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Earlier this year I was in a meeting in which a reading specialist in a primary school stated that the word through, “couldn’t be sounded out in a million years.”
This statement is very interesting. The idea the teacher was expressing so colorfully, namely, that attention to letter/sound relationships is of little or no use for learning how to read certain words, is a view shared widely not only by teachers who are skeptical of the use of phonics generally, but also by many enthusiastic proponents of systematic phonics instruction. Unfortunately, this view, though quite widespread, is demonstrably wrong. Examining why it is wrong it provides an excellent opportunity to highlight a prevalent misconception about the role of phonics in good reading instruction.
First, to the demonstrably false parts of the statement. The word through is composed of 3 isolated speech sounds, technically referred to as phonemes. These phonemes are /th/ /r/ /oo/. The relationship between the sounds /th/ and /r/ and how they are represented in this word, namely with the letters th and r, respectively, is quite regular. In contrast, the way the /oo/ sound is spelled in this word, namely with the letter string ough, is unusual. Indeed, it is the only common word in which the sound /oo/ is spelled this way.
However, just because a sound in a word is represented in an unusual way does not make the word impossible to sound out. It is a simple matter to say, “We spell the /oo/ sound in this word ough.” In ABeCeDarian, in fact, this word is initially presented in Unit 8 of Level B1 in an activity in which students sort words according to how the /oo/ sound is spelled in them. As you can see the spelling units (often referred to as graphemes) on this list of /oo/ words to sort is presented with space between each grapheme. For instance, the word food is displayed as f oo d. The word through, therefore, is presented as th r ough.
The student already knows the regular letter/sound correspondences th and r and there is only one other grapheme in the word. The student, moreover, has been told that all the words on the list have the /oo/ sound. With the help of the spacing between between graphemes and the knowledge of the common sound in these words, most students who have advanced to this level of the program have no difficult sounding out all of the word correctly. If a student does, however, get stuck on what sound to try for the ough, the teacher points to it and says simply, “In this word, that’s /oo/. You say /oo./“ Armed with this knowledge, the student then sounds out the word. It’s really that simple, no more difficult, indeed, than sounding out its more regularly spelled homonym, threw.
If it is really so easy to sound out a word with a highly irregular spelling such as through, then why do so many teachers think that it is impossible to do so? The heart of the error, I think, lies in a view that understands the utility of phonics in an overly narrow way. According to this view, learning basic letter/sound relationships provides a set of conversion rules that allows a person to translate strings letters into words. For example, the conversion rules specify that the letter m represents the sound /m/, the letter o represents the sound /o/, and the letter p represents the sound /p/. Thus, the student can figure out that the word mop is /m/ /o/ /p/, /mop/.
These conversion rules, however, are useful only to the extent that they are relatively simple and straightforward to apply. Or so the conventional wisdom goes. But as we all know, in English, there is usually not a simple relationship between letters and sounds. Letters and letter combinations can represent many different sounds, and when there are too many variations, trying out each variation when reading a word makes the conversion too cumbersome. The string ough, is an extreme example of this variability, representing different sounds in each of these 7 words: though, through, rough, cough, bought, plough, hiccough.
Because of this variability many people argue that words with highly irregular spellings must be taught in a different way, without recourse to letter/sounds. Thus, in many reading programs, the words with irregular spellings words grouped together, often often labeled with an slightly accusatory designation, such as “red words,” “outlaw words,” “rule-breakers” or “words that don’t play fair”. Some teachers don’t analyze these words at all and have students practice reading them as unanalyzed wholes or by embedding them in sentences in which, supposedly, context will allow them to be read properly. Other teachers do have students analyze the words by having them repeat the letter names and write the words saying the letter names in various exercises. And of course, some teachers do a little bit of all of these strategies.
To see the flaw in these approaches, it is necessary to tease apart some related but distinct goals of reading instruction. The act of translating print into speech is usually referred to, quite aptly, as decoding. When a person looks at the word “mop,” and says /mop/, for example, she has decoded the word.
There are two kinds of decoding. One kind is used for translating into speech a word that a reader is unfamiliar with. This is the type of decoding a person uses when she tries to read a word she has never seen before, such as the nonsense word fluntercumicality. Even experienced and highly adept readers will read this word relatively slowly and have to approach it by translating individual letters, or, in some cases, perhaps small groups of letters, into sound and then combining those sounds in the proper order. This type of decoding is often referred to as sounding out a word.
But as readers acquire increasing skill, they also acquire the ability to translate printed words into speech almost instantly, without overtly converting individual letters or sub-groups of letters into sounds first. The acquisition of this automatic word recognition for a large number of words is, of course, essential to the development of the ability to read well. Reading would be of little utility if a person had to sound out most of the words on a page.
Even for students who learn to read rather quickly and easily, however, the ability to read a particular word automatically generally requires that the student be exposed to the word at least several times.
As I’ve noted, many teachers maintain that teaching letter/sound correspondences to students might very well help them decode the word hand if they had never seen it before, but it will not help them read words such as through if they had never seen it before. This observation is probably true, especially in the case of very beginning readers who have very little experience reading.
But that is hardly the end of the story. In addition to being able to sound out unfamiliar words, as I just noted, students also need to be able to instantly recognize many common words. The word through, for instance, is a word we would want second graders to be able to read easily in text. Indeed, a large number of the most common words in English have somewhat irregular spellings.
Given that most students need repeated exposures to words, we are now confronted with an important pedagogical question: What sort of presentation and practice allow the student to instantly recognize irregularly spelled words like through, with the least number of repetitions.
