A reminder about the importance of tool skill fluency

Recently I started working with a new student, a seventh-grader who was in the regular level of math.  Although he had been getting solid B’s all year, he complained to his parents that he was having difficulties and he sensed that there were gaps in his knowledge that were not being addressed systematically in the class.

In our initial work together, it was easy to confirm his suspicions.  During our first lesson, he brought a sheet from school with 8 problems to work on as part as review in preparation for the annual state math exam.  Although he knew something about how to do each problem, he was able to get only one correct without any help.

It’s really no wonder that he did so poorly.  He doesn’t know all of his multiplication facts, he didn’t know how to divide fractions, he can’t read decimal fractions with place value (i.e., say that .014 is fourteen thousandths), he doesn’t know how to do any arithmetic with decimals, and he doesn’t understand the relationship between fractions, decimals, and percents.  These are all topics that were covered in his 5th and 6th grade math classes, but he had not yet mastered them.

One of the sad lessons we can draw from this situation is that this student’s course grades did not accurately reflect his mathematical abilities.  And this student is hardly unique.  We can see how common this situation is when we examine the passing rates of students on state exams and international assessments.  In New York, for instance, statewide, the 3rd through 8th grade passing rate on the state assessment was about 39%.  But we can be sure that relatively few of the 70% of the students who didn’t pass the exam were given failing course grades.  (I know that there are many legitimate criticisms of the state tests, but we can find many other evaluations that call into question the validity of the grades students receive in their courses.)

The problem, clearly is that in so many classrooms and so many schools many students do not have the opportunity to master critical tool skills.  Moreover, the curriculum, or at least the classroom assessments, compound the problem of these deficits by obscuring or ignoring them.

States have expended tremendous effort to prepare elaborate sets of standards for each grade, especially in the areas of reading and math.  There are some problems with many of these standards (the gobbledygook factor is quite high), but even when they are precise and clear, they are of little use if students are moved on to a new set of lessons even when they are not yet able to perform the skills just covered fluently.

I invite you to share your thoughts on this topic.

Happy Teaching!

What's Wrong with American Education?

With the recent contested confirmation of Betsy DeVos as Secretary of Education, I thought it would be timely to touch on the broad question of the state of American education and not just on a narrower issue with regard to teaching and learning.

Often, as in the debate about Devos’s nomination, the argument we have in this country centers on the benefits or limitations of public schooling and the extent to which market forces and competition can improve the teaching and learning of our students.

To be sure, there is good reason to be concerned with the quality of teaching in our schools.  I have seen too many students who have struggled needlessly or who have trudged through activities that never ignited their engagement in a topic.  These faults are real and, unfortunately, widespread.

But in my teaching career of 35 years, I have not found that public schools hold a monopoly on these problems. Quite to the contrary, it is in charter schools and private schools that I have observed some of the most distressing examples of poor teaching.  Some new evidence from Louisiana and Indiana, two states with large voucher programs, confirms this observation.  Researchers found that voucher students who transferred to private schools had poorer academic performance than peers who remained in public schools.  (I will list references to this research in the comments section.)

In light of these observations, it seems very fair to say that the debate about the virtues of public versus private education does not enlighten us very much about the most significant challenge we face in the effort to improve teaching and learning in classrooms across the country.  That challenge can be put quite simply:  We need to put in every classroom a teacher who uses good curriculum materials and uses them well.  Or, to rephrase the matter to answer the question I asked to begin today’s blog, “What’s wrong with American education,” the answer is, “Not enough teachers are using good curriculum materials.”

Given the billions of dollars spent for published curriculum materials and the countless hours that teachers spend developing and refining lesson plans, one might well ask, “Why is putting good curriculum in classrooms a challenge?”  The problem certainly hasn’t been due to a lack of funds or a lack of effort.

I suspect, rather, the problem is similar to the one raised by the old story of the blind men and the elephant.  In that story a group of blind men set out to describe an elephant.  Each blind man touches a particular part of the elephant and describes the elephant in terms of what he feels. But each blind man confines himself to just a single part of the elephant.  As a result, each describes a single feature of the elephant accurately, but mistakes that feature for the whole.  In some versions of the story, each blind man becomes irate at the descriptions provided by his fellows that are so different from his own experience.

Likewise, I think, significant and broad educational reform has remained elusive because it is easy for educators at all levels, from academic researchers to curriculum designers to administrators to teachers to parents, to concentrate on only a few of components of good teaching at any one time.  The weak curricula that we have in many classrooms often contain some elements of good teaching.   They might, for instance, present interesting and relevant activities, or provide precise directions, or keep students constantly engaged in relevant work, or show students how new material is connected to what they already know, or lead them in rigorous analysis to explore and reveal important patterns and relationships, or provide efficient practice, or include opportunities to apply and extend what has been learned, or help students think explicitly about their own learning so that they can learn how to learn.

But It is the rare curriculum that address all of these areas.  And the omission of any of these components makes learning unnecessarily cumbersome and difficult.

So to all of those who want to improve teaching and learning, I say, “Keep your eyes on the curriculum!” Genuine educational reform is curriculum reform.  And our ability to achieve serious curriculum reform rests on our ability to speak more clearly and more rigorousy about all the components of good teaching. 

ABeCeDarian Student Workbook A now available as an app for tablet computers

Student Workbook A is now available as an app for tablet computers.  The app replaces the online version that had been available for subscription.  (Those who have the subscription will still be able to access it, but we are not taking any new subscribers.)

The app features the same lessons in the same sequence that are found in the Student Workbooks A1 and A2.  The therefore include all of the Word Puzzles, the Reading Chains, Spelling Chains, reading pages, Tap-and-Say pages and handwriting practice pages.

Please pass the word and let us know what you think.

The new app joins the e-reader versions of the Storybooks for Level A in the digital universe.  And in the works are a Level A app for classroom teachers to use for whole group instruction, and an app version of the Level B materials.

The app is available on the Apple App Store and on the Google Play Store.

Key Features of Good Handwriting Instruction

Last week I wrote about the importance of handwriting instruction, even in our increasingly digital age.  Today I want to review some key elements of good handwriting instruction.

One of the most fundamental and important aspects of helping people learn is to help them analyze their experiences.  That means helping them take apart something compound and perhaps complex so that they can see the parts and how the parts combine to form the whole.  With regard to handwriting, this means helping students understand the particular strokes involved in forming each letter.  The best way to help students do that is to give them short directions specifying how to move their pencil step-by-step, including a simple way to indicate where to start when forming a given letter.   Thus, for example, in ABeCeDarian, teachers give the following directions to write /m/:  "Start at the dot. Fall down to the line, bounce up and over, fall down, bounce up and over, fall down.”

Another important pedagogical aspect of providing such clear instructions is that it helps the child with motor planning and helps build the habit of “self-talk” that is essential for building up skills at self-monitoring and self-correcting.  A nice trick that I learned years ago is to encourage young children to speak the letter-formation directions out loud directly to the tip of their pencil so that it knew what to do.  This technique makes the handwriting practice a great deal more enjoyable for the children and also helps them feel themselves more in control of what is going on.

Many handwriting programs, especially the inexpensive workbooks one finds readily in book stores, provide the letter formation directions in the form of numbered arrows superimposed over an already formed letter.  I suspect that most children find these lines more confusing than helpful.  Also, unlike directions that are provided verbally, they cannot be rehearsed and internalized as easily and are more difficult to use for planning letters made without the benefit of the arrows.

There is also a wide variety of additional rules available to help guide the writing.  A bottom rule, that is, a line on which to rest the bottoms of the letters, is essential.  I also like a mid-line that shows the correct height of the “short” letters such as a, c, e, and the short parts of letters such as b, d, and h.  I think the midline is extremely useful for beginners.  Many programs also provide a top line and some also provide a descender line, often in red, that indicates the boundary for letters such as g and y that have parts that descend below the base of the letters.  These probably don’t hurt, but I don’t know that they are necessary for most students.

