teaching tips

"Subtraction is killing us!"

When my son was ready to enter first grade many years ago, my wife and I visited a number of private schools to help us decide where to send him.  In one of the classes we visited the students were being introduced to subtraction.  The teacher had given each student some small objects they could use to help them count the totals, and she demonstrated how to use these counters to solve basic subtraction equations.  The children then worked independently and came up with their papers one by one to have them checked.  To the horror of the teacher, most of the calculations were incorrect.  At one point she looked at us and sheepishly acknowledged, “Subtraction is killing us."

Why were the students in this class having such difficulty? The problem, I’m quite sure, was NOT with the student’s ability to grasp the concept of subtraction.  The idea of removing objects from a group is something common and recognizable to six-year-olds.  Moreover, all of these children were good at counting.  If I had asked them to count out five gummy bears and to give me two of the gummy bears and then count how many they had left over, they all would have been able to do this task easily.

The difficulty they were experiencing in class that day lay not in the complexity of the concept of subtraction, but rather in how the teacher presented the mathematical symbolization for it. Specifically, she began by presenting an expression such as “5-2” and then demonstrated how to use the counters to find the result.  But by starting with the mathematical expression, she began with something that the students were not familiar with, and so it was extremely difficult for them to make sense of the what she was showing them, even though I'm sure the teacher thought that by using the counters she was making that procedure quite explicit and clear.

The lesson would have worked much better with a very simple change in sequence.  Instead of starting with the unfamiliar mathematical symbols, the lesson should should start with the familiar experience of counting various objects, such as books, or pencils, or the children themselves. After the students had counted an intitial set of objects, the teacher should remove some of the objects and then have the students tally the number of objects that remained.  At each step she should show the class how each action (counting an initial set of objects, removing some of the objects, counting the objects that remained) was represented with mathematical symbols.  Only after the students could easily go from performing these actions with objects to recording the action with the correct subtraction equation would the teacher then work on having the students do the work in the other direction, that is, present them with a subtraction equation and have them represent the equation with a set of counters. This last step, of course, is where the teacher we observed had started, and in doing so, she left her charges thoroughly confused.

It is important to point out that the problem with the lesson wasn’t simply that it was being executed by an inexperienced or a bad teacher per se.   I believe the teacher was faithfully following the routine recommended in the curriculum the school asked her to use.  Rather the problem in this lesson was a tendency all too common in math education to put the cart before the horse by rushing to present mathematical symbolization without adequately showing students what the symbols represent.   I’ll have more to say about this point in the next blog post, where I will discuss how this problem bedevils not only math lessons but other areas of the curriculum as well, and how we can derive an extremely important teaching principle from it.

Epilogue:  We ended up sending our son to public school, and he remained in public schools until he graduated from high school in 2011.   We made the decision for several reasons.  One important reason was that from what we saw, the teaching in the private schools was in general not noticeably superior to that in the public schools.  Our son graduated from college with honors in 2015 and just started his second year in law school.

Two Examples of Bad Teaching (and What We Can Learn from Them)

In my last post I discussed the importance of focusing on good teaching instead of just trying to identify good teachers.  One of the cardinal principles of good teaching is to provide not only positive examples of a new concept, but also negative examples as well.  Learners who are trying to categorize information according to a criterion or set of criteria new to them need to know not only what the new thing is, but what it isn’t as well.

It is with that thought in mind that I will devote the next couple of posts to the topic of bad teaching. I hope that is is clear from my earlier posts (and my general disposition, for those who know me) that my purpose is not to embarrass or make fun of the teachers responsible for the bad teaching.  As I have mentioned, I have done bad teaching myself, and I will include this in the current discussion.  Rather, the goal of this discussion is to use examples of bad teaching to flesh out a description of good teaching and suggest some important ways to evaluate the quality of teaching, whether one’s own or that of another. 

One of the worst examples of teaching that I’ve seen involved a homework assignment for spelling practice required in a 6th grade classroom.  The teacher instructed the students to rewrite each of their spelling words in a code, with a tall rectangle used to represent any “tall” letters like “l” and “d”, a short square to represent “short” letters like “c” and “v”, and a tall rectangle below the base line to represent “basement” letters like “g” and “p.”

Now, even in this very bad assignment there is a kernel of a good element, namely, an attempt to help students analyze the content they are learning.  As I shall argue over and over, helping students analyze what they are studying is a critical component of good teaching. 