The answer, perhaps surprising to many, involves showing the student quite clearly and explicitly how the letters in the word represent the sounds of the word. In our example of the word through this kind of presentation means showing students quite explicitly and clearly that the graphemes th, r and ough represent the sounds /th/, /r/ and /oo/.
While it might seem counterintuitive at first that a “phonological” presentation is key to developing rapid, automatic word recognition, It’s not difficult to show why this approach is more efficient than others.
As I mentioned earlier, one common “non-phonoogical” option is to read the word frequently in context. Reading the word frequently and in many different contexts will certainly help a student learn to read the word, but if the student doesn’t get help analyzing the letter patterns in the word, she will make many mistakes before learning the word correctly. The semantic information in a sentence, such as, “The children walked through the park” doesn’t provide sufficient information to narrow down what the word is. The words in, into, around, by, toward, from, and others could replace the word through in this sentence and preserve a grammatical sentence. Moreover, such practice leaves the student guessing at what the connection between the letters and the word are, and many students will not be able to analyze it without some explicit and clear assistance.
One can even better appreciate the confusing diffuseness of the information presented to the student with this approach when one asks, “How should the teacher correct a student who makes an error reading the word through in this sentence?” As I just noted, there is not enough semantic information in the sentence to determine that the word is indeed through and not some other similar preposition. If the teacher starts pointing out letters within the word to help the student understand why this word is through, and not into, or around, then one has to ask why not make sure the student is clear about all of the letter/sounds in the word.
This criticism of using context to help develop decoding skill is not to say that semantic information is irrelevant to decoding. Students should be told constantly that what they read should make sense. Good readers certainly do and should use this guideline to monitor whether they have decoded words accurately. But the role of the semantics of the sentence and other information related to the meaning of the text is properly information that a reader uses to monitor the accuracy of her decoding, but it does not serve well as the principal information used to practice, develop, and improve a student’s decoding.
Another option is to present the word in isolation and drill it along with other irregularly spelled so-called “sight words.” Because the student has to decode the word without any information from context, this approach probably does help focus the students attention a bit more on the letters in the word. But again, why should the student be left to analyze the word herself, when the letter/sound patterns can be shown to her and then rehearsed explicitly. Correcting any errors is very cumbersome without explaining clearly how the letters in a particular word represent its sounds.
The remaining option for teaching the word without reference to letter/sounds is to have the student spell the word out loud and then say the whole word: tee, aitch, ar, o, yu, gee, aitch, /throo/. This is the approach of some explicit phonics programs such as Orton-Gillingham, which adds motor gestures to the performance in the belief that this aids memory.
Of all the alternate methods of teaching irregularly spelled words, this is the best because it focuses the student on letters. But the focus is still too broad. It calls the student’s attention to individual letters rather than functional groups, and as a result, it doesn’t help to clarify how our spelling system works, nor does it help students make connections among words with similar patterns. The complexity of decoding through, resides primarily in the facts that ough, not only represents /oo/ exclusively in this word, but also is used in other words to represent so many different combinatins of sounds in other words. Nevertheless, this spelling does bear some connection to the spelling of other words. Specifically, the ou in through represents a similar sound to the ou in several other words, such as soup, group, route, you, and youth, and having a “mute” gh is a feature of many common words, such as sight, eight, and dough).
In short, words such as through do not have to be learned as discrete and isolated wholes, but can still be integrated with the patterns that appear in other words. Only when a student is helped to analyze explicitly how the letters in words represent sounds do these patterns and relationships become clear. And it is this understanding of the patterns and relationships which will generalize and help students read not only the single word through correctly, but make it easier to learn other words with ou and gh. The approaches using context only, or isolated, whole word practice or oral spelling don’t readily promote this kind of generalization.
These connections exist among the thousand or so most common words in English, that is to say, the words we want students to learn to recognize automatically in primary school. But what about less frequent words with very unusual spellings, such as colonel and yacht. Is it really possible to help people learn to read these words by pointing out letter/sound relationships?
I would argue with a resounding, “Yes!” Here, the description will often involve more than just the explication of letter/sounds, but delve into even deeper relations, regarding the etymology of the word and how the word was pronounced at various times in various different languages. For example, the word colonel, comes from the Italian colonnello, meaning a person who commanded a column of soldiers. The pronunciation changed in Spanish and Portuguese, with the l replaced with an r (coronel in both languages). The modified pronunciation was retained in English along with the older spelling. (A similar sort of change in which a changed pronunciation is matched with an earlier spelling, explains why the pronunciation of the word one doesn’t match its spelling.) This approach, which strives to explain letter/sound relationships clearly, not only helps students learnn to read the words easily, but exposes them to the richness of language and how words evolve and are connected across different languages..
I should add that it is very useful to pronounce words such as this to adopt a “spelling pronunciation” in order to help people learn to spell them. In other words, for most people, spelling performance is improved if the letters can be connected to individual sounds, thus ensuring that the person does not have to rely solely on visual memory to spell the word correctly.
Cognitive scientists who have developed neural networks to simulate reading have observed a similar pattern in these networks, namely, that information about letter/sound relationships, is discovered and utilized even for words with irregular spellings. For instance, the word give, seems to be an exception to the pattern of pronunciation found in dive, drive, jive, hive, thrive, alive, strive, but there are nonetheless words that have similar spellings yielding similar pronunciations, such as gift and river, as well as the dual pronunciations of live. So there is some good evidence from cognitive science that letter/sound knowledge can be utilized even for the decoding of words with irregular spellings.
