Logical Dominoes: Planning Saves the Day

Without problems to solve, there would be no need for new inventions or innovations.

Problem solvers are the creators of our tomorrow; the builders of new eras, the faces in our history books.

Engineers and mathematicians are often regarded as lacking in creativity and imagination. We are seen as analytical number crunchers swallowed whole by occupations requiring systematic, rigorous analysis. Our drive for functional, practical, and efficient solutions is assumed to preclude even the slightest flicker of fanciful curiosity.

To the contrary, inspiration, passion and ingenuity lie at the very heart of everything we do as STEM professionals. We push boundaries and drive progress into new imagined realities.

We are problem solvers, we cry!

Students often fail to appreciate the beauty that is the art and creativity of problem solving; they expect and desire a formulaic approach that they can memorise by rote and execute on-demand. They ask, “How am I supposed to know what to do?”

But you’re a born problem solver, we reply!

We are, indeed, all innate problem-solvers, predisposed to analytical thinking and ingenuity in our quest for solutions to the problems that we face. Critical thinking, plus a blend of structure and flexibility, allow us to approach issues from different perspectives and to develop novel solutions. As problem-solvers, we use our minds to "craft" solutions, capturing sparks of ingenuity and moulding them with lived experience, and then turning those thoughts into concrete resolutions.

It’s challenging; it’s exciting; it’s rewarding!

In the real world, solutions are almost always developed gradually. Challenges rarely have a single, obvious answer. Viewing a problem from multiple angles, and honing our flexible thinking skills are key to developing good solutions. There are often no wrong answers, only stronger and weaker solutions. But as GCSE & IGCSE maths students, we are only apprentice problem solvers: our training involves seeking out precise answers to often over-simplified problems. Our answers are usually either correct or incorrect. But these exercises should not be scoffed at for being unrealistic or manufactured in nature: as our skills grow, so does the realism of the problems that we can tackle. F = ma is not going to put humankind on Mars, but it is where all aerospace engineers start out.

We can grow from apprentices into masters of this rather wonderful art form, but we must hone our skills over time to do so. We must practise analysing problems, by breaking each one down and examining its components. We must bring our own lived experiences to the table at every opportunity in order to better grasp the nuances of a problem and gain access to different perspectives. We must expand our mathematical abilities, too – but that’s for another course.

We must let go of our fear of failure!

If we are to experiment with ideas and explore potential solutions, then we must brave enough to fail. We must pick up familiar tools, drag them from the pages of our textbooks, and see whether they fit the job at hand. We will sometimes make the wrong choices; we will unknowingly choose the hard road; we will have to double back on our thinking at times, and start over. But we will learn from our mistakes and become better problem solvers for having made them.

This Course

Having hopefully inspired you to want to become a problem-solving virtuoso, we’d  like to share just a few hints and tips regarding this particular course. We intend for our courses to be excellent value for money, but also equally excellent value with respect to the time that you invest in them. With that in mind, as you work through this course, we urge you to:

• remember that learning Maths is never a passive activity

  • Have a pen and paper handy so that you can attempt the problems yourself. Or maybe you like the ‘Tell it to the Duck’ approach and would rather speak your thinking aloud.

  • If you can spot a starting point, then have a go at the problem before you watch our solution – it will make our solution that much more meaningful when you come back to the lesson and hit play.

  • When you can’t see a way to approach the problem, then have a go on your own after watching our solution. Can you make sense of the steps that we took, do the logical dominoes all fall into place for you?

  • Give yourself time to think. Take the time to muse over each problem and its solution, and ask yourself how you might have plucked out those logical dominoes for yourself and lined them all up.

• remember that we’re focusing on building problem solving skills. We want to establish a strong framework to guide our approach and thinking no matter the problem before us.

  • We want to eliminate the chaotic approach that can ensue when we dive headfirst into a problem without first pausing to think, and instead take the time to understand what a problem is asking and plan how to solve it.

  • We want to clearly identify the data or inputs needed.

  • With only a couple of exceptions, we keep the actual maths involved to that found in lower KS3, or topics that are typically taught around the age of 12.

• never underestimate the power and clarity than you can bring to a written, worded problem by roleplaying a similar real-world scenario in your mind’s eye. It helps you to organise your thoughts and to forge a natural and logical sequence of steps through to a solution.

Video Itinerary

Twinned with Fractions Deep Dive: Preparing for Algebra

This course alone will add valuable tools to your mathematical toolbox. It is nevertheless worth noting that we intentionally wrote it as a sibling course to our Fractions Deep Dive: Preparing for Algebra course in which we tackle ALL of the confusions that we see students wrestle with when they graduate from numeracy into maths-proper and start working algebraically.

These two courses, when combined, become greater than the sum of their parts. Using JUST the skills that we teach across these two courses, we demonstrate – using 2023 past papers – how you can achieve 95% of the marks needed for a Grade 4 at GCSE! Watch the marks add up for yourself in Chapter 8 (Exam Questions) of Fractions Deep Dive: Preparing for Algebra.

A note for our neurodivergent learners

The problem solving in this course is all about logic and the sequencing of events; it involves organising and planning multiple steps that ultimately lead to a solution. At times it places a heavy load on our executive functioning skills, including our working memory and cognitive flexibility. These are skills that do not suit everyone’s brain. While we’d encourage everyone to give the course a go – because the approach we teach gives students every chance to scaffold their own thinking and build solutions one step at a time – we appreciate that once we hit those 4-step questions in particular, some students might face information overload.

Should you find yourself hitting that brick wall at any stage in the course, then be sure to check out videos 5 (Raising the Barn) and 12 (4-Step in the Cowshed) where we introduce Dimensional Analysis as an alternative problem-solving method. This is an all-round different approach that provides a Plan B for all students, or indeed for some it becomes their much-loved Plan A; a lifeline for tackling this type of question.

So, we’re now ready to begin. Are you ready to join us?

Let’s embed a plan-first habit into our problem solving. A lifetime supply of problems awaits us out there in the big wide world!

Course Author:  Louise

Course Presenter:  Michael

Video Editor:  Alex

Supplementary Information and Disclaimers:

• Free-to-view videos are intended to give non-members a taster of our courses.

• The course worksheets and accompanying worked solutions are available only to members of Gill Learning.