It is most efficient to teach students from the very beginning to associate a sound with a letter.  This practice will reduce the time it takes for students to learn basic letter sounds and will reinforce the general connection between letters and sounds.  Letter sounds are much superior in this work than letter names, because letter sounds can be used directly both to read and spell words.  So it is ideal to incorporate handwriting instruction as a part of beginning reading and spelling instruction.

Some programs incorporate a technique known as “sky-writing” into the instruction.  This involves having the student extend her arm with her pointer finger also extended and then tracing letters in the air.  The muscles involved in this activity are very different from the muscles involved in writing the letters with a pencil, so I doubt how much this practice actually helps students improve their writing with a pencil.  Furthermore, because the letters are traced in the air and are hence invisible, it is harder for the student to evaluate whether he has done a good job or not.  Nonetheless, if you are working with young students who need to move around, this might be an option to incorporate some more physical activity in a lesson.

Much better, especially for very young students, is to have students trace in a salt-tray or similar contraption that will allow them to make a visible mark and receive some tactile stimulation as well without using a pencil or marker or chalk.  There are many variations and lots of information about these on the internet.

So, in summary, here are the key things to look for in handwriting instruction:

  • simple and clear verbal directions for stroke formation
  • a simple mark to indicate where to start each letter
  • writing paper or boards with a baseline and a midline
  • teaching letter-sounds in conjunction with letter-formation

Until next time,

Happy Teaching!

The Importance of Handwriting Instruction

In an age dominated by digital devices the practice of handwriting might seem a quaint relic of times past, destined to fall out of use like quill pens and inkwells.  Indeed, in most schools nowadays time devoted to handwriting instruction has been reduced and in some cases virtually eliminated.

However, handwriting instruction doesn’t deserve this neglect.  On the contrary, there is ample evidence that fluent, legible handwriting is an important academic tool skill that enhances learning generally.  For one thing, there is still quite a bit of writing done by hand, both by students in school, in the home, and at work, even if it is much abbreviated from the lengthy hand-written documents more common in pre-computer times.  If students do not develop fluency and accuracy in forming letters without conscience thought, they will not be able to do their writing very efficiently.  Too much of their attention will be devoted to the act of forming the letters instead of on the content and stylistic form of what they are writing.  (This is precisely the point I was making in general about tool skills in recent posts.)

The exact same principle applies, by the way, with regard to fluent writing of the numerals.  Often when I work with students referred to me for tutoring because they are struggling in math, I find that they labor writing the numerals and can often not do so correctly and legibly.  The effort they require to form the numerals detracts from the attention they have available to focus on the higher level math concepts they are working on.  In addition, not surprisingly, they make frequent calculation errors because they cannot read their own writing!

Fluent and accurate handwriting is also a critical part of learning how to read and spell.  The motor engagement involved in writing words boosts the student’s ability to analyze and remember letter patterns more easily than if he examined the words solely by sight. If you have any doubts about this, try to learn to read Chinese characters.  Work on a set of characters solely by examining them visually, and then work on a similar set in which you also learn how to write the characters correctly and fluently. It's amazing how adding a motor component to the analysis of the visual form improves one's ability to remember it.

There is a related debate, even among proponents of explicit handwriting instruction, about whether or when to teach cursive handwriting.  Some educators maintain that cursive is easier for children to learn than manuscript handwriting, especially for students with various kinds of learning difficulties, and so this should be the first (and perhaps only) form of handwriting taught.  This view, however, as always remained a minority position, and it never made sense to me.  It seems to make much more sense and to be much more efficient to teach beginning readers a handwriting style that matches the form of the letters they see in text.

Learning cursive handwriting at around the age of 8 or so, that is, after one has acquired basic decoding skills, however, seems to me to be quite valuable. Most people (including especially people of my age, who had quite a bit of formal handwriting instruction in elementary school) associate the word “cursive” with a particular style of handwriting generally known as “looped cursive,” a category of styles that includes the well-known Palmer method.  As the name indicates, in "looped cursive," the joins between letters are often made with loops, creating a rather ornate script.  However, the term “cursive” is quite general, denoting merely a style of handwriting that “flows” because most of the letters are joined.  (The literal meaning of the root in “cursive,” cur-, is “flow.” The root also appears in the word “current.”)

The point is that “cursive” does not necessarily mean “looped cursive.”  Indeed, a number of educators prefer teaching students a modified italic script.  As with all cursive handwriting, using an italic script helps a person write with a consistent rhythm and greater speed than is possible with the block letters of typical manuscript handwriting.  Modified italic script has the additional benefit of representing letters with much the same stroke patterns as employed in standard manuscript, and so is easy to learn.

I happened upon one of these modified italic forms about 10 years ago and adopted it as my handwriting style.  I can now write much faster but with equivalent legibility as I could with the looped cursive I had been using since my elementary school days.  (Unfortunately, I have not yet had any luck convincing my 23-year-old son, who never uses the cursive he was taught in school, to give it a try.)

The benefits of good handwriting exist well beyond elementary school.  With the proliferation of laptops in universities, more students are taking class notes on their computers.  There is strong evidence, however, that students who do so are processing the class information at a shallower level than students who take manual notes.  The key element seems to be that those students who are typing their notes are simply transcribing what is said in class, which they can do with little attention to what the words mean, or how the concepts presented are related to one another.  In contrast, students who take notes by hand are much more likely to be engaged in thinking about these relationships as they think about how to record the information, especially if they are also using graphical elements in their note-taking. (This spring I plan to make some posts about note-taking.)

So students and teachers, don’t throw away your pencils or pens (or styluses) just yet!  You will be rewarded for learning how to use them well.

In my next post, I’ll go over the key elements of good handwriting instruction and list some good handwriting programs.

Until then,

Happy Teaching!

More about Tool Skills

In my last blog, about fluency, I said I would continue the discussion about tool skills by presenting the tool skills for a particular subject.  So here is a list of tool skills necessary to decode fluently.  Those of you who have used ABeCeDarian already will be very familiar with this list.

1. Students need to be able to segment a word into individual speech sounds.  For example, students need to understand that the word “mop” is comprised of the individual sounds /m/ /o/ /p/. This is the skill is the foundation of writing systems such as English that use an alphabet, that is characters that represent not whole words or syllables, but smaller sound units generally comprising a single mouth gesture.

2. Students need to be able to blend isolated speech sounds into syllables and words.  For example, students need to be able to take the string of isolated speech sounds /m/ /o/ and /p/ and recognize that they are the sounds that make up the word “mop.” This is the skill that allows people to rapidly acquire a large store of words that they can decode automatically.

3. Students need to know the major letter/sound correspondences.  For example, they need to know that we write the /m/ sound with “m” and the /p/ sound with “p.”  This so-called “code knowledge” is also essential to the rapid acquisition of a large store of rapidly decoded words.

4. Students need to know how to combine the skills of phoneme blending with their code-knowledge to sound out words and they need to get in the habit of using this strategy when they encounter words that they do not automatically recognize. Slight variations of the strategy need to be taught as well to deal with the situation in which a word has an uncommon spelling for a sound or when the student is trying to read a multi-syllable word.

5. Students need sufficient practice reading words to be able to retrieve them from memory almost instantaneously without overtly sounding them out.

Where did this list of tool skills come from?  The starting point is research.  But research findings are often too general to be distilled into a precise list of tool skills.  For example, research quite definitively underscores the importance of “phonemic awareness,” that is a broad ability to identify and manipulate speech sounds. There isn't, however, much research about the specific contributions of sound segmenting and sound blending, in particular, nor much research to help us weigh the relative importance of all the different aspects of phonemic awareness.

Much educational research in education yields various kinds of associations that suggest that students with one particular quality generally succeed (or fail) at a particular task.  But research findings of this type, while interesting and suggestive, do not specify what actually causes success.  Failure to understand this distinction has littered curriculums and district standards with inefficient goals.

