How to avoid unhealthy competition: Using the Wall Chart.

Competition may develop to unhealthy levels

When students begin to pass levels in Rocket Math, and color in the Rocket Chart in their folders, they naturally are proud of their accomplishments. Students want to tell me what level they are “on” when I visit classrooms.   Unhealthy competition may develop among students sometimes.  Some students begin to feel really bad about their slower progress, and students in the lead act arrogantly or disrespectfully.  The Rocket Math Wall Chart is designed to curb that competition.

The Wall Chart puts all the students on the same team.


Over 700 star stickers come with the Wall Chart.  Each time a student passes a level the teacher awards them with a star sticker, which they take up to the Wall Chart and put into one of the squares in the chart.  Students fill the chart from the bottom up.  The teacher sets a goal in a few weeks, which date is marked on the goal arrow, and the goal arrow is placed a couple of rows up from where the students are now.  (You can just see that in the picture above.)


Students develop pride in their whole class.

If the students fill in the squares up to the arrow–before the date specified on the arrow–they earn a group reward such as extra recess time, or music during math, or a congratulatory note home, or a popcorn party, etc.  Wall chart half filledIn this way, each time a student passes a level they are putting up a score for the whole team.  It is good for everyone.  The teacher is able to praise the class for their hard work and accomplishments, and the whole class is able to feel good about their collective effort.

The Wall Chart shows visitors (like principals) how well the class is doing.

Passers-by as well as interested administrators can praise the class as a whole for their successes with Rocket Math.  In many schools, classes post their completed Rocket Chart on their door with all 725 stickers in place!   The Rocket Math Wall Chart becomes a focus of pride and recognition for the whole class.  The price for the Rocket Math Wall Chart (#2005) of $20 includes directions, plenty of star stickers, four goal arrows, and the chart itself.   They are cheaper by the dozen, $155 for all twelve.

How to Grade 1-Minute Math Fluency Practice Tests

Katy L from Wilson Elementary asks: How can I keep up with everyday Rocket Math grading? Do you teach students to grade their own 1-minute math fluency practice tests?

Dr. Don answers:

Only grade 1-minute math fluency practice tests if students pass

An integral part of the Rocket Math Worksheet Program is the 1-minute math fluency practice test. One-minute fluency practice tests are administered every day, to the whole class, and only after students practice in pairs for two to three minutes each. Check out the FAQs page to learn more about conducting 1-minute math fluency practice tests in class.

Teachers do NOT need to grade, score, or check daily Rocket Math 1-minute math fluency practice tests unless the student has met their goal. Students do NOT need to grade their own daily Rocket Math fact fluency tests either.

Why grading each math test is not important

The important part of math fluency practice is the oral practice with the partner before the test–what’s going on in this picture. Because the students are orally practicing every day and getting corrections from their partners, there should be VERY FEW errors on the 1-minute math fluency written tests.  

Correcting written tests doesn’t help students learn anyway. Corrections are only helpful if they are immediate, the student has to acknowledge the correct answer, and remember it for a few seconds–all of which is part of the oral correction procedure. “Correcting” what’s on the paper takes a lot of time and does not help students learn more, so it shouldn’t be done. But you have to check them before declaring that the student has passed a level.

How do you know if a student passes?

Students should have a packet of 6 sheets math fact fluency sheets at their level. Each Rocket Math student has an individual goal. For example, if a student has a goal of 32 (based on their Writing Speed Test) and they only do 31, they know they did not pass. If the student does 32 or more, they pass!

What to do when a student beats their goal (passes)

If a student meets or beats their goal, then have them stand up, take a bow, and then turn their folder into a place where you check to see that all problems were answered correctly. When YOU check (after school?), make sure all of the completed problems were correct and the student met their goal. If so, then you put the unused sheets in that packet back into the filing crate and re-fill the student’s folder with a packet of 6 worksheets at the next level and hand the folder back the next day.

When students receive the new packet of worksheets, they know to color in another letter on the Rocket Chart (and maybe put a star on the Wall Chart).

What to do if a student doesn’t pass?

Students who don’t meet their goals, don’t pass. These students should put the non-passing sheet into their backpacks and take the sheet home for more practice.

The next day they will use the next sheet in their packet of 6. If you want to give them points, do that the next day after they bring back their worksheet where they did a session at home (signature of helper should be there) and all items on the test are completed. If that’s done, they get full points.

