Throughout this series, we’ve been exploring how the teaching techniques we use at Mentorhood Math, inspired by the highly successful Singapore Math method, focus on learning through intuition.
We refer to our students as “learners,” subtly encouraging them to see themselves as in charge of their own learning, rather than subordinate to instruction. We take our time, solidifying basic concepts before moving on to more advanced ones. We utilize visual tools, concrete objects, and fun to make learning more accessible and relatable to young minds. All of this helps students internalize what they are learning.
But what really is the difference between CPA learning (working with Concrete, Pictorial, and eventually Abstract representations of math) and rote training, also known as learning through repetition or spoon feeding? Are the differences that major? Are the advantages truly significant?
Leaving More than Potential
In much of the world, math is told rather than taught. Instructors state a new type of problem that the class will be tackling, and then go on to explain a series of mechanical steps to solve the problem.
Often, the example problems are stated in abstract, mathematical terms and are hardly connected to the real world at all. Ever heard a child say, “But when am I going to use this?” Perhaps you’ve said it yourself.
Once the mechanical steps have been sufficiently memorized by a sufficient percentage of the class, the real keeners are given the opportunity to apply the mechanics to word problems. Students have the added challenge of picking out relevant details, determining which operation they should use, and proceeding with the calculations.
The rest of the class is left stumbling, puzzling over which number goes where in the formula and trying not to invert steps while asking themselves in frustration, “Why on earth is Joan wasting time calculating where her baseball will land at all? Just hit it!”
Rote learning leaves more than potential for understanding, connection to the real world, and appreciation for math in daily life on the plate. Rote learning often leaves a lot of its learners behind.
Lap It Up or Be Lapped
In much of the world, courses race through math concepts at top speed. This is understandable: regional, national, and global mathematical standards have ever-growing standardized lists of concepts that should be mastered in order to compare the performance of one particular group of students to another.
This causes an interesting dynamic of quantity over quality in the classroom: teachers must emphasize getting through a certain number of topics and hope the students keep up.
As a result, math is often taught abstractly, with mechanics and symbols and numbers and letters being the only experience with math many students have. Teachers are tasked with connecting math to the real world, but this is seen almost as superfluous to knowing the mechanics. Students learn for the sake of learning—or, more accurately, memorize for the sake of learning, without understanding why. Those that do not have a naturally abstract mind will have trouble following and executing, while those that can pick up mechanics are at an advantage. This is why we see students label themselves into two groups: “good at math” and “bad at math.”
The problem with rote or mechanical training is that it takes independent decision making out of mathematical discovery. Students are given a problem and a set of steps to solve it and taught to look for identical circumstances in which to put their memorized steps into practice.
This is different from the way that math was first developed. People started with real world problems that they needed solutions for. They discovered universal mathematical truths as they came up against dilemmas, and used abstract language—numbers and symbols—as a way to efficiently represent and extrapolate these truths.
Abstract math language evolved as an efficient way to represent real-world needs for math. In short, it was an eventual destination on the journey of mathematical inquiry. With rote learning, this “destination” is used as a starting point. Students rehearse answers without any real investment in the problem: they are spoon fed, when they don’t even know they are hungry.
The Link Between Independence and Discovery Math is about discovery. It’s about representing the world through awe-inspiring universal truths and finding information that we need to make decisions. It’s about solving problems one dilemma at a time, using arithmetic tools at our disposal to learn and create. Taking an intuitive approach to math enables us to make decisions along the way that we think will drive us closer to the answer we’re looking for.
At Mentorhood Math, time is on our side. We with little mathematicians to understand the simple problems they face in class. We make sure that they are involved every step of the way in determining how a problem should be solved, and we don’t leave students behind.
Students who have studied at Mentorhood Math are gifted with understanding on a very foundational level, so that they are not only set up to pick up more advanced concepts, but they also understand the why they are doing what they are doing. We are equipping them with tools and growing their confidence that they can apply the right tool for the job.
A Review: VGCC
Last week, we looked at how one method we favour at Mentorhood Math can help students move from understanding concepts with concrete representation to advancing into the abstract world of symbols and numbers we call mathematics.
The method we highlighted was VGCC: Visualization, Grouping, Connection, Calculation.
Using this method, we might model two-digit addition and subtraction questions on the fingers and hands of a few student participants. The advantage of using hands is that students can immediately Visualize how larger numbers are broken up into smaller parts using base five or base ten. (For example, students can see visually on the hands of their colleagues that eight is five plus three, and thirteen is five plus five plus three.)
This helps students move to the next stage of the method, known as Grouping. By concretely understanding how larger numbers can be broken down, students can group together easier parts of the numbers, and finally make the Connection between all these number groups to arrive at a final answer.
