When to begin Rocket Math?
Answer: We don't keep customer contact information to give out--our privacy policy does not allow that. You want to know when in first grade to begin the program. The answer to your question is that students can begin math facts in First grade as soon as they understand the concept of addition. The following is from page six of our teacher directions which can be downloaded for free from on RocketMath.net - just look under Our Products and Rocket Math .
When are students ready to begin fact memorization in an operation?
When they “understand the concept” of the operation. “And how does one know that?” you might be asking. Well, we’re going to tell you. Drumroll, please. Children “understand” an operation when they are able to compute or figure out any fact in the operation. They can use their fingers to figure out the addition and subtraction facts. Or they can use successive addition to figure out the multiplication facts. Or they can use manipulatives and get the right answer. Or they can draw lines, or horses, or dots, or cookies
(We’ve seen it all.) and get the answer. Somehow, some way, given any fact in the operation, and unlimited time, the child can figure out the answer. Then the child is ready to begin memorizing.
This might be at the start of the school year, or might not be true until November--depending on your district's scope and sequence. Hope this answers your question.
Free Rocket Math Wall Chart
Answer: If you go to our website and sign up with our email contacts you will receive a welcome email that gives you a code. You can then go onto our website, www.rocketmath.net , order the wall chart, enter the discount code, and all you have to pay for is the mailing costs. The chart itself is free, but we do ask people to pay the mailing costs. Thank you for your interest in Rocket Math.
Flashcards: Helpful Hints for Happier and More Effective Practice or Flashcards Without Tears
Flashcards: Helpful Hints for Happier and More Effective Practice or Flashcards Without Tears
So you have tried flashcards and they didn’t seem to be making the difference you were hoping for. It’s not your fault. There are some things that many people don’t know about the use of flashcards. It’s not really an X-File, but there are some secrets out there. We put all that stuff into our flashcard strategies and wait until you see how well it works!
The proper use of flashcards guarantees the necessary conditions for learning to occur. (Read as: You can do this your way, but we’ve been there, done that and our way works better.) The best conditions for learning are having a few things to learn in a sea of mastered material!
Work with a fairly small deck, of no more than 12 cards at one time. This is called your working deck. (or you could just go sit on a deck with a beverage and … WAIT. NO. There will be time for that later.)
Only 3 of the 12 cards in your working deck should be “new” or “hard.” The other 9 should be things the student has already mastered. [1] [This is why the first 9 cards in the Rocket Math addition and subtraction flashcards just ask the student to identify the numeral on the flashcard.]
(Mix up the working deck periodically to change the order in which the facts are presented…just in case.!)
Students should practice by reading aloud the whole problem on the flashcard and then saying the answer from memory. Saying the whole problem and then the answer is very important as it creates the verbal chain. Eventually, after many repetitions, an amazing thing happens. Whenever the student reads “eight times seven” in any problem, (ever later, in multi-digit problems) the answer “56” pops into their minds unbidden. (We try to use the word “unbidden” at least once in everything we write – just because we can.) This automatic coming-to-mind is called “automaticity ” and is the goal of facts practice.
The student should say the answer without any hesitation. We really mean NO hesitation! ...
<< MORE >>How do I set up Rocket Math Facts Curriculum
- Copy 20-30 sheets of every letter A-Z in the operation and put into hanging files in your crate. Keep your originals in a binder.
- Copy a class set of these three sheets: Rocket Chart , goal sheet and Individual student graph . For each student create a folder with Rocket Chart on the front, and goal sheet and graph inside.
- ...
Home use of Rocket Math
Answer: Yes, the Rocket Math Individual Classroom License disc has the teacher directions included. So does the download only version (if you buy with a credit card) for only $65 and each separate downloaded operation (addition, subtraction, multiplication, and division) also includes directions for only $29. For home learning (one-on-one) we encourage you to consider math flashcards at $19 per operation. The directions for flashcards (and the worksheet program) are available for free under the "our Products" tab--so you can compare. Also we have some blog entries that address the issue of worksheets and flashcards for home use. See especially the third and fourth entries regarding children with disabilities:
- Rocket Math Worksheets Vs Flashcards
- Ordering Rocket Math for Home Use
- Getting Stuck- What Should you Do About a Lack of Success
- Special Needs Students and Rocket Math
When are Students Ready for Division Facts?
Program with my students for quite some time. I have 2 students who have
completed the addition, subtraction, and multiplication portions to mastery
and have been told that 2nd graders should not move on to division. I have
2 questions. What is the reason for not allowing these particular children
to move on to division if they are ready? What do you recommend that I do
with these students for the remainder of the school year if they should not
move on to division? They have completed each set for addition,
subtraction, and multiplication with a goal of 45 seconds. Thank you so
much for your help.
Answer: There should not be a hard and fast rule prohibiting second graders
from practicing division facts. If they are ready, they should be able to
begin. That said, I am skeptical (open-minded but need data to be
convinced) that they are ready and that they are truly at mastery at all
three other operations by the middle of second grade. That is so far in
advance of so many children that it would be wonderful if true.
You say they completed each set with a goal of 45 seconds. So a goal
of 45 seconds doesn't quite make sense for me. That doesn't tell us if they
were fluent. [The previous version of this math facts program had only 40
problems in the test part of the page, so to raise goals above 40 problems
per minute required cutting the time down to 45 seconds.] Weren't their
goals to answer a certain number of problems in 45 seconds. If students
were tested for 45 seconds, how many problems were they expected to do in
that time period?