Sometimes you’ll catch errors on sheets that students turn in as “passes.” If you see an error, the student doesn’t pass. As a result, the student keeps the old packet and has to continue with that same level worksheet.

For more information about conducting 1-minute math fluency practice tests in class and how to implement the Rocket Math worksheet program, visit the FAQs page.

The 100-point observation form: How well do you implement Rocket Math?

Use the 100-point observation form to evaluate your implementation.

Use the 100-point Rocket Math Observation form to self-evaluate, or have someone observe your class doing Rocket Math and use the form to evaluate you.  The form observes and evaluates seventeen different indicators of the quality of your Rocket Math implementation.

You or your observer begin by looking at four important indicators of the quality of student practice.  The quality of the paired practice of your students provides most of the value of Rocket Math.  Accordingly, these four indicators provide nearly half of the 100 points.  If one or two of these things are not in place (tutors aren’t listening carefully and correcting errors AND hesitations, for example) the implementation will not earn high marks, because students won’t be learning nearly as well as they should or could.

The other thirteen indicators are mostly about the efficiency with which Rocket Math runs.  If it takes more than 15 minutes a day to complete Rocket Math, it won’t happen every day.  If Rocket Math doesn’t happen every day, students do not learn nearly as well as they should or could.

Where can you find the 100-point Observation form?

There are three places you can find this handy form.  (1) It is included in hard copy form in the Administrator and Coach Handbook which we sell and ship to you.

(2) The 100-point Observation form is also available for free on the Resources/Educator’s Resource page on our website where you can find this link to its pdf.

(3) And finally in the Rocket Math subscription filing cabinet, in the Forms and Information drawer, there is a section (pictured on the left) that is devoted to all the information in the Administrator and Coach Handbook and near the bottom you’ll see the 100-point Observation form for you to print out.

Why is a gifted student having trouble with Rocket Math?

Question: Hi, Dr. Don! Just had a question recently from a parent of a gifted child whose son is having a lot of difficulty doing Rocket Math! He understands almost everything conceptually in math (in the 99% on national testing) but he is not being successful working with a partner on his math facts. Have you had this problem in other places? I’m not sure if the problem is he really can’t focus on the facts, he’s stubborn and doesn’t like details (big picture thinker), etc. He’s a very social kid so the partnering doesn’t seem to be the problem. I would greatly appreciate any suggestions you might have that I could give this mother. She says that he is fine at home doing his facts with her without a timer. But I don’t like the idea of excusing any student from doing this valuable practice. Thanks for your thoughts. Linda

Answer: I’ve blogged a bit on some of these issues elsewhere on the Rocket Math website, but let me try to be more specific here. First, gifted kids are stunned to find out that they have to work hard to memorize math facts. They probably need three or four days of practice—which to them seems like failure.  They are like an athletic kid who excels easily at every sport but finds he needs to work out with weights as much as a klutz to get to be able to lift heavy weights—his natural talent doesn’t help in this instance. So kids who’ve never had to work to learn things before, really are annoyed by having to practice several days in a row.  But it is really good for them!

How is mom practicing with him at home? Can she video him doing the test “untimed?”  If the child is “writing facts” and “without a timer” then he may be figuring out facts over and over—but is not getting to instant recall. That’s why the oral peer practice is so critical—if there is even a slight hesitation the child is to repeat the fact three times, back up three problems and come at it again—until the answer comes with no hesitation. There is a fundamental difference between instant recall of facts from memory and strategies to come to the answer by thinking it through. My parent letter addresses how to practice.  On the other hand, if the student is able to write the answers to math facts at a fast enough rate to complete 40 problems in a minute, but only when he thinks he is not being “timed” then he needs to learn how to do the same thing when he is being timed.

If he is not learning with the daily practice, we have to ask, “Why not?”  Social kids sometimes socialize instead of practicing. Social kids also can convince their partner not to do the correction procedure. Or they just say the answers instead of the whole problem and the answer. Any of those things would result in not successfully learning the facts. The teacher would need to monitor the quality of the practice. My experience has been that when students are “stuck” or “having difficulty” even just one session of practice done the right way rigorously (with me) and they suddenly improve enough to pass or to recognize they can pass the next day with another session of rigorous practice.