This difficulty is well-illustrated by the practice of teaching letter-names in kindergarten. There is a very strong correlation between a student’s knowledge of letter names in kindergarten and her reading skills later in elementary school.  That is, students who know a lot of letter names at the beginning of kindergarten are usually good readers by the end of second grade, and students who don’t know many letter names tend to be relatively poor readers by the end of second grade.  Because of this correlation, many schools require kindergartners to learn the letter names.  But the research looking at the association between the teaching of letter names and student reading performance is not especially strong.  That is, teaching letter names does not seem to produce superior reading performance.  The explanation for these seemingly conflicting results is that the knowledge of letter names probably is a signal that the student has had a variety of experiences with printed words and word sounds that help prepare her to learn how to read.  It isn’t just knowledge of the letter names per se, but the various other bits of word knowledge that the student acquired while learning letter names.  When the letter name instruction is distilled into classroom instruction, therefore, the other associated things aren’t present.  And on careful analysis what is functional, what actually helps students decode words accurately, is not knowledge of letter names, but knowledge of the sounds the letters commonly represent.  I know this because if I teach letter names to students without also teaching letter sounds, the new knowledge doesn't translate into greater word reading.  But if I teach letter sounds (along with sound segmenting and sound blending), students soon learn to read and spell words.

So research is the starting point.  But to refine the general lessons of research into a set of tool skills useful to teachers and students, one requires in addition a commitment to identify functional subskills and develop activities that help students rapidly develop them. One might think that such a commitment is an ordinary characteristic of curriculum design and of the typical teacher's approach to her subject, but that hasn't been my experience. A very large number of the lessons that I have seen, whether they have come from established publishers or were the handiwork of a single teacher, have lacked this orientation.

I remember, for instance, conducting a workshop for some kindergarten teachers and I asked what they did for literacy instruction in their classrooms.  One said, "In my classroom, each student gets to play King and Queen for a week."  I asked her to explain some more. She went on to say that she picked and boy and a girl whose names began with the letter "a," and they get to wear and crown with the letter A on it, and the class paid special attention that week to words that began with the letter "a."  A different letter of the alphabet was featured in this way each week.  While this activity does ask students to pay some attention to letters, it is an extremely indirect way to do so, and doesn't clearly show the students the steps necessary to learn how to read.

Published and well-established curriculum materials can also suffer the same defect.  I'm a big fan of the upper level math textbooks written by Harold Jacobs, who began writing textbooks in the 1970's. Although he doesn't use the term "tool skills" explicitly, it is clear as one goes through his lessons that he had a keen understanding of how to break of the complex skills involved in learning algebra and geometry into smaller parts so that his students never had to make conceptual leaps that they were not prepared for.  As I work in my private tutoring practice with math students using a variety of more modern materials, it is striking how little most of these lessons break down the complex tasks they are presenting into smaller subskills.

The moral of this story for parents and teachers, I think, is to make sure that the teaching materials that you use make a serious attempt to identify the relevant tool skills and to make clear how the activities and sequence of the curriculum develop these skills. Because educational research remains incomplete, there will always be room for some disagreement about the exact tool skills that comprise different subjects.  Nonetheless, If we as teachers are committed to identifying as best we can the subskills necessary to master what we are teaching, I think we will find quite a bit of inefficient teaching that we can eliminate.

I would love to hear from you about what you think of the decoding tool skills I have listed and whether you find them useful to think about in your own teaching.

Over the coming months, I will return to the question of tool skills and propose a list of tool skills for other subjects.  Also, I want to explore various aspects of developing good practice routines.  Even when one has properly identified a tool skill, there are often a dizzying number of different ways of providing practice for the skill, and an equally dizzying number of debates about which practice techniques are best.

Until next time,

Happy Teaching!

Fluency

In my most recent posts I have talked a bit about the importance of making sure that students develop fundamental skills to a level of mastery.  But what is the best way to measure mastery?

In many schools and educational settings, the measure used most frequently is accuracy. We give students tests and see how much of the work they got right. Of course, accuracy is a part of mastery, but it is not a sufficient measure of mastery by itself. To see see what is missing, let us consider two students who read a paragraph of 100 words without any oral reading errors. The first student read the passage in 1 minute, while the other did so in 3 minutes. If we look solely at the accuracy of the reading, these students look equal. But when we examine the rate at which they performed the task correctly, we see that they are quite different.

What this example suggests is that mastery involves fluency, that is, the ability to perform a task correctly at a certain rate. Rate is important because it signals how much conscious attention and effort a person needs to do something.  A skill that is mastered can be done automatically. Indeed, in the cognitive psychology literature, “automaticity” is the term generally used to refer to mastery. This “automaticity” of component skills is necessary so that a person can use her limited resources of attention to solve some more complex task. If we return for a moment to consider the two students who read a passage accurately but at very different rates, we can easily imagine that the student who read the passage in one minute recognized most of the words almost automatically and so could pay attention to what the sentences meant.  The student who read much more slowly, in contrast, probably had to spend a considerable amount of effort sounding out the words or using other tools to figure out what the words were, and so had much less attention available to understand what she was reading.

There is quite a bit of evidence that the attainment of fluent levels of performance is important. As part of a large meta-analysis of reading research, for instance, The National Research Panel identified oral reading fluency as a critical component of successful reading. There are numerous studies in math, also, that show a connection between fluency in component math skills and understanding of math concepts. We know from other fields, as well, such as music and sports, that fluent performance of component skills is essential for success.

In spite of this evidence, some schools and teachers de-emphasize the development of component skills, arguing that skill practice makes lessons boring, reduces student engagement and motivation, and fails to develop creativity.  Indeed, this dispute is one of the central points oflongstanding contention between so-called traditional and progressive education.

It is possible, of course, that in some classrooms, skills are practiced incessantly, with little opportunity for students to use them to pursue more complex and challenging investigations or problems. There are first and second grade classes, for example, that are consumed by the so-called “mad minute” tests on arithmetic facts, where math seems only to be about the automatic recall of facts and not about more involved exploration of number relationships. But this over-emphasis of practice is an error in judgment about how to organize a student’s time in the classroom. There is really no necessary, fundamental antagonism between the work necessary to develop fluent skills and work that is more “creative.”  

On the contrary, creativity should be understood properly as the combining of existing behaviors in a new way. And that means that a person’s creativity requires a stock of well-developed, that is, fluent, skills. Moreover, far from killing student motivation, as is so often claimed, helping student’s to perform “tool skills” fluently keeps them engaged and enthusiastic as they see, over and over again, that they are able to do more things and do them well.

Another problem to be alert to is giving students various timed tests on tasks for which they have been inadequately prepared. Many kindergarten students, for instance, are required to memorize several dozen so-called “sight words.”  It is true that students need to learn to read most words automatically, but the acquisition of this ability usually requires quite a bit of knowledge about letter/sounds, and the development of segmenting and blending skill. Another example is the “mad minute” of math fact practice that I just referred to. Yes, indeed, it is important to be able to retrieve these facts from memory automatically, but for most students, this involves extensive opportunities to explore numbers and number relationships in a variety of ways. The moral of the story is that when accessing a student’s fluency, it is important to make sure that she has the necessary component abilities. Skill development is exactly that, a kind of development, and hence, something that takes time, experience, and practice.  Giving students tasks they are not ready for is just bad teaching.

In my next blog, I will take a stab at identifying some key “tool skills” and sketch out a little bit of a road map for the proper sequence of addressing them.  In the meantime, I leave you with some resources with some additional information about fluency assessment.

______________________________________________________________

There is a very interesting program called Precision Teaching developed a while ago by a psychologist named Ogden Lindsley and his associates, and their are numerous resources on the web with information about this program.

A useful resource for teachers based on the ideas of Precision Teaching is One Minute Academic Functional Assessment and Interventions by Joe Witt and Ray Beck.

