After 17 years as a teacher and 8 as a researcher in education, I have become increasingly aware of a “gravitational force” pulling me to instruct with little attention to the most ambitious goal a teacher, parent, or administrator can aspire to—inciting curiosity. I say ‘incite’ because it seems counter-culture to do so. My goals here are to illustrate briefly this force and to provide one simple way we can begin to counteract its pull.
When I discuss the matter with colleagues, I see that we all feel the same way. Few of us seem to know why we’re teaching what we’re teaching, how to get students to be interested in it or what to do about it. After years of thinking about this, I’ve come to understand that confusion is often a precursor to learning.
Communication of ideas is a central part of learning. Language matters – especially in mathematics. So far, I’ve said nothing new. However, let’s examine a way in which typical instruction attempts to provide language for students. Even the very attempt to provide language for students can be misguided and can be seen as the source of many learning difficulties. I will attempt to illustrate with an example.
Consider a typical lesson from middle school pre-algebra classes—this is a scenario that plays out across grade levels and content areas. I have chosen this example to illustrate how a common, well-intended teaching practice (pre-teaching vocabulary) can squelch curiosity, contribute to anxiety in the learner, and ultimately turn students into what I call “Do-Monsters.” (Even if you feel anxious reading this and start to break out in math hives, I encourage you to persist!)
Teacher: Today we are going to learn about: like terms, monomials, binomials, trinomials, and ultimately polynomials. Please take out your notebook and write these terms down together with the definitions I will show you. (Definitions are copied…)
Teacher: Now I will show you an example of a polynomial. Please identify two like terms in this example. Example 1: 2x – 3 + x
Take a moment to remember a similar occasion in your learning experience (if you can): your lack of curiosity or need to think critically, and the aimless feeling of the activity. Remember the worksheet with 25 mindless problems you worked on for the rest of that class?
The situation I am referring to is, generally, one in which the topic is foreign to the students and the teacher. The reason for discussing the vocabulary (or even the concept) is that there is an impending test, a social contract that a teacher must cover the book, or a belief that knowing the vocabulary is crucial for critical thinking to begin. Learning vocabulary out of context becomes the purpose of the lesson rather than asking how these words can help us solve problems or think critically.
Questions, Problems, and Meaning
Consider an alternative. Instead of giving students meaningless terms upfront (pre-teaching vocabulary) so they can think about a question, why not give them the question first and deal with vocabulary within a context? I have found that the most meaningful lessons are those in which a student has something worth discussing; one in which there is a problem to solve. The problem with this is that the kinds of questions I have been trained to ask and those primarily found in textbooks are not really central to the topic I teach. They make language acquisition the end—instead of the means to an end.
I have also come to believe that imagination is a necessary ingredient in learning. Once it’s in play we can ask different kinds of questions that leverage existing language to create a need for new language. We might ask questions like, “Do we all imagine the same things happening in this situation? How do you see it? How do our images differ? How are they the same? What would happen next? Can we play out the scenario? What if we changed the situation?”
When a context is present, there is a chance that I can play! Playing is good. Unfortunately, in our desire to ease students’ frustrations/suffering in the learning process we (teachers/parents/administrators) often seek to give them the words they need before they need them, creating a situation in which students don’t know what to do with what has been given to them. We expect a child to wait patiently with a word for the right time to spring into action and use it.
So, what might a lesson look like in which we try to put students in a problematic situation before introducing the vocabulary necessary to describe the phenomena we want students to reason about? Consider a potential revision to the previous scenario in which the goal of the lesson was to teach students about: like terms, monomials, binomials, trinomials, and ultimately polynomials. Changing the goal from language acquisition to critical thinking might play out like this.
The Birth of Puzzlement
Teacher: I’m thinking of a number…so that 3 less than twice my number plus my number again equals 15. What number am I thinking of?
Did you find yourself trying to answer the question? Maybe you did and maybe you didn’t. The point is that students have a starting point to imagine what could happen. A debate could break out about what that number is and students could talk to each other. The situation could be changed enough to make them think again until they are comfortable with the phenomenon – searching for an unknown number that satisfies the given condition.
Eventually, when the situation gets complicated enough, we might need the terms: like terms, monomials, binomials, trinomials, and ultimately polynomials. We know we are at that point when students see the need to refer to 3 less than twice my number as one entity instead of two. What would we call that thing? It is incumbent on us as teachers to bring students to this point through a careful selection of tasks. There is no algorithm other than asking the question, “How can I puzzle my students the right amount today?”
One argument to continue pre-teaching vocabulary is that a student (a second language learner in particular) might not know what a word means and might not have a sense of the question at hand. In that case, isn’t it better to pre-teach vocabulary?
There are no hard and fast rules. Sometimes yes, but usually no. In fact, I have come to see that puzzlement goes hand in hand with confusion in the beginning and that this is a sign that learning can occur. It represents the possibility that something can be learned and this is what excites me most about teaching. I WANT students to raise questions about words they don’t understand and begin to ask questions about them spontaneously. I WANT students to play out the scenario under their own interpretations—whatever those interpretations are. I WANT debate.
Unfortunately, too often we see no possibility of debate because we spend far too much time focusing on the wrong lessons. The irony about language acquisition is that it happens best as we use the language we have, not when we are taught words out of context. To really learn a term, we must first have something to talk about that requires the new word itself. In short, here is where we lose our way. To quote a colleague, “We are so fearful we won’t cover the material, that we fail to uncover something meaningful.”
Harel, G., Fuller, E., & Soto, O., (in press), Determinants of a DNR expert's teaching, In Transforming Mathematics Instruction: Multiple approaches and practices, (Li, Y., Ed.), Springer.
Harel, G. (2008). DNR Perspective on Mathematics Curriculum and Instruction: Focus on Proving, Part I,ZDM—The International Journal on Mathematics Education, 40, 487-500.
Harel, G. (2008). DNR Perspective on Mathematics Curriculum and Instruction, Part II, ZDM—The International Journal on Mathematics Education.
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