We then ask students to repeat this representation using numbers and symbols. Little do they know that they are building a three-dimensional understanding behind every two-dimensional number they are penciling onto their page.
Highlighting IMMC: Visualizing Numerical Interactions
This type of approach is known as the CPA approach: walking students through advancing levels of understanding math problems Concretely (using objects and manipulatives), Pictorially (using visuals), and Abstractly (using numbers and symbols), developing a robust intuition behind the digits we see on the surface of math. This method was first developed in Singapore and made popular in a number of countries known for their high-performing math students.
IMMC is another CPA method developed by Mentorhood Math, which focuses on helping students visualize how numbers interact with and relate to one another using comparisons.
IMMC stands for Imagery, More or Fewer, Model Fill-in, and Calculation. Under IMMC, a teacher will present a word problem in subtraction or addition and model out the components of the problem through Imagery, using visual rectangles. The key component of the method is the next step: More or Fewer. Students are asked to analyze the text for the key words that indicate what kind of operation is being performed. By doing this analysis, students are already building a connection between the model they are seeing, the text in the problem question, and the math that will need to be carried out.
When students answer, the next step is to fill in the Model, or label the rectangles and their differences in size, with the numbers from the question. This type of modelling can help students tackle simple problems, such as adding up how many apples Sally and her friend have all together. But it possesses a ton of power with all kinds of comparisons, and can be used to solve problems such as figuring out how much money James would have spent if he started with twice as much money as Casey, bought a shirt that was half as expensive as the one she bought, made a few other purchasing decisions, and had three times as much money left over.
Students have now done the legwork of breaking down the problem, and have a head start in seeing the problem come together by identifying key interactions and relationships between the numbers. Students can now advance to analyzing what Calculations need to be done, and in which order if there are several.
Students are embracing a highly intuitive way to solve these kinds of problems, encouraging the mental visuals that many of us might eventually stumble on naturally. They will be able to build on that intuition to mentally model how numbers interact with each other when given more complex sets of instructions.
We’re Vocal About the Hidden Benefits
As hinted, the golden nugget of this method lies in the second step: More or Fewer. Students are asked to analyze the text for clues that indicate which operations are being performed. We’ve already talked about how models can help students understand relationships between numbers visually, but by starting off in a word based problem, we are helping students understand numerical operations linguistically.
We talk in comparisons all the time: ratios, portions, percentages, doubles, halves, triples. We don’t always take the leap in our minds from vague comparisons to concrete math. Using IMMC and methods like it, we are helping students recognize math in everyday language.
Not only this, but especially in English, we represent a lot of operational concepts in many different ways. Just look at all the ways we can say “subtract” in English: minus, less than, take away, fewer…not to mention less-than-obvious ways that only make sense in context. (Three penguins walked away. How many are left?)
For students for whom English is already a fluent tongue, we are teaching them to think about their world mathematically: to take an instant leap between conversation and calculation, dialogue and operations. For students who are learning English, we are giving them an expanded, flexible vocabulary, enabling them to grasp the nuances of math spoken about in English.
Intuition, Intuition, Intuition
For a lot of the world, math is learned through repetition, repetition, repetition. At Mentorhood Canada, we believe in intuition, intuition, intuition. We believe in building a solid understanding of basic mathematical concepts that more advanced concepts can be built on later, saving students from having to be re-taught earlier concepts.
At Mentorhood, learning math can really be as simple as 1, 2, 3…so long as it’s spelled C, P, A.
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:
Sounds simple, right? Or maybe a bit too abstract?
Let’s break it down into something more concrete.
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.
Singapore Math: New Trend?
The Singapore Math method is not new. Concepts and teaching styles that closely resemble Singapore Math are the preferred method of instruction in countries like Singapore and Japan, whose students have historically outranked their global counterparts in math performance. In 2018 Singapore students ranked highest by a landslide in the Pisa Global Competence Test, conducted every three years by the Organization for Economic Cooperation and Development (OECD) to test for math, science, and reading skills. About 46% of Singaporean students achieved the highest levels of competency--more than three times the global average by percentage. And Singapore was no stranger to the top of the pile, ranking 1st in the overall Pisa assessment program in both 2012 and 2015. That’s enough to make any instructor sit up and pay attention to exactly how math is being taught in Singapore and what principles can be drawn from it.
More than a curriculum or a set of performance standards, the Singapore Math method is a philosophy. Singapore Math focuses on helping students understand and then to master root concepts, and then builds from the conceptual to the practical. Singapore Math unapologetically starts out slow but boasts of great efficiencies as students progress. Students do not need to be retaught previous concepts and can advance to more difficult problems without hesitation. Moreover, students are subconsciously developing a profound intuitive understanding of the math concepts they are learning, setting them up to take leaps and bounds in complexity and mental modelling in more advanced stages of their learning career.
