The litmus test for any proposal in business is return on investment, or ROI for short. This means that if Joseph the widget-maker proposes a change in the widgets his employer makes, perhaps adding rubber feet to them, that change must return more money than it costs. The increase in profit from rubber-footed widgets must outweigh the cost of gluing on the rubber feet. This makes perfect sense, and no right-minded businessperson would argue otherwise.

I’m an educator, a profession nearly entirely devoid of business sense, so I feel no such obligation to agree with this principle. Here is the flaw, as I see it, followed by a real example.

ROI requires that we know the effect of a change before it is implemented. Joseph the widget-maker is guessing that the rubber feet will make the widgets better and that consumers will buy more of the new and improved product. He is predicting that the increase in sales will be so great that it will justify the added expense of glue, rubber, labor, and changes to the production process. Notice the verbs here; he is guessing and predicting. At the moment the boss decides to make the change or not, ROI is just a guess or a prediction. More accurately, the decision will hinge on whether Joseph is persuasive enough to convince his boss that the addition of rubber feet will return more than they cost.

I recently worked for a business that produced educational materials. Having many ideas as to what works in education, I tried to implement some changes. One particular change was to include computer games, the likes of which you can see in this website. My experience using such games in the classroom told me that they are a popular, practical, and effective addition to the usual means of teaching.

The bosses wanted justification in business terms. In other words, they wanted me to state that the games passed the ROI test, that they would return more than they cost. The games were rather expensive to produce as I was the only person who could make them and I had to handcraft each game from scratch. The reason for this is simple. Just as every lesson is unique and every learning objective different, so must each game be one-of-a-kind. I tried to convince those above me that our clients would so appreciate the inclusion of the games as part of our educational materials, that they would gladly absorb the added expense.

Businesses, however, have many competing voices. One such voice argued that one-of-a-kind games, one-offs he called them, were too inefficient to make and therefore too costly. His thesis was that we needed to produce materials cookie-cutter fashion with templates that anyone could fill in. He also argued that games need to resemble the video games kids play for entertainment, with robot avatars, racing cars, shooting and such. This, he asserted, would make the games more appealing to our clients. In other words, the games would be cheaper to produce and return more for the investment.

So ROI boiled down to who could convince management that his idea was best for the company. In the end, I lost out. Management agreed that games, at least the way I make them, cost too much and would return too little.

But the point isn’t that “return on investment” is a flawed tool. I’ll leave that concern to the businesspeople who rely on it. The point is that, from a business perspective, proper educational design will always be too expensive and return too little. The amount of time a teacher spends preparing materials, delivering the lesson, and assessing learning will always outweigh any dollars-and-cents return. Every lesson is different (or should be), a costly one-off, so to speak. Results are given in incremental gains in understanding, a measure that is hard to translate into dollars. Even if we could agree on a dollar figure for the gradual, layered widening of a learner’s perception of the world, it would have to be sky-high to exceed the total amount of money invested. An educator knows that the return on this investment does exceed the investment. The businessperson will always be blind to the true ROI of education.

Trainers steep us in mantras of good teaching. Whether we believe them or not, whether we practice them or not, they are the mantras of a teacher, so we repeat them. Chief among them are two mantras that, if a person has any hope of success in the world of education, he must learn and keep ever on his tongue. They are 1) that every student wants to learn, and 2) that every student can learn.

A casual observer of the typical high school classroom might conclude that the first of these chief mantras—that every student has a desire to learn—might fall into the category of wishful thinking. Teachers are hard pressed to keep students from daydreaming, shooting rubber bands, and whispering to their neighbors. If the mantra is correct, then the students are acting against their own desire to learn.

Every teacher will repeat the mantra that students want to learn, but not all teachers believe it. How do I know? I used to run a computer lab in a public high school. Teachers would bring in their classes. Sometimes I would conduct class while the students’ regular teacher graded papers or sipped coffee in the back of the lab. Other times the students’ teacher would be in charge while I went station to station making sure the students could log in and access the technology. It was during these times when I was relegated to the responsibilities of a computer technician that I got to witness what few teachers ever do: another teacher delivering a lesson.

If we accept the idea that every student desires to learn, what I saw should resemble the ice cream man’s stop in a suburban neighborhood. Kids should be clamoring for what he’s vending, elbowing each other for the chance to be first to enjoy the goods. What I in fact saw more closely resembled a lion tamer forcing unwilling beasts through a series of hoops. Not always, but more often than not the teacher’s approach was to coerce students to participate in activities by means of threats or just through a sense of that’s what they are supposed to do.

