TEACHING MATHEMATICS WITH REAL LIFE MODELS Dr. Renan Sezer LaGuardia Community College

Miss Match Between Professors and Students Difference in : Learning style Interest Appreciation Ability to deal with abstraction

Most professors teach mathematics in the style that they prefer, not in the style the students’ prefer.

Effective pedagogy must include motivation of students. What motivates a student to learn is the belief that the material he/she is about to learn is useful. “Useful” should mean more than “is required in order to graduate.”

What Makes a Material “Useful”? Related to one’s major Increases one’s understanding of inter- disciplinary problems Used in daily life in order to be a citizen who can fully participate in the decision making process be a knowledgeable consumer

First students must encounter a “real” life problem, that requires the particular concept they are about to learn, in its solution.

However, the application problems, intended to show the relevance of mathematics, are often concocted in nature (age problems, coin problems, train and pool problems) confirming students’ beliefs that mathematics has no relevance to real life.

The premise of this talk is that students learn mathematics best by doing mathematics in a meaningful context. This approach is the key to bridging the gap between application and abstraction and a basis for transfer learning.

The aim is to develop mathematical intuition and a relevant base of mathematical knowledge, while gaining experiences that connect classroom learning with real-world applications.

Other Desired Outcomes Are increasing knowledge of mathematics through explorations with appropriate technology developing positive attitude about learning and using mathematics building techniques of reasoning for effective problem solving working independently as well as in collaborative groups

How Can We Reach These Goals? By teaching mathematics through models that emphasize understanding, reflection and enable students to see mathematics in a concrete context that serves as a bridge to abstraction.

Advantages of This Approach Steers students away from memorization and following recipes Helps them formulate what they observe in a language that they understand Helps them generalize concepts and thus move closer to abstract thinking

Open Ended Problems & Learning to Organize Information Open ended problems can be unsolved problems or problems where you need to make some assumptions, that lead to multiple solutions Not necessarily require advance topics P:1-3 Time Management

Real Life Application I Super Bargain p:4-5 Composition of functions using percentage Having two discounts of 25% off and 40% off f(x)=0.75x, g(y)=0.60x and getting g(f(x))

Real Life Application II Per Capita Income P: 6-7 Social Concepts Solving Quadratics Numerically and Graphically Real life is messy Use technology as needed

Real Life Application III Blood-Alcohol Level P: 8-10 Rational Functions Linkages between numeric, graphical, algebraic approaches

Connecting Verbal, Numerical, Algebraic, Graphical Representations Charity Event P: Starts with verbal scenario Requires a simple algebraic model Numerical Representation Graphical Representation

The Interpretation What Does It All Mean? The answer is “2.5 buses” From beginning algebra to higher level math, interpretation is a must. Super Bowl Commercial P:14-15 Ask not what the intercepts are but what they signify?

At Calculus Level (Hughes-Hallett) The cost of extracting T tons of ore from a copper mine is C=f(T) dollars. What does it mean to say that f’(2000)=100? Since f’(T)=dC/dT is measured in dollars per ton, it means that when 2000 tons of ore have already been extracted from the mine, the cost of extracting the next ton is $100.

Calculus Hughes-Hallett (From an Article in the Economics) “Suddenly, everywhere, it is not the rate of things, that matters, it is the rate of change of rates of change. Nobody cares much about inflation; only whether it is going up or down. Or rather, whether it is going up fast or down fast. “Inflation drops by disappointing two points,” cries the billboard. Which roughly translated means that prices are still rising, but less fast than they were, though not quite as much less fast as everybody had hoped.” In the last sentence there are three statements about prices. Rewrite these as statements about derivatives.

Decision Making Cancer patients using a certain treatment have an average survival time of 2 years with a standard deviation of 4 months. A physician claims that his new method of treatment increases the average survival time of patients, while keeping the standard deviation the same. 100 randomly chosen patients who received this new treatment had an average survival time of 26 months.

1) What is the probability that 100 randomly chosen patients had an average survival of 26 months or more? 2) At the 95% significance level, test the claim that the new treatment is more effective. 3) Write a sentence indicating your conclusion in layman’s terms.

Projects Have them play the whole game Income and Expenses (Tax) P: Calculus p:126

CONCLUSION Such an approach prepares students effectively for further work in mathematics and related disciplines, while developing the skills required in solving mathematical problems arising in real life.