In Rocket Math we set goals based on the writing speed we find as we
test the children. Once we know their writing speed we set goals in terms
of the number of problems students are to complete in one minute. We don't
have to do the 45 second thing anymore, because this version of Rocket Math
has more problems on a page.
Students should answer math facts as fast as they can write--but
goals should not exceed their writing speed. Goals for second grade students
(who don't yet write as fast as older students) might typically be in the
30s in one minute. The two minute timings should be in the neighborhood of
60 facts answered in two minutes--with no errors.
If the students have passed all the sheets in all three operations
with say 24 problems in 45 seconds (or whatever their personal goals are)
then they might be ready for division. If not, they might need to do a bit
more practice--with you making sure that they can meet their goals on every
timing without errors.
Even if the previous three operations are at mastery--they can answer
all the facts about as fast as they can write--they might not be ready for
division for a different reason. Students may not be ready for division if
they don't yet understand the concept. It is very important to take the time
to teach students the concept or the process of computing math facts before
they begin to memorize the answers. Otherwise, the memorization will be
simple rote memorization and won't be supported by an understanding of the
process.
To measure if students "understand" the concept of division you give
them fact problems and ask them to "figure out" the answers without a time
limit. If they can correctly figure out the answers to any division fact
you give them (without a time limit), then they understand the concept. If
you give them the problem seven divided into sixty-three and the student can
correctly work out that the answer is nine--then that student understands
the concept of division. In that case, assuming all the other three
operations are fluent, then those students are ready for division facts and
should begin memorizing the answers to them. Hope this helps.
Lack of success with flashcards. What should you do?
program. My typically developing 7 year old is coming along well. However,
I have a 6 year old with some mild learning disabilities, and he is really
struggling. I am only having him do 10 cards, but we have been on 7-17
forever...weeks. He is discouraged, and I can hardly blame him. He can get
the answers correctly, for the most part. However, he cannot do it without
hesitating. Any suggestions? He does have expressive language problems
also, so I wonder if I am expecting too much... Thanks!
Answer: If he is on cards 7-17, then can I assume he was able to do cards 1-10
without hesitation? If so, then it isn't a speech problem. He may need a
good bit more practice to achieve the ability to answer instantly than your
other child. That's OK, just be sure to give him that practice and he'll be
able to answer without hesitation. Here's how to give him more practice.
1) Back him up to where he can read the problems and say the answers for the
problems without hesitation--all the way back to the first set of cards if
need be. You may need to explain that "we have to start over because we
went too fast. You're smart and you shouldn't be having this much trouble,
so we're going to start over."
2) After he does a set without hesitation, praise the daylights out of him.
But, do not add anything new until he goes through the stack without any
hesitations for say three days in a row.
3) Each day give him two more practice sessions (two minutes in length or
so) than you have been doing. This means he may do nine or ten sessions
without any hesitations in any of the cards. This is important because it
establishes that this is how one is supposed to answer math facts--and it
will allow you to praise him a lot and for him to be successful a lot. (And
of course, to get more practice.)
4) Only add one new card at a time instead of three at a time.
5) Build up to 15 cards in the deck. (Cutting back to only ten in the deck
wasn't good. It gives him less practice with the older facts and he needs
more.)
6) Once he's up to 15 cards in the deck, remove only one at a time as you
add one at a time.
7) This will take him longer to get through the operation--but he will still
get through it in a year, so that's OK.
If you find that he is ridiculously successful, then you can perhaps begin
to add two cards at a time. Or cut back to two days in a row without
hesitation before adding a card.
The key is to get him to a high level of success, by providing a lot more
practice, then keep him there as you gradually and carefully progress
through the facts in the operation. The standards and the expectations
can't be lowered--or the facts won't be automatic. Instead you just
increase the amount of practice. Hope this is clear.
Questions about the Format of Rocket Math
Answer: Why yes, there is a rationale for our new formats, and thanks for asking. We found that a lot of 4th graders and older could do quite a few more than 40 facts in a minute. If you keep raising their goals as they demonstrate the ability to write faster (they always need to meet their previous best) you can give them a much more rigorous learning experience. A student who is learning the facts fast enough to do 50 or 60 facts in a minute (as long as they can write that fast) is learning them better and therefore will know the facts with greater automaticity--and thereby find higher math much easier. So we wanted to create a format that would allow a few more facts in a minute. We also found a lot of sites where students practiced once through the 40 facts on the top half of the page and stopped. Although we emphasized the "take another lap" in our training sessions the format didn't lend itself to continuous practice--so we decided to make it look more like laps. Finally, we found that often the first errors or hesitation would occur on the first one or two problems and no one knew how to "go back three" in the old format. Now, it is quite obvious and works well for practicing.
Memorizing Math Facts—is it really necessary?
Memorizing Math Facts—is it really necessary?
In today’s society with computers and calculators ready at everyone’s fingertips—is memorizing math facts really that important? To be clear, we are not talking about whether students should spend a lot of time practicing calculation. While one could make a case that a lot of practice getting fast at long division, or even accurate at long columns of addition problems, is no longer valuable, quite the opposite is true for memorization of single digit math facts. Memorizing math facts is probably even more important today than it was 50 years ago.
Using calculators and computers to do complex calculations for us is smart. That’s why a lot of time practicing how to do this by hand may no longer be necessary. Using a calculator saves time and it’s more accurate—except when we make an error in data entry or in the formula we have used to do the calculations. At that point, we must have already done a quick and unconscious mental calculation of the probable answer, so that we see the error. Catching errors in a calculator’s answer requires a ready knowledge of math facts. If you can’t catch your calculator errors then you’ll continue to make more and more of them. Furthermore, if you must use a calculator to compute single digit math facts (because you don’t know them) you will be incredibly inefficient at all math operations. So the ready availability of calculators makes the need for quick mental math facts more important than ever.
