Number sense is an important outcome of the Alberta math curriculum. Yet many of our students find arithmetic so vexing and unnatural that they need calculators to perform even the simplest operations. This lack of arithmetic fluency puts students at a big disadvantage when they move on to tackle algebra and other abstract math concepts in higher grades.
Why, with so much focus on developing number sense in elementary age students, are we seeing such mixed results with students graduating from our school system?
It may come down to how arithmetic is taught in today’s classrooms.
New math vs. old math
Old math emphasized memorization (think times tables) and constant practice (think worksheets) to develop number skills. New math holds that “[a] true sense of number goes well beyond the skills of simply counting, memorizing facts and the situational rote use of algorithms.”
New math sees rote learning as the worst type of learning, bordering on not learning at all. It’s dismissed as drill and kill, a sure way to choke off interest and motivation. With rote learning, students end up being able to do things they don’t understand and can’t easily explain. Any knowledge they gain is brittle, incoherent, and easily forgotten.
For new math advocates, memorizing standard algorithms and basic math facts is a one size fits all approach that doesn’t address the “diverse needs and learning styles of today’s students”. They instead favour more flexible approaches such as mental math (“alternative algorithms for calculating without pencil and paper”) and estimation (“techniques for determining approximate values using benchmarks and referents”) to develop a “deep understanding” of numbers.
3 good things about old math
Old-fashioned math has three virtues when it comes to preparing younger students for the study of higher level mathematics: 1) it emphasizes that problems in arithmetic have a right answer; 2) it defines a simple set of rules that, properly understood and followed, will always produce the right answer; and 3) it shows students the pleasure in not only getting the right answer but in being confident that the answer can’t be anything else.
1. Math problems have a right answer. 7 * 4 is always 28 no matter how you arrive at the result. You can estimate the answer to be 30, but 30 isn’t the right answer. Old-fashioned math draws a student’s attention to the fact that right and wrong in K-12 math isn’t controversial. You don’t debate a problem like 7 * 4. You solve it.
Understanding and fluency comes from tackling and solving problems, making and correcting errors. Whereas new math is an education in critical, open-ended thinking (questioning and often rejecting rules), old math is an education in correct thinking (understanding and following rules). Thinking correctly comes before thinking independently, creatively, or even critically.
2. Math is a set of rules. Taken from the Alberta curriculum’s beliefs about students and mathematics and learning: “Students learn by attaching meaning to what they do, and they need to construct their own meaning of mathematics. They must realize that it is acceptable to solve problems in a variety of ways and that a variety of solutions may be acceptable.” The emphasis is on developing “personal strategies” (which leads to deep understanding) rather than memorizing and applying fixed rules (which presumably leads to shallow and incomplete understanding).
While it’s true that mental math offers a variety of ways to solve arithmetic problems, it doesn’t really do away with memorization. In the case of multiplication, it replaces one general purpose algorithm (which always works) with many limited purpose algorithms (which work in specific cases). Mental math thus imposes an additional cognitive burden on the student attempting to perform arithmetic: she not only has to remember each mental math algorithm but she also has to determine which one to apply for any given problem.
New math has the admirable goal of helping students attach meaning to mathematical thinking. But it overlooks the fact that while we can follow rules without knowing what they mean, we can also discover meaning by following the rules and observing what happens. Keith Devlin talks about this in his excellent post How Do We Learn Math?:
3. Math is funner when you “get” it. It’s easy to fall behind in grade school math. Each new concept builds on an earlier concept, and concepts tend to become more abstract and sophisticated. If you haven’t mastered arithmetic with numbers, you’ll likely be baffled by arithmetic involving variables. Many students resort to blindly doing what they’re instructed to do. They can solve familiar problems, but they struggle when the same concepts are presented in new contexts. New math advocates see this as an easy trap for students to fall into, easily missed because high marks in solving familiar problems suggest that they’re doing well.
So why does basic math understanding elude capable and motivated students? The typical response is that it doesn’t, and that the explanation lies in some failing on the students’ part. They’re not motivated or they don’t have the talent for math. Another suggestion is that the failing lies with schools. Many teachers don’t understand what they’re teaching, and they confuse students with their lessons and explanations.
I don’t claim to have an answer here. But I’ve observed what students do once, having solved many familiar problems of a certain type, they solve a related problem that’s a little different and they just know they got it right. They’ve understood what they’ve been asked to do, they’ve seen the components of the problem, they’ve taken the right approach to solving it, and they’ve carried out the procedure correctly. And, once they’ve done all that… they smile.
Growing up liking math
Timothy Gowers, a Cambridge mathematician, writes that “any child who is given one-to-one tuition in mathematics from an early age by a good and enthusiastic teacher will grow up liking it.” Deep understanding, if that means thinking mathematically, takes time (take a look at the Wikipedia page on natural numbers and try and get your bearings). As a first step, it’s important to know and apply mathematical rules correctly. Here’s Gowers’ explanation of the benefits of this abstract approach.