Last of all, sometimes the writing goals are off because of some glitch in how you gave the writing speed test.  So the student might know the facts well enough but not be able to write them fast enough to pass the tests.  If the student can answer 40 facts in a minute in the current set (just saying the answers without having to say the problems) then the facts are learned to automaticity—and the goal in writing should be lowered to whatever the student has done to this point.

Hope this helps. You are right not to excuse this student from learning math facts to automaticity. He might be a stellar mathematician someday if he learns his facts well enough that math computation is always easy for him. If math computation remains slow or laborious he won’t like it enough to pursue it as a career.

Without the directions you may get lost!

What happens when teachers don’t have a copy of the Rocket Math Teacher Directions?  Bad things!  

When teachers don’t have the written directions to Rocket Math, the essence of the program usually gets lost.  Procedures get modified and modified over the years until they are not even close to what should be occurring. Sometimes we have found schools that are not even providing daily oral practice.  Other schools don’t give the answer keys to the peer tutors.  Other schools don’t give the writing speed test and make up impossible-to-reach goals for students.  We often see teachers implementing the “Rocket Math” program incorrectly and wondering why it doesn’t work.  We ask them if they have read the teacher directions, and they say they didn’t know there were any.  When teachers have never seen the directions, is it any wonder they don’t know what they are supposed to be doing?  Hear-say directions handed down over the years from one teacher to another just don’t convey all the important details.  Teachers need the directions!

This is why I’d like you to have my complete directions for free. Even if you purchased Rocket Math ten years ago and haven’t gotten the updated versions since then, you can have these directions for free.  I have them in three places.  I have the directions broken out into FAQs on their own web page here.  That’s easy for quick reference.

The second place I have the Teacher Directions is as a downloadable booklet you can print out and distribute.  The Rocket Math Teacher Directions for the worksheet program booklet is here.   Please print this out and give to your teachers, especially in schools that began implementing several years back.  Read them and have a discussion at a professional development time.  You will be astounded at how much your implementation differs.

The third place I have the Teacher Directions is in the “filing cabinet on the web” for those of you who have the subscription. In the “Forms and Information” drawer we have the booklet and the FAQs which can be opened and printed out.

In school-wide implementations of Rocket Math, principals or math coaches need to take a leadership role.  The Administrator and Coach Handbook gives you forms with what to “look-for” in a Rocket Math implementation.  If you use that to observe Rocket Math in your classrooms you’ll quickly see whether or not things are going the way they should.   If you have a subscription to Rocket Math you’ll find all of the chapters of the Administrator and Coach Handbook in the “Forms and Information” drawer of our filing cabinet on the web.

Please take the time to see that you or your teachers are implementing Rocket Math according to the directions.  Trust me, it works SO MUCH BETTER if you do.  I wouldn’t steer you wrong!


Rush help to those who need it with an aimline

The sooner you provide extra help the easier it will be to catch them up.  

How can you know when students need help to meet expectations?  Use the graph above, which is available from the Educator’s Resources page or here: One Semester Aimline.  It is also available in the basic subscription site, Forms and Information Drawer as an optional form. It is an “aimline” for finishing an operation (Sets A-Z) in one semester.  Schools that don’t start Rocket Math in first grade need students to finish addition in the first semester of 2nd grade and subtraction in the second semester.  This means that students who get stuck on a level for even a week need to be helped.

If you indicate on this graph the week in which the student finishes each set in Rocket Math you can tell if the student is making enough progress, or if he/she needs to be getting extra practice sessions each day. If the student is working on a set above the line of gray boxes or on the line then progress is adequate–they are on track to finish the operation by the end of 18 weeks of the semester.  But if the student is working on a set that is below the line that means he/she needs intervention.

In the example above the student whose progress is shown in red is above the aimline.  That student has been passing at a rate that means he or she will finish the operation by completing Level Z by the end of the semester.  That student does not need any extra intervention.  In the example above the student in blue is falling behind.  By the fourth week that student has only passed Level C and so he needs to have extra help.

The first step would be to ensure this student has a good partner and is practicing the right way.  Sometimes students don’t stay on task or do not listen and correct their partner.  If hesitations are allowed (while the student figures out the answer) and not corrected the student will not improve.  Fix the practice in class first and see if the rate of passing improves and the student starts to get up to the aimline.

