Charlotte Mason's 12th Principle of EducationEducation is the science of relations.
A Christian classical education approached with Charlotte Mason’s pedagogical insights can’t help but prioritize comprehension when designing student learning objectives. It’s for this reason (and many others) that Clapham uses Singapore Math in our Lower School. Singapore Math teaches the why before the how. Singapore aims to equip students to master how to execute mathematical functions by spending time in the why. At Clapham, sometimes we refer to this process as going slow to go fast.
Jane Phillip, Clapham Lower School PrincipalSingapore at Clapham School encompasses the component of the why before the how. This provides students the opportunity to dive into discussion and pull from prior knowledge as they actively solve and discover with curiosity and engagement. Singapore Math often encourages the idea of “getting messy” in math and emphasizes problem solving skills as crucial to learning. Through the use of manipulatives and mastering the concrete concepts, it emphasizes the importance of thinking and mastering each concept before moving onto the next. Teachers play a vital role in asking questions that elicit discovery, explanation, and the demonstration of concepts... Within the classical tradition, we want to equip our students to seek understanding of what quantitative terms describe, what they are asking, what principals apply, and the why before immediately solving an algorithm.
Building Comprehension through Sequencing and Discussion: Incorporating the CPA (Concrete, Pictorial, Abstract) Approach with a Charlotte Mason Philosophy
At Clapham, a math lesson is driven by discussion. The teacher facilitates this process through posing a question (often in the form of a word problem) along with several follow-up questions: What information do we have, and what information do we need to answer this question? These questions naturally steer the class toward problem-solving.
If Eric has five apples and Alice has three, and they place all the apples in a box, how many apples are there all together?
Solving a Problem: The First Step
Singapore Math (so-called because it closely follows the math education methods of Singapore) begins with concrete (also known as the object itself) in introducing concepts. For instance, students may use miniature bears, beads, or linking rods to count or illustrate a number bond before even realizing it is a number bond.
Number bonds demonstrate both addition and subtraction simultaneously. Number bonds demonstrate that numbers are made up of other numbers; in effect, they are the fruit or sum of a relationship. Sometimes number bonds are referred to as fact families.
The Second Step
Following the concrete stage, students transition to the pictorial stage of illustrating a number bond, showing the various parts of a whole. As students see the parts of a whole represented, they are prepared to engage in addition or subtraction. They also visualize number bonds in such a way that they are prepared to master math facts, leading them to facility in calculating mental math, or the automaticity required to simply solve an equation quickly because of their level of comfort and familiarity in prompt mental strategizing.
The Final Step
In the final stage of a Singapore math sequence, students compose and solve equations, ending the sequence with the abstract; the written algorithm. Singapore refers to this sequence as the CPA approach (concrete, pictorial, and abstract). The CPA approach strengthens student comprehension because it roots math in concrete, three-dimensional manipulatives children can understand.
Charlotte Mason, Home Education. . . having used eyes and fingers upon ten balls or twenty balls, upon ten nuts, or leaves, or sheep, or what not, the child has formed the association of a given number with objects, and is able to conceive of the association of various other numbers with objects. In fact, he begins to think in numbers and not in objects, that is, he begins mathematics.
CPA and More Complex Operations
As students move into multi-digit multiplication or division, the CPA approach may shift the way parents are used to performing certain math functions. Consider this example:
Julia and her brother are picking strawberries on their grandmother’s farm. They each fill one basket. If each basket holds exactly 46 strawberries, how many strawberries did they gather in total?
Concrete
For instance, with a problem such as 46 x 2, instead of handling 92 individual pennies, students may use color-coded place value discs (white 1 discs, red 10 discs, yellow 100 discs) or base-ten blocks. They may lay out two rows, with each row containing 4 ten discs or blocks and 6 one discs. To multiply, they group the tens together and the ones together. If they have more than 10 ones, they swap 10 ones out for a single ten disc, concretely demonstrating the notion of carrying, or renaming, as they say in Singapore Math.
Pictorial
In the pictorial stage of the problem, Singapore instructs students to draw a bar model to further solve the equation. At this point in discussion, a teacher or parent may ask, “Tell me what steps we need to take to solve the problem. How can you be confident in your solution?” A bar model is how students in Singapore show their work in understanding a word problem.
Abstract
In the abstract stage of the CPA sequence, students write the algorithm out:
Finally, students will conclude a word problem with a cursive, complete sentence, ultimately answering the initial question posed: Julia and her brother collected ninety-two strawberries.
Singapore’s sequencing approach helps students build confidence and problem-solving skills. As a classical, liberal arts school, Singapore’s mathematical philosophy closely aligns with Clapham’s goals and vision in student formation. We trust learning about Singapore’s unique features will equip our school families to better understand and appreciate our approach to math education in our Lower School.
Jane PhillipAt home, we encourage [parents] to use discussion and ask interactive questions such as “Can you explain that? What information do we need to answer the question? How do you know if your answer is correct? Tell me how you solved that and why you solved it that way.” So much of what we do at Clapham School revolves around rich discussion, this includes Singapore Math. Mathematics plays a crucial role in our lives, and equipping our students with numeral fluency and literacy through discovery and discussion is vital in our technology saturated society.