Another reason for knowing math facts fluently has to do with fractions. Understanding the manipulations of fractions that should be learned in upper elementary or middle school depends upon automatic recall of multiplication facts. Students who don’t know the multiplication facts fail to see when they should reduce facts like 8/24 or 12/16. They don’t recognize that 6/9 and 16/24 are equivalent fractions, or see why they are when it is pointed out to them. They struggle figuring out the lowest common denominator between thirds and twelfths, let alone between thirds and fifths. Many children are doomed to failure in learning fractions, decimals and percents simply because they lack a fluent knowledge of multiplication facts and the relationships built upon them. That failure makes it nearly impossible for them to succeed in algebra. And we all know that if you can’t “pass” algebra your chances of getting into a four-year college are slim to none.
Instantaneous recall of math facts is also important because it enables students to see patterns in numbers. We know that recognizing patterns is essential in math, but few teachers realize that recognizing patterns in numbers is dependent upon knowing math facts. The pattern 2, 4, 8, 16, 32, 64 is readily obvious to students who know the multiplication facts—but not at all obvious to those who don’t. The pattern of 49, 40, 32, 25, 19, 14, 10, 7, 5, 4 is obvious to students who can mentally subtract, but not to those who can’t.
So there are several reasons that knowing math facts to a level of automaticity is important to future success in higher levels of math. But is it really necessary to embark on an organized process of memorization? Won’t students just naturally become more and more fluent with the facts—once they’ve learned how to figure them out? The answer is no for many children. Because there is less emphasis on calculation in today’s math, students have less opportunity to practice using math facts on arithmetic worksheets than children did 50 years ago. Without practice to build up that immediate recall it becomes more important than ever to have in place a good method of memorizing those facts.
Rocket Math Worksheets vs Flashcards
Question: My son is in third grade and is still counting on his fingers to figure out math facts. I was wondering if I could order Rocket Math to help him learn math facts at home.
Answer: Yes, parents are welcome to purchase the download version of any operation in Rocket Math off our website for $29. BUT, you might want to consider using our flashcards instead, which are less expensive at $19.
If Rocket Math is good enough for teachers to use in schools, why isn’t it a good idea for me at home?
You should know that Rocket Math is set up on worksheets so that teacher scan give a one minute timed test to all the students in class at once. Rocket Math involves a lot of copying. The worksheets save a lot of time for the classroom teacher. But it might not be necessary for you to do at home.
Let me explain the point of the one-minute tests on the worksheets in Rocket Math. The teacher needs to make sure that each student can instantly recall all the facts learned so far, before giving them any more facts to learn. Each child has individual goals to “pass” a worksheet. The goals are based on the child’s writing speed. The plan is that the goal will be just right for all the students, requiring them to write answers to the facts, just as fast as they can. When a student can answer facts as fast as he or she can write, this lets the teacher know the student is not hesitant on any of the facts learned so far. That tells the teacher that the student should “pass” on to the next worksheet, to get some more facts to learn.
All the work of sheets and testing is designed to find out if the student is hesitant on answering any of the facts learned so far. If not hesitant, the student is ready for more facts to learn. If still hesitant on some of the facts, the student needs more practice before getting any more new facts. You can easily determine this in seconds with a one-on-one oral test using flashcards. The worksheets are needed to deal with a bunch of children at once. If you have your child with you at home, one-on-one, the flashcards make more sense. You are welcome to download the directions for the math facts flashcards for free off our website. Then you’ll know how to use them most effectively. We mail a box of flashcards to you (first class) for only $19 plus $5 shipping and handling.
The only time I’d recommend parents order the Rocket Math worksheet curriculum is if they need to back-up Rocket Math in school. And even then I’d rather see parents working off the sheets already run off in the school. If you’re not trying to help your child “pass” Rocket Math in school, the flashcards are more fun, more intimate, and more flexible.
I hope this helps.
Ordering Rocket Math for Home Use
Question: Hi, my daughter is in second grade and her school has started using Rocket Math. I was wondering if there was any way possible to purchase or download the practice test sheets for extra practice at home?
Answer:
Yes, you are welcome to purchase the download version of Rocket Math for whichever operation your daughter is learning. It’s $29 on our website. But, you may not have to do that. (I know, I should be telling you to buy our product, but I’m a teacher first and a salesman second.) While we’re at it, at $19 the flashcards are a less expensive alternative. We made them available for people who want to work one-on-one teaching the facts to their children at home. However, if you are trying to support the school’s implementation of Rocket Math (so your daughter passes quicker), it makes more sense to use the same worksheets she is using in school.
But before you purchase the program, I have another question first.
Why can’t your daughter just bring home a copy of the worksheet she is working on at school currently? (There is one advantage to purchasing the program for use at home, but we’ll come to that later). The main reason you want the sheets is to provide extra practice—and the best practice is oral. Once your daughter has completed the test on the day’s worksheet, it can still be used for practice—by orally reading and answering the facts around the outside. In our extensive teacher directions, we encourage teachers to send the used worksheet home each night with their students—to be made available for just the kind of practice you want to do. So why aren’t those sheets coming home?
Perhaps your daughter’s teacher thinks that after the test is completed on a worksheet it won’t be able to be used at home. Some teachers think that students will “cheat” by looking at the test for the answers to the problems around the outside—and therefore a worksheet with the test completed shouldn’t be sent home.