The problem with this particular assignment, of course, is that besides being tedious and requiring a significant amount of time to complete, it really has nothing at all to do with spelling. The kind of analysis required of the students to perform this activity involves absolutely no attention to the underlying patterns and relationships of our writing system.  These fundamental patterns and relationships include how individual letters and groups of letters represent individual speech sounds, how certain functional units such as “un” and “ed” and “ion”, known as morphemes, are used over and over in the formation of words, spelling patterns such as when to double a consonant when adding a suffix to a base word, how both the pronunciation and spelling of words can change as various morphemes are added to base, and how knowledge of a word’s origin can often provide insight into why it is spelled the way it is.  In other words, there is plenty of appropriate analysis that students can do to help them learn to spell words correctly. Moreover, this productive analysis helps students recognize fundamental patterns that reveal the architecture (and, dare say it, the beauty)  of our writing system.

A similar sort of problem bedevils a technique used in several classrooms I know of to teach 6th graders how to divide fractions. This practice entails having students memorize the phrase “Keep-Change-Flip” to help them perform a calculation such as 1/2 ÷ 3/4.  What it means is that the students are to rewrite the the equation by modifying it in the following way.  They are to keep the first number (unchanged), change the operation from division to multiplication, and then flip, meaning, write the reciprocal of, the second number, thus yielding the expression: 1/2 x 4/3.  

There are several problems with this approach.  First of all, it shares a problem that is very common in American mathematics instruction, namely,  it focuses student attention initially on a procedure for calculation without adequately developing an understanding of the underlying patterns and numerical relationships that make the procedure work.  Now, this technique is effective if the goal is to get a student to do a set of problems exclusively involving division of fractions as quickly as possible.  Only a few minutes of instruction is required to accomplish this objective.  But that objective is very useful.  Students will rarely ever confront a series of similar division problems all in a row.  It is much more likely that they will have to perform an unpredictable mixture of various arithmetic calculations, in which case, they have to remember which calculation procedure goes with which operation.  And there is nothing in “Keep-Change-Flip” that designates it as the procedure for dividing fractions.

This approach makes the procedure more difficult to memorize for another reason as well.  It is presented as an isolated fact about arithmetic, rather than as something firmly embedded within a set of fundamental patterns and relationships.  To see this limitation clearly, consider an alternative approach, namely, having students learn that “dividing by a number is the same as multiplying by its reciprocal.”  This statement, like “Keep-Change-Flip” provides all of the information students need to perform the procedure for dividing fractions correctly.  But unlike “Keep-Change-Flip,” it clearly identifies this procedure as something involved in the division (and multiplication!) of numbers.  It also includes another key concept, the idea of reciprocal.  And it places the procedure within the context of a fundamental pattern of arithmetic which the student can (and should) explore.  Indeed, it is very easy to give the student a set of investigations in which the concept of reciprocal is examined and then this fundamental relationship between division and multiplication is made apparent.  When taught correctly, in fact, the students use the experiences and analysis they have done in these preliminary investigations to state the procedure themselves.  All the teacher does is to give them the precise wording for a relationship the students already understand, as well as to provide sufficient practice to perform the calculation correctly.   (I will have lots to say in the future about both of these points, namely, providing vocabulary AFTER students have the relevant experience and hence know what the vocabulary refers to, and the characteristics of efficient practice.)

Now some readers may be thinking, “But the presentation of ‘Keep-Change-Flip’ is something like a mnemonic, that is,  a device that helps us memorize things.  Are you saying that mnemonics are bad?”  No, not at all. Mnemonics have a very important place in good teachingand I will talk a lot about such tools when I devote some blog posts to the topic of memory and helping students memorize. 

But the purpose of mnemonics lies primarily in helping people learn things that are related in an arbitrary way, such as the digits of pi, or the order of the planets in the solar system, or the order of the colors in the rainbow.  In contrast, the procedure for dividing fractions (as well as the skills involved in learning how to spell words correctly, in the first example in today’s blog post) do not involve isolated and arbitrary information.  On the contrary, both involve concepts that are connected to other material according to a set of underlying relationships.  And when we want students to learn this type of material, our job is to direct their attention first to these patterns and principles.

