Teach math symbolization as a method for recording actions
Many years ago my wife and I visited a number of schools when we were trying to decide where to send our son for first grade. One of the schools we visited was a prestigious private school in our area where the first grade class was just introducing the students to subtraction. The teacher made a brief presentation and then distributed a worksheet and some counters the students could use to do the calculations on the worksheet. In a few minutes students were coming up to the teacher to have their work reviewed, and, to the great surprise of the teacher, student after student had done the calculations incorrectly. After a few minutes of this, the teacher looked up at my wife and me and said plaintively, “Subtraction is killing us!”
What was going on? These kids were well-behaved and attentive. The teacher managed the group ably. The classroom was well-equipped and well-organized. The children had adequately developed counting skills and other background knowledge. And, for goodness sakes, they had physical counters to help them with their calculations!
This troubled lesson presents, I think, an example of an extremely common situation, namely focusing on math symbolization too soon, and failing to clearly show that it is a means of recording particular actions.
In this particular lesson, for example, the teacher wrote a few subtraction expressions on the board, such as 4 - 1 = , and then proceeded to say something to the effect of, “Today we are going to work with subtraction. Here is an example of a subtraction number sentence. To figure out the answer, I’m going to take 4 counters and then I’m going to take away 1 counter. How many do I have left? Yes, I have 3 left, so that is my answer. Four take away one is three.
In other words, the actions with the counters were presented as a way of solving a particular kind of calculation, rather than as an example of a common concept, “removal,” which we can record with certain math symbols. The distinction may seem subtle or overly nuanced at first, but it is extremely important.
To begin a lesson with the math symbolization the way this ill-fated first grade teacher did is to begin with the thing the students are likely least familiar with, so it is the hardest thing for the students to attend to and to put into any sort of familiar context. The steps of how to use the counters then become very abstract and decontextualized steps to memorize, rather than components of a sensible, logical and familiar experience. As a result, the steps become very hard to remember and execute in the correct sequence without quite a bit of practice.
A better approach when introducing the symbolization of subtraction is to start with an experience of counting some common objects, such as pencils or books, and removing some, and then counting the number than remained. As the students do these activities with the teacher, the teacher should then write down the equation, saying something such as, “We started with 4 things, so I’ll write a number 4. We took some things away. Here is a symbol we write to show we are taking away. We took away 1 thing, so I’ll write a number 1. Now what remains is 3, so I’ll write “equals or is 3. Now I’ll read my whole number sentence for what we just did: Four take away three is one.”
In this sequence, it is important to note, the symbols were presented AFTER the concept was exemplified with the manipulation of physical objects. In this way, it is clear that the symbolization is a code, a recording system. If a teacher starts a lesson with the symbolization first, however, the concept remains obscured and the role of the symbolization unclear.
After a few examples, the teacher should then ask the students to write down the number sentences that go with a few more examples she performs with the entire class, continuing until everyone can record these actions with the correct symbolization.
It is not difficult to show how all of the symbolization covered in the K-8 math curriculum can be presented in this manner and following this sequence, with the new symbolization presented only after what it represents has been shown.
There are several important lessons to draw here. First of all, just because the shelves in a classroom are groaning under the weight of math manipulatives doesn’t mean that math concepts are being introduced as actions on quantities. It is certainly possible that the various physical things to count are being presented primarily as tools for calculation, as in the example I shared at the beginning of this blog post.
Second, many math expressions, even ones related to very basic concepts, are used to record many related, but subtly distinct actions. For example, we can use an expressions such as 4 - 3 to calculate the remainder if we remove 3 objects from a set of 4 objects. But we can also use it to answer a questions such as, “If one person has 4 books and another has 3 books, how many more books does the first person have?” In this situation there is no removal as in the first example, but a comparison. Likewise, we can also use subtraction to think about the question, “If I need 4 chairs at the table and there are already 3 there, how many more chairs do I need?” This is also a type of comparison, but slightly different than the previous example.
If instruction starts with equations instead of actions on quantities, then these distinctions can remain hidden to the students and it will usually take some time to completely fathom. However, if one starts with a proper variety of common situations and then shows how they can be symbolized, the range of information that is generally packed into or associated with the symbols is much more transparent, and it makes it far easier for the student to apply her math knowledge to solve problems that come up in word problems and in, even more importantly, in day-to-day life.
In many math programs it is common to teach a new calculation procedure and then end the unit on this procedure with a variety of word problems. This approach, I think, has the sequence exactly backwards. The introduction of new calculations should START with word problems, that is, questions about the manipulation of quantities investigated initially with ordinary language. (I will have much more to say about word problems and their proper role in instruction in future posts.)
Doing so allows the teacher to start with the familiar and then introduce the new material tightly connected to the familiar. In this way the student is readily able to embed the new information within the network of associations she already has, rather than lingering in some isolated recess of her mind, disconnected from experiences she has outside of math class. Embedding this new information in an already existing set of associations improves both her retention of the new material as well as accelerates her ability to apply it correctly in various situations.
As I mentioned in the earlier blog posts, the concepts of arithmetic and basic geometry, when presented as actions, are not very difficult for children to understand. Their confusions and frustrations with math, therefore, usually are NOT due to any inability to grasp the underlying concepts being investigated, but because they do not adequately understand what the associated math symbolization represents. In short, the most likely challenges and confusions children will have with elementary and middle school math have to do with understanding the symbols, and these potential confusions can be almost completely avoided by following the simple precept presented here.
In the first two posts in this series, I’ve discussed some principles for organizing how to introduce new concepts to students. After students are introduced to new concepts, however, they have to learn how to do associated calculations, and they have to learn how to do these quickly. In the next few blog posts, I’ll talk a bit about how to help students become fluent with the calculation procedures they need to learn in K-8 mathematics.