That kind of cheating is not really a problem at home, and here’s why. If your daughter has to scan the test to find the answer to a fact, it will be obvious to you that she doesn’t know that answer instantly like she should. You can tell because there will be what we call a “hesitation” in answering. And you know that isn’t good—she’s supposed to know that fact instantly. You will provide a correction, helping your daughter to learn that fact, so she won’t have to “look.”
Once you’ve taught your daughter that if she “has to look” for the answer she doesn’t really know that fact, she’ll get it. After that, she will try to answer without looking. You will have taught her that there is no “honor” in looking—and she’ll become proud of the fact that she doesn’t have to look. This is easy for a parent to do. Make sense?
So please ask your daughter’s teacher to let her bring home the worksheet any day she doesn’t “pass.” And you can guarantee the teacher that you’ll make sure that your daughter doesn’t “cheat” when practicing.
On our website we have posted an “Open Letter to Parents,” that explains how to practice in a way that will support your daughter’s success. And that explanation will work if, god forbid, you find it easier to spend $29 to download the Rocket Math worksheets than to ask the teacher to send them home with your daughter. (Yikes, I hope that isn’t true, but just in case!)
If you download the sheets, you will have to rely on your daughter to tell you which sheet she is “on” at school—so you practice on the right one. With the worksheet in hand, you’ll be able to give your daughter the extra practice session or two each evening that will insure that she passes each set relatively quickly.
Now, let me tell you the one advantage of purchasing the worksheet program. If you print out your own worksheets from the Rocket Math pdf files, in addition to the oral practice you do each evening, you’ll also be able to give your daughter a practice written test each night. A practice test is not as helpful to your daughter as doing the oral practice, but it doesn’t hurt. You’ll be able to see if she is having troubles with the writing goals or is getting distracted during the tests. When she “passes” her practice test in the evening with you, she’ll be confident that she’ll be able to pass in school the next day.
Hope this helps. Thanks for being so involved in your daughter’s education.
Special Needs Students and Rocket Math
Answer: Rocket Math was designed to be effective as it is with special needs students--but only if it is done according to the directions. Both Randi and I have used it successfully in special education classrooms. All the details of how it should be used are especially critical for special needs students. Some of the aspects are especially important.
One key with special needs students is to monitor their writing speed carefully. One should be sure to give the writing speed test and make sure that they follow the time limits. It’s not unusual for special needs students to try to squeeze in a few more responses on timings after time is up, because they are used to not being able to perform as expected. Of course, if a special needs student does that on the Writing Speed Test their goals would end up being impossible to meet. So be careful there. This may involve consultation between the special education and general education teachers so that goals don’t get too high causing lack of success.
Writing speed is an issue for many special needs students, and they often have great variation in how well they can perform from day to day. I would recommend caution about moving “up” the goals for special needs students. Perhaps you could wait until they have beaten their previous goal two or three days in a row, before raising the goal. You just want to be sure they can consistently write that quickly.
It is important not to give lower goals to special needs students, as they need to reach automaticity the same as everyone else. What will be different is the amount of practice they will need to achieve the goal. Where other students can develop automaticity with the four new facts in a couple of session a special needs students might need ten or fifteen practice sessions. Rather than spread that practice out over two or three weeks, special needs students should get more than one practice session (of two or three minutes duration) each day. Remember, don’t make sessions much longer than three minutes or students will burn out.
I would encourage special education teachers to provide their students with an extra practice session each day in the special education room as well as the one the students have in their regular classroom. I would also encourage special ed staff to work with the parents (or siblings) to show them how to do another practice at home each evening. If the parents of special needs students can be recruited and trained to provide extra practice at home--done positively—it can make a huge difference in the rate of learning. Three short sessions each day would enable a slow performer to be able to pass in five or six days—within the expectations for all the other students.
It is very important that practice procedures for special needs students be monitored and done exactly as written. It is hard to overemphasize the importance of the proper and complete correction procedure for special needs students. Besides teaching the parents or siblings how to do the practice, it might be valuable for special education staff to monitor how the practice sessions in class are going. Often special needs students are not good at self-advocacy or leadership and if their in-class partners are not following the procedures the special needs students will need help to correct the problem.
Finally, we know that special needs students have had a history of failure at academic tasks. Therefore they often lack perseverance and give up rather more easily than we’d like. This implies that special needs students are more dependent upon the motivational procedures to keep them going. Unfortunately not all general education teachers make full use of Rocket Math’s built-in motivational procedures— such as coloring in the rocket chart, using the Rocket Math Wall Chart, or using the achievement awards. Special needs students may need all of these things to keep them going and not giving up.
The special education staff should work to provide some extra reinforcement if the homeroom teacher is not doing a lot. Even if there is no Wall Chart being used in the regular classroom, one could be put up in the special education room, and all the students on that teacher’s caseload could come in and put up star stickers as they pass levels in their regular classrooms. The special educator could set goals and have celebrations with his or her special needs students when the stickers pass the goal mark. In addition, a special education teacher can give out achievement awards to his or her students when earned, even if the general education students don’t normally get them in the classroom. One of the most important would be the “helper award” if the special needs student is getting practice at home.
While none of these things involves modifying the directions for special needs student, it is important to use ALL the tools provided in Rocket Math to ensure the success of special needs students. The extra effort involved in using all of the tools carefully may need to be undertaken by special education staff to make sure it all happens.
Why Are Math Facts Important?
There are several reasons why mastery of math facts to fluency is a critical skill.
- Calculators expect users to have a ready knowledge of simple facts in order to know when errors entering data return incorrect answers.
