Math Games - The 31 Game

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.  

Math Games - The Factor Game

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:

How to Play The Factor Game


  • Factor Game sheet enclosed in a clear plastic protector sheet

  • dry erase pens

  • paper or dry erase board for keeping score

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.

Unlock your child's understanding of math, part 8

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

    Unlock your child's understanding of math, part 7

    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.

    Unlock your child's understanding of math, Part 6

    Help students develop fluent performance

    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.

    Until then,

    Happy Teaching!

    Michael Bend

    Unlock your child's understanding of math, Part 5

    Normalize error   

    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.

    Until next time,

    Happy Teaching!

    Michael Bend

    Unlock Your Child's Understanding and Enjoyment of Math, Part 4

    In order to learn calculation procedures, give students ample opportunity to explore and analyze key patterns and relationships

    Here’s a nifty trick.  Take a look at the following calculations:

    1/3 + 1/5 = 8/15

    1/2  + 1/13 =15/26

    For most people, addition calculations involving numbers with unlike denominators (bottom numbers) are somewhat difficult to do.  But calculations like the one above are actually quite easy to do in one’s head.  All you have to do is add the denominators to get the numerator of the sum and multiply the denominator together to get the denominator of the sum.  In other words:

    1/3 + 1/5 = 3 + 5 / 3 * 5

    1/2 + 1/13 = 2 + 13 / 2 *13

    With just a little practice, one can get good at doing these calculations mentally.  Try calculating these sums:

    1/10 + 1/9

    1/100 + 1/3

    (I’ve put the correct sums at the end of this blog post.)

    Learning this technique is quite useful, and, moreover, it definitely possesses some sort of whizz-bang appeal because it greatly simplifies an otherwise complex calculation.   For most people, however, I suspect that following the description of this technique was a bit difficult.  And that, in fact, is why I presented it, because I actually want to discuss its serious limitations in spite of its obvious attractions. 

    In particular, I want to point out that nowhere in the short-cut procedure I presented is there any need to understand the underlying number relationships that make this technique possible.  In this regard, my presentation of this technique was similar to lessons that still predominate in math classes throughout the world, whether the subject is adding multidigit numbers with carrying, or subtracting multidigit with borrowing, or long division, or dividing fractions, or solving rate problems or using the Pythagorean Theorem.  Teachers often present new calculation procedures by saying, “Follow these steps,” without allowing the students sufficient opportunity to explore in some depth the number patterns that make efficient calculation possible.

    When students learn calculation procedures without having sufficient time to explore fundamental concepts in depth, several problems arise.  One of the most significant is that students become needlessly confused and make numerous errors. I’ve already mentioned one example of this problem in my previous post when I spoke about the the problems of the first grade teacher introducing subtraction to her students.  There is a considerable literature on common and persistent errors of students, and skipping steps or not remembering sequences properly is extraordinarily common.  Anecdotally I can share that many of the students I see for math tutoring find the presentation of a procedure in school difficult to follow.

    Another common problem with this method of instruction is that students have limited opportunity to develop and use problem-solving and analytical skills during math class.  The lion’s share of their time is spent on memorizing and practicing calculation procedures.  As a result, students remain highly dependent upon their teachers and have difficulty making even simple judgments about whether a particular calculation they have made is correct.  They thus have less time to develop their number sense, and without adequately developed number sense, they have difficulty applying what bits and pieces of math they do know.  For many students, even when they learn to apply calculation procedures correctly, math remains a subject that frequently doesn’t make much sense to them, a set of rules and procedures whose internal logic remains opaque.  Indeed, many studentts learn to de-activate their sense-making skills in formal math classes because what they do day in and day out rarely makes much sense to them.

    Fortunately, there is another way to help students learn efficient calculation procedures, namely, to give them ample time to explore number patterns using calculation tools that they already have.  For example, to help students learn how to subtract multidigit numbers efficiently, it helps to have them use snap cubes to represent 2-digit quantities, with a “stick” of 10 cubes representing a ten, and individual cubes represent one.  Students can practice representing numbers with the cubes and then taking away quantities and recording these.  It's worthwhile doing such calculations using other models as well, such as number lines.  After students have had the opportunity to perform and record many such calculations, the teacher should then help the students analyze any patterns in their calculations and give them the challenge of figuring out how to do a subtraction calculation without using the snap cubes.

    When a teacher helps students acquire new calculation skills in this way, many good things happen.  First of all, the work gives students important experiences that deepen their understanding of important number patterns and relationships.  Adults often forget or are oblivious to the tremendous amount of experience with numbers it takes for a person to see important patterns.  Having good models helps, certainly, but there is no substitute for time exploring numbers with tools which one is already familiar and comfortable.  Additionally, this approach requires that students engage with the task actively looking for patterns and trying to solve problems based on the patterns they have observed.  Thus, they build up their problem-solving skills.

    Their view of math as a subject changes in the process as well, because their work is something they have to puzzle over and make sense of, rather than merely memorize.  Students who learn calculation techniques this way will also be less confused and better able to apply their new knowledge in different contexts.  This is not to say that students will not make errors.  But when they do make errors, the teacher will be able to help them identify the error with models and calculations that make sense to them and the error correction will be integrated into the overall sense-making quality of the investigations.

    So if you want your students to learn calculation techniques efficiently, use curriculum materials that give them lots of opportunities to explore and analyze the relevant number relationships, and keep your lessons more about making sense than about memorizing steps.

    In my next blog post I will talk about the role of error in lessons.

    Until then,

    Happy Teaching!



    Here are the answers to the beginning problems.

    1/10 + 1/9 =19/90

    1/100 + 1/3 = 103/300

    Unlock Your Child's Understanding and Enjoyment of Math, Part 3

    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.

    Happy Teaching!