Furthermore, we have ample evidence that good decoders differ from poor decoders primarily with regard to their letter/sound knowledge and ability to sound out unfamilar words. Therefore, it is important to make sure that all of students understand how letters represent sounds, and not rely on trying to read words by treating them as unanalyzed or poorly-analyzed wholes. This approach reduces guessing, and, generally reduces student confusion and stress.
So, it is indeed possible to sound out the word through and others like it that have irregular spellings. Moreover, if teachers fail to make the letter/sound relationships explicit in these words, they cheat their students of a valuable tool to acceleratte their reading growth.
Here is a simple game that is wonderful for building perseverance and problem-solving skills. There are many ways to play the game. The simplest, and the one show my students first, requires 7 counters. These can be snap cubes, paper clips, or pennies. Anything small, relatively uniform, and easy to pick up will work.
This game is best played with 2 players. The object of the game is to be the player who removes the last item from the pile. Players alternate turns and can remove one or two items each turn.
There is a winning strategy for the game, which most children will figure out after playing the game a number of times. Once they have figured out how to win the game with these original characteristics, change either the number of items that can be removed at a single time, or the total of number of items in the original pile, and play repeatedly until the student figures out the winning strategy for this configuration.
When a student has figured out a winning strategy, help her to put the strategy in words and write it down. You can present the task by saying, “If you wanted to teach a friend how to win at this game, what would you tell her?” As I’ve mentioned in other blog posts, this sort of writing is extremely complex and demanding, so I would recommend that you as the adult write down what it is the student says. Take the student’s original words and make suggestions about how they could be more precise by pointing out any expressions that are unclear or ambiguous. Once you’ve got a precise statement, copy it onto paper, so you can compare it with the winning strategies of other variations (i.e., when you are starting with a different number of blocks or are allowed to take away a different number of blocks).
Once students have played a number of variations using counters, try the variation here in which players advance on a “ladder,” and the winner is the person who reaches the top rung first. You can find another version of the game, this one using playing cards, in Math for Smarty Pants by Marilyn Burns, p. 64.
If you are working with students in middle school or beyond, challenge them to come up with a winning strategy that would apply to any configuration.
This game, as well as the Factor Game, which I wrote about in my previous blog post, are both very engaging and can help students to think analytically. But even more than that, they both nicely model an excellent structure for presenting almost all new material in math, certainly up to and including beginning algebra. In both games I’ve described, students begin by doing a certain activity several times. After they have had some experience with the activity and have a beginning sense of how the activity works, they are asked to analyze it, that is, to figure out in much more detail, exactly how it works, which means, understanding the fundamental relationships at play. The activity is governed by rules that make it clear whether a suggested solution is adequate or not. The student’s analysis will involve some trial and error and will benefit from some way of recording their observations. The teacher’s role throughout this process is primarily to serve as a coach who can help the student record her work, break it into smaller parts, and help her evaluate the adequacy of her solutions. Although previous knowledge is useful, if not essential, to perform the task, the student’s job isn’t simply to parrot some previously learned information, but to apply it to a particular task.
In my next few blog posts, I will share you some math games that I use with students. Games are used in many classrooms by many teachers, but very often these games are designed primarily to make practice more palatable. There is certainly a place for games of this type, but I am especially interested in another type of game that involves not just a practice of some skill, but requires an analysis of strategy. For this reason they serve as a natural extension of the investigation and problem-solving that should be at the heart of most math lessons,
One of best-designed and most interesting games I’ve come across is called The Factor Game. It is perfect for 4th grade through 6th grade students who know the multiplication facts well. Here are the rules:
This is a game for 2 players. Player 1 selects a number between 1 and 30 inclusive and circles it. This number represents the score for Player 1 on her turn. Player 2 then circles all of the factors of Player 1’s number that have not been circled. The sum of all these factors is Player 2’s score.
Player 2 then circles a number and Player 1 circles all of the uncircled factors. Play continues with players alternating circling an initial number for the round and the factors of the initial number.
If a player who is setting an initial number for the round circles a number for which there are no remaining uncircled factors, that player receives a score of 0 for that round and the number she chose remains uncircled.
Play continues until there are no remaining legal moves. The player with the highest score wins.
Teachers and tutors should play the game by themselves at first and figure out the best first move, and then the best second move, if the first player makes the best first move. Then use what you have learned to figure out how to determine the best move at any given time during the game.
When you understand the winning strategy, you can then play the game with your student. Read the rules with the student and play a few games together, but on your turn, make more or less random moves, without using the optimal strategy. The student will need a little bit of time to experience the game in order to begin to understand the consequences of certain moves.
After the student has played the game 2 or 3 times, ask her to figure out what the best first move is and to prove it. The best first move, of course, is 29, because that is the greatest prime number on the scoreboard. If a player chooses 29 on her first move, then her opponent gets only 1 point, because 1 is the only remaining factor of 29.
If your student claims that a non-prime number as the best first move, tell her first that there is a better first score than the one she proposed, and then ask her, “What is the lowest score possible for Player 2 as a result of Player 1’s first move?”
If she selects a prime number greater number other than 29 as the best first move, ask her why she selected that number and why it is a good move. Then tell her that there is a prime number on the board that is greater than the one she picked.
I offer these teacher responses because It is important that the student figure out the best moves by virtue of her own analysis. Helping students become fluent at identifying the factors of a number is a useful by-product of the game, but the real purpose of it is to give students work at critical analysis and problem-solving. Therefore, your job as a teacher is to point out any flaws in her thinking (i.e., that there is a better move than the one she proposed) and to ask questions in order to help her clarify the important relationships and patterns she needs to understand in order to be able to come up with the optimal strategy. You should never just tell her the best strategy, or even the best move at a particular point in the game. That is always her job.
