Challenging the young learners in secondary school to determine the minimum capacity of a tank that can be filled up using different jars, one jar at a time, with a capacity of 8, 10, and 15 litres, failed to meet the condition for correctness and significance of the mathematical LCM.

The maths contest was supposedly scripted by examiners from a higher learning institution in Kenya, laying bare the need to repurpose the contests and enhance the quality of moderation if the contests are to do justice to the young minds they intend to inspire and develop.

The Critical Question on Teaching for Developing Thinkers

Does it matter much to a teacher nowadays to underscore First Principles, treating as mere parentheses the popular shortcuts, such as product divided by the sum or the difference when dealing with pipes filling up or emptying tanks, F = ma, and so on? Do learners have the self-drive or create the time to ask why the simplified formulae are what they are? What does a rainfall of 1000 mm per annum mean and its implications for rainwater harvesting? What is the practical significance of probability or the Least Common Multiple (LCM) applied to real life?

The Crucial Case in Kenya

Fresh off the page of the IBD Thought Leader’s Diary is a recent striking example of a maths contest for secondary schools held at the Kenya High School in May 2024. I looked at the first question, supposedly set by examiners from a higher learning institution, and got a rude shock. The mistake was glaring, inviting questions about the quality of moderation and the purpose of such examinations, especially when dealing with young minds who should be inspired to be critical thinkers and innovative solution-finders. Challenging the young ones to determine the minimum capacity of a tank that can be filled up using different jars, one jar at a time, with a capacity of 8, 10, and 15 litres, failed to meet the condition for correctness and significance of the mathematical LCM. Applying First Principles makes it evident that the remainder in each jar after filling up the tank alone must be a common number for all the three jars, but the question specified a remainder of 2 litres for only the jars of 8 litres and 10 litres, confusing the learners with a remainder of 13 litres in the 15-litre jar – instead of 2 litres (or 13 litres to be poured out of it in the last lap to be correct). If the aim of such an exam is to discover talent among learners and inspire them to develop their genius, then the answer lies in quality moderation and rewarding procedures grounded in First Principles, not just answers from cramming simplified formulae.

Back to the Basics – First Principles

Applying First Principles involves breaking down a complex problem into its fundamental parts and understanding it from the ground up. This method encourages critical thinking and a deeper understanding of concepts rather than relying on memorised shortcuts. Of late, the mode of instruction at the basic education level has been rushed to complete the primary and secondary school syllabi and equip learners with shortcuts and formulae that help them to score high marks in national examinations so as to elevate the stature of the schools they enrolled in. Little surprise, then, that few graduates can be good examples of critical thinkers and complex problem solvers.

Here’s why this approach is crucial in basic education:

1. Encourages Deep Understanding:
   • Students learn the ‘why’ and ‘how’ behind a solution, which fosters a more profound comprehension of the subject matter.
2. Promotes Problem-Solving Skills:
   • By dissecting problems and building solutions from fundamental principles, students develop strong problem-solving skills that are applicable in various contexts.
3. Enhances Creativity and Innovation:
   • Understanding the basics allows students to think creatively and come up with innovative solutions to new problems.
4. Builds a Strong Foundation:
   • A solid grasp of first principles provides a robust foundation upon which more advanced concepts can be built, making learning more progressive and cumulative.
5. Develops Critical Thinking:
   • Students are trained to question assumptions, analyse problems logically, and approach solutions systematically, which are essential skills to both academic and real-world settings.

By applying first principles in education, we not only equip students with knowledge but also empower them with the ability to think independently and tackle challenges confidently.

Bringing out the Meaning: LCM, Mixed Fractions, and Improper Fractions

In the stated problem, we should apply fundamental principles to derive the minimum tank capacity that can be filled up by jars of 8 litres, 10 litres, and 15 litres, each jar remaining with 2 litres in it after completing the job of filling up the tank alone. It must be emphasised that if the remainder in a jar is 2 litres, then the LCM as the upper limit for no remainder necessitates that any of the three jars must remain with 2 litres after filling up the tank. Why? The LCM is the minimum tank capacity that will leave no remainder in any of the jars, or that will ensure the last lap takes in a full jar in each case. This is a perfect case of positive integer multipliers. The remainder in the filling far is a complement of the mathematical remainder used in mixed fractions. Again, the remainder of 2 litres means that the number of whole jars will be one less than the number of times needed for any of the jars to fulfil the LCM. Therefore, since the LCM here is 120 = 8*15 = 10*12 = 15*8, then the number of whole jars in this case will be 14 for the 8-litre jar, 11 for the 10-litre jar, and 7 for the 15-litre jar. Respectively, the last laps will help define the mathematical remainders/complements of the reminders in each jar (2 litres), as follows: 6 for 8-litre jar; 8 for 10-litre jar; and 13 for the 15-litre jar. This approach returns the following non-simplified mixed fractions (do not simplify to avoid the risk of reducing the target capacity): 14⁶/₈ or 11⁸/₁₀ or 7¹³/₁₅ and it can be seen in the resulting improper fractions that the common minimum tank capacity for fulfilling the condition is the numerator, 118, in each case divided by 8, 10, or 15 to leave remainders. At this point, having applied the first principles to arrive at the correct answer of 118 L, it is timely to tell the learners that the simplified formula is LCM minus the common remainder in each filling jar. This should be shared only after understanding the basics.

It must be noted that the remainder in the filling jar/divisor is common for such a condition to be met, but the remainder to be added in the last lap to fill up the tank varies for each jar. It is the latter that should be used in the mathematical expressions or other more complex formulae, such as modular arithmetic, congruences, or Chinese Remainder Theorem.

Conclusion

The example shared demonstrates the importance of a methodical approach to problem-solving, highlighting the value of first principles in fostering a deeper understanding and robust critical thinking skills in education. Moderation of content and contests for quality assurance remains a key requirement for modern learners to grow into innovative thinkers and leaders. Teaching and mentoring learners to think, critique, and challenge the status quo, as such, should be given top priority right from the basic education level.

By Nashon Adero

IBD.

IBD empowers youth with the knowledge, international exposure, and digital fluency they need to be emancipated global citizens with borderless influence for sustainable development.