Research Mathematics

Learning to think like a mathematician

This course introduces students to mathematical investigation. Instead of solving only narrowly formulated exercises, students learn to explore what happens when the conditions of a problem change. Students learn that mathematics is not only a collection of established methods. It is also a process of experimentation, observation, conjecture, proof, and refinement.

• asking meaningful “What if?” questions;
• recognising patterns and formulating conjectures;
• testing ideas through examples and counterexamples;
• organising the results of an investigation;
• using Desmos, GeoGebra, spreadsheets, AI;
• distinguishing experimental evidence from proof;
• presenting mathematical conclusions clearly;
• extending elementary problems into deeper investigations.

Example Investigation
Five turtles are standing at different points on a number line (see picture below). Where should they meet so that the total distance they travel is as small as possible?

Believe it or not, this problem comes from a 4th-grade Kangaroo Math competition. You could try different points where the turtles must gather and count the total distance covered each time. Sooner or later you would find out that they all have to meet at the point with coordinate 3. It gives a minimum total distance of 13 units. Is it a coincidence that the minimum occurs at the position of the middle turtle? What pattern might this suggest?

But now let's ask more questions that turn an ordinary problem into a mathematical investigation: What if?

What if there are 3, 7 (odd) turtles? Or 4, 6 (even)? Or 100, 101?
What if some of the turtles start from the same point?
What if each turtle has a different "cost" of moving, so that some distances count more than others?
Can we reformulate the problem?
Can we use Desmos/GeoGebra to investigate motion of many turtles instead of doing everything by hand?
And what if the turtles move on a plane instead of a number line?

A single elementary problem can grow into a deep mathematical investigation.

We can model the problem algebraically by introducing a variable x for the point where all the turtles meet. Referring to the figure above, the blue turtle travels a distance of |x - (-3)|, the green one will cover |x - 1|, the orange one: |x - 3| and so on... The total distance covered by all turtles is now given by a function D(x) = |x+3|+ |x-1|+ |x-3|+ |x-4|+ |x-7|. It is time for Desmos or GeoGebra to come on the scene!
Look at the following graph of D(x) plotted on GeoGebra.

Notice that the graph of D(x) reaches its minimum at x = 3. This means the minimum possible total distance is achieved at x = 3, so it is the optimal meeting point for the turtles. We have found a new tool for mathematical investigation: graphing software such as GeoGebra or Desmos.

I am looking forward to seeing what would happen if there were 4 turtles instead of 5. Let’s remove the blue turtle located at -3. Then D1(x) = |x-1|+ |x-3|+ |x-4|+ |x-7| and its graph drawn now on Desmos is below:

Wow! It turns out that for the case with four turtles they can meet at any point between x = 3 and x = 4. This opens up an entirely new line of investigation. We now have a powerful tool for investigation, but also many questions to answer.

I will stop here and leave you to explore the problem further.

What I offer in this course is more than 20 genuine investigations on a range of topics, progressing from accessible introductory problems to genuinely challenging investigations.

This course is suitable for:

• curious students who enjoy asking questions;
• students preparing for mathematical projects or investigations;
• strong school students who need more than routine exercises;
• learners interested in mathematical modelling and technology;
• students who want to develop independent mathematical thinking.

Probability Theory

Understanding randomness, uncertainty, and risk

Probability is one of the most important areas of modern mathematics, yet in many school programmes it appears only as a relatively small topic within a much broader curriculum. Students learn a few standard rules and solve several familiar types of problems, but often do not have enough time to understand how rich, useful, and widely applicable the subject really is.

For some students, probability and mathematical statistics may eventually become part of their future profession. They are fundamental to data science, artificial intelligence, machine learning, actuarial science, quantitative finance, economics, risk analysis, medical research, engineering, and many areas of business analytics. An early introduction allows students to discover whether they enjoy this type of thinking and gives them a valuable foundation before university, where probability and statistics courses often progress very quickly.

The course is also an excellent preparation for AP Statistics and for more advanced study of mathematical statistics, data analysis, and stochastic models.

We study the core topics of a standard probability syllabus, but the course goes beyond learning formulas and completing routine exercises. Students solve a large and varied collection of problems, compare different methods, investigate surprising results, and learn to recognise probability structures in unfamiliar situations.