The University of Oregon Center for Teaching and Learning has a useful website with information on the Big Ideas in Beginning Reading. Here is a link to their content regarding fluency:  http://reading.uoregon.edu/big_ideas/flu/index.php

Tool Skills

In my last post I talked about Mastery Learning, that is, the idea that it is important to train people to a high level of performance before advancing to a new topic or skill. The idea behind Mastery Learning is that there is a hierarchical nature to many academic skills.  Relatively simple skills, such as learning how to tally a group of objects, or to write letters and numbers easily, or learn basic letter/sound correspondences are the foundation for more complex skills, such as adding and subtracting, and reading and writing words and sentences.  And these more complex tasks in turn become tool skills necessary to accomplish even more complex tasks, such as solving algebra equations or writing an essay.

The sub-skills that are needed for a task are often referred to as “tool skills."

There are, I think, two especially interesting and fundamental aspects of cognition that help explain why developing tool skills is so important.

First of all, there is a relatively small cap on how much a person can consciously attend to at any given moment.  Therefore, if a person requires quite a bit of effort and attention to perform a particular task, she will not be able to perform another one very well at the same time.  For example, If a student is still doing a basic arithmetic calculation on her fingers, it will be very difficult for her to attend to the new patterns and relationships involved in multi-digit addition, and almost impossible to think in a very sophisticated way about fractions.  The same applies, of course to reading.  If a student doesn’t know a large number of basic letter/sound correspondences and  how to break a word into individual speech sounds and how to blend those sounds together, it is very difficult to learn to recognize a large number for words instantly.

Secondly, when a person is trying to solve a problem, he brings to the task the tools that he has available.  If a person has a well practiced skill (such as the ability to tally a group of objects) then that skill is readily available, it can not only be used but combined with other skills when working on new tasks (such as learning how to add and subtract small quantities).

Furthermore, if all the necessary tool skills that are needed to perform a task have been well-developed, a person often will be able to integrate them in order to perform a new, more complex task with little or no instruction about how to perform the new task.  This phenomenon is known as “generative learning,” because the learning is to large extent self-generated.  All it requires from the outside is a task to accomplish.  The rest is internal, in the form of well-developed sub-skills available to the person to use and combine as needed.  And as you can imagine, generative learning greatly accelerates the rate at which a person can learn new skills.

This general learning principle has been well-established in the lab with pigeons.  In one ground-breaking experiment years ago, pigeons were trained in 3 separate skills:  pushing a box to a particular spot, to climb onto a box and peck a facsimile of a banana hanging overhead, and not to jump or fly toward a banana when it was out of reach.  These pigeons were then able to perform the novel behavior of pushing a box so that they could climb up on it and peck the banana without receiving any training to combine these skills.  Pigeons who had received training in only 1 or 2 of the sub-skills, however, did not spontaneously perform the novel, integrated task.

I have seen this process in action many times in my own teaching.  For example, when students acquire a variety of sub-skills involving fractions, such as being able to add and subtract fractions with like denominators and  having skills both visualizing and calculating equivalent fractions very easily, they often figure out how to add and subtract fractions with unlike denominators with very little additional guidance.

In my next posts I’ll continue on the theme of tool skills, addressing issues including the importance of fluency, and how fluent tool skills improve student motivation.  I will also present a tentative list of some basic tool skills and resources for investigating tool skills in more depth.

Mastery Learning

I just finished watching a very interesting and pertinent TED talk by Sal Khan, the founder of Khan Academy.  He does a very nice job of describing Mastery Learning and contrasting it with the typical model for pacing instruction in classrooms.

In most classrooms, what is held constant for students is the amount of time that they have to study a particular topic, and what varies is the performance that students achieve at the end of that time.  Some students will perform at a very high level, some will perform at a very low level, and a fair number will perform at a middling level.

While we take this arrangement for granted in a school setting, Khan gives a clever example to suggest just how unusual it is in many other contexts.  For instance, he muses, what would happen if we applied this system to home building?  A homeowner would tell her contractor, “Please build my foundation.  You have 2 weeks.  Do the best you can.”  There is some rain, and some supplies don’t arrive when expected.  The inspector comes to evaluate the foundation after 2 weeks and finds some problems.  “I’ll give it an 80%,” he says, and work on the rest of the house continues using the same system of evaluation.  Of course, eventually the problems at each step add up and the house tumbles down before it was completed.

Khan adds that it is not only in home-building but in many other areas of endeavor that mastery of a lower level is required before a person moves on to the next, more complicated levels.  He offers the study of martial arts and music as examples.

What would happen, though, if we adopted mastery learning in schools, if we made what varied was the amount of time the students took to learn some material, but what was constant was that they achieved a very high, nearly perfect level of performance?

There would be some dramatic changes.  First of all, achievement would soar.  Learning material at higher levels would improve dramatically because students wouldn’t be encountering the material with gaps in the set of skills they need to perform the more complicated tasks.  Also, the learning would be easier and take less time.

In addition to improvements in performance, there would also be a profound change in students’ attitudes toward their own learning, what Khan refers to as “mindset.”  Too often now poor performance is viewed by both students and parents (and often, unfortunately, by teachers) as a matter of biological destiny.  That is, there is a belief that students who perform poorly don’t have the cognitive equipment to perform at a high level. But there is little evidence that this is really the case. Indeed, on the contrary, there is considerable evidence that putting in effort at a task (if it is leveled and structured properly, of course) yields high levels of skill.  If virtually all students were expected to master each level of a subject, they would come to view their own learning much more in terms of their own effort and perseverance.  They would come to see failure at a task not as a sign of lack of ability, but simply as an indication that they need some more study and practice. 

Mastery Learning is hardly a new idea.  I first encountered it years ago when learning about the Morningside Academy in Seattle and it  is a central part of a fairly old and well-established program called Precision Teaching.

Those of you familiar with Precision Teaching know that there are a couple of additional points that need to be made to round out the excellent introduction to Mastery Learning that Sal Khan presents in his video.  One is the idea of tool skills, that is, fundamental skills that are combined to allow for a person to perform more complex tasks.  Identifying these tool skills properly is essential.  The other important idea is that of fluent performance.  It turns out that what is critical in helping people acquire a set of necessary skills for complex tasks is not that they can perform the sub-tasks without error (i.e., achieve mastery) but that they can perform the task correctly at a certain rate, which indicates that the task is performed automatically, with little or no conscious attention required.

I will write some more about these concepts next week.  In the meantime, I hope you will look at Sal Khan’s video and let me know what you think.

ABeCeDarian materials on Amazon.com

ABeCeDarian materials are now available on Amazon.com, and I encourage customers making small orders to buy the materials there.  Moving the bulk of our small orders to Amazon will help us improve our service as the number of orders we receive increases.

One important improvement is that there are several options for expedited delivery on Amazon that we are not able to provide on our own store.  Another is that because materials on Amazon are warehoused throughout the country, customers in the midwest and west will be able to receive their materials more quickly no matter what level of shipping service they choose.  Finally, when using Amazon, customers will be able to receive tracking numbers for their orders, a valuable service that is difficult for our small operation to provide on our store.

To cover the cost of using Amazon to distribute materials, it is necessary to increase prices slightly.  Fortunately, there are a number of options available on Amazon to reduce the shipping costs (and in many cases to eliminate them entirely). Because of these shipping discounts,  I believe that many customers using Amazon will end up paying a total similar to the total charge on the ABeCeDarian webstore. 

We still welcome purchase orders from institutional customers.  As before, please e-mail a purchase order to info@abcdrp.com or send it by U.S. mail.

Of course, we are also happy to continue to take phone orders at 607-266-3310. 

The shopabcdrp.com website will remain active indefinitely and you are certainly welcome to continue purchasing materials on that site as well.  However, there will no longer be a link to this store on the main ABeCeDarian website.

Thank you for your cooperation in making this change.  We remain committed to offering the ABeCeDarian materials at a reasonable cost and getting them to you as quickly and efficiently as we can.