- Children who don't have a quick grasp of the facts do not generate correct estimates or make good guesses in complex math problems. Their problem solving ability is hurt because they cannot tell if an answer "makes sense."
- Lack of fluency with math facts limits the number of more advanced math problems students can and will do on a daily basis. Because doing math problems are so slow and difficult students resist doing their math assignments and thereby learn less.
- Students who are being distracted because their fact knowledge is not automatic find it difficult to learn complex math algorithms. Students who are automatic with their math facts find it much easier and are much more successful at higher level math.
- Once students reach fractions, they need to figure out common denominators, equivalent fractions, reduce fractions and cancel out like factors. All of these procedures depend upon instantaneous recognition of multiplication facts. Students cannot easily learn, nor easily understand fractions procedures without a ready knowledge of multiplication facts.
- Mastery of fractions, decimals, percentages and ratios is required for students to be ready for algebra. Algebra is the gatekeeper to all higher math classes and success on the SAT. Higher math is the gatekeeper to 4-year colleges and all of the more technical professions. For some students all of these opportunities are denied due to lack of mastery of multiplication math facts!
How Fast is Fast Enough for Math Facts?
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. Sixty problems per minute is exactly in the middle of the range of 40 to 80 problems per minute shown by adult teachers in this author’s workshops. 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 fewer errors is appropriate (Mercer & Miller, 1992, p.23).”
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.
One goal is to develop enough math fact skill in isolation, so that math fact speed will continue to improve as a result of using facts in more complex problems. Miller and Heward discussed the research finding 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 (1992, p. 100).” If a range from 30 to 40 problems per minute is recommended the higher end of 40 would be more likely to continue to accelerate than the lower end at 30, making 40 a better goal.
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.
In summary, if measured individually, a response delay of up to 1 second would be automatic, but when students are reading the problem aloud after presentation of the stimulus there should be no delay in answering. In writing the range of 30 to 40 seems to be the minimum, although a good case can be made that setting an expectation of 40 would be more prudent. However, students can be expected to be able to develop their fluency up to about 60 per minute, for students who can write that quickly. Sadly, many school districts have expectations as low as 16 to 20 problems per minute (from standards of 50 problems in 3 minutes or 100 problems in five minutes). Children can count answers on their fingers at the rate of 20 per minute. Such low expectations “pass” children who have only developed procedural knowledge of how to figure out the facts, rather than the direct recall of automaticity.
REFERENCES
Ashcraft, M. H. (1982). The development of mental arithmetic: A chronometric approach. Developmental Review, 2, 213-236.
Ashcraft,M. H., Fierman, B. A., & Bartolotta, R. (1984). The production andverification tasks in mental addition: An empirical comparison. Developmental Review, 4, 157-170.
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.
Hasselbring, T. S., Goin, L. T., & Bransford, J. D. (1988). Developing math automaticity in learning handicapped children: The role of computerized drill and practice. Focus on Exceptional Children, 20(6), 1-7.
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.
Can Parents Buy and Use Rocket Math in the Home?
Answer: Yes, the Rocket Math program is appropriate to buy to use at home with your child. The download version has everything a parent or classroom teacher needs to run the program--it's less expensive when we don't have to bill, burn a CD, and ship it. But...
That being said, a parent at home may want to consider using Rocket Math flashcards instead of the worksheets in the original Rocket Math program. Flashcards are designed for one-on-one where the worksheets are designed to run an entire class at the same time. You can buy a set of flashcards for one operation for less than the cost of the worksheet program. You can download the Flashcard Directions for free--and I highly recommend you doing that, so you know exactly how to work with your child effectively to learn math facts from flashcards. We really like the watch-your-favorite-TV-show-together-and-do-flashcards-during-all-the-commercials plan.
If you are teaching your child math facts at home, you definitely wouldn't want to work on more than one operation at a time; addition in first grade, subtraction next in second grade, multiplication in third, and division in fourth grade.
The practice procedures are very similar between the two forms of Rocket Math--flashcards and the original worksheet program. In both cases the student is to read aloud the problems and say the answers from memory without hesitation. The person listening (tutoring) provides the same correction procedure--saying the correct fact and answer, having the student repeat the fact and the answer three times, then doing two more problems before revisiting the target fact (the one on which there was an error or hesitation). The difference is that in the worksheet program students are reading facts from the worksheet, while in the flashcard program the student is reading the facts off the flashcards.
With the worksheet program you will have to print out the worksheets, the writing speed test, the goal sheet and the rocket chart. Each time you give the student the one minute test (to see if they are ready to move on to the next sheet) you'll use up that sheet and have to print a new one. When your student passes the set of facts on that sheet, you'll need to print the worksheet for the next set. With the flashcards, no additional printing is required. That alone is reason to use flashcards in my mind.
There is one very special circumstance in which it might be important to use the original Rocket Math worksheets at home. If your child is using Rocket Math in school, AND if the program is not being run correctly, AND if your child is being frustrated--then you might want to get the download of the operation they are doing in school. You could read the full 40 page directions, set up the program carefully and correctly and you'll find that your child will be able to succeed without frustration when the program is run according to the directions. You will be able to discover what is wrong. It may be that not enough time is spent practicing, or practicing the right way. It may be that your child's handwriting speed was not taken into account when setting their goals. It may be that your child's student partner in school is not correcting errors or hesitations in the right way. In any case it would be very important to show your child that he or she CAN in fact learn math facts successfully (all children can) and to overcome the frustration that improper use of the program is causing.
