In this post, we look at the third “R” of education: ‘Rithmetic. In our next post, we’ll include a discussion of science and logic at Trinitas to complete the picture of how our classical school measures up in the area of STEM. At the outset, we should be honest that classical schools may often be thought of as great books programs or humanities schools. From the time of its founding in 2006, however, Trinitas was committed to providing our students an excellent education in the sciences. Not content to simply adopt the curricula many classical schools were already using, we were somewhat “cutting edge” for classical, and even non-classical, schools.* We think our math and science programs map nicely onto the grammar, logic, and rhetoric stages of the Trivium.
When choosing a math curriculum, it seemed to make some sense to find out what the best students in the world are using. Math students in Singapore are consistently at the top of The Trends in International Mathematics and Science Study (TIMSS). These students use the Primary Mathematics Series (otherwise known as Singapore Math). Lest we commit a logical fallacy, we won’t assume that simply using the same curriculum will produce the same results (there are very likely multiple factors at work), but it is probably fair to be optimistic about the possibilities. When we examined Singapore math pilot studies in the US, we found that students were, on average, a grade ahead of their peers. And, they liked math more. We’ve found this to be true of our math students as well over the last fifteen years—they are typically advanced in math, do very well on standardized tests, and go on to honors and AP math tracks in high school. And, importantly, a large percentage enjoy the subject.
How does Singapore math fit with the classical model? The subject of math lends itself much better to showing than to telling, but I’ll do my best to try to describe what we do. At the grammar stage, our students, like most math students, memorize their math “facts,” but they often do this “classically,” through the use of songs. More substantially, Singapore math builds comprehension through a concrete-pictorial-abstract (CPA) approach. That means that in the grammar stage, students first observe concepts being modeled by the teacher, then they imitate, practice, and discover with hands-on/concrete materials (e.g., groups of objects for addition, subtraction, multiplication, and division, as well as other manipulatives like 10-frames and solid figures). As they progress toward the logic stage, students learn to model mathematical concepts and investigate relationships with drawings (you may have heard your student talk about ”unit bars”). As they enter the rhetoric stage, they work with concepts on a more abstract/symbolic level (e.g., variables in algebra).
To illustrate, think back to learning subtraction. Imagine you were given the problem: 93-7 = ?. “Start in the ones,” your teacher probably said. Stuck on how you were supposed to subtract 7 from 3, you waited until your teacher instructed you to cross out the number in the tens column and put a little 1 next to the numeral in the ones column and then told you to subtract. The Singapore curriculum builds a concrete, visual foundation so students really understand what is happening in subtraction. Place-value disks and charts show what actions are needed to solve such problems. Students first build the total number 93 with 9 “tens” disks and 3 “ones” disks. They will try to subtract 7 disks but very quickly see that they cannot do this with the disks they have laid out. They’ll need to figure out what action to take: trade in a “tens” disk for 10 “ones” disks. Then they’ll have 13 “ones” disks. From there, they can subtract the 7 and easily finish the problem. After working with physical disks, students move from handling the disks to representing them with drawings. Trading in physical discs and drawing disks helps to build understanding that students take with them into the traditional (abstract) subtraction phase. They’ll understand and be able to articulate where the 13 “ones” came from and why. This CPA method is used throughout the curriculum to build comprehension of the various mathematical concepts.
In addition to modeling concepts and guiding students with opportunities to imitate, practice, and discover, teachers also employ the Socratic method, asking questions like: How did you solve the problem? Why did you solve it that way? Can you solve it another way? How do you know your answer is correct? Skills in rhetoric are built throughout the process. This approach to math, which teaches the “why”, not just the “how”, leads to better long-term retention and deeper understanding; and, it equips students to apply their knowledge to new situations.
As mentioned above, one of the benefits of the Singapore curriculum is improvement in students’ attitudes toward math. For many, it brings increased confidence–an “I (or we) can solve anything” attitude. Students learn to be willing to play with numbers while trying out different ways to find a solution. Story problems are a good example. You may remember the feeling of reading through all the details of a problem and thinking, “Oh no! I have no idea how to find the answer. There isn’t enough information, and I can’t remember the formula.” In Singapore math, students learn to begin by drawing unit bars and simply filling in what they know from the information in the problem. They then start asking questions to see what other bits of information they can discover that will help them fit the puzzle together. Singapore math utilizes a lot of story problems. These reinforce reading comprehension, grammatical thinking, logical reasoning, and problem solving. Facing a problem without fear by just starting with what you know, asking questions, and making observations and logical connections is a good skill for life.
This is the Logic stage of the Trivium, the investigation and discernment part where students get good at seeing the relationships between mathematical concepts. It’s fascinating how the pictorial stage of Singapore math prepares their minds for more abstract/symbolic math in the rhetoric stage. I remember when I first clearly realized this—I was the one who usually provided my son with help for his math homework. One night, my husband (who was not familiar with Singapore math) came to his aid instead. When I asked my husband how the homework had gone, he said, “It was pretty hard. I had to teach him how to use two variables to solve his story problems.” I laughed. I showed him how to solve the problem the Singapore way–just draw some unit bars, fill in what you know, and start looking at the mathematical relationships, do a few computations, and Voila! It can all be done without any variables at all. When students start using symbols later, they’re better prepared, have a deeper understanding, and, hopefully, have more fun.
With this foundation not only in math but also mathematical thinking, many Trinitas graduates have pursued math-related fields such as engineering, computer science, economics, and medicine. We’ll end this post with a “testimonial” from one of our National Merit Scholars who will be starting graduate school at the University of Chicago in the fall: “One thing I found (and still find) particularly helpful is how Singapore math is structured so that you actually comprehend and visualize what’s going on in a problem. Instead of remembering little tricks that help you quickly solve a specific type of problem but don’t expand your mind, Singapore math ensures that you have the tools to unpack a problem which equips you to engage with new material much better–like drawing unit bars–I actually used them on the GRE for a few problems!”
*Already in 2006, we employed Singapore math and FOSS inquiry-centered science modules. More recently, other classical and non-classical schools have moved in this direction.
Reference: Why Before How: Singapore Math Computation Strategies, by Jana Hazekamp