Making a difference in Mathematics Teaching and Learning Jozef Hvorecký Vysoká škola manažmentu, Bratislava, Slovakia University of Liverpool Liverpool, UK

Motto: It is a miracle that curiosity survives formal education. Albert Einstein

To Remember or To Comprehend? To remember: To be able of reproducing the learned fact To comprehend: To be able to apply the learned fact in a case of need Pupils are requested to remember but not to comprehend

Consequence Public consensus: Mathematics has no use for common people Good students’ opinion: If I’d know everything, I could be a Mathematics teacher. Silently presuming: But no one else.

Consequences of the Consequence 1.Economists, physicians, engineers do not expect that mathematicians could cooperate in solving their problems. 2.Many job opportunities for mathematicians are lost (by their not-creating).

Can someone comprehend the role of mathematics without remembering everything ? Mathematics as a goal Mathematics as a tool Examples: Payments in shops Surfing the Internet

Hyperbolic functions sinh x = cosh x = Why do we have them in our curriculum?

Excursion to history Hold the end of a chain in your hands in such a way that it is freely hanging. Which curve does it form?

And what is the truth? Galileo Galilei: Parabola Jungius (1669): Catenary i.e. c osine hyperbolic Hurray! Mathematicians also make errors!

Curing by shock A model of logistics?

Mathematics education today Typical formulations of problems to solve: Find the maximum of the function: F(x) = 693,8597 – 68,7672.cosh 0, x Solve the equation: 693,8597 – 68,7672.cosh 0, x = 0

Dreaming about future In 1965 in Saint Louis (USA) a huge arch was built. It tracks the curvature of (inverted) cosine hyperbolic described by the formula F(x) = 693,8597 – 68,7672.cosh 0, x How high is it? How far are its pillars from each other?

Right moment for applying information technology Intelligent calculators CAS Specialized software Max(693,8597 – 68,7672.cosh 0, x) Height of Arch: 192 m Root(693,8597 – 68,7672.cosh 0, x) Distance of its pillars: 225 m

Not the end yet Supporting curiosity: Why was the catenary chosen? –Set up hypotheses –Search through information sources (books, journals, Internet) –Set up discussion Local international

To be discussed among us How much do teachers/students have to know about the background of the problem before starting its solution? What relationships between Math and real life do clarify its real-life role in the most appropriate way? How to change: –the classroom atmosphere, –teachers, –pupils, –school administration, –parents?

Naïve vs. formal solution A shoemaker has been asked to make 100 special-purpose shoes. During the first week he produces nine of them, on the next week eleven, on the third week thirteen. He sees that due to his growing experience and improved skills he will be capable of producing two shoes more during every next week compared to the previous one. How long will it take for him to produce all pairs?

“Naïve” Solution Step 1 Step 2

Analytical Solution

Back to Our Calculator Setting parameters Getting 2 roots Roots: and Is naïve solution better than the formal one?

Real-life problem formulations Let us assume building a structure with many floors. The basic floor is built for a certain (basic) price. Building higher structures becomes more expensive as the material must be carried higher, the safety precautions must be stronger, and the risk is higher. For these reasons, insurance companies usually request higher payments to balance the risks. The ARITHMETIC insurance company gives you its proposal: Our basic insurance per floor is $15. For each floor above the ground you pay $2 more than for the previous floor. How much is the insurance cost for a 10- level building (i.e. for a building with a ground floor and 9 floors above it)?

Continuing Series of Problems To have a choice of offers, we visited another insurance company named GEOMETRIC. Its manager gave as a different proposal: Our basic insurance per floor is $1. For each floor above the ground you will pay 2-times more than for the previous floor. Their offer starts at much lower level than the one from the ARITHMETIC. Will we pay more or less for the whole building? When is it cheaper to take an offer from the ARITHMETIC and when from GEOMETRIC?

Motivating formulation A student body discusses a possibility to organize a fund-raising dinner. The presumed price of a ticket is $24. One member of the organizing committee has found an appropriate space for the event which can be rented for $350. Another one knows a company providing chairs for $1.50 per night plus free tables. The committee needs to know the minimum number of people which has to come for covering all these expenses.

Capability of Reading Drawings Costs: Space rental $350 Chairs for $1.50 Revenue: People for $24 Break-even point

Interpreting results How many must come to the party to raise our profit to $500? The party should be visited by 37,77 persons. 

Non-linear Models Two girls want to make money to buy Christmas gifts for their relatives and friends. They see their opportunity in making and selling necklaces from glass beans. They realized that first they have to invest $50 to various tools. For each necklace they also need a set of beans. The supplier offers them for the basic price $2, but the price declines by 1 cent per set.

Costs Fixed cost: $50 Variable cost per set: 2–0.1*x (x is the number of sets)

Revenue and Break-even point Revenue: $3.99 per necklace Intersection of costs and revenue

Further Analysis of the Solution Why are we interested in the x-coordinate? What is the meaning of y-coordinate? What if we would start selling with discounts, too?

Why Should We Ever Mention the Word “Quadratic”? Fixed cost: $50 Variable cost per set: 2–0.1*x (x is the number of sets)

Mathematics, Models and Reality Colette Laborde (20 June) –New types of tasks –Joining mathematics and technology We (22 June) –New types of tasks –Joining mathematics, models and reality

Examples of a new category of tasks About a model: A horse runs with the speed of 20 km/hour. What is the speed of 20 horses? About the interpretation of results: Our TV set has collapsed, we could not watch any programs. So, daddy went to the shop and bought 16 TV’s. Mommy went to the shop and bought 27. How many TV sets do we have?

Questionable interpretation Peter went to the forest behind his house and saw 5 giraffes, 4 gazelles and 7 lions. How many animals did he saw? In Mathematics, everything is absolute. In real life, everything is relative. No surprise that Mathematics does not attract pupils.

Thank you for your attention