Why should teachers organize lessons around analytical questions?

In a recent blog post I argued that one important characteristic of good teaching is to ask students analytical questions.  In contrast with rehearsal or review questions, which ask students to recall previously learned information, analytical questions help a student break down some experience so that she can see the relevant patterns and relationships in the topic you are investigating.  Today, I want to expand a bit on that topic and discuss the benefits of organizing lessons around analytical questions.

There are several benefits to giving this type of question a central role in a lesson.  First of all, it forces the student to think about what she already knows and how it can be applied to the task at hand.  Does this question resemble other questions I’ve had?  If so, what is the connection?  What do I already know that can help answer this question?  Do I need more information? By virtue of connecting new information with what she already knows, this cavalcade of questions makes new ideas and new concepts both more approachable and usable.  The subject of the lesson isn’t something to be filed away for future use simply, but something to be investigated at that moment.

This engagement helps students attend to the lesson.   Because they are being asked to prepare a response, they have to assimilate and interpret the situation the teacher is presenting quite actively.  In contrast, when a student listens to a lecture or observes a demonstration, there are usually no demands for an immediate response and so it is much easier for her mind to wander.  The active engagement of students who have to respond to analytical questions also keeps the energy of the lesson and the interest of the students high.  The analytical questions present the topic as a kind of puzzle, and solving a puzzle is much more exciting than listening to someone explain something.  In addition, something that is considered a problem elicits more of our attention than something that doesn’t present itself to us as problematic.

Another benefit of asking analytical questions during a lesson is, not surprisingly, that it helps to develop a student’s analytical skills.  If a teacher constantly asks, “What patterns and relationships do you see,” then the students get tremendous practice taking things apart to see what makes them tick.  Indeed, they come to see that as a central aspect of learning.

With this approach, moreover, it doesn’t take long for the student to understand the active role he plays in his own learning.  The teacher’s job is to provide activities that help clarify the underlying patterns of a thing that are obscured by ordinary and random experience, but it is the student’s job, in the end, to find the connections herself.  What is truly learned isn’t given, but earned.

In upcoming posts, I’ll continue discussing the benefits of organizing lessons around analytical questions by showing how this technique is used in the ABeCeDarian Reading program.  I also plan to give some examples about how this technique can be applied to math instruction as well.

"Bugs"

In my recent post about foreign-language learning, I recommended that students and teachers embrace errors.  This recommendation, however, shouldn’t be restricted to the study of another language but should be applied to almost all learning.   Unfortunately, we all too often treat errors as failures rather than as necessary and important ingredients of learning. 

There are several important benefits to treating errors as an essential part of learning.  First of all, the goal of learning should be some kind of performance, such as doing a certain kind of mathematical calculation, writing an insightful, interesting paragraph, playing a piece of music on a particular instrument, or executing some maneuver in a sport. When a person who is attempting any of these endeavors commits an error, that means, of course, that she has not yet achieved the desired level of performance.  But it rarely means that the person should abandon the endeavor!  An environment in which errors are understood as a necessary part of attempting something interesting or important encourages a person to continue with his efforts until he has achieved the desired outcome.   In short, it helps to build perseverance and resiliency.

Another benefit of  to treating errors as an essential part of learning is that it helps a student develop analytical skills.  If after making an error, one repeats the effort without any reflection, it is very easy to make exactly the same error again.  To avoid that fate, it is necessary to analyze one’s own performance to figure out what went wrong.  Thus, it is important for teachers to help students learn how to monitor and reflect on their performance so that they can determine the cause of any errors and consider how to avoid the error in the future.  For this reason, in ABeCeDarian teachers usually respond to errors not by saying or writing a word correctly, but telling the student where in the performance the error lay.  It is then up to the student to use this information and try again.  For example, if a student says /map/ when he is trying to read the word “mop,” the teacher responds, “You said /map/, with an /a/ here <pointing to the “a.”>  But that isn’t /a/.  It’s /o/.  Try again please.”  As students progress, the teacher shifts more responsibility for analyzing the student’s performance to the student himself by responding to an error simply by asking him to explain why the word he wrote or said couldn’t be correct.

We often think of practice as a phase or step of learning that is separate from the kind of investigation and reflection involved in exploring something new.  Moreover, the practice phase is frequently presented as inherently tedious.  Critics of traditional phonics programs, for example, often speak of “drill and kill,” meaning, students in such programs receive so much drill in decontextualized subskills that their interest and motivation in reading suffer.  However, if errors are treated not just as something for teachers to tally, but as something for students to analyze, then practice becomes infused with the same problem-solving focus as other parts of instruction.  Both student engagement and performance improve, creating a powerful virtuous circle.  Practice becomes more engaging, and so students do more of it, and so their performance improves, which motivates even more practice, and so on.

One of the most illuminating discussions of how to treat errors productively is in the superb book, Mindstorms, by Seymour Papert.  In that book, written in the 1970’s, Papert talks about how the personal computer can transform education, not as many educators believe, as a tool for providing more individualized practice to students, but as a tool that the students can use to make things by means of computer programming.  As students engage with the computer in this way, almost immediately they encounter the concept of “debugging.”  Rarely is it the case that a person writes computer code that performs exactly as the programmer expected.  After writing code, the programmer has to evaluate it, and when some part of the program doesn’t perform as expected, the programmer has to analyze the problem.  The relevant questions for the programmer when analyzing these errors are:  Can the problem be fixed, and if so, how?  “Bugs” are failures, in a sense, but they are both ordinary and, in most cases, temporary.  They aren’t faults that should be punished, but obstacles to be overcome.

The concept of “debugging” in computer programming thus provides an accessible and evocative metaphor for how to treat errors in general that can help prevent both teachers and students from embracing the false and debilitating notion that challenging and worthwhile tasks can be accomplished without making mistakes.

Resources for foreign language learning

As promised in my last post, here are some excellentl resources for foreign language learning.

Assimil

Assimil, published in France, is one of the best language learning programs I have found.  The general lesson structure follows the principles that I advocated in my last blog post.  The core of each lesson is a dialogue spoken by native speakers.   A transcript is provided along with a translation.  Grammar points are introduced as necessary to understand expressions in the dialogue.  The dialogues are short and witty.  In addition to listening to, reading, and reciting the dialogue, there are only two other exercises during the first half of the program: translating 5 sentences each lesson from your target language into English, and filling in missing words in 5 sentences written in your target language.  A third exercise is added to each lesson after the student has completed the first half of the program, namely, returning to the earlier dialogues in turn and translating the English into the target language.

Foreign Service Institute Language Courses

The Foreign Service Institute is the office in the United States government responsible for training the country’s diplomatic corps and others who work for the U.S. State Department to advance American interests overseas.  In the 1950’s, 60’s, and 70’s, this office developed intensive courses to learn many languages.  These materials are now in the public domain and available for free. The lessons are generally rather dense, but there is a tremendous amount of material contained in them.  Like the programs in Assimil, they contain dialogues spoken by native speakers, accompanied by transcripts and translations.  Unlike Assimil, though, each lesson includes a large number exercises.  One frequent exercises is a replacement drill, in which a basic sentence is given and then the student has to say a new sentence that involves a slight modification of the preceding one, such as using a different pronoun or verb tense.

https://www.livelingua.com/fsi-language-courses.php

I have come across 2 polyglots who provide excellent recommendations and resources for people learning a foreign language.