The Singapore Math Three-Dimensional Approach: CPA
Singapore Math teaching breaks down any mathematical concept into three dimensions, building the students’ understanding of the problem from something relatable to something more abstract. This is known as the CPA approach, and goes like this:
**What is meant by manipulatives are concrete objects that we can move around and “manipulate.” These might be blocks, playing cards, fingers, coins, or even little action figures that the children can move around to model addition, subtraction, multiplication, and division. In an online setting, it is typical to move around objects on a screen such as circles, squares, or little pictures.
Up Against Algebra
Where this differs from traditional Western mathematical teaching is in leaving the abstract symbolic and numerical representation of problems and answers until last. This difference unlocks the power of Singapore Math instruction, and is also sometimes a contention point for criticism.
Typical Western mathematics focuses on symbols and numbers. You won’t get through too many math classes without encountering a unit on algebra. This, unfortunately, is where some students get lost. Algebra, the language of math, is truly a separate and abstract language in the same way that musical notes differ from the sound we hear. Students found wanting for the abstract language skills to master it are left believing that they are equally unable to master the concepts behind the symbols.
Singapore Math instead teaches students using concrete, relatable objects, problems, scenarios, and visuals, and leaves the language of math as just that: a symbolic representation of a very real concept. That is, of course, how mathematics first emerged: mathematicians needed ways to efficiently and universally explain and extrapolate realities they were seeing around them.
“But,” one may argue, “when my future engineer is trying one day to calculate the forces of speed, inertia, and stability on an industrial crane picking up a precarious load, or when my future finance major is attempting to determine the historical statistical deviation of a group of stocks from index returns, do you really expect him or her to be counting out blocks? I mean, how many fingers do you think he has?”
A fair point. But it is a point that misses the power of Singapore Math completely. Singapore Math does not downplay the importance of mathematical language that has taken humanity’s most brilliant minds centuries to develop—and they continue to develop it today—but rather, it allows students to take their time in understanding the concepts that the symbols depict, empowering them to unlock the full power of the math language tool.
The Power of Intuitive Understanding
This hits on the crux of Singapore Math’s strength: the more intuitive and relatable the understanding is, the more flexible a student can be to manipulate variables to better reflect, model, hypothesize over, or predict real-life situations of vast complexity. Singapore Math’s focus on intuition not only enables the youngest students to understand how to apply the simple concepts they learn over and over again, but it empowers a generation with mathematical agility so that they can navigate the obstacle courses of today’s modern math problems.
Engineering. Computer science. Space exploration. Biological research. Economic management. Sports math. Agricultural innovation. Automation.
Today’s modern world is changing at a faster rate than we have ever seen before. It is crucial that the math specialists on the frontiers of new and exciting fields of study not only have a deep understanding of what math has already been shown to do, but can adapt and stretch that knowledge to dig into never-before-seen problems. Intuition is key to that kind of mathematical flexibility, and it all starts for your young child in a classroom setting where they can learn, for the first time, how they relate to the math problems they face.
Next week, we’ll start to explore exactly how the CPA method works and how it differs from other teaching methods, such as spoonfeeding and rote training.
Davie, S. (2020, October 22). Singapore's 15-year-olds top OECD's Pisa global competence test. The Straits Times. https://www.straitstimes.com/singapore/parenting-education/singapores-15-year-olds-top-oecd-global-competence-test.
Dean, M. E. (n.d.). Singapore Math vs. Common Core: What’s the Difference? Argo Prep. https://argoprep.com/blog/k8/singapore-math-vs-common-core-whats-the-difference/.
Morin, A. (2021, May 21). What Is Singapore Math? Very Well Family. https://www.verywellfamily.com/singapore-math-pros-and-cons-620953.
Teng, A. (2019, December 3). Pisa 2018: Singapore slips to second place behind China but still chalks up high scores. The Straits Times. https://www.straitstimes.com/singapore/education/pisa-2018-singapore-slips-to-second-place-behind-china-but-still-chalks-up-high.
Addition with in 10 (this printable in the video)
Subtraction within 10 and Addition within 20 printable are also available.
Check facebook post to get all 3 printable:
1. Print out the printable and prepare a deck of Math Cards
2. Sort out Math cards, we only need 0 - 5 in this game
3. Shuffle the cards, each player takes turn to take 2 cards
4. Do the sum and find the numbers in the circle, colour it.
5. The player wins when he/she colours 4 circles in a row horizontally, vertically or diagonally.
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