Outline Creativity Good, . . . And Yet, and Yet . . . Informal (Qualitative) . . . Creative Discussions Formal (Quantitative) . . . or Memorization? We Must Alternate . . . Which Topics Should . . . Maybe Both? Other Applications of . . . Applications Beyond . . . (on the example Home Page of teaching Title Page Computer Science) ◭◭ ◮◮ ◭ ◮ Olga Kosheleva 1 and Vladik Kreinovich 2 Page 1 of 19 1 Department of Teacher Education Go Back 2 Department of Computer Science University of Texas at El Paso, El Paso, TX 79968, USA Full Screen olgak@utep.edu, vladik@utep.edu Close Quit
Outline Creativity Good, . . . 1. Outline And Yet, and Yet . . . • We all strive to be creative in our teaching. Informal (Qualitative) . . . Formal (Quantitative) . . . • However, there is often not enough time to make all We Must Alternate . . . the topics creative fun. Which Topics Should . . . • So sometimes, we teach memorization first, under- Other Applications of . . . standing later. Applications Beyond . . . Home Page • We do it, but we often do it without seriously analyzing which topics to “sacrifice” to memorization. Title Page • In this talk, we use simple mathematical models of ◭◭ ◮◮ learning to come up with relevant recommendations. ◭ ◮ • Namely, all the topics form a dependency graph. Page 2 of 19 • The most reasonable topics for memorization first are Go Back the ones in the critical path of this graph. Full Screen Close Quit
Outline Creativity Good, . . . 2. Creativity Good, Memorization Bad And Yet, and Yet . . . • Modern pedagogical literature is very convincing: Informal (Qualitative) . . . Formal (Quantitative) . . . – creative discussions lead to a better understanding We Must Alternate . . . – than memorization. Which Topics Should . . . • Gently guided by an instructor, students Other Applications of . . . Applications Beyond . . . – solve interesting problems and Home Page – uncover – themselves – the desired formula. Title Page • This is great: ◭◭ ◮◮ – the students fell good about it, ◭ ◮ – they remember it better, Page 3 of 19 – they use it more creatively. Go Back Full Screen Close Quit
Outline Creativity Good, . . . 3. And Yet, and Yet . . . And Yet, and Yet . . . • Some students of introductory CS cannot move forward Informal (Qualitative) . . . since they forgot a formula for the log of the product. Formal (Quantitative) . . . We Must Alternate . . . • Some forgot even how to add fractions. Which Topics Should . . . • Yes, we can stop and let them recreate this formula – Other Applications of . . . but: Applications Beyond . . . Home Page – do we really want to teach a few weeks less com- puting and a few weeks more math? Title Page – and are we, CS folks, the best teachers of math? ◭◭ ◮◮ ◭ ◮ Page 4 of 19 Go Back Full Screen Close Quit
Outline Creativity Good, . . . 4. What We Do And Yet, and Yet . . . • What many of us do is: Informal (Qualitative) . . . Formal (Quantitative) . . . – have students memorize the needed math and We Must Alternate . . . – use the remaining time to be creative in computing. Which Topics Should . . . • Even in computing: Other Applications of . . . Applications Beyond . . . – we ask students to memorize patterns correspond- Home Page ing to sum, maximum, etc., Title Page – instead of having them re-create all these codes cre- atively every time. ◭◭ ◮◮ • We do it, but we do it shamefully: should not every- ◭ ◮ thing in education be creative fun? Page 5 of 19 • Our point is: maybe we should not feel guilty. Go Back • In this talk, we justify our point by analyzing simple Full Screen mathematical models of teaching. Close Quit
Outline Creativity Good, . . . 5. Informal (Qualitative) Analysis of the Problem And Yet, and Yet . . . • Our first argument is that: Informal (Qualitative) . . . Formal (Quantitative) . . . – while creative teaching is good, We Must Alternate . . . – it is often slower. Which Topics Should . . . • In most classes, there is a dependence between mate- Other Applications of . . . rial: Applications Beyond . . . Home Page – to study some topics, Title Page – students need to know some previous ones. ◭◭ ◮◮ • In the resulting dependence, there is often a critical path. ◭ ◮ Page 6 of 19 • Along this path, it may be better to use memorization first – and get a deep understanding later. Go Back • Another argument is that we want to optimally use the Full Screen student’s brains. Close Quit