Fluent Forever

In this book author Gabriel Wyner provides recommendations for an overall approach to foreign language learning as well as enormous number of invaluable resources, including a list of the 600 most common words in various languages and guidance about how to make outstanding flash cards using Anki, a spaced-repetition computer-based flash card program.  Wyner also provides recommendations about outstanding grammar books and dictionaries for several languages as well as information about various web-based services to find people to correct one’s writing and to provide opportunities for conversation.  

https://fluent-forever.com/

LingQ

This site has been developed by the polyglot, Steve Kaufman, who also blogs frequently about themes related to foreign language learning.  The LingQ website is a tremendous resource of text and audio from many different languages.  A special feature of the site is the ability to keep track of new words in passages as well as how many words one knows.  I have been especially persuaded by Kaufman’s strong advocacy of focusing on maximizing input, which in practice means lots and lots of reading and listening, followed by frequent writing and speaking.

https://www.lingq.com/

News in Slow Spanish, French, Italian, and German

These websites provide weekly news articles spoken slowly.  Each news story is followed by commentary spoken slowly.  Users also have the option to hear the text read at a normal speed.  New vocabulary is highlighted and a translation of that vocabulary is immediately provided when one places the mouse cursor over the highlighted text.  In addition to the podcasts on current events, every week a new dialogue is presented on a theme of general interest that contains many examples of a particular grammar point.  The programs also provide a dialogue each week repeatedly using a common idiomatic expression.  This is an absolutely amazing resource, a virtually inexhaustible supply of high-quality, interesting text and audio with numerous, clever aides for the foreign language student.

https://newsinslowspanish.com/

https://newsinslowfrench.com/

https://newsinslowgerman.com/

https://newsinslowitalian.com/

Linguee

There are many useful online dictionaries now, but Linguee I think stands out.  It not only provides definitions and audio with proper pronunciation, it also access an enormous data base to provide examples of words and phrases as they are used in text.

www.linguee.com

Phrase Books

Phrase books are generally designed for travelers who want to learn a number of useful phrases in a language, but who do not necessarily care to study it at length in order to be able to speak and read at a high level.  Nonetheless, they can be very useful tools to help a beginning student get started with his serious and more extensive language study.  Two excellent series are those by Rick Steves and those by Lonely Planet.

There are many other excellent resources that I haven't mentioned, including many that deal with learning one language in particular.  But one of the overwhelming aspects of beginning to learn another language is figuring out where to start, and these resources will quickly help you begin efficient and organized study.

Thoughts about good foreign language instruction

Although I studied Latin, German, and ancient Greek in high school and college, I never achieved anything beyond rudimentary abilities in these languages.  For many years after college I had no strong motivation to pursue any additional language study.  Over the last 10 years, however, I have had the opportunity to travel to a number of other countries, and to prepare for these travels I have done quite bit of foreign language study on my own.  

I would like to share some lessons I’ve learned from these recent experiences as a language student. I hope they may be useful both to adults who want to learn another language and to teachers who want to teach foreign languages to children as effectively as possible.

Focus on reading and listening at the beginning with just enough grammar to allow for understanding

As I discussed in previous posts, a good lesson structure involves providing students with some relevant experience and then analyzing it.  For foreign language instruction, the relevant experience, of course, is listening to speech or reading text. Grammar instruction is necessary, but it is best addressed in the second step, the analysis step.  For example, when introducing a new tense to a student, I wouldn’t begin by saying, “Here are the endings used in the future tense.”  Rather, I would have the student read some text with this construction and then point out the words in the future tense and analyze how they are formed. 

Focusing on listening and reading thus keep the focus on developing the student’s ability to comprehend the new language, which, of course, is the goal of language study. It is all too easy for grammar to assume a role not as a tool used to acquire understanding, but almost as an end in itself.  I know this was the case with much of the foreign language study I did in school.  When grammar becomes the focus of lessons, valuable time is taken from working to understand speech and text, and students often become overwhelmed and confused. Grammar points are much easier to learn when they are directly related to a student's experience with the language rather than in isolation and out of context.

A initial focus on listening and readiing also allows the student to postpone significant work in speaking and writing until he has acquired a vocabulary of several hundred words and the ability to understand simple expressions using basic sentence structures.  Postponing speaking and writing has several benefits. I have worked with a few programs that emphasized developing speaking skills from the outset, but they did so at the expense of developing oral language comprehension.  As a result, I could express some simple ideas quite early in my studies, but I was not able to understand what a person was saying to me.  Receptive tasks, such as oral comprehension, are generally easier than expressive tasks, such as speaking, so it makes sense to start with the easier tasks that are the foundation of the more difficult ones. 

Another benefit of postponing speaking and writing is that it can reduce the anxiety often associated with foreign language learning.  It is psychologically quite demanding to try to say something when one has very words in one's vocabulary and very little knowledge of how to string words together in the language. If extensive work on speaking is delayed until the student has a more extensive vocabulary and ability to understand more speech and text, that is, when he has more experience with the language, it is much less daunting to speak and much easier for the student to be able to express a reasonably complex idea.  Indeed, I suspect students who follow this sequence not only often find themselves ready to say quite a bit but can surprise themselves as well with how eager they are to do it.

Train with audio (and lots of it)

I have tried using some books that didn’t have any associated audio. This approach is effective if the only goal is to read the language.  However, if one wants to be able to understand when others speak and to be able to speak oneself, it is absolutely essential to listen to lots of the language spoken by native speakers. Fortunately, thanks to the wonders of digitized sound, the personal computer and the internet, there is a tremendous amount of audio readily available, much of it free.

The most efficient way to learn from audio is to study it in conjunction with a transcription.  An excellent routine is for the student to listen to the audio a couple of times without reading the text, trying to understand as much as possible.  After these initial trials, the student can then read the transcript in order to understand the passage thoroughly.  During the reading the student can figure out, moreover, why he wasn’t able to understand certain parts.  Was it because the speakers used vocabulary new to the student, or was it because he wasn’t able to recognize the pronunciation of a known word.  This analysis is tremendously useful.  Then the student can go back and listen to the audio again without looking at the text.  Students should continue this process until they can understand the spoken text easily.  When I follow this routine, I not only comprehend a new passage thoroughly by the end of the routine, but also I find that my ability to comprehend new speech improves quite dramatically.

Once a student is familiar with a particular text, he should then try to read the text out loud along with the native speaker.  At first it is advisable to play a phrase, pause the audio, and then repeat it.  Once the student is able to pronounce all of the words easily, then he should go back and try the exercise again, but this time he should try to read the text at the same time as the speaker. It is an extremely challenging but powerful exercise.

A final step is to for the student to memorize a few sentences or a paragraph and recite them without looking at the text, repeating this exercise until he can recite the text at a normal conversational rate.

Embrace errors

The more errors a student makes, the faster he will learn. This counterintuitive phenomenon occurs because the best way to commit something to memory is to try to recall it.  In other words, the best way to practice is to test oneself constantly.  However, it takes many repetitions to remember new information and how and when to apply it correctly, especially when the learner is embarking on a new area of study.  Therefore maximizing repeitions, which should be a primary goal of instruction, naturally leads to many errors.

There is, of course, another condition that needs to be met to make this practice as efficient as possible, and that is that the student needs to receive feedback about his errors, including help to analyze them so that he understands why his mistakes were mistakes.  Those of you familiar with the ABeCeDarian Reading Program know how much I discuss the nature of good error correction and how important it is to the program.  I’ll have more to say in future blog posts about specific techniques for dealing with errors, both in foreign language instruction in particular and in other areas as well.

 

In my next post I will present a number of resources I have found invaluable in my own foreign language studies.

 

 

Two key components of good lessons

If you asked people what makes a good lesson good, I suspect most of them would identify the clarity with which the teacher explained the subject.  A good lesson, they would maintain, has a clear theme, appropriate background information, illuminating examples, precisely connected points.  I’m sure we have all suffered through enough lessons that didn’t have these features to appreciate just how much they contribute to our understanding of a topic.

I would argue, however, that this emphasis on clear presentation assumes that the general structure of an academic lesson involes a person explaining or demonstrating something to some students.   I think it is more useful, however, to think of lesson in broader terms, as a set of experiences that help students learn something.  When we expand our conception of a lesson in this way, I think we will find that explaining or demonstrating, while certainly one kind of experience students can have, is a rather limited and generally overused technique.