As you continue to analyze the game together, help your student put the optimal strategy in words. Over the last 30 years or so there has been a lot more emphasis in math instruction on having students explain their thinking. In general, this has been an important and welcome emphasis. However, in many lessons I’ve seen, teachers have students write down their thinking on their own with little guidance or support. Teachers always need to remember that expressing one’s ideas about mathematical patterns and relationships is quite difficult, and expressing oneself in writing is much more difficult than expressing oneself by speaking alone. Both are important, but they require years of practice, with significant support from teachers. So as you have the student put the strategy for the Factor Game into words, I suggest you do it collaboratively, with you writing down and rephrasing as necessary what the student says. You should also do it iteratively, that is, writing something down, evaluating its adequacy, revising, and then repeating the process until the strategy is both complete and precise.
I also recommend that at first, in order to keep the student’s attention focused on the game, you, the adult, should keep score. But once the student begins to understand the key strategic elements of the game, you should have her keep score. It’s a wonderful opportunity to encourage the use of and to provide excellent practice of mental math skills for doing multi-digit addition.
If you are interested in playing a non-competitive version of the game, you can simply ask the student to figure out the optimum play for each player for a whole game!
As I mentioned, in the upcoming blog posts I will share some other games I use with students. I hope you will share with me games that you have found to help students develop their analytical skills.
The Factor Game was originally published in "Prime Time: Factors and Multiples," Connected Mathematics Project, G. Lappan, J. Fey, W. Fitzgerald, S. Friel and E. Phillips, Dale Seymour Publications, (1996), pp. 1‑16.
An Examples of efficient practice: Providing interleaved practice with spaced repetition when practicing the addition and subtraction facts
In my last post I discussed in general terms the importance of interleaving topics during practice and of using spaced repetition to keep information readily retrievable. In this post I’ll give you some details about how practice of some particular math skills would look when organized according to these principles.
One aspect of math instruction that students, teachers, and parents spend considerable time on is developing automatic recall of the so-called “arithmetic facts.” Here is a good way to organize this practice of the addition and subtraction facts.
First of all, it is important to emphasize that efficient practice comes only after the student has had sufficient time to explore the relevant concept with various models. For basic addition facts, that means that students have had considerable experience combining quantities of objects and removing some objects from a quantity of objects and recording the results using the appropriate math symbols. Students also need extensive experience with a variety of materials to explore how a single quantity can be broken up in various ways. Doing activities in which students have to move on a number line is also very useful, as is using fingers readily to perform calculations.
After the student has developed an understanding of what addition and subtraction are through this experience, then it is important that she commit the basic facts to memory. This is not only to allow her to perform multi-digit calculations fluently, but it also helps to firm up her understanding of key number relationships that will allow her to grasp multiplication, division, and fractions more easily.
To begin recall practice, it is useful to start with a selection of just 3 facts. These facts should NOT be in the same family. I begin with 7 + 5, 9 + 7, and 4 + 4. I present these on a sheet in which the facts are repeated multiple times along with their “twins,” 5 + 7 and 7 + 9. I do not let the student write the answer, but rather just say the sum out loud. (Many students when writing answers on timed fact practice sheets as these will copy previous answers when a sum is repeated rather than attempt to recall it from memory.
I introduce the work by telling the student that she has made great progress in learning about addition and subtraction and is very good at figuring these out by counting. Now it is time to move to the next step and help her learn the basic sums so she can remember them automatically. I tell her that today she will be working on memorizing 3 addition facts.
I show the student the first 3 calculations and ask her to calculate the sums. After she correctly does a sum, I have her repeat the entire number sentence immediately (e.g., she should say, "seven plus five equals twelve.") After she has stated the three sums in this way, I tell her that the rest of sheet has only those 3 problems in mixed-up order. Then I ask her to continue with the rest of the sheet, saying the correct sum out loud.
If the student doesn’t say the correct sum in 3 seconds or less, I tell her the sum and have her repeat the whole number sentence before going on to the next one. I continue either until we have completed the whole sheet or she can do a couple of rows quickly and without any mistakes.
To provide an initial dose of spaced repetition, I have the student do some other work for 2 or 3 minutes and then do the sheet again. I stop once she can do several rows easily and then come back to the sheet at the end of the lesson.
Most students will quickly be able to provide the sums from memory by the time they are finished with a sheet that contains 36 sums. If they did still have difficulty recalling the sums automatically, I would repeat the sheet during the subsequent lesson.
When the student can recite the facts on the first sheet easily, I would then have her use the same routine to practice the 5 related subtraction facts, namely 12 - 5 = 7, 12 - 7 = 5, 16 - 9 = 7, 16 - 7 = 9, and 8 - 4 = 4. The procedure for this practice is the same, except that after presenting 18 problems containing only the subtraction equations, I mix or “interleave” them with the addition facts she had practiced in the previous lesson. I continue in subsequent lessons with this sheet until the student can easily recall all of the addition and subtraction facts on the sheet.
Once the student can do this sheet easily, I move on to the third sheet, which introduces three new addition facts. Again this sheet presents 18 sums to calculate containing only these 3 calculations, followed by a mixture of these calculations with the other calculations that the student has done.
Ideally the sheets should be practiced 3 or 4 times a day, with each practice sessions lasting just a couple of minutes.
Students find this method of practice quite motivating because by focusing on just 3 new facts each new lesson, they find that they can remember the new facts easily with just a little bit of concentrated practice. Moreover, as they develop a foundation of automatically recalled facts, they will find the future facts easier to learn because they can see more readily how they are related.
