In the last blog I talked about a lesson in which the teacher rushed to teach students how to manipulate symbols without first making clear to them what the symbols represented. The students became so confused that the teacher llamented, “Subtraction is killing us.”
This problem is quite common in math education. One of the reasons it arises is because math calculations can be performed correctly by memorizing a number of correspondences (such as the arithmetic “facts") and a number of procedures (such as how to multiply a multi-digit number by a multi-digit number, or how to do long division) without understanding how the procedures work or what the symbols mean. In addition, those of us who know the meaning of basic math symbols understand these so thoroughly that it is often difficult to remember or acknowledge how meaningless the symbols are to the novice. As a result, there is a great temptation to focus instruction on manipulating the symbols without taking time to help students grasp what the symbols represent.
Teachers can avoid this problem by beginning math lessons with a relatively familiar and "unmathematized” experience. In the case of introductory lessons on subtraction to first graders, this experience would involve removing objects from a group. When the lesson is begun in this way, students immediately have a context for the discussion, analysis, and new information that arise during the class. This context makes it easier to learn new math symbols and to apply them appropriately, and almost always will help them learn new calculation procedures more quickly than if the teacher had begin focusing just on the symbols and the calculation procedure. They are learned faster and retained more easily precisely because, instead of being isolated facts to memorize, they are embedded within an existing web of known relationships.
This problem of focusing too soon on symbols is prevalent in math education, but it exists as well in other parts of the school curriculum. Those of you who are familiar with the very beginning lessons in the ABeCeDarian Reading Program know that, in contrast to the beginning of most phonics programs, the very first task the students do is a spelling task, in the form of a Word Puzzle. The rationale for beginning with a spelling task is that letters, like numbers and math signs, are symbols for a certain kind of thing. In the case of letters, they are symbols for the speech sounds with which words are formed. To understand the logic of our writing system, therefore, involves understanding, first of all, that words can be broken into smaller units of sound, and, secondly, that these smaller speech sounds can be represented by letters and combinations of letters. Almost all reading programs, even relatively thorough, explicit phonics programs, begin their instruction backwards, that is with the symbols first. A common introductory lesson might have the teacher present the letter “m” to her students and then have her say, “This is the letter “m”. It makes the sound /m/, as in “mouse,” and “motorcylce,” and “monkey.”
But this introduction does not do a good job at all of showing where the “m” comes from and how it functions. In ABeCeDarian, in contrast, the teacher says in her introductory lesson, “Today you will help me spell the word “mop.” By beginning with a spelling task, we are presenting initially something that is familiar to the student, a familiar word, “mop.” The teacher then proceeds by saying, “What is the first sound you hear in the word ‘mop?’” There are a variety of supports the teacher provides to help the child understand what this question refers to and how to answer it. After the sound /m/ has been identified as the first sound, only then is the letter “m” referred to. Specifically, the teacher will ask, “Do you know what letter we use to write /m/?” If the student knows, she identifies it (the letters needed to spell the word “mop” are on the work space in mixed-up order). If she doesn’t know, the teacher shows her.
I’m not saying, of course, that children are unable to learn how to read using what I’m calling the “backwards” approach of a traditional phonics program. Most children who receive reading instruction do learn to read pretty well. But this instruction is relatively inefficient and tends to obscure rather than to clarify what the child needs to know to understand how the code works. In the ABeCeDarian approach, every bit of the logic of the code is embedded within an introductory activity that takes just a couple of minutes to conduct.
A similar sort of “backwardness” prevails in much instruction in which a teacher introduces some new vocabulary. In thousands or perhaps tens of thousands of classroom each day, a teacher will greet her students with the announcement, “Today we are going to learn about _______.” But even if the teacher immediately provides a definition, the students do not yet really grasp what ________ means, and so they cannot readily place it within a network of known relationships. If instead teachers would withhold the name for the new thing until AFTER students had a chance to experience it and to explore it, I am quite sure there would be a dramatic improvement in how well students retained and used their new vocabulary.
I remember as a child playing a variety of made-up games with my brothers, and one of the key activities was coming up with new names for the events, steps, or procedures that we developed or encountered. Any group of people sharing similar experiences, whether athletes, or shopkeepers, or parents, have similar experiences. We love giving names to things we observe and interact with and use. Indeed, we cannot stop ourselves from doing so. But the names originate because of an experience that needs naming, and they are vivid and useful because we know exactly what experience they refer to.
I think all of these examples suggest the tremendous power of beginning lessons with some kind of experiences that are readily accessible and available to be explored and analyzed. As students come to understand the “thing” they are exploring, it is a relatively straightforward matter to attach to it some symbolic representation, whether in the form of a math symbol, a letter, or a word. This sequence returns us to what is the natural order of abstraction, namely, things first, names second. And this order allows us to give new expressions to our students just when they are ripe for them.