Outline Creativity Good, . . . 6. Analysis of the Problem (cont-d) And Yet, and Yet . . . • Yes, it would be nice if we could keep the brains in the Informal (Qualitative) . . . permanent state of active creative fun. Formal (Quantitative) . . . We Must Alternate . . . • However, brains get tired, they need rest. Which Topics Should . . . • Here, memorization helps. Other Applications of . . . • To solve a non-trivial problem, we use creative thinking Applications Beyond . . . Home Page to find known patterns for solve it. Title Page • Then we “switch off” the active brain and use memo- rized techniques to solve the resulting subproblems. ◭◭ ◮◮ • If we end up with a quadratic equations, we do not ◭ ◮ want to recall the tricks that lead to the formulas. Page 7 of 19 • We just want to plug in the numbers. Go Back • Meanwhile, the active brain rests and gets ready for Full Screen new creative activities – and everyone benefits! Close Quit
Outline Creativity Good, . . . 7. Formal (Quantitative) Analysis of the Problem And Yet, and Yet . . . • Let us denote the total amount of creative effort that a Informal (Qualitative) . . . student can perform during the learning period by E . Formal (Quantitative) . . . We Must Alternate . . . • We want to have the best overall learning result. Which Topics Should . . . • What is the proper way to distribute this amount be- Other Applications of . . . tween different moments of time? Applications Beyond . . . • Let n denote the overall number of moment of time. Home Page • Let e i denote the amount of creative effort that a stu- Title Page dent uses at moment i . ◭◭ ◮◮ • Let r ( e ) denote the amount of learning that results ◭ ◮ when a student uses a creative effort e . Page 8 of 19 • In these terms, we want to maximize Go Back – the overall results, i.e., the sum r ( e 1 ) + . . . + r ( e n ), Full Screen – under the constraint that the overall creative effort Close e 1 + . . . + e n is equal to the given amount E . Quit
Outline Creativity Good, . . . 8. Solving the Problem And Yet, and Yet . . . • We want to find the values e 1 , . . . , e n that Informal (Qualitative) . . . Formal (Quantitative) . . . Maximize r ( e 1 ) + . . . + r ( e n ) We Must Alternate . . . under the constraint e 1 + . . . + e n = E. Which Topics Should . . . Other Applications of . . . • Lagrange multiplier technique leads to Applications Beyond . . . r ( e 1 ) + . . . + r ( e n ) + λ · ( e 1 + . . . + e n − E ) → max . Home Page Title Page • Differentiating relative to e i and equating the deriva- ◭◭ ◮◮ tive to 0, we get F ( e i ) = 0, where we denoted ◭ ◮ def = r ′ ( e ) + λ. F ( e ) Page 9 of 19 • Intuitively: Go Back – small changes in the amount of creative effort e Full Screen – shouldn’t drastically affect the learning result r ( e ). Close Quit
Outline Creativity Good, . . . 9. F ( e ) Should Be Analytical And Yet, and Yet . . . • Therefore, it is reasonable to assume that the function Informal (Qualitative) . . . r ( e ) is smooth. Formal (Quantitative) . . . We Must Alternate . . . • r ( e ) is probably even analytical (i.e., can be expanded Which Topics Should . . . in Taylor series). Other Applications of . . . • In this case, the function F ( e ) is also an analytical Applications Beyond . . . function. Home Page • It is known that an analytical function F ( e ) �≡ 0 can Title Page only have finitely many roots on an interval. ◭◭ ◮◮ • Thus, all the optimal effort amounts e i must belong to ◭ ◮ the finite set of these solutions. Page 10 of 19 • For usual analytical functions, this set of solutions is Go Back small. Full Screen • Indeed, an arbitrary analytical function, by definition, is equal to its Taylor series. Close Quit
Outline Creativity Good, . . . 10. F ( e ) Should Be Analytical (cont-d) And Yet, and Yet . . . • An arbitrary analytical function, by definition, is equal Informal (Qualitative) . . . to its Taylor series. Formal (Quantitative) . . . We Must Alternate . . . • It can therefore be approximated, with an arbitrary Which Topics Should . . . accuracy, by a polynomial. Other Applications of . . . • A polynomial of degree d can have no more than d Applications Beyond . . . roots; so, e.g.: Home Page – if a cubic polynomial is a reasonable approximation Title Page for the function F ( e ), ◭◭ ◮◮ – then, in this approximation, the function F ( e ) has ◭ ◮ no more than 3 roots. Page 11 of 19 • So, we use no more than three different levels of cre- ative effort. Go Back • A 7-th order polynomial is usually enough for most Full Screen known analytical functions such as sin, cos, etc. Close Quit
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