Many students, teachers, and parents are familiar with somewhat similar fluency practice in many classrooms and tutoring services under the guise of the “mad minute,” in which children fill out a sheet of calculations in a minute and are timed on them. Some students find the timed minute assessments intimidating. Fortunately, when working one-on-one with a student, it is not difficult to assess their fluency adequately enough without the pressure of formerly timing them. Because in this format the teacher provides the correct sum or difference if the student cannot provide it in about 3 seconds, all she has to do is count the number of times she provides the answer. If it is more than 10% of the time (4 or more times on a sheet with 36 problems), it is a good idea to review the sheet again before going on.
To summarize, the key elements of the practice are:
to introduce a small number of randomly selected facts to learn
to have the student recite the facts, rather than write them on paper
to provide the answer if the student isn’t able to come up with it in about 3 seconds
to interleave addition and subtraction
to interleave calculations the student has done previously
to have the student do 3 or 4 practice sessions of three to five minutes on the same material each day
Practice should be done using a spaced repetition schedule and it should be interleaved
In my last post I discussed the importance of developing fluent performance of basic skills. The next question to address is: What are the most efficient means of achieving this goal? Or, in other words, what are the characteristics of efficient practice?
In order to be able to recall new information easily, the best time to practice remembering it is when one is on the verge of forgetting it. Initially, the interval between remembering something and forgetting it is quite small. With children learning something like arithmetic facts, it can be just a few seconds. But once some new information can be recalled correctly after a very short delay, the amount of time until one is about to forget it increases. Moreover, the very act of trying to remember something is the primary means of strengthening the memory. Therefore, the most efficient way to commit something to memory is to try to recall it at increasing intervals.
There are various recommendations for the best interval schedules, but a rough rule of thumb is to double the amount of time of the previous interval after each successful attempt to recall the information, from seconds to minutes to hours to days to weeks to months. If at any point a person cannot recall the information, she should begin the process over again, returning to a very short recall interval. This routine of increasing the interval between successive correct attempts to recall the information is referred to as “spaced repetition.”
The general idea behind spaced repetitions is quite easy to understand. Our brains are inundated with an enormous amount of information every day, much of which is not very important. It would be quite daunting if we remembered all the irrelevant details of our day, such as what we wore or had for lunch or who we passed on the street every single day. It is important, therefore, for the brain to store only those memories that are important, and the way it does that is to make stronger connections with information that is repeated, that is, that appears again and again in our environment. Once a memory is reasonably well-established, it requires very infrequent review to remain relatively strong and easily accessible.
Learning new material well, though, involves not simply remembering it as an isolated, decontextualized fact. Rather, learning something well requires that a person apply the information in the right way and under the right circumstances. There is overwhelming evidence that the best way to achieve this goal involves doing a variety of different tasks during a different practice set instead of practicing just a single skill.. In a primary math class, for example, that might mean shunning review that focuses on calculations with just a single kind of calculation, such as a page with just addition calculations on it, and instead doing a randomly presented mixture of addition and subtraction calculations, and perhaps within each operation, a couple of different types of problems, such as those with just single digits and those with two-digits.
Unfortunately, teachers and students often avoid this sort of “interleaved” practice because in the short run, it is much harder, and therefore the student performs the task more slowly and with more errors than a practice session involving just a single skill. It is slower, of course, because the student has to make more judgments about the situation in order to recall the correct information. However, in the long run, a student’s ability to apply the new skill in a variety of appropriate contexts is greatly enhanced with interleaved practice.
The reason is that what really needs to be learned isn’t only HOW to do the task, but also judgment about WHEN to do the task. And when this judgment or discrimination step is a regular part of practice because it is interleaved or varied, learning is more robust and durable and it is more readily mimics how the person will apply the knowledge outside the classroom or practice session.
So, to make practice as efficient as possible, space the intervals in which the student is asked to recall the information in increasing intervals, and also make sure that practice time is spent with “interleaved” material, in which several related skills are practiced at the same time.
In my next post, I will give examples of how to apply these principles to prepare practice routines for helping students develop rapid performance of some particular math tasks.
Not long ago I worked with a very intelligent and charming high school student who needed help in his algebra course. I remember very vividly watching him work through one problem that required him to calculate 9 x 5, and he proceded to use his fingers to skip count by 5’s nine times. He was fairly dextrous in extending his fingers while doing this calculation, and he got the correct value for 9 x 5. Now, using one's fingers to do calculations is a very useful step for students learning beginning arithmetic. But it's use by a high school student in an algebra course put on clear display the primary source of his difficulties, namely, he had never advanced from counting on his fingers to memorizing the multiplication facts.
In the first blog posts of this series on unlocking your child’s math abilities, I emphasized the importance of grounding lessons in student exploration of counting both physical things and counting using drawn models. Such exploration, I believe, is necessary to develop genuine understanding of basic mathematical patterns and relationships, and this understanding is the basis for mastering rapid calculation procedures, applying math skills to solve problems, and enjoying and appreciating math.
While exploration of counting both using things and models ought to be the foundation of math lessons, it is, however, in and of itself neither sufficient to develop a student’s proficiency with basic calculations nor to develop the student’s skills in such a way that she can learn new concepts as quickly and efficiently as possible. If we want students to develop advanced skills as well as to enjoy math, we need to make sure as well that they develop fluent performance of basic math skills, and this means that students have to be able to recall basic relationships and procedures rapidly and with little effort.
First of all, rapid recall of basic number relationships helps students do many calculations easily. Even though we now have at our disposal electronic calculators, there are still many calculations, such as adding 16 and 5, or multiplying 16 by 5, that a person can do much more quickly in one’s head than by punching the numbers into a calculator, or even asking a digital assistant such as Alexa or Siri.