The second step is to include this student in a group of students who get a second practice session each day.  They would work in pairs and do another Rocket Math session each day.  Whether or not they take tests is unimportant.  What is important is that they do the oral practice with a partner who corrects their hesitations as well as their errors.  This could be done by a Title One teacher or assistant or a special education teacher or assistant.  It should only take ten minutes.

Another step is to involve parents if that’s possible.  Another practice session (or two) at home each evening would make a big difference.  Parents will need to know how to correct hesitations, but there’s a parent letter in the Forms and Information drawer for that.  Also note that siblings can do this practice as well, as long as they have an answer key.

You will be pleasantly surprised at how an extra few minutes a day of good quality practice can help students progress much faster at Rocket Math.  The sooner you intervene, the easier it will be for the student to catch up.

NOTE: There is an aimline for finishing one operation in a year.  It is also in the Forms and Information drawer and on the Educator’s Resources page of our website.  If you follow recommendations and do addition in first grade, subtraction in second, and multiplication in third you can use that aimline.  It won’t require intervening on so many students.



What about students who can’t pass in 6 tries?

A teacher writes:

Help! I’m feeling bogged down in Rocket Math. I have some students who have been working on the same sheet for over 10 times and are no closer to passing. What am I doing wrong?

Dr. Don answers:

The problem could be one of several things.  You have to diagnose what it could be.  I am assuming you have students practicing orally in pairs, with answer keys, for at least two minutes per partner every day (as shown in the picture above).  I am assuming you already have students, who do not pass, take home the sheet on which they didn’t pass and finish it as homework and practice with someone at home.  The extra practice session at home each day can be a big help and the students should be motivated to do that.   If this is the case and you still have a problem, below are two possible things that may be needed.

(#1) Need to improve practicing procedures.  Pick one of the students who is stuck and be that student’s partner while they practice orally.  Make sure they are saying the whole problem and the answer aloud so you can hear what they are saying.  Correct even any hesitations, not just errors.  Correct the student by saying the correct problem and answer, having them repeat the correct problem and the answer three times, then back up three problems and move forward again.

Diagnosis.  If, after practicing with you, the student does much better on the one minute timing and passes or nearly passes (this is what I usually found) then you know the problem is poor practicing procedures.  If your work with the student makes no difference (they don’t do better on the one-minute timing) and they seem equally slow on all the problems then it is not practicing procedures at fault.  Try #2

Solution:  Monitor your students closely during oral practice to see if they are all following the correct practice procedures.  If you have quite a few students who aren’t practicing well you may need to re-teach your class how to practice.  [Note: Even if they know how to do it but aren’t doing it right, treat it as if they just don’t know how to to do it correctly.]  Stop them and re-do the modeling of how to practice and how to correct for several days before allowing them to practice again.  If your students haven’t been practicing the right way, they won’t be passing frequently, and they will be unmotivated.  You have to get them practicing the right way so they can be successful and so they can be motivated by their success.

Solution:  If you have poor practicing with only a handful of students you might assign them to more responsible partners and explain to them that they need to practice correctly. During oral practice monitor them more carefully the next few days to be sure they are practicing better and passing more frequently.

(#2) Need to review test problems also.  The problems practiced around the outside are the recently introduced facts.  The problems inside the test box are an even mix of all the problems taught so far.  If there has been a break for a week or more, or if the student has been stuck for a couple of weeks, the student may have forgotten some of the facts from earlier and may need a review of the test problems.

Diagnosis.  Have the student practice orally on the test problems inside the box with you.  If the student hesitates on several of the problems that aren’t on the outside practice, then the student needs to review the test items.

Solution. If you have this problem with quite a few students (for example after Christmas break) then have the whole class do this solution.  For the next three or four days, after practicing around the outside, instead of taking the 1 minute test in writing, have students practice the test problems orally with each other.  Use the same procedures as during the practice—two minutes with answer keys for the test, saying the problem and the answer aloud, correction procedures for hesitations, correct by saying the problem and answer three times, then going back—then switch roles.   Do this for three or four days and then give the one-minute test.   Just about everyone should pass at that point.

Solution.  If you have this problem with a handful of students, find a time during the day for them to practice the test problems orally in pairs.  If the practice occurs before doing Rocket Math so much the better, but it will work if done after as well.  They should keep doing this until they pass a couple of levels within six days.