But every student does want to learn. Every student does walk through the classroom door with the hopes that the teacher will be selling ice cream, or at least something that has real value to it. A master teacher relates what the students are learning to the real world and packages it so that the students want to learn it. She’ll make polynomials seem like a natural part of the world around us, one that we all should be familiar with. And the learners, in the presence of a master teacher, will want to know about these curvy, wavy polynomial things that relate temperatures to sounds and baseballs to gravity. The teacher who believes that all student desire to learn, and thus presents desirable lessons, will make learning the joyful adventure that it should be.

But as soon as a teacher starts passing out worksheets and telling students to quietly circle the nouns in a passage of Pilgrim’s Progress, it becomes clear the students don’t desire to learn that. “Do the odd numbered questions on page 536 and then trade papers so the person sitting next to you can see whether you got the answers right.” “Take notes while I tell you what happened in the Battle of Antietam. You can use your notes during the test on Friday.” “Christina, is chlorine a base or an acid?” These are the utterances of a lion tamer. Do it… because I said so. The student did walk through the door desiring to learn, but the teacher soon made it clear that the material wasn’t worth learning.

Students will desire to learn what’s worth learning. Most of what we are supposed to be teaching is worth learning. We need to believe that and then present lessons so that the students believe it too.

The Lewis and Clark expedition was a military operation to explore and lay claim to whatever land lay between the Mississippi River and the Pacific Ocean. Except for a slave, a couple of civilians they picked up along the way, and a baby that was born during the expedition, the group of explorers were all military men under the joint command of Captain M. Lewis and Captain W. Clark. Back then, as now, commanding officers told their subordinates what to do; the decisions were not up for debate. For the first half of the expedition, men who dared to argue or contradict decisions were punished accordingly.

But when they reached their destination, the two commanders did something extraordinary. A decision had to be made whether to cross to the south shore of the mouth of the Columbia River, or camp in a more familiar location further back along their path. With winter already upon them, this could be a life or death decision. Rather than ordering the men to do what the commanders wanted, as they had done many times in the past, they took a vote, counting equally the choices of every military man, at least two civilian interpreters, York the slave, and Sacagawea the native civilian. When they tallied the votes, the majority felt that it was best to cross to the river’s south bank. That’s what they did, building Fort Clatsop to house them through the winter of 1805-06.

Hold that thought. This article is about modern education, not American history. The relevance of the Lewis and Clark vote will, if I wield my pen properly, become apparent in the end.

Most people today hold this model of education: A teacher possesses a certain body of knowledge. Students don’t. The teacher reveals the knowledge to students who then, to varying degrees, remember what the teacher revealed. Good students pay close attention, record the teacher’s revelations, study, and can eventually demonstrate that they now possess the new knowledge too. They have learned, and the knowledge has passed from one person to many.

This is a good, practical way to view what happens between teacher and learner. It matches what we see in classrooms around the world and across time. It is economic as well because one person can reveal knowledge at once to large groups of students—30, 40, or more.

But that is only one way to describe learning. Another way, one that has been popular with staff development folks in the past ten or fifteen years, favors the teacher’s role as a guide. Learning, in this model, is an adventure shared by both the students and the teacher. Though the teacher has been on the adventure before, he explores things together with the students. Learning happens when students make sense of some newly encountered situation. They work out how to maneuver a dowel around a bend in a hall; they realize that Shakespeare wrote some R-rated scenes; they debate the motives behind labor unions; they discover the secrets of inheritance; etc. In this model, learners don’t simply absorb information, they seek it out. They form an initial understanding and refine it through trial and error and continued investigation. They help each other, referring to books, or the Internet, or direct measurement, or any other resource that is available to them. A student of mine even held an instant message conversation with the author of the story we were reading.

Luckily, this has been my way of interpreting the complex, indefinable magic that happens between teacher and learner. I would not have survived as the only Academic Decathlon coach of a large urban high school if I tried to convince students that I knew the information they needed to master. The Academic Decathlon’s competitions require students to know in depth and at a college level all of these subjects: math, science, literature, art, music, history, writing, oratory, and the art of the interview. Each year, the first six of these categories focuses on new pieces of art, novels, musical compositions, branches of science, etc. At least in a school that can afford but one coach, the only way to teach Academic Decathlon is to explore the topics together with the students, not deliver the knowledge to them.

What happens between teacher and student cannot be put into a box. It defies definition. It is a nebulous, complex interaction that takes a master to both guide the students and inform them, and to have the wisdom to know which to do when. I personally like the idea that I lead students to discover things about the world rather than telling them how the world works.

