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A BALANCE BETWEEN CONCEPTUAL UNDERSTANDING AND PROCEDURAL SKILLS IN MATHEMATICS - A PROPOSED MODEL by Paul Vaz Following the presentation Revisions to MAT 270 by Jay Abramson and Pat Thompson, School of Mathematical & Statistical Sciences, ASU Fabio Milnor, Director, Mathematics for STEM Education, School of Mathematical & Statistical Sciences, ASU, sent out an email asking anyone interested, to send: COMMENTS/OPINIONS ON THE FOLLOWING: A balance between conceptual understanding and procedural skills in the math classes; How How to achieve this balance; to assess the accomplishment of reaching such a balance. THE DISCUSSION IN THIS PRESENTATION LEANS TOWARD CALCULUS CURRENT STATUS OF CALCULUS IN THE DEPARTMENT Calculus classes offered in the department are Calculus with Analytical Geometry (MAT 270, 271, 272) Calculus for Engineers (MAT 265, 266, 267) Calculus for Business (MAT 210, 211) Calculus for Life Sciences (MAT 251) More than 75% of our calculus students are not mathematics majors. DESCRIPTION IN THE ASU CATALOG OF FIRST –SEMESTER CALCULUS MAT 210, MAT 251: Differential and Integral Calculus of Elementary functions and Applications MAT 265: Limits and continuity, differential calculus of functions of one variable, introduction to integration. MAT 270: Real numbers, limits and continuity, and differential and integral calculus of functions of one variable. CORE TOPICS: FIRST–SEMESTER CALCULUS Compute limits Determine whether a function is continuous at a given point. Determine the interval(s) of continuity of a given function. Determine the derivative and antiderivative of standard functions, like polynomials, trigonometric, exponential, logarithmic, hyperbolic, and the standard inverse functions. Geometrical and Physical Applications of the derivative and the integral. NEEDS OF THE CLIENT Client Departments stress more on the need for functional understanding or procedural fluency (how to do) than on conceptual understanding (what to do). Client Departments seem to prefer accelerated programs and in some Universities, have designed their own Mathematics Classes. WHAT RESEARCH HAS TO Investigations SAY have shown that conceptual knowledge has positive effects on student learning. Students exposed to conceptual knowledge had greater retention and were more likely to use the ideas in new situations. Conceptual and procedural knowledge developed inter-dependently A REFERENCE ARTICLE “The conflict between conceptual and procedural knowledge: Should we need to understand in order to be able to do, or vice versa?” By Lenni Haapasalo University of Joensuu The author utilizes a recent analysis of conceptual and procedural knowledge, to present models for planning and evaluating of learning processes based on a large empirical project. THE DISTINCTION BETWEEN CONCEPTUAL UNDERSTANDING & PROCEDURAL SKILLS - AS STATED IN THE REFERENCE ARTICLE Conceptual knowledge denotes knowledge of and a skillful “drive” along particular networks, the elements of which can be concepts, rules (algorithms, procedures, etc.), and even problems (a solved problem may introduce a new concept or rule) given in various representation forms. Procedural knowledge denotes dynamic and successful utilization of particular rules, algorithms or procedures within relevant representation forms. CLOSING REMARKS IN THE ARTICLE An appropriate assumption seems to be that doing should be cognitively and psychologically meaningful for the pupil (team). A more important thing than to do is to understand what, why and how you are doing. This basically calls upon and considers the following questions: do I know that, do I know why, and do I know how I know MAIN GOALS IN THIS PROPOSAL Foster conceptual understanding without sacrificing procedural fluency and viceversa. That the learner achieves an optimum understanding of the concepts. That the learner can eventually apply the concepts productively. A MODEL FOR DELIVERY & ASSESSMENT This model proposes the consideration of the following elements in the plan of teaching and assessment: a. b. c. d. e. f. g. Conceptual Understanding Procedural skills Collaborative learning Multiple Representations Interdisciplinary connections Reading and Comprehension Challenge AN EXAMPLE OF A LESSON PLAN TANGENT LINES CONCEPTUAL UNDERSTANDING/PROCEDURAL SKILLS Definition of the tangent: Let y = f (x) be a differentiable function. A straight line is said to be a tangent to the curve at the point (c, f (c)) on the curve if the line passes through the point (c, f (c)) on the curve and The line has slope f '(c) where f ' is the derivative of f. NOTE You require the knowledge of the derivative to find the tangent line – this generates a large number of procedural problems in this unit. Differentiability (smoothness) implies the existence and uniqueness of the tangent line The tangent line as the limit of secant lines is motivational in defining the derivative. LESSON PLAN CONTINUED CONCEPTUAL UNDERSTANDING/PROCEDURAL SKILLS CONTINUED Finding the tangent line to a curve: Secants on a curve to find slopes Slope of a curve using the derivative Equation of a line – Review (PS & SI) Equation of a tangent at (c, f (c)): y f (c) f (c)( x c) WHERE DOES THE METHOD FAIL? The method assumes the existence of the derivative and so fails where the derivative does not exist: At points where the tangent is vertical At corner points At discontinuities “PROCEDURAL” IS NOT ENOUGH by B. Abramovitz, M. Berezina, A. Berman, L. Shvartsman 4th MEDITERRANEAN CONFERENCE ON MATHEMATICS EDUCATION University of Palermo - Italy 28 – 30 January 2005 PROCEDURAL FORM Let g ( x) 3 sin 5 x and f (u ) e (4 u 12) / 5 Find the tangent line of h( x) f ( g ( x)) at x 0 . CONCEPTUAL FORM 2. Let y 5x 3 be the tangent line of the function Let f ( g ( x)) e 4x g ( x) at x 0. for every real number Find the tangent line of the function x. f ( x) at x 3. LESSON PLAN CONTINUED MULTIPLE REPRESENTATIONS (RULE OF 4) Estimating the slope of a curve using a graph paper Numerical calculation of the slope of a curve Definition of the slope of the curve – the derivative Connection between slope of the curve and the rate-ofchange of the function. COLLABORATIVE LEARNING In-class activities: These activities reflect a mixture of exercises to assess the conceptual understanding and the procedural fluency of the student LESSON PLAN CONTINUED INTERDISCIPLINARY CONNECTION The connection between the slope of the curve and ‘rateof-change’ of the function opens the doors to a wide range of applications in various fields Linear Approximations: The tangent line at a point on the curve is the best straight-line approximation to the curve at that point. CHALLENGE “Differentiable implies continuous” “Discontinuous implies non-differentiable” Existence of the slope of a piecewise defined function at a dividing point. Constructing a piecewise defined function that is differentiable (smooth) at the dividing point ASSESS WHETHER WE HAVE ACCOMPLISHED THE GOAL OF REACHING SUCH A BALANCE By way of: In-class Group Activities Quizzes Online Homework system Midterms Final Exam THINGS TO BE CONSIDERED Is class time sufficient for learning that fosters conceptual understanding and procedural skills without sacrificing one for the other? Some learners end up with a concept different from the intended one. Do we need to consider time for the refinement or correction of the learner's understanding of the concepts