worksheets ( • Intro Stats • 1200 students per semester in our gen ed course • 4 stage instruction • Lecture – Applets – Computer – Assessment • Lecture and Assessment are traditional • Applets prepare for graded Minitab activities • Technology for extended projects

  33. Technology as aid to Student Research • Allegory – Introduction to research problem, where problem is an extension of known result • Integration -- Student uses Maple, NetLogo, etc. to reproduce or simulate known result • Analysis – Predict an answer to problem using technological extension of known result • Synthesis – Proof or otherwise solution of given research problem

  34. Research Examples: • M.P. began by reproducing a well-known agent-based army ant raiding pattern model en route to agent-based model of division of labor in social insects. • P.C. began with simple implementation of classic Neural Net algorithm en route to using neural nets for data mining micro-array data • A.T. began with simple curve-fitting algorithm en route to proving a version of the C.H. theorem for a class of elliptic operators

  35. Summary Allegory (intuition) that leads to integration, perhaps via technology and interactive assessment, so that they are ready for lecture-based analysis that fleshes out the concept, and then can begin to synthesize their own ideas But students can’t create their own allegories or coach themselves as synthesizers. Thus, the model ultimately predicts the necessity of the mutually-beneficial student-teacher relationship.

  36. Thank you! Any Questions?