The problem with direct explanation or demonstration is that helping students understand something new cannot be accomplished simply by adding information to their brains.  The development of understanding requires a further step, namely, integrating the new information with what is already known.  Now,  a good lecture attempts to do this, but it can do so only in a rather crude way, making assumptions about what the person already knows and thinks. Moreover, the connections a lecturer makes in his lecture enter the mind of a student in a weakened form because the student aquires them, in a sense, too easily.  They are predigested and second-hand, and not hard won through the student’s own active exploration.

In this light, I would argue that one key component of a good lesson is selecting the right task or tasks for the student to do at the very beginning.  These tasks must be familiar to the student, or at least have a structure that provides her with all she needs to perform the task.  In addition, they should require the student's active participation and engagement.  The final challenge in developing suitable tasks is to carefully embed something new within the familiar elements.  The something new, of course, is the focus of the lesson.

Certainly, giving students experiences is, by itself, an inefficient and indirect way to develop a student’s understanding.  To allow the experience to yield new connections and new understanding the teacher needs to include the second key component of a good lesson, and this is to ask the student questions to help her analyze some experience. 

The questions I am referring to here are not those that ask a student to recall specific facts per se, but rather the kind that focus on identifying patterns and relationships.  These questions include, “What’s the same between this and that,” “What’s different,” “What patterns do you see” and “How is this connected to something you already know?"

For those of you already using ABeCeDarian, I encourage you to review lessons you have done to see this general structure: giving the students a task in conjunction with analysis of the task.  To assist you, let me point out three examples of this structure in ABeCeDarian lessons.

As I’ve discussed in an earlier blog post, the very first thing that a beginning student does is to spell the word “mop” as part of an activity called a Word Puzzle.  After the teacher states very briefly that the student is going to help spell the word “mop,” the teacher then asks, “What is the first sound you hear in the word ‘mop?’”  The activity includes a number of supports that help the student answer this question.  The lesson thus begins with the experience of hearing a familiar word “mop,” and the teacher prompts the student to analyze this experience by asking “What is the first sound?”

Here is another example, this one from Level B-1.  When a student has learned the one-letter consonants and vowels, she then begins to learn to read 1st grade level words with vowel digraphs such  as “boat,” and “rain”  (Digraphs are two-letter spellings used to represent a single sound, such as the “oa” in “boat,” and the “ai” in rain.)  The student's initial task at this level is to examine a list of words that all have the so-called “long-o” sound and sort the words depending upon how that sound is spelled.  Students begin the task by reading the words.  If they don’t know the word immediately, they sound it out, a task that they have done hundreds of times in the earlier level.  What is new is a spelling for the long-o sound.  Most can figure this out without help from the teacher.  But if they don’t, the teacher simply tells the student the sound.  That is the whole activity, reading and then sorting.  Again, the student begins with a task, reading the word, and then analyzes it by sorting, prompted by the teacher’s question, “How is the /oa/ sound spelled in this word?"

My final example comes from Level C of ABeCeDarian, a level suitable for students at a 3rd to 4th grade reading level. This level address word parts, namely, prefixes, suffixes, and root words. Many programs covering this material begin with a statement such as, “Words are made up of parts that we call prefixes, suffixes, and roots.”  In ABeCeDarian, in contrast, students are are introduced to the topic by reading the following list of words:  helpful, unhappy, landed, rebirth, starting, refill, landing, started, careful, and untruthful.  Then the teacher asks them to identify all of the syllables that appear more than once in the words.  As in the other examples, the lesson has the same basic structure:  the student is given a task that requires them to analyze and identify a key feature of words. 

As I have mentioned before, I am currently preparing a series of math lessons to add to the ABeCeDarian lineup.  These math lessons are also organized in the same way:  Students are given tasks and then, with the aid of precise questioning from the teacher, they analyze what they have done in order to develop new skills and to see new patterns and relationships.  I will be writing more about these new math materials in the coming months.

In the meantime, I hope you will reflect a bit about how close or far apart other lessons you have your students do are from the structure I have just described.  And if you are not happy with how your students are doing in those lessons, one area to examine is the extent to which the student performs some relevant task and then analyzes it with your help.

More tips to improve practice routines

Last week I talked about the importance of providing short practice sessions spread out over time (distributed practice and spaced repetition) and making sure that practice sessions involve a variety of the skills (interleaving practice).  Today I want to touch on a couple of other points to keep in mind when setting up practice.

When the content allows, it is extremely useful to present practice activities that reveal the underlying structure of what they are investigating. For example, a difficult part of learning how to spell the word “president,” is to know that the vowel in the second syllable is written with an “i.”  This is difficult to remember because the sound we say in the word in the second syllable is a little “uh," a vowel sound called a schwa that is common in the pronunciation of unaccented syllables.  Unfortunately, this sound can be spelled with any of the one-letter vowels.  It so happens, however, that there is a form of the word in which the second syllable is accented, namely, “preside,” in which we say the so-called “long sound” of “i.”  If the student’s practice routine involves connecting the word “president” to the word “preside,” he will be able to recover the underlying sound of the vowel and thus find it much easier to remember the spelling. Moreover, the practice will have reinforced some of the structural ways that words are related in English.  This same relationships can be used to help remember the spelling of hundreds of words.

Practicing letter/sound relationships as students do in phonics programs likewise helps them connect the reading and spelling of  particular words with other words and reinforces a central part of the structure of our writing system.

The study of elementary math provides many opportunities for enhancing practice by highlighting critical structural relationships, although most curriculum materials do not take advantage of these opportunities.  When learning how to do calculations with decimals, for instance, it is extremely useful to investigate the same calculation when using common fractions.  Even something like memorizing the multiplication facts can be made much easier by practice that stresses relationships.  For instance, when I am reviewing multiplication facts with students and they have difficulty recalling a fact, I ask them to connect it to something that they know.  If a student forgets the product of 6 x 6 but knows the product of 5 x 6, he can talk himself through the problem by adding 6 to the product of 5 x 6.  Another option is to draw an array on a grid board that corresponds to the multiplication problem.  When trying to figure out 6 x 6, for example, this means drawing a 6 by 6 rectangle.  The student can then divide that shape into 2 or 3 parts and then calculate the product by adding the parts.  Of course, the ultimate goal of this practice is for the student to have rapid and accurate recall of the facts.  But by emphasizing the way every fact is part of a broad network of relationships, you will accelerate how quickly your students achieve fluency and, at the same time, help them develop a sense that the various things they are learning in math aren’t random and isolated but tightly interconnected.

Of course, not everything we want to learn has the kind of structure we find in our writing system or in mathematics.  Some things, such as learning the order of the colors of the rainbow, the order of the planets in the solar system, or the capitals of the states, are associated in arbitrary ways.  Many of us have learned such things with brute-force memorization, simply repeating the order of the items and their connections over and over.  This technique often works eventually, but in most cases, there is a faster and easier method.

Since antiquity there have been people interested in developing systems to help with memorization. What these memory system have in common is that they provide ways to impose a structure on abstract, random information by associating it with something you already know.  As Harry Lorayne and Jerry Lucas write in their classic, The Memory Book, “In order to remember any new piece of information, it must be associated to something you already know."

This is the idea behind mnemonics such as the sentence “Every good boy deserves fudge,” to learn the order of notes represented by the lines on a scale.  This mnemonic works because it is much easier to remember common, familiar words organized in a simple sentence than to remember the order egbdf.  Memory trainers such as Lorayne and Lucas have developed flexible systems that utilize the same underlying principle and that can be applied to memorizing any content. The key to most of these successful memory systems involves developing ridiculous and wild images that connect objects.  In the Lorayne and Lucas book, they offer the example of trying to learn a random lists of 10 objects.  If the first 2 objects were airplane and tree, the person should make a crazy image in which the two are together, such as a tree that has small airplanes for leaves, or an airplane that has a variety of trees as passengers.  There are a variety of elaborations or extensions of this basic technique, but all successful memory systems rest on this fundamental strategy.