So you can buy and use the original Rocket Math worksheet program at home, but think about whether flashcards would be easier than the worksheet program. Teachers can't effectively use flashcards in their classrooms because they can't monitor the learning of that many students at once without the testing procedure. But you can when you are home alone with one child at a time--so flashcards can work for you.
Why Are There Errors in the Rocket Math Division Section?
Answer: We get this question a lot. It makes us a little sad,because it tellsus that people aren't reading every single word of our wonderfully charmingdirections before they begin using our program. We tried to make thedirections interesting enough to bring you to the very end, but alas, wearen't that stimulating. Way back on page 29 of the Rocket Math directions we show thesequence of facts for division. At the bottom on the page we have thefollowing footnote:
* Please note that Set L problems are problems where the divisor (what youare dividing by) is larger than the dividend (what you are dividing into).These kinds of problems occur in long division. The answer is 0 for ourpurposes here. We know that in a long division problem there would be moresteps, but for now we simply want students to recognize that, for example,“nine doesn’t go into eight” and therefore the answer is zero.
We have taught long division. We know the look that kids get in their eyeswhen they first run into something like four divided into two in a longdivision problem. They are stunned. They have no idea what to do. They gooff task. They run amuck. So we are trying to prevent that from happeningin your classroom. [See how considerate we are. We have your best interestsat heart. Really we do!] Now your students will know what to do. Theyknow that you answer with a "0" because seven doesn't divide into three.
So, no, there aren't a bunch of errors in division starting in Set L. Wejust thought that students should get a chance to learn what to do when thedivisor is larger than the dividend. Thanks for asking!
Three Keys to Practicing Math Facts, Cont.
The third key - How to Practice
To review, the first key to practicing math facts is for the learner to be practicing only three or four new facts at a time. The second key to practicing math facts, lies in knowing how to correct students who answer incorrectly or with a hesitation. The third key is the method of practice—both how the learner practices and how long to practice.
When committing something to memory, such as a math fact, like “eight times seven is fifty-six,” the key is to practice saying the whole thing aloud while retrieving it from memory. The point is to create a “verbal chain” just like we use to memorize the words to songs or people’s first and last names. By saying all the parts together in a set sequence over and over we get to the point where once we’ve said the first part, the final part pops into our head, unbidden. [We like that word unbidden, so we use it as often as we can.] After saying the whole fact and the answer many times a student can just say “eight times seven is…” and the final word “fifty-six” comes to mind immediately without the student having to think about it. So saying the whole fact and the answer every time the student practices is critical.
Equally important is that the student be retrieving the answer from memory every time. As long as a student does at least two or three different problems in between instances of the target problem, the student has to be pulling the answer from memory. Simply saying the same fact over and over would not be nearly as effective as correctly answering the fact, doing a couple of other facts in between, and then going back and recalling the target fact again. That’s why a mixed set of problems is important to building the neural connections for each of them. Every time the student retrieves the correct answer from memory the connection is strengthened and it becomes more automatic.
The final part of how to practice is to know that spaced practice—short sessions with a few minutes or a few hours in between is the most effective way to learn. Spreading thirty minutes of math fact practice across ten sessions over ten days is the most effective way to practice. Students who practice for two or three minutes each day at school and two or three minutes each night at home will learn at an optimal rate. After about three or four minutes at one sitting the student has learned as well as possible for that time period. Any further practice should be done a few hours later or perhaps the next day. Practicing for a half hour at one time is painful, punitive and not very effective.
So the third key to learning math facts is to practice a couple of minutes at a time, once or twice a day and then to spread your daily practice over the next several weeks. Be sure to be saying each whole fact and its answer very time time and to be retrieving the answers from memory correctly.
Be sure you are practicing with small sets of facts and only three or four new facts at a time. Whenever you hesitate or make an error, be sure to get the right answer, try again to commit it to memory by repeating it three time, do a couple of problems and then try again on the fact that caused the trouble. Finally, be sure to practice by remembering the answer and then saying the whole fact and the answer aloud, practice for two or three minutes at a time but do it once or twice every day for weeks.
Rocket Math, either the original worksheets for school, or the flashcards for home are a great tool to manage your practice so that it is effective. Why not check out the Race for the Stars math game by Rocket Math to find an alternate way to review once you have learned the first few sets of facts?
Three Keys to Practicing Math Facts, Cont.
The first key to practicing math facts is for the learner to be practicing only three or four new facts at a time. The second key lies in knowing how to correct students who answer incorrectly or with a hesitation. Even if students answer correctly, they need more practice on any fact they can’t answer immediately after reading or saying the fact. So how does the person helping the student correct the error or hesitation?
The tutor should begin the correction by stating the target fact and the correct answer. Even if the student just hesitated it is important to clarify the correct problem and answer. Next the tutor should ask the learner to repeat the fact and answer. At this point the learner should be trying to commit this to memory, so it is helpful to ask the learner to repeat the problem and answer three times in a row. Now comes the best part. The tutor should give no more than two or three problems before coming back to the target problem. This is critical because the learner should still be able to remember the problem and answer, and answer correctly and students only learn from answering correctly. Remember, practice only makes perfect if it’s perfect practice!
After the tutor corrects an error or hesitation in the original classroom Rocket Math worksheets, the tutor asks the student to go backwards three problems prior to the target problem and begin again. The student will then get to say the problem and its answer before forgetting it. When using Rocket Math Flashcards, we recommend that the tutor put the target flashcard back about three cards after correcting an error. This ensures it will come back to the front before the learner forgets the fact.
Correcting errors or hesitations in this fashion helps students learn. This procedure gives students a chance to redeem themselves by practicing the facts and then answering them correctly a few seconds later. They learn the point of the exercise—learning facts.
