math facts

Without the directions you may get lost!

Posted on by Dr. Don

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!

 

How fast should students be with math facts?

Posted on by Dr. Don

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.

References

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?

Posted on by Dr. Don

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.”

Can a few minutes of fact practice each day be harmful?

Posted on by Dr. Don

Practice is not harmful as long as students are successful.

The best way to practice math facts is by saying them aloud to a person who can tell you if you’re wrong or hesitant in your responses.  If you are wrong or hesitant, you should practice on that particular fact a bit more until you know it well. This is an effective way to learn anything, including math facts.  It is especially valuable if students are given a limited set of facts to learn at each step so they develop and maintain mastery as they learn.  If practice is set up carefully, and students get positive feedback showing they are learning and making progress, it is enjoyable and motivating for students.  This is the essence of Rocket Math.  How in the world could this be harmful?    Only by doing it wrong, and doing it wrong specifically in a way that students are not successful.

If teachers skip the practice and learning part and just give the tests–that would be harmful.  Students won’t get a chance to learn and will experience failure.  The daily oral practice is the heart of Rocket Math–it can’t be skipped!

Daily tests in Rocket Math determine if a student has learned the set of facts he or she is working on, and learned them well enough to have a new set to be added to memory.  If students are not proficient in the facts they are working on now (proficient means being able to say a fact and its answer without any hesitation) then they will become overwhelmed with the memorization and will not be successful.  So it is critical that teachers are certain (based on the daily tests) that students can answer all the facts up to that point without hesitation.  Otherwise they will not be successful and it won’t be enjoyable.

Goals for those daily tests must be based on how quickly students can write.  Slow writers must have lower goals. Fast writers must have higher goals.  Every student’s goal should be “as fast as her fingers can carry her” and no faster.  Arbitrarily raising those goals (expecting faster performance than possible) or arbitrarily lowering those goals (moving students on to the next set before they have mastered the previous set) will cause students to be unsuccessful.

If the checker does not listen and correct errors or hesitations, a student can practice incorrectly and learn the wrong fact.  They can also fail to get the tiny bit of extra practice they need on a fact that they can’t quickly remember yet.  If practice does not proceed as it should, then students will not learn as they should.  Lack of success will make facts practice onerous or counterproductive.  The teacher has to monitor students practicing carefully to make sure they are doing it the right way to be successful.

Rocket Math has very explicit instructions here and answers to FAQs here.  I have a 3 hour training DVD here.  I am available at don@rocketmath.com  to answer questions.  Practicing math facts ten minutes a day is NOT harmful, if we do it in the way that students are successful.

Will finger counting ever go away?

Posted on by Dr. Don

Some “experts” in education think that teaching and practicing rote information, like math facts, is unnecessary.  Just let students do math or do games and they will learn facts well enough to get by.  That is true for a few students, but many students continue to count on their fingers up into junior high and high school if we don’t help them commit these facts to memory!  So the short answer to the question of whether finger counting will ever go away, is “No!” unless we do something.

One of the things that is unique about Rocket Math is that students begin to learn facts well enough so they have instant recall.  By practicing orally with a peer, they are saying the facts and the answers aloud, and from memory, over and over again.  By doing that, students come to the point that, when they say that problem to themselves, the answer pops into their heads without effort, like the words to an advertising jingle.  When the answer occurs to them instantly, they realize they know the answer before they can count on their fingers, and they stop.  This is how finger counting goes away.  Students recall the answer before they have to start counting fingers.  The end of finger counting comes with the kind of daily oral practice that the procedures of Rocket Math provide.

Summer School? Use Rocket Math!

Posted on by Dr. Don

Hi Dr. Don,
We plan on using Rocket Math with our Summer School students in grades 3rd-8th. I know you recommend Multiplication for these students. We will only have these students for 12 days over the course of 3 weeks. Any thoughts on how you would implement it? Your opinion is greatly appreciated.
Yvonne K. Colland ELD – Instructional Technology Coach

Thank you, Yvonne,
Twelve days is VERY short, however students enjoy Rocket Math, if they don’t have to do it too long at one time. I would recommend three ten-minute sessions each day spread out as much as possible. So one at the start of the day, one in the middle and one just before going home. If there’s an hour or more between sessions students won’t mind. They can take it home and practice at home if they like.
The regular practice with a peer for 2-3 minutes, switch roles and take a test can be done in ten minutes.
The next thing you could do is set up the Game Center with the Race for the Stars Multiplication game set out. The Game Center has a poster for students names and they get to post their best time on each board of Race for the Stars. Students can put the 24 problems cards on race-for-the-stars-product-imagery-multiplicationeach board down next to the answers as fast as they can with a partner timing them on the included (Silent!) stopwatch. When they beat their best time on the board they get to post the new time and put a star sticker over their previous time. That would stimulate a lot of practice. Right now we have a coupon code FreeGameWithCtr that gives you the Race for the Stars game free with the purchase of item #2112 Game Center with stopwatch ($49).

Finally, if your school has iPads, or if you can have students bring in iPads or iPhones, there is the Rocket Math Multiplication App that sells for $2.99 (half that if you purchase through the Apple Education VPP).  Students can practice on that app for five minutes at a time and each device cRocket Math iOS Multiplication App Top Rated by Balefirean support up to three students.  Students love to play that game and you can print out Rocket Charts for them to keep track of their progress.  If they can take it home they’ll be sure to practice more there.