Special attention is given to carefully designed competition problems, including problems from AMC contests. These questions are often elegant, unexpected, and mathematically deeper than ordinary textbook exercises. They help students develop counting skills, logical reasoning, mathematical intuition, and the ability to construct a solution when no standard method is immediately visible.

We also explore connections between probability and other areas of mathematics. For example, in financial mathematics a future payment is not always guaranteed. Once we introduce the probability that a payment will occur, questions about expected value, risk, insurance, investments, and financial decisions become much more realistic and interesting.

The exact depth of the course depends on the student’s age and mathematical background. Younger students may begin with games, coins, dice, counting, and simulations, while more advanced students can progress towards conditional probability, random variables, expected value, distributions, and applications of mathematical statistics.

Financial Mathematics

Mathematics for personal finance and investment decisions

Financial mathematics connects school mathematics with decisions students and adults make in everyday life.
The course introduces the mathematics of saving, borrowing, investing, and evaluating financial alternatives. Students learn not only to apply formulas, but also to interpret financial information and understand the assumptions behind calculations.

The course provides mathematical and educational foundations. It does not offer personal investment advice or recommend particular financial products.

Topics of the course:

• percentages and percentage change;
• simple and compound interest;
• inflation and purchasing power;
• savings plans;
• loans and amortisation;
• effective and nominal interest rates;
• present and future value;
• annuities;
• mortgages;
• stocks and dividends;
• bonds and bond pricing;
• investment returns;
• risk and diversification;
• using a financial calculator or spreadsheet.

This free "BA Financial Calculator" is one of the main tools we use in the course:

Below is a selection of problems from the course. They illustrate the range and depth of the topics we explore.

#1. A shop advertises everything is "half price in today's sale." In addition, a coupon gives a 20% discount on sale prices. Using the coupon, the price today represents what percentage of the original price?

#2. The annual compound interest rate was 13% in 2014, 11% in 2015, and 15% in 2016. Find the effective annual compound interest rate that produces the same return over the three-year period.

#3. A bank offers a three-year loan with interest compounded quarterly and a two-year loan with interest compounded monthly. In both cases, the nominal annual interest rate is 20%. Which loan is more advantageous for the company? Compare the effective annual interest rates in the two cases.

#4. You purchased a small shop, including all its equipment, for $50,000. You expect to receive a profit of $10,000 after one year, $12,000 after two years, $15,000 after three years, $20,000 after four years, and $25,000 after five years. Determine:
a) When will the initial investment in the shop be recovered?
b) What rate of return do you expect to earn from the shop over the five-year period?
c) The future value of the profits if the bank interest rate is 12%;

d) The present value of the profits at the time of purchase.

#5. A loan of $60,000 issued at an annual interest rate of 6% is repaid through quarterly payments of $8,000. Calculate the time required to repay the loan.

#6. Prepare an amortisation schedule for a $5,000 loan repaid through equal annual payments over four years at an annual interest rate of 9%. How would the total interest paid change if only the interest were paid each year and the entire principal were repaid in a single payment at the end of the term?

#7. A 14-year bond with a face value of KZT 100,000 and an 8% coupon rate, with interest paid quarterly, is selling for KZT 92,000. What is its nominal annual yield?

#8. Karl van Loon takes part in a quiz in which he has to answer true-or-false questions. The questions are difficult and unfamiliar, so Karl is forced to guess. If he answers all five questions correctly, he will receive €2,000, while four correct answers will earn him €400. Karl had to pay €100 to enter the game. Is it financially worthwhile for Karl to participate?

#9. How much money would you be willing to give someone today if they promised to pay you KZT 30,000 at the end of each month for two years, given that in each month there is a 5% probability that they will stop making payments? You require a 13% rate of return. What important condition must also be satisfied for your conclusions to be valid?

#10. Don Jorge Valdano purchases bonds issued by his old rivals, Barcelona. The yield at the time of purchase was 19%. However, the bond is callable before maturity, which would reduce the yield to 8%. The probability of an early redemption is 40%.

Don Jorge also has the option to sell the bond before Barcelona calls it, earning a 15% return. Should he exercise this option?