In short, good teaching helps students make connections.  The corollary is also true.  If some teaching does not involve helping students make connections, then there is almost always a way to improve it!

Verification -- a powerful teaching concept

One day, many years ago now, when my wife was picking up my son from kindergarten, he was very excited to read to her a story posted in the front of the room.  His rendering was true to the meaning of the passage, he did decode a couple of words incorrectly.  My wife was not going to say anything, but several of the other kids chimed in to correct him.  Unfortunately, having been taught a curious mixture of whole language and phonics in kindergarten, at that time he did not possess the tools to verify whether he was correct or whether the other students were, and he had to take them at their word (or not).

I have had similar experiences with several of my math tutoring students when I first met them.  I would ask them how to do some calculation, and they would light up and say that they had done that in school and they would proceed to go on their merry way manipulating the numbers, confident that they were accurately replicating a procedure that they had practiced at some point earlier.  The problem was that what they were doing made no sense at all.  The manipulations were not based on genuine number sense, but were arbitrary and misremembered procedures.  And like my kindergarten son, they did not possess the tools to verify whether their answer was correct, and indeed, they didn’t even have a sense that after they offered an answer it was their responsibility to review it to make sure that it was sensible.

These examples, I think, demonstrate a critical component of lessons that is generally not discussed very thoroughly and explicitly, a concept I refer to as “verification.”  By verification I mean the ability of a person to prove that what he has done is correct without reference to authority.

Now, of course, there is a fair amount to learn that involves arbitrary and solely conventional associations and therefore cannot be verified without recourse to some expert.  The shape we use for the letter “m” for instance, is just a matter of convention, as is the fact that we use it to represent the /m/ sound.  The association between this letter and its sound can be confirmed only by someone who already has learned this association of letter and sound.  (I will have quite a bit to say about how to learn such conventional material in future blogs.)  But the pronunciation of a particular written word is another matter.  For example, the word “mop,” is spelled the way it is because it has three individual speech sounds or phonemes, /m/, /o/, and /p/, articulated in that order, and the letters “m”, “o”, and “p” represent those sounds.  If a person understands this spelling architecture, then she can confirm the association between the written and spoken word.  The same type of confirmation is available in the domain of mathematics.  For example, the fact that 8 x 7 is equal to 56 is something that can be proven in any number of ways, such as counting the value of eight 7’s.

Understanding when verification is possible allows the teacher to correct errors in a very sophisticated and powerful way.  If a child reads the word “mop” as /map/, for instance, it is not especially helpful to say simply, “No, the word is /mop/.”  It is much better to point out that the middle sound she said doesn’t match the middle letter of the word and have her try again now armed with this additional information.   (Detailed directions for this kind of error correction, and many others, can be found in the ABeCeDarian Error Correction Guide.) This approach treats the error as a “bug” in her reading procedure that needs to be corrected.  The teacher doesn’t give the student the correct answer, but rather points out explicitly what the problem was and has the student try again.

This approach yields several benefits.  First of all, it makes the material easier to learn.  Instead of having to use brute force memorization to remember hundreds or thousands of things, the student learns how to use a relatively small set of known material in a relatively straightforward procedure to produce a correct response, a procedure that can be applied to get a correct response in any similar situation.  

What’s more, the material is not only easier to learn in the first place, but it is much more likely to be retained.  As long as some learned response is isolated and disconnected from other knowledge,  it is easily forgotten.  But when something is embedded within a rich network of associations and relationships, it is anchored in a way that helps keep it from drifting into oblivion.

Another benefit of this approach is that it encourages self-monitoring.  The student does not always need to rely on a teacher to know if she has provided a correct response.  She can determine this for herself because she is capable of figuring the thing out for herself. And it keeps the responsibility for doing the task firmly with the student.  The teacher serves as a coach, providing guidance and support, and when a student makes an error, it is not an occasion for her to give up and have the teacher do it for her, but to evaluate what the problem was and try again.   It is difficult to overstate how dramatically this improves a student’s attitude, her ability to apply herself, and her active engagement in her lessons.

Let me leave you then with two important questions to ponder as you think about your own teaching:   Are you aware of the parts of your lesson that allow for verification, that allow students to determine on their own whether they are right without relying on your authority?  And, when your students are learning content that can be verified, are you correcting errors not simply by providing the correct answer, but by helping identifywhere the “bug” in a faulty procedure was and having them try again.