A more important reason, though, is that rapid recall of basic number relationships helps deepen a student’s understanding of math. It does so in two ways. First of all, the ability to see patterns requires seeing and investigating lots and lots of examples. If students have to do problems by counting on their fingers, they will not be able to do as many problems in the same amount of time as a student who can recall basic facts readily. Furthermore, doing a calculation on one’s fingers is much more demanding and tiring than retrieving a known fact from memory, and so a student’s ability to attend to what he is doing is diminished.
A related benefit of automatic recall of basic number relationships is that the student is able to recognize new number patterns more quickly. For example, when a student is learning about multiplication, the ability to rapidly add a one-digit number to a two-digit number will help the student become familiar with skip-counting patterns more readily, and this familiarity deepens the students understanding of just what multiplication is, and accelerates her acquisition of the multiplication facts.
Likewise, a student who knows the multiplication facts can do some fraction calculations using drawn models and discover that 1/2 of 2/3 = 2/6 and 3/4 of 2/5 = 6/20, and then readily see the relationships between the numerators and denominators on each side of the equal sign. Without knowing the multiplication facts, it takes much longer to see this relationship.
So there are clear cognitive benefits to help students develop fluent, automatic recall of basic number relationships:
It helps them do calculations rapidly
it increases the number of math experiences they can have a given time
it improves their ability to identify new patterns more readily.
In addition to these cognitive benefits, there is as well a very important affective or emotional benefit to helping students acquire rapid recall of basic number facts. Without this ability, doing any sort of math investigations will remain unnecessarily laborious. In general, people like to do things that are relatively familiar and easy, and would rather avoid things that are difficult, especially if the difficulty persists for months or years. So if a student, even a student who is otherwise quite successful in school, is still counting on her fingers to do a basic multiplication problem into 5th or beyond, she will be taking much longer to do the work than her fellow students who have memorized these patterns. Such struggles often lead students to conclude that they are “just not that good at math,” or don’t have “math brains” or some such thing. And they certainly contribute to the unnecessary scene I sketched at the beginning of today’s blog, the situation in which an otherwise successful and motivated 14-year old was struggling in his algebra class.
Now, few would disagree that automatic recall of basic number relationships is a good thing. So, why are there so many students who never develop this automatic recall? One culprit is essentially administrative: it is much easier to evaluate or grade the performance of students after a certain period of time than it is to make sure that a sufficient amount of time is provided for virtually all of them to acquire fluent performance.
Another important reason has to do with the inefficiency of most of the practice students receive. All students in schools get some kind of practice to memorize basic math facts, but much of this practice is extremely inefficient. So not only is it difficult to provide sufficient time for all students to develop fluent skills, the practice given usually does not use the readily available time very well.
So, what are the characteristics of efficient practice. That will be the topic in my next post.
When a person writes a computer program, she has to run small parts of the program at various points along her work to see if the code does what she intended it to do. Frequently the programmer finds a “bug,” that is an error of some sort. The programmer then has to “debug” her program by correcting the error. Usually this involves evaluating what the program actually did and comparing that with what the programmer intended the code to do, and then analyzing the discrepancy to identify what changes he needs to make. Unless a programmer is working on a trivially simple program, she expects that the process of writing a computer program will involve debugging. In other words, identifying and correcting “bugs” or errors is understood as a normal part of computer programming.
This attitude toward errors is a very productive and useful way to think about mistakes. In many math classes, however, there is a very different attitude: Mistakes are treated as failures and students are penalized for them in one way or another. If you ask students what the difference is between a good math student and a bad one, the chief criterion is usually whether or not the student makes errors.
One of the main reasons that errors are often treated in math class this way, is that, as I have discussed in earlier blog posts, the dominant method by which teachers present new material to students is to demonstrate a new procedure and have the students parrot it.
When a teacher exposes students to new material primarily by means of demonstration, she is circumscribed in how she can respond to a student error. She must focus on helping students memorize decontextualized steps that often don’t make much sense to the students. Thus in many classrooms students learn various mnemonics such as DMSB (for, divide, multiply, subtract, bring down, i.e., the steps involved in standard long division) or "keep-change-flip" a set of symbol manipulation procedures for dividing a number by a fraction (i.e., keep the first number unchanged, change the operation from division to multiplication, and flip, or use the reciprocal of the second number). Techniques for solving words problems are also frequently taught in a similarly mechanical way, with students taught to translate certain words into certain operations.
If a lesson has been presented in this manner and a student makes a mistake, when the teacher works with a student to correct the error, it is rarely about whether the student’s answer makes sense based on an understanding of basic number patterns, but whether the student has implemented the series of steps correctly or not. The consequence of this approach is that in the minds of many students, doing math means the memorization of steps that make little sense, and many students, even many who have been generally successful in school, graduate with weak math skills and great trepidation when they are required to do any math.
There is an alternative. It turns out that students can have experiences exploring fundamental patterns and relationships that lead them to understand how to use new techniques and perform new tasks. An essential part of such experiences and investigations is an analytical stage in which students try to explain new patterns and relationships that they are observing. As they analyze their new experiences and try to come up with general observations, they will inevitably make mistakes. These mistakes, it is important to stress, aren’t because of some failure or defect of the person’s mind. They are an essential part of how humans learn. That is, learning is a process that involves repeated doing followed by evaluation and modification. Learning most things, one might say, involves lots of little bites instead of one big gulp.