Oh, I almost forgot about Lewis and Clark. What does the vote at the mouth of the Columbia River tell us about the relationship between officers and rank-and-file? Should I tell you? Or should I lead you to the edge of discovery and let you step over the threshold? ☺

People learn new words and phrases by hearing them, reading them, and using them in context. A learner will hear the word elm applied to the tall branching thing in the front of the house. He tries to use the word elm for the tall branching thing in the backyard, but is told that that is not an elm; it is a eucalyptus tree. Later, he hears someone say that they should trim the tree in the front yard. Human brains being what they are, he figures out that the tall branching thing in the front is both a tree and an elm, while the tall branching thing in the backyard is both a tree and a eucalyptus. Tree must mean a tall branching thing. Elm must be a kind of tree. Eucalyptus must be a different kind of tree. This is confirmed when he hears other trees, similar in appearance to the one in the backyard, also referred to as eucalyptus.

In a book he sees a picture of what he thinks is a eucalyptus tree, but the caption says that it is a red gum tree. It’s possible that a red gum tree looks like a eucalyptus tree, or it is possible that red gum and eucalyptus mean the same thing. The learner files that information away for later clarification.

That is how we acquire words and build our functional vocabulary. It is a slow process. From the time that we first encounter a word to the time that we have a clear understanding of its meaning may be very long, even a year or more. In some cases, we may carry that imperfect definition around with us forever. For instance, a person who rarely messes with mechanical things might go to his grave confusing a socket with a ratchet.

In school though, a student can’t wait for multiple in-context encounters with a word before she figures out what it means. If today’s lesson is on metonymy, then the first encounter with the word had better be a clear definition of metonymy. Otherwise the rest of the lesson won’t make any sense.

Lessons often rely on several key vocabulary terms. In addition, these new words are embedded in a context containing academic vocabulary and conventions that are still being defined in the learner’s mind. A learner, for example, might be asked to contrast the author’s use of figures of speech in the poem “Do Not Go Gentle Into That Good Night” to those found in an excerpt from Titus Alone. Not only does she have to know the definition of the phrase figures of speech, but to fully understand the task, she also needs to know what it means to contrast two things, what an excerpt is, what is meant by the author’s use of something, and why Titus Alone is italicized while “Do No Go Gentle Into That Good Night” is in quotation marks.

We must use some sort of direct instruction for vocabulary so that students don’t have to slowly absorb all the terms for our lessons. Of course they will hear us saying the words and they will see the words in textual contexts (hopefully rich media). Right up front, though, we need to provide them with clear definitions of the key terms.

Here things tend to go terribly wrong. As educators, we should know that the dictionary is not full of clear definitions. It's full of precise definitions. A precise definition is not what the students need. When I was doing my internship, the class I was observing was reading a book that used the word manta, which became one of the vocabulary words for the day. Before the students started reading, they had to look up the vocabulary words in a dictionary and write the definitions in their journals, supposedly so they could glance at these definitions as they read. That weekend I was grading journals and saw that about half the students had copied something similar to this: “Manta, n., A square piece of cloth or blanket used in southwestern United States and Latin America usually as a cloak or shawl,” while the rest had copied something like this: “manta, n., The Spanish-American name for a fish of the genus ceratoptera, also called a devil-fish.” Which kids should get credit? Which definition was correct? (I was told to award credit if the students copied the second definition.) How would the students know which was correct if they hadn’t seen the vocabulary word in context yet? Is the correct definition even clear to students living in oceanless Arizona? Worse yet, the students had proceeded in their reading as if they knew what manta meant.

At best, the misguided activity of copying definitions from the dictionary wastes time. If you think about it though, it actually harms the learners by giving them a false sense that they now know something. In spite of this, it is a very widespread practice. When I was manning a computer lab at Camelback High, I watched a social studies teacher direct his students to look up a list of words including colonialism in an online dictionary and paste the definitions into a document. He then went around the lab awarding credit if they had done it. His approach was slightly better than the one I witnessed as an intern in one respect: copying and pasting wastes much less time than transcribing by hand. From talking to teachers, I would guess that the dictionary activity is used more than any other for “teaching” vocabulary. Sometimes it is modified by asking the students to translate the dictionary definitions into their own words, but if the dictionary definitions are unclear in the first place, then how can a student give clear definitions in their own words?

If we need to directly teach vocabulary and the dictionary activity is no good, what then?

