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Rod and Spring Approximation

Rod and Spring Approximation. Steve Gutstein Eric Freudenthal Ali Jalal-Kamali Vladik Kreinovich David Morgenthaler * University of Texas at El Paso and *Lockheed-Martin.

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Rod and Spring Approximation

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  1. Rod and Spring Approximation Steve Gutstein Eric Freudenthal Ali Jalal-Kamali Vladik Kreinovich David Morgenthaler* University of Texas at El Paso and *Lockheed-Martin This work is supported by the NSF, DHS, Texas Instruments, Microsoft, Calculex, and the Computing Alliance of Hispanic Serving Institutions. Our opinions are not necessarily shared by our funders.

  2. Goals of STEM courses • Procedural Fluency – The ability to execute (mathematical) procedures with competence • Conceptual Understanding – A sufficiently reflective understanding of which (mathematical) procedure is suitable for a given problem and why to facilitate discovery and understanding of additional problems. • Productive Disposition – The confidence to apply and develop mathematical solutions for novel problems, when suitable

  3. Linear Best Fit w/ Least Square Error(linear regression) • High school curriculum requires student use • But algebraic proof is painful and doesn’t enlighten • Exposes dilemma. Either • Expose students to proof. Lessons: • Math is hard, proofs don’t explain. • Only geniuses understand them • No advantage to following proof, just memorize. • Tell students to consult magician within the calculator • Worse: memorize an opaque & intricate algorithm • Lesson: math is magic spells. Only wizards understand magic. • Both messages disempower

  4. Rod-Spring Approximation – an Accessible Precursor to Least Squares X1 X2 X3 10 p = Raster((20,15)) samples = [( 5.0, 4.0), (10.0,10.0), (15.0,10.0)] n = Sx = Sy = Sxy = Sxx = 0 for x,y in samples: x *= 2; y *= 2 Sx += x; Sy += y Sxy += x*y; Sxx += x*x n += 1 p.set((x,y), blue) m = (n*Sxy – Sx*Sy) / (n*Sxx – Sx**2) b = (Sy – m*Sx) / n for x in range(0, 40): p.set((x, m*x+b), red) Y2 = Y3 4 Y1 b = 2 5 10 15 m=0.6 mxi+b - yi -1 +2 -1 Fi = Ti = xiFi -5 +20 -15 Balancing forces ΣFi = 0 Σ(mxi+ b – yi) =0 mΣxi+Σb - Σyi=0 Let Sx=Σxi ;Sy= Σyi mSx+bn – Sy=0 Balancing torques ΣTi = 0 Σxi(mxi + b – yi) =0 mΣxi2+ bΣxi - Σxiyi=0 Let Sxx=Σxi2;Sxy=Σxiyi mSxx+bSx – Sxy= 0

  5. Why does this matter? • Data from our intervention • Media-Propelled Computational-Thinking http://iMPaCT-STEM.org • Engages students in quantitative problem solving • No sexy graphics. Just simulation & plotting of kinematics • All math & physics transparent & accessible • From surveys of college freshmen:

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