Can you remember your basic algebra days? The endless equations asking you to solve for X or Y? Maybe you were assigned 40 or 50 problems a night, Y+2 = 5, solve for Y, or 4X-7 = 33, solve for X. I must have performed thousands of these equations and never really got the purpose. Just a math teacher drilling us in a seemingly endless list of equations, with no real meaning, and no real understanding of the critical importance of a linear equation. Perhaps, like me, this is your version of a 7th or 8th grade math class. Perhaps you even forgot it was called a linear equation.
The problem, of course, is that being able to solve a linear equation is a foundational mathematics skill that is used in nearly every in-demand profession, from financial analysts to architects and even doctors. For any student who wants to pursue a science, technology, engineering or math (STEM) related career, even through a vocational school degree, solving for X is something they must know. Sadly, the system in Thailand fails to offer kids the opportunity for engaging, hands-on learning. When teacher after teacher uses ineffective methods, like rote memorization and drill and practice, it shouldn’t surprise us when Thai students (and ultimately workers) do not grasp what a 'variable' is (that letter we need to solve for, be it X or Y) and its importance, or how and when linear equations can help us solve real-world problems.
In twelve years of working to improve STEM education in Thailand, we at Kenan have learned that our curricula works because we address two of the critical issues in the Thai educational context: knowledge gaps of both students and teachers, as well as differentiated learning needs of students. First, many students have significant knowledge gaps that prevent them from learning the next lesson, which severely limits their ability to progress and may derail their potential. As such, each lesson must be delivered to students without the assumption that they have mastered previous lessons. At the same time, our curricula offers engaging instruction for advanced students who quickly master a lesson, rather than requiring them to spend unnecessary effort drilling 50-60 similar problems. Second, teachers lack opportunities to develop their instructional skills, and may even have gaps in subject matter knowledge themselves. If we want them to break away from rote learning, we must develop their ability to teach complex matters in simple ways, and in some cases improve their subject matter knowledge.
Let’s take a look at a math lesson for linear equations developed by the world-renowned, Japanese curriculum company Tokyo Shoseki, and adapted by Kenan’s education experts for use in the Thai education system. Like every lesson in our math modules, this one has been matched carefully to the Thai curriculum, which allows teachers to replace the outdated materials used in most government-run schools with lessons that teach kids key concepts through hands-on, high-impact methods.
Through our program, teachers learn to use this lesson to make the abstract concept of linear equations tangible and easy for all kids to understand, even if they don’t enter the classroom as algebra whizzes. This means we make it simple for both the teacher and the students. We do this through the use of pattern recognition, repetition and application of formulas, so that kids can learn this foundational math skill at their own pace.
At the beginning of the lesson, kids are given a bundle of matches and asked to create five attached squares as in the diagram below. We use matches because they are cheap, and it’s easy to distinguish one end of the match from the other. Furthermore, the use of matches shows teachers how to take an otherwise abstract topic and allow for a blend of experiential and conceptual learning.
Each student recognizes that it requires 16 matchsticks to create five boxes. But the fascinating part comes when the teacher asks the students to explain how they figured out the answer. Some will say they counted around the outside first, while others will count the vertical matches first. Still others will share that they made a square with four matchsticks and then subtracted the overlapping ones. It becomes apparent to all that there are numerous ways to solve the problem. Here are some of the solutions written out arithmetically.
Outside First: 12+4 = 16
Vertical First: 6+10 = 16
Five boxes minus overlapping matchsticks: 4+4+4+4+4-4 = 16
Now if I asked you, how many matchsticks would you need to make 100 boxes? What would you do? Would you draw 100 boxes, or find more matchsticks? What if I asked you to make 1,000 boxes? How would you figure it out? Could you examine your five boxes and come up with an answer?
Let’s do some simple math. If I gave you seven matchsticks, how many boxes could you make? The answer is two. If I gave you 10 matchsticks, how many boxes could you make? The answer is three. Something is going on here that we can use to our advantage. Did you notice anything about the pattern in the boxes above? Each box uses the side of another box. Clearly, we need a formula (a linear equation, in this case) to express this pattern. But how do we find a formula?
Well, we know that each box has four sides, but one of those sides is shared when we add another box. It means that, when examining the boxes from left to right, the top, bottom, and left matchsticks are unique to each box, while its fourth side comes from the adjacent box. In other words, each box has three matchsticks to itself and shares one with the next box.
A formula that will allow us to find the number of matchsticks needed, even for a huge number of boxes, is 3N+1 = X, where N is the number of boxes and X is the number of matches. The “3” comes from the number of unique sides for each box and the “1” comes from the final box, which will need an extra matchstick to close in the box. Now let’s put the formula to use:
3*(box)+1 = matches
3*1+1 = 4
3*2+1 = 7
3*3+1 = 10
3*5+1 = 16 (above)
3*100+1 = 301
3*1000+1 = 3001
The formula always holds true. Similarly, we could use a formula to find the number of matchsticks needed when the boxes are not attached. This time the formula is 4N = X, because each box requires “4” unique matchsticks, so you simply multiply the number of boxes by “4” without the need to add “+1.” Using the formula, you can quickly find the number of matchsticks needed for two boxes (4*2 = 8), 4 boxes (4*4 = 16), and so on.
This lesson leads students to discover a simple way to recognize patterns and then express those patterns through formulas to find the number of matchsticks needed without the time consuming and inefficient process of building the boxes and counting the matchsticks individually. Even better, because students are learning through practical application and critical thinking, the same thought process and skills can be used in future careers.
A simple example is a business owner who makes widgets. Say she has a fixed cost of 100,000 THB for renting the machine that produces widgets and an additional cost of 100 THB to produce each widget. With only this small bit of information, she can create a formula that will accurately predict the costs for any amount produced. The formula is: Total Cost = 100X + 100,000, where X is the number of widgets produced. This process of analyzing relationships and identifying patterns is used by meteorologists to predict the weather, technicians to calibrate machinery, and doctors to prescribe medicine. The real work applications are endless.
The linear equation module is just one example of the exciting work conducted by Kenan Foundation Asia to deliver 21st century education in Thailand. Our model combines teacher training, mentorship, and the necessary curriculum and materials to help develop skilled teachers and inspire all students to achieve their dreams. You can learn more about our work here.
John DaSilva is Kenan's Head of Partnerships and can be contacted at [email protected]