At a recent conference, an alternative vocabulary activity was touted. The students were to fold a piece of paper in quarters and fill those quarters with these pieces of information on each vocabulary word: a definition in their own words, a drawing illustrating what they think the word means, a list of related words including synonyms and antonyms, and an example of how it is used in context. The students could fill in this information as the lesson progressed. I like this approach as the students gradually construct their own understanding of what the term means based on its actual use. (This activity is not really as new as the presenter made it seem. I’ve seen variations of it several times throughout my career.)

But that 4-square vocabulary activity is decidedly elementary. I’m a high school teacher. Although it would work for high school students, it is not a very efficient way to absorb the large number of vocabulary terms a high school student needs to learn, nor is it particularly suited to the self-image of high school students, who associate folding paper and drawing pictures with younger grades.

We will all have different solutions to the problem of how to directly teach vocabulary terms in a way that is meaningful for the students. My preference is to treat it as an extension or acceleration of the natural way we acquire vocabulary. I will tell the students a definition as it comes up in a lesson, illustrating it with an explanation or a drawing on the board or perhaps a pantomime. Then I would supply examples. Often I would supply examples only, and then ask the students to come up with a definition based on what the examples show. I would reinforce the vocabulary terms at a later date by playing a game (what else?!). The games I used had names like Vocabulary Baseball, Scattergories®, and Concentrati. They were all whole-class adaptations of existing games. Every student got involved. Each game covered dozens of vocabulary terms in one activity. That's what worked for me.

I’m not trying to provide a solution for teachers. This article is more about what not to do, what to avoid. If you keep the following essentials of vocabulary acquisition in mind as you develop your own lessons, you should do well:

• People naturally learn vocabulary terms from context.
• Contextual encounters are not enough for academic vocabulary.
• Dictionary definitions confuse learners more than they clarify.
• A learner needs a clear definition up front.
• A learner needs to see key terms used in context during the lesson.
• Play a game on Friday to help reinforce the vocabulary terms.

Okay, just kidding on that last one… or was I?

Math is about numbers. Everybody knows that.

But it’s not. The numbers just happen to be the symbols we use in math. Just as writing is not about the letters of the alphabet, math is not about numbers.

In reality, it is about relationships. It answers questions like these: What is the relationship between a shadow and the object casting the shadow? How far can a balcony extend before the support beams become unstable? If I weigh 180 pounds, how much cough syrup should I take? What is the relationship between a sound traveling over water and a sound traveling over land? What is the relationship between the size of the sidewalk rectangles and their temperature?

No one really cares that if \(\frac{x}{68} = 144 (9.4\times 10^{-6})\), then \(x = 0.92\). By themselves, the numbers are meaningless. A person trying to nail a 12-foot span of copper sheeting could use this particular relation to determine that his material will expand about a tenth of an inch when the sun heats it to 100°F. That’s important because a tenth of an inch is enough to buckle the sheeting and pull it away from the underlayment.

So that begs the question, why do we spend so much time learning to juggle numbers? In fact, my observation is that that is very nearly all we do in math classes nowadays. We calculate and calculate and calculate, trying to learn tricks and memorize algorithms that produce right answers. A teacher might mention the relationships. She might even ask the students to complete a few word problems that illustrate the relationships. But students’ grades rely on whether they can juggle the numbers in such a way that produces right answers.

As a byproduct of this type of math training, we as a society believe that to be good at math means to be extremely meticulous. A mathematician is someone who can write a long string of numbers and symbols without omitting a negative sign or misplacing a decimal point. What a pity.

Stephen Wolfram, the inventor of WolframAlpha and its related technologies, argues convincingly that this preoccupation with calculating is holding us back. We have calculators that can do that, and computers. There is no need, his argument goes, to spend hours doing long division or expanding polynomials by hand. Instead we should be spending our precious classroom hours learning about the relationships that the calculations represent.

He is right, we focus too much on calculation. That being said, calculating does develop a familiarity with the mathematical relationships that studying them directly does not. A wrestler who has spent hours on the mat practicing moves with a sparring partner will gain a feel for the moves that the coach could never impart. Calculating also develops mental agility with numbers, and the ability to recognize when algorithms allow shortcuts, or when different algorithms might be useful. This is similar to what a language student learns from exercises in sentence combining or parsing.

My games are, for the most part, calculation practice. They are not meant to provide insights into the relationships that math reveals. A person needs to do that—a teacher or parent or tutor. The games do provide practice calculating, a necessary part of the learning of math. One advantage of the games is that the student can play them outside of the classroom, giving the teacher time to delve into the more meaningful parts of math instruction.