Thinking about student errors in this light gives teachers a powerful framework for understanding how to respond to them. The teacher should understand that her role in many cases isn’t simply to give the student the correct answer in response to an error, but to show her why her answer is incomplete or inaccurate so that she can figure out what the “bug” in her thinking was. This sort of response keeps the responsibility for “fixing the bug” with the student and ensures that she is always trying to make sense of the material she is investigating.
This approach also helps students develop strong self-correction skills because they are constantly in an environment in which they are testing the adequacy of their responses. This in turn helps build confidence because the student amasses countless experiences in which she had to continue to analyze some task until she herself comes to recognize and understand a new pattern.
Student engagement in math is also improved with this approach. One of the joys of doing mental work is, as it were, putting the pieces together oneself. The student has legitimate pride of ownership about her new understanding because it was the result of her own initiative and effort. Knowledge acquired in this way is much more satisfying than knowledge that is simply “given” to one by someone else.
I owe much of the previous thought to the ideas of Seymour Pappert in his classic book, Mindstorms. I highly recommend that book. I owe a debt as well to my training twenty years ago in Lindamood-Bell reading programs. Their mantra, “respond to the response” helped me understand the importance of precious error correction and to see it as an essential part of good curriculum design.
I know it is difficult to grasp the abstract points I'm making without examining specific interactions between teachers and students. I do have videos on the ABeCeDarian YouTube channel with regard to correcting oral reading errors, but I don’t yet have any for math. The general principles, however, are the same: the goal is to provide the student enough information that he can figure out what he has to adjust to perform a given task correctly. Over the next months, I’ll try to make some videos of good error correction in math.
Teach math symbolization as a method for recording actions
Many years ago my wife and I visited a number of schools when we were trying to decide where to send our son for first grade. One of the schools we visited was a prestigious private school in our area where the first grade class was just introducing the students to subtraction. The teacher made a brief presentation and then distributed a worksheet and some counters the students could use to do the calculations on the worksheet. In a few minutes students were coming up to the teacher to have their work reviewed, and, to the great surprise of the teacher, student after student had done the calculations incorrectly. After a few minutes of this, the teacher looked up at my wife and me and said plaintively, “Subtraction is killing us!”
What was going on? These kids were well-behaved and attentive. The teacher managed the group ably. The classroom was well-equipped and well-organized. The children had adequately developed counting skills and other background knowledge. And, for goodness sakes, they had physical counters to help them with their calculations!
This troubled lesson presents, I think, an example of an extremely common situation, namely focusing on math symbolization too soon, and failing to clearly show that it is a means of recording particular actions.
In this particular lesson, for example, the teacher wrote a few subtraction expressions on the board, such as 4 - 1 = , and then proceeded to say something to the effect of, “Today we are going to work with subtraction. Here is an example of a subtraction number sentence. To figure out the answer, I’m going to take 4 counters and then I’m going to take away 1 counter. How many do I have left? Yes, I have 3 left, so that is my answer. Four take away one is three.
In other words, the actions with the counters were presented as a way of solving a particular kind of calculation, rather than as an example of a common concept, “removal,” which we can record with certain math symbols. The distinction may seem subtle or overly nuanced at first, but it is extremely important.
To begin a lesson with the math symbolization the way this ill-fated first grade teacher did is to begin with the thing the students are likely least familiar with, so it is the hardest thing for the students to attend to and to put into any sort of familiar context. The steps of how to use the counters then become very abstract and decontextualized steps to memorize, rather than components of a sensible, logical and familiar experience. As a result, the steps become very hard to remember and execute in the correct sequence without quite a bit of practice.
A better approach when introducing the symbolization of subtraction is to start with an experience of counting some common objects, such as pencils or books, and removing some, and then counting the number than remained. As the students do these activities with the teacher, the teacher should then write down the equation, saying something such as, “We started with 4 things, so I’ll write a number 4. We took some things away. Here is a symbol we write to show we are taking away. We took away 1 thing, so I’ll write a number 1. Now what remains is 3, so I’ll write “equals or is 3. Now I’ll read my whole number sentence for what we just did: Four take away three is one.”
In this sequence, it is important to note, the symbols were presented AFTER the concept was exemplified with the manipulation of physical objects. In this way, it is clear that the symbolization is a code, a recording system. If a teacher starts a lesson with the symbolization first, however, the concept remains obscured and the role of the symbolization unclear.
After a few examples, the teacher should then ask the students to write down the number sentences that go with a few more examples she performs with the entire class, continuing until everyone can record these actions with the correct symbolization.
It is not difficult to show how all of the symbolization covered in the K-8 math curriculum can be presented in this manner and following this sequence, with the new symbolization presented only after what it represents has been shown.
There are several important lessons to draw here. First of all, just because the shelves in a classroom are groaning under the weight of math manipulatives doesn’t mean that math concepts are being introduced as actions on quantities. It is certainly possible that the various physical things to count are being presented primarily as tools for calculation, as in the example I shared at the beginning of this blog post.
Second, many math expressions, even ones related to very basic concepts, are used to record many related, but subtly distinct actions. For example, we can use an expressions such as 4 - 3 to calculate the remainder if we remove 3 objects from a set of 4 objects. But we can also use it to answer a questions such as, “If one person has 4 books and another has 3 books, how many more books does the first person have?” In this situation there is no removal as in the first example, but a comparison. Likewise, we can also use subtraction to think about the question, “If I need 4 chairs at the table and there are already 3 there, how many more chairs do I need?” This is also a type of comparison, but slightly different than the previous example.
If instruction starts with equations instead of actions on quantities, then these distinctions can remain hidden to the students and it will usually take some time to completely fathom. However, if one starts with a proper variety of common situations and then shows how they can be symbolized, the range of information that is generally packed into or associated with the symbols is much more transparent, and it makes it far easier for the student to apply her math knowledge to solve problems that come up in word problems and in, even more importantly, in day-to-day life.