Three Keys to Practicing Math Facts
Everyone seems to know that children ought to practice math facts in order to learn them. And almost everyone has worked with some students that don’t seem to learn them. Often people ask us, “Are you sure that all students can learn math facts?” Absolutely they can, but not if we practice the wrong way. So how should we practice for students to be successful? This is part one of a three part series detailing the three keys for all students to be successful learning math facts.
The first key idea is that students should not be practicing more than three or four facts that they have to learn. The rest of the facts that they should be practicing should be ones they already know instantly. The known facts provide a short break in between the new facts the learner is trying to master.
The point of practice is for students to get it right. The saying, “Practice makes perfect,” should be amended to: “Only perfect practice makes perfect.” In other words, students should be mostly practicing facts they already know, with just three or four that they are trying to remember.
This creates an immediate problem. What do you do when the student doesn’t know any math facts yet? Answer: You either practice with a set of three or four facts, or you add in some things to practice that the student already knows. For example, in the original Rocket Math classroom program the first worksheet has only four facts to practice. In Rocket Math Addition Flashcards the first nine cards simply ask students to identify numerals displayed on the card. The first practice deck will consist of those nine already-known numerals and three facts to learn. Now the learner has a fighting chance because he or she only has a few facts to learn in the deck of facts.
Once the first set of three or four facts are learned, a new set of three or four facts can be added without causing frustration or failure. This is first of the three keys to successful math facts practice.
How to Grade the Rocket Math Program
The question of How to Grade Rocket Math always catches us by surprise. We really see little need for the grading of Rocket Math and it's even harder for us to imagine how to grade Rocket Math appropriately. Rocket Math is math mastery program and as such it is designed for students to progress at their own pace but always to be at mastery. Some students are able to develop fluency in a set of math facts in three tries while others need six tries. If students need more practice to develop fluency then the teacher should arrange for them to get another practice session each day, or to encourage students to also practice at home (short, three minute sessions) the same math facts being practiced during the day at school.
You could imagine that students might be graded down for not practicing the way they should. But that is the teacher's responsibility. Student’s should be trained how to practice with lots of teacher modeling. Student’s should be monitored closely while practicing with their partners and praised by the teacher for practicing the right way. Students should be motivated by teacher praise, by coloring in their rocket chart as they move up the rocket, and by receiving certificates to take home to celebrate their small successes.
With all of Rocket Math procedures properly in place, students should be trying their best each math practice session to pass that set of math facts. As long as students are practicing the right way (getting corrections for every fact on which they hesitate) they will learn the facts as fast as they can. The fastest learning students may complete the sequence in two or three months, while others may take four or five months--but if they are doing what they are supposed to be doing each day--the teacher should not mark students down for needing more time to master a set of facts. There is plenty of time in elementary school to learn all the math facts to fluency, even if it takes a year to master each operation. Why in the world would a student get a low grade, let alone a failing grade, in Rocket Math?
If you must give a grade, then every student should be getting a good grade as long as they are working hard at Rocket Math and doing what they are supposed to do. If they are not practicing the way they should, the teacher needs to model how it should be done, explain why it needs to be done that way, and then motivate with praise and recognition those who practice the right way. Soon after practicing the right way, students will pass a set of facts and in a couple of days they can be praised and recognized for their success.
The regular steady success with the small steps in Rocket Math will motivate the students (and their parents). It is a serious mistake to expect that poor grades will motivate elementary school children to try harder. In stead the teacher should follow our procedures for how to motivate children to try harder, so they can get certificates, recognition, and fill in their rocket chart.
Don Crawford
3439 NE Sandy Blvd. #359, Portland, OR 97232
Phone: (888) 488-4854 or (410) 960-0596
www.RocketMath.net Fax: (443) 708-4050
Kids Who Write Their Numbers Backwards
“My school is using the Rocket Math program. We are wondering if you would accept reversals on Rocket Math papers? By the way, I love your program.”
Becky, thanks for your question. Many teachers wonder whether they should accept reversals or backwards numbers, when children are writing out answers to math problems. The answer depends upon the type of reversed numbers. Bet you didn't know there were two distinct types of reversals, did you?
The benign reversals are single digits written backwards. A backwards 7 or 2 or 4 or 6 should not alarm you greatly. Up until the school years children have learned that a thing has the same name regardless of its orientation. A chair turned upside down or turned left to right is still a chair. Upside down toys are still the same and you learn to recognize things as being the same regardless of how it is facing. Then suddenly in school some things, particular symbols, have to be facing a certain direction. Learning which way the seven has to face, is best accomplished by a patient teacher who points out which way sevens have to face. Extra practice making 7s the right way would be a good idea.
Pre-correcting for the error by writing a model at the top of the paper for students who are persistently getting it wrong can be quite effective. And of course, the quickest learning results from lots of praise for "making these sevens face the right way." But refusing to accept a 7 as the correct answer because it is facing the wrong way is at best unnecessarily discouraging and at worst, may be confusing. Some students may think they got the wrong answer and conclude that 3 plus 4 is not seven!
On the other hand, there is another kind of reversal that must be treated as an error. The unacceptable reversal is when the digits are written in the wrong order, as when fourteen is written as a four and a one (41) and nineteen is written as 91. This kind of reversal represents a misunderstanding about place value. In addition the student has actually written the wrong number, so he/she must learn that is a different number. "No, that's not right. You wrote twenty-one and three times four is twelve. Twelve is written like this, 1-2." Again, it would be smart to give some practice writing numbers from dictation--especially those pesky teen numbers. Modeling on the board or overhead first, followed by practice of a small set of numbers would be most effective.