If neither the first or the second solutions seem to work, write to me again and I’ll give you some more ideas.

Are students really “friends?”

I hear teachers calling their students “friends” quite commonly these days.  While the use of the term “friends” is certainly harmless enough, it reminds me that there are extremely important distinctions between the way a person should treat friends and the way a teacher should treat students.  I don’t want to stop teachers from calling their students “friends” but I do think it is critical for teachers to know why and how they should not treat their students as friends.

The main reason that teachers should not treat students as friends concerns expectations.  With friends you’re nice to them and hope that makes them like you.  Then if they like you, they will be considerate of your feelings and treat you well.  Many beginning teachers expect that a classroom of students will be like a room full of friends.  If you are unfailingly nice to them, they will in turn be considerate of you and attempt to acquiesce to your wishes.  Unfortunately, this does not work.  Why?  Primarily because a teacher has to ask students to do things they’d rather not do and has to keep their attention on things to which they’d rather not pay attention.  In short, teachers are authority figures rather than friends.  Friends can get up and leave when they aren’t interested in what you’re doing, but students are required to stay.  Therefore teachers must treat students differently than they treat friends.

The first way that treating friends and students should be different concerns how a teacher reacts to student academic errors.  When a student answers a question incorrectly it shows they have a misunderstanding.  For example, a student says that the sun orbits around the earth.  That misunderstanding needs to be corrected to set the student “straight.”  A teacher who allows a student to continue with a misunderstanding is doing that student a disservice.  Errors should be corrected immediately, in a nice way, but as clearly as possible.  For example, the teacher says that although it appears as if the sun rotates around the earth, actually the earth orbits around the sun.  A good teacher may even take the opportunity to model how a spinning globe creates the illusion that the distant sun is going around us.  The student should be taught/told the correct understanding in as unequivocal a manner as possible and the teacher needs to check to be sure that the student learned the correct information both immediately after the correction and a few minutes later to see that the correct answer is retained.

When a friend makes a factual error, it is socially expected that you will not make a big deal of it.  It is socially inept to clearly and loudly correct errors of fact among friends.  At best one can simply not confirm an incorrect statement, but pointing it out as incorrect is just rude.  Teachers who treat their students as friends will make light of or gloss over errors, and they fail to teach students as a result.

Another way treating friends and students should be different concerns how a teacher reacts to student behavior.  Teachers need to learn to “catch ‘em being good.”  Teachers should look for students who are doing the right thing and should praise/recognize them by name, make eye contact and name the behavior they are doing that is exemplary.  “Alan has his desk clear, his textbook out and he’s ready to start learning.  He’s looking ready for college.”  Praising and recognizing appropriate behavior in the classroom helps prompt other students towards what they should be doing as well as reinforcing Alan.  It sends the signal of the behaviors the teacher values in the classroom and teaches students what’s expected.  At the same time the teacher should deliberately not give any attention to students who are not doing the right thing, who have not gotten ready to start.

With friends we are expected to give non-contingent attention.  We give them love and attention because of who they are, not based on how they behave.  One doesn’t turn away from a friend and deliberately pay attention and begin talking to someone across the room because you approve of their behavior more.  If you did that it would be too rude to your friend and it might hurt your friendship.  Instead, if your friend misbehaved at a party you would begin by attending to your friend, to see what’s wrong, or find out what you can do for them.  That attention reinforces your friendship and proves you’re a good friend.  In a classroom, teachers who respond to misbehavior as they would to a friend end up reinforcing the inappropriate behavior and they get a lot more misbehavior from all of their students.

There is a role for non-contingent reinforcement of students.  They need to know that the teacher cares about them as people.  The time for that is at neutral times when the student is not misbehaving, such as when entering the classroom, out on the school grounds not during class, or even when circulating the room.  Giving appropriate and friendly social attention to the student at times when they aren’t in crisis or off-task helps create good relationships within the classroom and is valuable.  In that circumstance “friends” is just what is wanted.