In many math programs it is common to teach a new calculation procedure and then end the unit on this procedure with a variety of word problems. This approach, I think, has the sequence exactly backwards. The introduction of new calculations should START with word problems, that is, questions about the manipulation of quantities investigated initially with ordinary language. (I will have much more to say about word problems and their proper role in instruction in future posts.)
Doing so allows the teacher to start with the familiar and then introduce the new material tightly connected to the familiar. In this way the student is readily able to embed the new information within the network of associations she already has, rather than lingering in some isolated recess of her mind, disconnected from experiences she has outside of math class. Embedding this new information in an already existing set of associations improves both her retention of the new material as well as accelerates her ability to apply it correctly in various situations.
As I mentioned in the earlier blog posts, the concepts of arithmetic and basic geometry, when presented as actions, are not very difficult for children to understand. Their confusions and frustrations with math, therefore, usually are NOT due to any inability to grasp the underlying concepts being investigated, but because they do not adequately understand what the associated math symbolization represents. In short, the most likely challenges and confusions children will have with elementary and middle school math have to do with understanding the symbols, and these potential confusions can be almost completely avoided by following the simple precept presented here.
In the first two posts in this series, I’ve discussed some principles for organizing how to introduce new concepts to students. After students are introduced to new concepts, however, they have to learn how to do associated calculations, and they have to learn how to do these quickly. In the next few blog posts, I’ll talk a bit about how to help students become fluent with the calculation procedures they need to learn in K-8 mathematics.
This is the second post in a series addressing the topic, “What to Look for In Your Child’s Math Materials and Classes.” In the first post, I provided some preliminary discussion of the principle, "New concepts in arithmetic should be presented initially as manipulations of quantities and described in ordinary language.”
I suspect that it is relatively easy to think of some of the very elementary concepts of arithmetic, such as basic addition and subtraction, in terms of actions done to a quantity. Simple addition, for example, can be demonstrated by having a student count out a small number of objects, then count out another small number of objects, and then count the total number of objects in the combined groups. However, because the math instruction that most of us received didn’t focus on this sort of physical representation, it may be more difficult for many readers to understand the underlying action on quantity represented by more abstract concepts that we meet at a higher level of arithmetic.
For example, what is the underlying action on quantity involved in simplifying fractions? Here is one good way to explore this action. Count out six red cubes and three yellow cubes. (If you don’t have snap cubes or unifix blocks, you can use any counter, so long as they can be sorted according to readily identifiable characteristics, such as color.) Lay them out in a row as shown in the diagram below:
R R R R R R Y Y Y
What fractional part of the cubes is red if we are thinking about individual cubes only? Well, there are 6 out of 9 cubes that are red, so 6/9 are red. However, the cubes can be arranged into equal, uniformly colored stacks. Rearrange the cubes so that you have the greatest number of cubes in each stack possible if all the stacks have exactly the same number of cubes and each stack is composed of only one color of cube. It turns out that we can rearrange the stacks into 3 stacks of 3 as shown in the following diagram.
R R W
R R W
R R W
If we count stacks or columns now (instead of individual cubes) we can say that 2 out of the 3 stacks or 2/3 of the cubes are red. We have just simplified the fraction 6/9. This sort of task is easy for students to do, and the result of the manipulation, the calculation, is obvious given the arrangement and not something mysterious and arbitrary.
It turns out that all of K-8 mathematics can be presented in this way, and when one does so, the student is rarely confused. There are several reasons that such presentation provides so much clarity. First of all, Children have many experiences grouping, sorting, and counting objects in their lives outside theh classroom, and these experiences give them a substantial amount of math knowledge about basic math concepts. By focusing on the manipulation of things, a teacher allows students to connect new, more formal math ideas with their extensive informal math experiences.
Another reason this form of presentation works so well is that the basic patterns and relationships the student is learning can be recovered relatively easily if she forgets. All learners forget some new information, but if they have some sort of physical relationship or activity to refer to, they always have a ready means for recreating and recovering the concepts they have forgotten.
Finally, this method of presentation makes it easy to visualize key patterns, and visualization provides the foundation to develop more sophisticated and potent mental models as one’s understanding becomes deeper and more abstract. As we saw in the case of simplifying fractions, the physical representation makes the result of the calculation obvious and in some sense necessary, rather than something elusive and mysterious.
Those of you who have been in math classrooms recently or who have looked through education catalogs with math supplies, know that there is an enormous number of physical, manipulable materials in many math classrooms, especially at the primary school level. However, to present new math concepts as a manipulation of quantity requires more than merely having counters and models available. It is also critically important to use ordinary language at first to describe these manipulations. The reasons for doing so should sound familiar. Using ordinary language roots new information and new concepts in something that is familiar to the student, and, moreover, helps them to visualize the new patterns and relationships she is learning about.
Students, of course, do need to learn academic math vocabulary, such as “plus,” “minus,” “times,” “denominator,” “reciprocal,” etc. But they don’t have to learn these terms when they are first learning these concepts. For example, talking about “4 groups of 3” to second graders first learning about multiplication will lead to much quicker learning and retention than talking initially about “4 times 3.” (Indeed, the word “times” is especially diabolical for many students because it is a word they are very familiar with a completely different context.) Precise instruction initially uses familiar and functional words such as “groups of,” "put more on", "take some off" or “bottom number” rather than the more formal mathematical terms.
In my next post, I will talk about the principle, “Math symbolization should be taught as a means of recording actions.”