So, single digits facing the wrong way can be accepted as the right answer, even though you work on correcting the way the numeral faces. But two digit numbers must be written with the tens digit on the left ALL the time.
I can't leave this topic without pointing out that your students may also be reversing their zeros, ones and eights--but you just can't tell!
Don Crawford
R&D Instructional Solutions
www.RocketMath.net
3439 NE Sandy Blvd. #359, Portland, OR 97232
Phone: (888) 488-4854 or (410) 960-0596
Fax: (443) 708-4050
Rocket Math - Blasting off to Math Facts Fluency
By: Randi Saulter, M. S. & Donald B. Crawford, Ph.D.
ItsRandi(at)aol.com & dc0843(at)aol.com
We know, as teachers, that we can trace our students’ difficulties with higher order math algorithms to a lack of fluency with basic math facts. We also know that we need to do something about our students’ lack of fluency without sacrificing loads of time that needs to be spent on the core math program. What do we do you ask? Here is the answer!
Rocket Math is a structured program for the sequential practice of math facts. The program is structured in a consistent manner (Let’s hear it for consistency!) through all four operations so teachers can implement a simple daily routine. (Simple is good!) The key uniqueness of the program is the slow introduction of new facts and the structure that will allow students to progress towards mastery at their own pace. This has proven to be a powerful ingredient in the success experienced by the students and appreciated by the teachers as students become fluent in math facts. This way of organizing math facts is both research-based and has had many years of on going field success. Allowing quick learners to master higher operations while students who require more time on an operation proceed at their own rate, builds confidence along with mastery. Each day’s routine takes no more than six or seven minutes of class time…Really, six or seven minutes!
The program’s daily component consists of one page practice sheets lettered A-Z. The outside of each page gives practice focused on 2 facts and their inverses. (Only four new facts at a time you ask? Yep!) The inside is a timed test on the facts learned thus far. A timed test every day and YOU don’t have to create it! (Sounds good to us too!) Students who meet or beat their goal get "glory" and a new sheet to work on (with two new facts and their inverses of course). Students who do not pass, take home the test and practice and repeat the page again the next day. Same thing happens the following day: practice on the outside on the facts learned thus far, then take a timed test on the inside. Students record their efforts on a rocket chart on which they color in lettered squares for each sheet they pass.
There are placement tests for accelerating students into an operation in which they already have some proficiency. There are tests of writing speed so that adjusted goals can be created for students who can't write fast enough to answer 40 problems in a minute. Students with such adjusted goals are expected to always meet or beat their previous best to pass. How smart does this sound? Sounds smart to us too!
Within each operation there are also five progress-monitoring tests (2-minute timings of ALL the facts). Every week or two students take these two-minute tests and graph the results on blank graph forms provided. This enables the teacher and students to keep track of their progress in memorizing these facts. These progress-monitoring tests are powerful tools for encouraging students as well as perfect data collection and management tools.
New Voluntary Math (and Reading) Standards Being Established
It has recently been announced that a new team of people is working on writing new, better and more rigorous K-12 standards for both language arts and math. This being a math oriented blog; we are more interested in the math standards. This new effort at writing standards is an attempt to create model standards that make more sense,are more measurable and are more rigorous than the hodge-podge of current standards. These will not be "national" standards because they won’t be imposed by the federal government, but will be voluntary for the states to adopt.However, 49 of the states have signed on and are involved.
So the question is,will these standards be better? And where do they stand on the "math wars?"There is reason for hope, because at least some of the people involved have clearly articulated positions on what is wrong with the current standards and what would make good standards. The executive director, Gene Wilhoit, is quoted as saying, "Fewer, clearer, and higher standards will help us get there [where every child is successful]." We are in favor of all three:fewer standards, clearer standards and higher standards. For the record, we are also in favor of motherhood and apple pie.The official announcement from the NGA (National Governor’s Association.) is available at the bottom, but we want to direct you to a couple of resources about some of the names on the list.
Not so long ago,three of the members of the math feedback group (Jim Milgram, of Stanford;William Schmidt of Michigan State and Stephen Wilson of Johns Hopkins), spokeat a Leading Minds forum on K-12 Math Education. The whole forum is available at www.baltimorecp.org/leadingminds/index.htm.In the forum, all three were clear that the math standards in most states were too diffuse, did not make pedagogical sense, and did not achieve a world class mathematics understanding. William Schmidt was interviewed in Class Notes—the Baltimore Curriculum Project newsletter where he was very clear that standards needed to be more focused, more rigorous and more coherent. The full and informative interview is here, http://www.baltimorecp.org/newsletter/BCPnews_spring09.htm#schmidt.
Of course our hope is that sensible standards will put more emphasis on learning fundamental skills as the prerequisite to more advanced math concepts. And the most fundamental skill is the recall of math facts. Like the NCTM Curriculum Focal Points state: children need to develop quick recall of addition and subtraction math facts by grade 2, www.nctm.org/standards/focalpoints.aspx?id=326,and quick recall of multiplication and division math facts by grade 4 www.nctm.org/standards/focalpoints.aspx?id=332. It would be a huge step forward if the new voluntary, not-really-national standards brought back sanity to math standards. It would good for all of our country’s children.
The official announcement from the National Governor’s Association, www.nga.org/portal/site/nga/menuitem.6c9a8a9ebc6ae07eee28aca9501010a0/?vgnextoid=60e20e4d3d132210VgnVCM1000005e00100aRCRD.
Welcome