A third and final way that teachers should not treat students as friends is when students break the rules.  To establish order in a classroom there needs to be rules and consequences for rule-breaking.  Consequences need not be major or draconian, but they do need to be applied consistently.  If a teacher says, “Wait to be called on before you speak,” the teacher needs to not answer or engage with students who call out without raising their hand and waiting.  The teacher should ignore the student calling out and call on someone who raised their hand.  That needs to be consistently applied, no matter who the student is who calls out.  Students only learn to follow the rules when the consequences are consistent.

I wouldn’t recommend treating friends in this manner.  If friends blurt out and interrupt your turn speaking, we generally tolerate it.  When a friend breaks a rule, we don’t apply consequences.  We might complain to them.  We hope that our friendship will cause them to re-examine their behavior, but we’d rather “ask” them not to do it than apply swift consequences.  That is because we are ultimately not authority figures with our friends.  But teachers are authority figures and they therefore have to treat their students differently than they would treat their friends.  As long as teachers understand this, they can certainly call their students “friends.”

How fast should students be with math facts?

Students should be automatic with the facts.  How fast is fast enough to be automatic?

Editor’s Note: “Direct retrieval” is when you automatically remember something without having to stop and think about it.

Some educational researchers consider facts to be automatic when a response comes in two or three seconds (Isaacs & Carroll, 1999; Rightsel & Thorton, 1985; Thorton & Smith, 1988).  However, performance is not automatic, direct retrieval when it occurs at rates that purposely “allow enough time for students to use efficient strategies or rules for some facts (Isaacs & Carroll, 1999, p. 513).”

Most of the psychological studies have looked at automatic response time as measured in milliseconds and found that automatic (direct retrieval) response times are usually in the ranges of 400 to 900 milliseconds (less than one second) from presentation of a visual stimulus to a keyboard or oral response (Ashcraft, 1982; Ashcraft, Fierman & Bartolotta, 1984; Campbell, 1987a; Campbell, 1987b; Geary & Brown, 1991; Logan, 1988).  Similarly, Hasselbring and colleagues felt students had automatized math facts when response times were “down to around 1 second” from presentation of a stimulus until a response was made (Hasselbring et al. 1987).”   If however, students are shown the fact and asked to read it aloud then a second has already passed in which case no delay should be expected after reading the fact.  “We consider mastery of a basic fact as the ability of students to respond immediately to the fact question. (Stein et al., 1997, p. 87).”

In most school situations students are tested on one-minute timings.  Expectations of automaticity vary somewhat.  Translating a one-second-response time directly into writing answers for one minute would produce 60 answers per minute.  However, some children, especially in the primary grades, cannot write that quickly.   “In establishing mastery rate levels for individuals, it is important to consider the learner’s characteristics (e.g., age, academic skill, motor ability).  For most students a rate of 40 to 60 correct digits per minute [25 to 35 problems per minute] with two or few errors is appropriate (Mercer & Miller, 1992, p.23).”   This rate of 35 problems per minute seems to be the lowest noted in the literature.

Other authors noted research which indicated that “students who are able to compute basic math facts at a rate of 30 to 40 problems correct per minute (or about 70 to 80 digits correct per minute) continue to accelerate their rates as tasks in the math curriculum become more complex….[however]…students whose correct rates were lower than 30 per minute showed progressively decelerating trends when more complex skills were introduced.  The minimum correct rate for basic facts should be set at 30 to 40 problems per minute, since this rate has been shown to be an indicator of success with more complex tasks (Miller & Heward, 1992, p. 100).”   Rates of 40 problems per minute seem more likely to continue to accelerate than the lower end at 30.

Another recommendation was that “the criterion be set at a rate [in digits per minute] that is about 2/3 of the rate at which the student is able to write digits (Stein et al., 1997, p. 87).”  For example a student who could write 100 digits per minute would be expected to write 67 digits per minute, which translates to between 30 and 40 problems per minute.    Howell and Nolet (2000) recommend an expectation of 40 correct facts per minute, with a modification for students who write at less than 100 digits per minute.  The number of digits per minute is a percentage of 100 and that percentage is multiplied by 40 problems to give the expected number of problems per minute; for example, a child who can only write 75 digits per minute would have an expectation of 75% of 40 or 30 facts per minute.

If measured individually, a response delay of about 1 second would be automatic.  In writing 40 seems to be the minimum, up to about 60 per minute for students who can write that quickly.  Teachers themselves range from 40 to 80 problems per minute.  Sadly, many school districts have expectations as low as 50 problems in 3 minutes or 100 problems in five minutes.  These translate to rates of 16 to 20 problems per minute.  At this rate answers can be counted on fingers.   So this “passes” children who have only developed procedural knowledge of how to figure out the facts, rather than the direct recall of automaticity.