I’m not certain you could get all the students through Level Z in the three weeks you have, but you could make a big dent in it.  Most importantly they would know as many facts as they could learn which would put them ahead in learning the rest.

Do students practice sums to 12 or 18?

Posted on by Dr. Don

Julie asks:
For addition and subtraction, do students practice with sums through 12’s or 18’s? For example, 12+ 6. Thanks.

Dr. Don answers:
Hi Julie!  The Rocket Math basic program (and basic subscription) is 1s through 9s both for addition and subtraction–meaning students practice single digit sums up to 9+9 and subtracting single digit numbers up to 18-9. The Rocket Math Universal subscription provides access to the Add to 20 program which includes 7 + 12 and the Subtract from 20 program which includes 19-7. The Add to 20 and Subtract from 20 programs were added because the Common Core recommends students be fluent at adding and subtracting these numbers mentally without manipulatives.

As the picture above suggests if a student knows 7+2 is 9, it doesn’t take much to learn that 7+12 is 19. However, the Add to 20 program will give students practice with those facts. Especially for students who master the 1s through 9s facts quickly, these additional programs will cement in the basic facts learning by extending them to some teen number addends. Conversely, for students who struggle with learning the 1s though 9s facts, these extra facts should be considered optional or enrichment, in my opinion.

So I recommend first graders learn the 1s-9s addition first followed by the Add to 20. Then in second grade the 1s though 9s subtraction followed by Subtract from 20. Not everyone will get through both, but kids will see the connections doing the higher problems if they have completed 1s through 9s first. They won’t need to practice to Z before they see the patterns and can do the problems without much practice. I am happy to provide this extra option for those who can take advantage of it.

Are you ready for summer?

Posted on by Dr. Don

Preparing now can insure that students will maintain their Rocket Math learning over the summer.

(1) The simplest and most important thing you can do to get ready for summer is to save those Rocket Math folders at the end of the year. The folders can then be given to the next year’s teacher, so he or she knows where the student left off. Given special practice techniques at the start of fall (outlined below), students do NOT have to go back or start an operation all over again the next year. Some students take months to get where they are in an operation, and it is a terrible waste of their time to start them over. Especially if they have new faster writing speed goals, now they really have to work hard to master each set and it may take them quite a while.

(2) Make sure to take a few days to re-teach your students how to correct and when to correct (errors and hesitations).  Teach this by modeling errors and hesitations and have students be your checker and model how to correct for the other students to see.  Keep working with that student until you get perfect corrections even on hesitations.  Then “rinse and repeat” with another student.  Do this teaching and modeling for ten minutes each day for the first week or so.

Two students participating in one of Rocket Math's math fluency programs(3) Start students practicing on the last set completed (passed) the previous year but for the first five practice sessions, practice on that set in a special way. First practice in partners around the outside for two or three minutes. But then, instead of taking a written test, have students practice in pairs orally with the test (inside the box), for two minutes. Practice the same way as around the outside. Have the student read each problem aloud and answer it from memory. The checker will need to have the test answer key. Practice for two to three minutes and then switch roles. This practice will provide the necessary review of all the facts learned so far, and will bring them right back up to speed.

(4) After a week of oral practice sessions with the test, then allow students to take the written test. Evaluate students based on their writing speed goals from last year (don’t re-test and raise them). Arrange for extra oral practice on the test for anyone who doesn’t pass. In the extra practice, make sure they orally practice the test in the center as well. Keep up the extra practice, on that same set until they pass. They should get there in a few days. They already learned this, they are just bringing it back. They haven’t forgotten it, the connection just needs a little strengthening.

(5) If students finished an operation before leaving, you can start them on the next operation appropriate for their grade. Second graders who have finished addition, for example, would start with subtraction (1s – 9s), and then go on to Subtract from 20, then Skip Counting.  Third graders need to be taught the concept of multiplication first, but then should begin multiplication, regardless of what they completed earlier.  Multiplication is so critical for future success in math you cannot let any child in your room (if you are in 3rd grade or above) leave it without learning those multiplication facts.  Best thing you can do for their math careers.

Now that you know what to do–enjoy the summer!

What is the Rocket Math “filing cabinet on the web?”

Posted on by Dr. Don

Jo asks:
Can you explain the “filing cabinet on the web” a little more? Is this a place a teacher can print copies for the class or does each student have to have subscription?

Dr. Don answers:
A subscription gives you access to our “filing cabinet on the web.” This is a place on the web where we keep all the worksheets and a teacher goes there to print out what is needed. You just click on what you want and print it out. Each operation has its own drawer. Each drawer looks like this:

SubscriptionLayout

There are five drawers that can be accessed with the $29 basic subscription: Forms and Information, Addition, Subtraction, Multiplication, Division.

But wait there’s more!
There are currently 8 more drawers, the contents of which are only available to those who have the $49 Universal subscription. The 8 programs added in the Universal Subscription are: Rocket Writing for Numerals, Skip Counting, Add to 20, Subtract from 20, Multiplication 10s, 11s, and 12s, Division 10s, 11s, and 12s, Factors, and Integers. Click on the name of any of these programs and you can get more information on that program.
You can preview (before you buy) the whole subscription site here: Rocket Math Subscription preview site.