A number of years ago I had the opportunity to travel to China and wanted to try to study the language a little bit before the trip.  As you may know, it is quite a challenge to learn to read Chinese characters, because it is necessary to memorize thousands of them in order to attain even the most basic level of literacy.  In the course of my studies I found a really interesting book to learn the 800 most basic characters.  The technique used in each book was to break each complex character into parts, and have each part represented by a visual image.  They then constructed a little story that connected these images with additional clues about the words meaning, its pronunciation, and its tone.  Thus, for each character, one had an easy to visualize story that connected all the essential elements needed to read the word.  I found this approach very useful and much easier than the brute-force approach I had tried previously.

The Chinese book gave me the idea to try to adopt this technique to prepare an English spelling book, and thus I developed the ABeCeDarian Spelling Book B-1.  This book addresses first grade words such as “boat” and “rain.”  The first skill a person needs to spell well is an ability to break a word into individual speech sounds.  To spell a word such as “mop,” requires a person to break the word into its three sounds and then remember how each of those sounds is spelled.  This a a relatively easy task because there is not much ambiguity about how to spell each of these sounds.  But a word like “rain” offers an additional challenge because there are several common ways to spell its vowel sound.  Is it “rain,” or “rane,” or perhaps “ran” or “rayn”?  To overcome this confusion, the ABeCeDarian spelling book presents all new words within sentences that have silly clues about the correct spelling of any ambiguous part.  Students learn, for instance, that the word “ape” is used to signal the letter “a,” and the the word “I” to signal the letter “i.”  When learning to spell “rain,” they then read, analyze and copy the sentence, “The ape and I sit in the rain.”  When they practice initially, they don’t write the word in isolation but the entire sentence with the clue words to associate the spelling "ai" (the ape and I) with the word "rain." 

I'll try to make some short videos demonstrating some of these techniques.  It's much easier (and more compelling) to demonstrate them than to write about them.  And in a future blog post, I'll show you a technique for memorizing the first 20 digits of pi.

So, here are the key points for today.  When the content you are investigating with your student has structure, make sure that the practice activities of your students help them organize new information within a web of central relationships and connections.  And when the content that your students are trying to master has a more arbitrary relationship, help your students use the techniques of memory experts and impose a structure on the material with silly, and thus vivid, associations.

Two tips to improve practice activities

We are all familiar with the expression, “Practice makes perfect.”  Almost all teachers take this aphorism to heart and rightly include practice activities as part of their lessons.  But just including practice isn’t enough.  How the practice is presented makes a big difference in how quickly students will master new material.  Here are a few tips that can help you make practice activities as effective as possible.

 Distribute practice

It is common for teachers to have students practice a new skill many times just after they have been introduced to it.  For instance, I remember from my school days having to write spelling words five times, one right after the other.  I’m sure you have seen students do 10 or 20 or even 30 addition calculations involving carrying just after they have had a lesson on this procedure. This kind of massed practice can feel very rewarding because at the end of the effort, most students perform the task successfully, often relatively rapidly.  This success, however, is deceptive. 

 Every waking moment our minds are bombarded with an enormous amount of sensory information.  If you walked into the room where I am working now, you would see, for example, that I am wearing a blue sweatshirt, and if I asked you, you could tell me what I was wearing.  That information would reside in your working memory, the memory we use to engage our immediate environment, where we have the thoughts we are thinking about at the moment. But if I asked you weeks, or months, or years from now, “What was I wearing on September 14, 2016?” (and you didn’t have access to this blog!) it would be extremely unlikely that you would remember.  The information would long have vanished from your working memory without being stored in long-term memory.  This same forgetting, in fact, happens to most of the thousands of experiences we have each day.

But of course, even though we forget much, we also are able to remember many things.  What mental operations do we perform on the things that we remember that are different from the things that we don’t remember?  At heart, it is that we try to remember them!  When something first enters the mind, it starts to fade from memory fairly quickly.  But when you try to retrieve this information from your memory, the effort strengthens the ability of your mind to remember it in the future.  This system works rather well, in that the fact of trying to remember something is a good indication that it is important and worth being remembered.

 The deceptive limitation of massed practice is that the new information or skill is practiced only in working memory and isn’t recalled from long-term memory, which is our ultimate goal.  While massed practice may have a valuable role at the very beginning of learning something new, forming an enduring memory requires another very important step:  trying to remember the new information once some time has elapsed.

And the best time to try to remember the new information is the moment when we are about to forget it.  When we are first learning something, that moment is quite close to the time we first processed the information, perhaps seconds, certainly no more than minutes in most cases. But as we successfully remember something, the interval necessary to keep the memory accessible increases quite dramatically, until we are able to retain the information for years.  So the most efficient way to commit something to memory is to try to recall it across many intervals of time, ideally, intervals that increase in length.  Because this ideal practice is spread out over time, psychologists generally refer to it as "distributed practice."

There is quite about of information about the ideal intervals needed to remember material well, much of it referred to as “spaced repetition.”  I encourage you to investigate this technique, and if you are not familiar with computer flash cards programs such as Anki that use spaced repetition, I recommend that you learn more about them.  They are an invaluable learning tool

Interleave practice

I’m sure that you have seen practice sessions in which students had to 10 or 20 or 30 problems of an identical type, such as adding 2-digit numbers with carrying, or writing plurals for words that end in “y.”  As I mentioned above, with any massed practice like this, the students usually perform the task at a high level at the end of the practice session, and so they and their teachers have a sense of accomplishment.

If the ultimate goal werre, for example, to have a person be able to solve problems when they are told in advance, “All the calculations are sums with carrying,” or “All the words you have to spell involve the pattern of pluralizing a word ending in ‘y,’" then the practice routines I just described might very well be satisfactory.  Such a limited goal, however, is almost never what we have in mind.  We want people to be able to do all sorts of arithmetic calculations depending upon the situation, and to be able to apply a variety of spelling patterns correctly whenever they write.  In other words, the ultimate application of the skill involves a great deal of judgment about what is happening in a particular situation.  When very narrow skills are practiced without having to make judgments about the environment,  however, students continue to struggle using their new skills appropriately.

There was a fascinating study on this topic done with baseball team at California Polytechnic State University.  It is typical for a players batting practice to consist of 45 pitches, 15 fastballs in a row, 15 curveballs in a row, and 15 changeups in a row.  At the end of each practice session for a particular pitch, the batter would have “timed up” the pitch and would usually be able to hit the ball impressively.   In the Cal Poly study, though an experimental group still received 45 pitches of batting practice, but the pitches were presented in random order.  In other words, the batter did not know which kind of pitch he was going to receive.   As you can imagine, the immediate performance of the batters in the experimental group suffered because they were not only having to hit a fastball, but having to discriminate whether a particular pitch was a fastball, a curveball, or a change-up.  However, their batting during the game improved significantly!   I think it is easy to see why.  What they really had to do in the game wasn’t just hit a fastball, but to recognize a pitch as a fastball and then hit it.  That is to say, there were two things involved in doing the task well, a judgment and a performance.  When the judgment was omitted from the practice routine, they weren't practicing the complete skill they needed during a game.

So, as you structure practice routines, you will dramatically improve your students' learning if you interleave different types of items in a practice session.  For example, it is much better for students to have practice involving, at the very least, addition and subtraction, and not just one of these operations.  This not only helps to distribute practice of material learned earlier, but it forces them to pay attention to the relevant details of the situation (such as what the operation sign is in the equation!).  If you want examples of excellent interleaving, take a look at the math textbooks by Harold Jacobs.  He wrote most of his books in the 1970’s, before cognitive scientists had done much work on “interleaving,” but he knew nonetheless what a powerful tool it was.

So the lessons for today:  to make your practice as effective as possible, make sure that you spread it out over time and that you interleave various kinds of tasks in each practice session.

I'll have some more to say about improving practice routines in my next blog post.  See you then!

What's in a name?

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.

"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!