Spoon-feeding: Leaves Too Much Potential on the Plate: Part 1

The Singapore Math method is a teaching method that first connects students to concrete and relatable scenarios, empowering them to develop understanding, intuition, and mastery in new mathematical concepts, and then uses that understanding as a springboard into more complex, abstract, and flexible ideas.

Singapore Math teaches through a signature three step method, called the CPA method, presenting math concepts in the most relatable dimension first and then progressing to the abstract: 

1. Working with Concrete objects and manipulatives to model a simple problem.
2. Representing the model Pictorially, enabling students to connect their understanding of the physical problem to visual and mental representations.
3. Translating and applying these model-based concepts into the Abstract symbolic and numerical language we recognize as mathematics.

Sounds simple, right? Or maybe a bit too abstract?

Let’s break it down into something more concrete.

Spoons Everywhere
Across a lot of the world, students are expected to master mathematical concepts through rote learning. Picture this: you're learning subtraction with two-digit numbers for the first time. Your teacher writes a question on the board, stacking one number on top of the other, with a big line for the answer underneath them both. 

Thirteen minus eight. And because you already know a little bit about subtraction, you're not allowed to use your fingers. 

But, your mind protests, along with all the other young and hungry minds in the room, you can't take away eight from three!

"So, here's now we do it," your teacher says. "This is the ones column, and this is the tens column. Now we're going to learn the concept of borrowing." 

"All you have to do," she continues, "is cross out the tens, subtract one out here, write the new amount of tens, put a tick mark here, subtract as normal, and voila! There's your answer!" 

You do a second worksheet, trying desperately to remember where all the cross outs and tick marks go, and hiding your fingers under the desk, hoping you won't get caught counting on them. 

Also known as spoon-feeding, rote learning involves"feeding" students a series of mechanical steps to solve a problem, and having them repeat them over and over again. They practice the mechanics until they have them memorized and can perform them (hopefully) without error. 

Students might later be given word problems to connect the mechanical, memorized steps to an "applied" situation. Sometimes, word problems are even seen as an advanced stage of learning. For students struggling to keep up with their abstractly-minded counterparts, they can have trouble breaking down the sentences and paragraphs into mathematical realities. “Just give me the numbers!” they may say. “I can solve it that way.”

Our approach at Mentorhood under the Singapore Math method is entirely different. Singapore Math doesn’t start with a set of mechanical steps and then tack on the real world as an afterthought; it starts with the real world and lets its students zoom out, one step at a time, so that they can see exactly what is being represented by the numerical tools they are using to solve the problem. In turn, there is no distinction between understanding what a math problem means in an applied sense and learning how to solve the problem.

A “Fingery” Alternative

Let’s take a look at how we at Mentorhood use one of our signature teaching methods to apply the CPA philosophy to the same thirteen minus eight subtraction problem with borrowing.

The method we will tackle is one of our most-used methods for early arithmetic training:  VGCC. VGCC stands for Visualization, Grouping, Connection, Calculation, and is one of our preferred methods for starting with something Concrete, like fingers, and walking students through logical steps to arrive at the answer. 

Two students would be chosen to represent thirteen on their fingers, ten from one student, three from the other. This helps them Visualize the numbers they are working with. The teacher might hold up eight fingers. “Can I take away eight from the student who has three?” The class answers in the negative. “What about from the student who has ten?” Affirmative. The teacher might hold up her eight fingers to the student who has ten, who would then proceed to lower ten fingers.

The student doesn’t realize that they are being taught Grouping. This involves breaking numbers down into smaller components, usually with a base of five or ten, so that they can better envision how numbers in the problem interact with and affect each other.

One student is left holding up two fingers, and the other still has his or her original three. The student is then asked to make the Connection between these two sets of numbers left over. Two and three makes five.

In the final step, students are asked to apply this to the original question, unlocking the power of Calculation: thirteen minus eight equals five.

Using a method that feels as easy as counting on fingers, these students have actually avoided the count-down approach and started to train their minds to think of numbers visually and in groups. The VGCC process for problem solving with concrete objects is similar to another favourite of Mentorhood: the IMMC method for working with pictorial representation. Both methods focus on teaching that is easy to catch on to and is highly rooted in visualization. Methods like these empower students to start to build a flexible landscape of numbers in their minds that they will eventually be able to use to model more complex numerical interactions.

We Prefer Fingers to Spoons
Regardless of which technique is being employed, Mentorhood has always favoured infusing our in-house methods with CPA philosophies, because ultimately, the objective is intuition. While practice and memorization does have its place for improving speed and accuracy, this is embraced on top of a foundation of solid, logical understanding. We want to show that picking up a new math technique can be as simple for your little one as picking up another carrot off the plate!

Next week, we’ll explore IMCC and its strengths, and we'll build on the picture of how your child can expect to learn in one of our friendly Mentorhood classes.