Ashcraft, M. H. (1982).  The development of mental arithmetic: A chronometric approach.  Developmental Review, 2, 213-236.

Ashcraft, M. H. & Christy, K. S. (1995).  The frequency of arithmetic facts in elementary texts:  Addition and multiplication in grades 1 – 6.  Journal for Research in Mathematics Education, 25(5), 396-421.

Ashcraft, M. H., Fierman, B. A., & Bartolotta, R. (1984). The production and verification tasks in mental addition: An empirical comparison.  Developmental Review, 4, 157-170.

Ashcraft, M. H. (1985).  Is it farfetched that some of us remember our arithmetic facts?  Journal for Research in Mathematics Education, 16 (2), 99-105.

Campbell, J. I. D.  (1987a).  Network interference and mental multiplication.  Journal of Experimental Psychology: Learning, Memory, and Cognition, 13 (1), 109-123.

Campbell, J. I. D.  (1987b).  The role of associative interference in learning and retrieving arithmetic facts.  In J. A. Sloboda & D. Rogers (Eds.) Cognitive process in mathematics: Keele cognition seminars, Vol. 1.  (pp. 107-122). New York: Clarendon Press/Oxford University Press.

Geary, D. C. & Brown, S. C. (1991).  Cognitive addition: Strategy choice and speed-of-processing differences in gifted, normal, and mathematically disabled children.  Developmental Psychology, 27(3), 398-406.

Hasselbring, T. S., Goin, L. T., & Bransford, J. D. (1987).  Effective Math Instruction: Developing Automaticity.  Teaching Exceptional Children, 19(3) 30-33.

Howell, K. W., & Nolet, V.  (2000).  Curriculum-based evaluation: Teaching and decision making.  (3rd Ed.)  Belmont, CA: Wadsworth/Thomson Learning.

Isaacs, A. C. & Carroll, W. M. (1999).  Strategies for basic-facts instruction. Teaching Children Mathematics, 5(9), 508-515.

Logan, G. D. (1988).  Toward an instance theory of automatization.  Psychological Review, 95(4), 492-527.

Mercer, C. D. & Miller, S. P. (1992).  Teaching students with learning problems in math to acquire, understand, and apply  basic math facts.  Remedial and Special Education, 13(3) 19-35.

Miller, A. D. & Heward, W. L. (1992).  Do your students really know their math facts?  Using time trials to build fluency.  Intervention in School and Clinic, 28(2) 98-104.

Rightsel, P. S. & Thorton, C. A. (1985).  72 addition facts can be mastered by mid-grade 1.  Arithmetic Teacher, 33(3), 8-10.

Stein, M., Silbert, J., & Carnine, D.  (1997)  Designing Effective Mathematics Instruction: a direct instruction approach (3rd Ed).  Upper Saddle River, NJ: Prentice-Hall, Inc.

Thorton, C. A. & Smith, P. J. (1988).  Action research: Strategies for learning subtraction facts.  Arithmetic Teacher, 35(8), 8-12.



How should students practice math facts?

Students should practice with a checker holding an answer key. 