    Michael Bend

    Unlock Your Child's Understanding and Enjoyment of Math - Part 2

    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.” 

    Until then,  Happy Teaching!


    Unlock Your Child's Understanding and Enjoyment of Math - Part 1

    In my own private tutoring I often work with students who are confused by their math lessons.  These students are in grades 1 through 8 and come from both public and private schools.  Some of these students have clear learning difficulties such as a language-processing problem like dyslexia, or attention-deficit disorder.  But many have no such difficulties, and indeed, have mastered much other academic material easily and so should be relatively easy to teach.

    This anecdotal and limited clue that there are some problems with math instruction generally is echoed by several broader measures, including a failure to improve 12th grade math scores over the last 20 or so years in spite of considerable national efforts to do so, persistent mediocre to poor performance of U.S. students on internationally administered math-tests, and the content of remedial math courses offered at colleges and the number of students who have to take these courses.  I might add, as well, that of all the school subjects, it is when helping their children with math that there is the most difficulty and confusion and, let’s be honest, tears.

    In order to help my struggling math students, I continually study the research literature regarding learning and cognition, and I also study and use a wide variety of specific math programs.  I have distilled what I’ve learned from this study, as well as from the successes and failures that I’ve had with my students, into 8 guiding principles for evaluating and constructing efficient and engaging math lessons.

    I’ve been thinking about these principles almost daily over the last couple of years as I have been writing the new ABeCeDarian Fractions Books.  As part of the launch of these new materials, I will be writing a series of blog posts to share these principles.  This blog is the first of this series. I hope you will find that these ideas give you a useful perspective for understanding any struggles your math students might encounter, as well as give you some tools for helping overcome their confusions or avoid unnecessary confusions in the first place.

    Principle # 1:  New concepts in arithmetic should be presented initially as manipulations of quantities and described in ordinary language

    Mathematics consists of abstract patterns and relationships.  The equation 2 + 3 = 5, for instance, doesn’t mean simply that if we combine 2 pencils with 3 other pencils we have 5 pencils, or if we have 2 dogs and 3 more come along, there are are dogs.  It means that if we combine 2 things of any type with 3 other things, we invariably have a total of 5 things. 

    One of the amazing and powerful aspects of mathematics is that we can represent these abstract patterns by symbols that can be manipulated without reference to physical things.  

    For instance, when people use the procedure for “borrowing” when doing pencil-and paper shortcuts to do calculation, they are following a procedure, a set of steps, that allows them to manipulate the symbols productively without paying attention to the quantities the symbols represent.  For instance, I will describe the steps of the standard procedure for calculating the difference, 63 - 24

    We can’t take 4 from 3 so we add 10 to the 3 and calculate 13 - 4 and write the result, 9 in the ones place. Then we look at the tens place, where we no longer have 6 tens but 5 tens because we’ve put one 10 with the 3 ones, so we calculate 5 - 2 and put 3 in the tens place.

    All of these steps, of course, are based on valid number relationships, but the number of steps and the abstractness with which they are expressed is, I hope you can appreciate, rather daunting, especially for a 7-year-old.  We know that this is the case, moreover, when we look at the common errors students make when they are taught this procedure and the length of time and amount of practice they need to stop making these errors.  An extremely common error, for example, is to come up with 41 as the difference between 63 and 24.  The error here, of course is that the student subtracts the 3 ones from the 4 ones, even though she should do the opposite.

    The relationship and sequence of these steps are much easier to understand for students if before confronting this procedure they have ample opportunity to perform this type of subtraction as actions on models such as snap cubes, base ten disks, and number lines.  Throughout exploration of this kind, the quantities involved are always at the fore, and the operation is simplified to 2 steps:  removing some cubes and then counting the remainder.  Not only are these steps simpler to follow, they are much more familiar and hence comprehensible to the student because it is very much related to some of their ample, non-school experiences.

    There is another crucial difference between focusing too soon on presenting calculation procedures on symbols versus allowing students ample time to explore a type of calculation by manipulating quantities, a crucial difference in the relationship of the student to the subject matter.  In the former type of instruction, the teacher demonstrates a procedure, and the students mimic it, while in the latter, the students are given tasks (count out a certain number of cubes, remove some cubes, etc.) and then, to reflect on their experiences.  In the one case, the emphasis is on remembering without much opportunity for sense-making, and in the other, the emphasis is precisely on doing things and then analyzing them,  looking for (and uncovering!) patterns and relationships.  I will be talking more about this crucial difference in the future blog posts as well.

    The difficulties I have outlined with regard to teaching the “borrowing” procedure, exist as well for most of the K-8 curriculum, including multi-digit multiplication, long division, and all the calculations regarding fractions, decimals, and percents. 

    I suspect it is not too difficult for readers to see the underlying action of subtraction, namely, the removal of a quantity.  Identifying the underlying manipulations of quantity at the heart of some more complex mathematical ideas in the K-8 math curriculum, however, may be less obvious or familiar.  For example, what manipulations of quantity are represented by the concept of simplifying fractions or the concept of dividing a number by a fraction?  I am not referring here to understanding a procedure to correctly calculate how to simplify a fraction or divide a number by a fraction, which most adults know how to do, but rather, understanding what specific manipulations of quantity these concepts involve, in the form of sorting, grouping, categorizing, and counting things.  I suspect that for many readers this task will be difficult.

    In my next blog post I will share some ways to represent the underlying actions of simplifying fractions and dividing fractions.  This is extremely important because not only these concepts, but indeed all of the content of K-8 mathematics can be represented in similar ways, and, if I'm right, this sort of representation is critical to teaching the ideas efficiently, with little or no student confusion.

    Until then,

    Happy Teaching!

    Michael Bend

    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 conscious 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!


    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:

    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.