  • One student has a copy of the PRACTICE answer key and functions as the checker while the practicing student has the problems without answers. The practicing student reads the problems aloud and says the answers aloud. It is critical for students to say the problems aloud before saying the answer so the whole thing, problem and answer, are memorized together. We want students to have said the whole problem and answer together so often that when they say the problem to themselves the answer pops into mind, unbidden. (Unbidden? Yes, unbidden. I just kinda like that word and since I am writing this, I get to use it.)
  • A master PRACTICE answer key is provided—be sure to copy it on a distinctive color of paper (yellow in the picture) to assist in classroom monitoring. The distinctive color is important for teacher monitoring. If you are ready to begin testing and you see yellow paper on a desk, you know someone has answers in front of him/her. When you make these distinctively colored (there, I said it again) copies, be sure to copy all of the answer sheets needed for a given operation and staple them into a booklet format…one for each student who is working in that operation. For some reason (I actually know the reason) teachers always want to find a way to put the answer keys permanently into the students’ folders. DON’T. Students need to be able to hold these in their hot little hands, outside of their folders. Then answer keys will be the same regardless of the set of facts on which a student is working. So students working on multiplication will have the answers to ALL the practice sets for multiplication. This allows students from different levels to work together without having to hunt up different answer keys.
  • The checker watches the PRACTICE answer key and listens for hesitations or mistakes. If the practicing student hesitates even slightly before saying the answer, the checker should immediately do the correction procedure, explained below. (Let’s stop here. This is critical. Critical, I tell ya. This correcting hesitations thing is sooooo important. I mean really important. You can probably guess why. We need students to be able to say the answer to these problems without missing a beat — not even half a beat. So students must be taught that there is no hesitation allowed. Really.) Of course, if the practicing student makes a mistake, the checker should also do the correction procedure.
  • The correction procedure has three steps:
    1. The checker interrupts and immediately gives the correct answer.
    2. The checker asks the practicing student to repeat the fact and the correct answer at least once and maybe twice or three times. (I recommend three times in a row.)
    3. The checker has the practicing student backup three problems and begin again from there. If there is still any hesitation or an error, the correction procedure is repeated. Here are two scenarios:

Scenario One
Student A: “Five times four is eighteen.”
Checker: “Five time fours is twenty. You say it.”
Student A: “Five times four is twenty. Five times four is twenty. Five times four is twenty.”
Checker: “Yes! Back up three problems.”
Student A: (Goes back three problems and continues on their merry way.)

Scenario Two
Student A: “Five times four is … uhh…twenty.”
Checker “Five times four is twenty. You say it.”
Student A: “Five times four is twenty. Five times four is twenty. Five times four is twenty.”
Checker: “Yes! Back up three problems.”
Student A: (Goes back three problems and continues on their merry [there is a lot of merriment
in this program] way.)

  • This correction procedure is the key to two important aspects of practice. One, it ensures that students are reminded of the correct answers so they can retrieve them from memory rather than having to figure them out. (We know they can do that, but they will never develop fluency if they continue to have to “figure out” facts.) Two, this correction procedure focuses extra practice on any facts that are still weak.
  • Please Note: If a hesitation or error is made on one of the first three problems on the sheet, the checker should still have the student back up three problems. This should not be a problem because the practice problems go in a never-ending circle around the outside of the sheet. Aha…the purpose for the circle reveals itself!
  • Each student practices a minimum of two minutes. The teacher is timing this practice with a stopwatch…no, for real, time it! After a couple of weeks of good “on-task” behavior you can “reluctantly” allow more time, say two and a half minutes. Later, if students stay on task you can allow them up to about three minutes each. Make ‘em beg! If you play your cards right (be dramatic), you can get your students to beg you for more time to practice their math facts. I kid you not. I’ve seen it all over the country…really!
  • After the first student practices, students switch roles and the second student practices for the same amount of time. It is more important to keep to a set amount of time than for students to all finish once around. It is not necessary for students to be on the same set or even on the same operation, as long as answer keys are provided for all checkers. If students have the answer packet that goes with the operation they are practicing and their partner is on a different operation, they simply hand their answer packet to their partner to use for checking. I know what you are thinking. Yes, I realize that “simply handing” something between students is often fraught with danger. I was a teacher too. All of the parts of the practice procedure will need to be practiced with close teacher monitoring several (hundreds of) times prior to beginning the program. Not really “hundreds,” but if you want this to go smoothly, as with anything in your classroom, you will need to TEACH and PRACTICE the procedural component of this program to near mastery. Keep reading. I will tell you HOW to do this practice. (This is VERY directive.)
  • The practicing student should say both the problem and the answer every time. This is important because we all remember in verbal chains.
  • Saying the facts in a consistent direction helps learn the reverses such as 3 + 6 = 9 and 6 + 3 = 9.
  • To help kids with A.D.D. (and their friends) the teacher can make practice into a sprint-like task. “If you can finish once around the outside, start a new lap at the top and raise your fist in celebration!” Recognize these students as they start a second “lap” either with their name on the board or oral recognition — “Jeremy’s the first one to get to his second lap. Oh, look at that, Mary and Susie are both on their second laps. Stop everyone, time is up. Now switch roles and raise your hand when you and your partner are ready to begin practicing.”