learning

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’t I copy answer keys for half the students?

Posted on by Dr. Don

Shane asks: After the answer keys are copied onto colored paper, can’t I just make enough copies of answers for half the students? It seems that they will only be using the answer keys while working with a partner and therefore will only need 1 set of keys per pair.

 

Dr. Don answers: Lots of people think this, but here are four examples of issues that make it preferable for each student to have their own answer key, and yes, it should be on colored paper.

1) When students are absent you must pair two students but under the one-answer-key-per-pair both students could be “without” answer keys!  In both cases, their partner has the answer key and that folder is in their desk.

2) When someone comes in to help or volunteer, you want Johnny to practice Rocket Math with that person–but Johnny doesn’t have an answer key–his partner does. So Johnny has to go searching for an answer key.  If Johnny had his own answer key he could just get out his Rocket Math folder and go to work.

3)  The Title 1 or Special Ed teacher or instructional assistant might offer to do extra practice with a student, the student takes his/her folder down to the a place to practice–but doesn’t have an answer key.

4) Alex moves up to division, but his partner doesn’t have an answer key to division–another example where Alex needs his own answer key.

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.

student counting on their fingers because they do not meet math fact fluency expectations by grade level

Facts practice: does it belong in middle school math?

Posted on by Dr. Don

It sure does, if you’re seeing this happen in your class!

Most middle school math teachers confide to me that their classrooms are negatively impacted by the number of students who stop to count out facts on their fingers.  Their issue was always what to do during facts practice with the other students who do know their facts.  It has taken a couple of years but I have put together a package of pre-algebra skills that are worth middle school students’ time practicing which are available in the Universal Subscription. Because the routine of Rocket Math is the same whether the students are practicing basic multiplication facts or learning equivalent fractions you’ll be able to manage all these different levels during the same ten-minute session.

Teachers know it is imperative that finger-counting middle schoolers get practice learning their facts.  Rocket Math is an excellent way to do that.  They will develop fluency and automaticity with the basic facts in an operation in a semester and from then on your lessons will be much easier.  Not only that, but a much higher proportion of the students will be finishing assignments.  There is a “Placement Probe” that can identify students who know their facts in about one minute. The students who know the basic facts of multiplication and division can be placed into the pre-algebra practice programs.

Factors Answers AFACTORS. Students probably ought to begin with the Factors program. What are the factors of 24? Answer: 1 and 24, 2 and 12, 3 and 8, 4 and 6. This is what students learn by memory from doing this program. Students practice with a partner, take a daily one minute timing, fill in a Rocket Chart, just like regular Rocket Math. Students learn all the factors for these numbers in this sequence: 12, 36, 24, 48, 18, 32, 16, 64, 10, 40, 20, 72, 8, 25, 50, 6, 21, 30, 60, 15, 45, and 100.

 

 

Fraction Number Line GEQUIVALENT FRACTIONS.  Students need to know that six-eighths is equivalent to three-fourths and that four-twelfths is equivalent to one-third.  While they can calculate these, it is very helpful to know the most common equivalent fractions by memory.  One of the most common problems students have in fractions is not “reducing their answers to simplest form.”  Equivalent fractions will help students commit 100 common equivalent fractions to memory.  Each set (A through Z) has four fractions which are displayed on a fraction number line.  Students frequently learn fractions equivalent to one,such as ten-tenths, as well as fractions that can’t be reduced, for example three-fourths is equivalent to three-fourths.  Using the fraction number line will help with student understanding of why those fractions are equivalent.

Integers ArrowsINTEGERS (Adding and subtracting positive and negative numbers).  Integers displays problems on a vertical number line and then teaches students two rules about how to solve problems that add or subtract positive and negative numbers.

Rule 1: Go up when you add a positive number OR subtract a negative number.
Rule 2: Go down when you subtract a positive number OR add a negative number.

Students gradually learn several variations of all four types of problems.  They practice with the number line on each page and then have a chance to build fluency on the top half of the page as they work with their partner.  You will probably not be surprised that there is a one-minute test on each set.  The goals are slightly different than before.  Students are to be 100% accurate and to complete at least 80% of their rate at answering simple addition and subtraction problems.

10s, 11s, 12s Multiplication and 10s, 11s, 12s Division facts are also available in the Universal Subscription.  If you have students who think they know the basic facts, but need review, putting them into either of these programs will review the 1s through 9s facts, teach them new ones and allow them to save face.

Among these five programs there are good things for ALL middle school math students to learn, even the more advanced students.  This will enable a math teacher to devote ten minutes a day to fact practice without holding anyone back.  Everyone will have something meaningful to practice during that time.  I think this could be a huge step forward for a lot of middle school MATH classrooms.

 

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.

Why do multiplication facts have priority after 3rd grade?

Posted on by Dr. Don

Because older students CANNOT succeed in math without multiplication facts.

Am I sure? Yes, I’m sure. “But,” you say, “my students are still counting addition and subtraction on their fingers.”

I know. And I am still sure—fourth grade and up—multiplication. Why? Once children are in fourth grade it is critical that teachers make sure they memorize multiplication facts—primarily because you can’t be sure of how much help they will get later to learn the math facts. Sadly, your students may only learn one operation to fluency. If so, multiplication facts have priority over addition and subtraction. Besides complex multiplication and division, the multiplication facts are needed for success in fractions and ratios. Students have to immediately see the relationships between numbers in order to understand topics like equivalent fractions, reducing fractions, combining unlike fractions, as well as ratios. Let’s be honest here…those are the things that state tests LOVE to ask about. Not to mention, these are the pre-algebra skills students need to be successful in algebra and the rest of math.

If you have the students for long enough (at least one year) you may find that they finish and have mastered both multiplication and division facts. Then you can go back and have them learn addition and subtraction facts as well.

Don’t get me wrong — I know that addition and subtraction facts are VERY IMPORTANT — it’s just that multiplication is MORE IMPORTANT.

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!

Challenge your students with the Race for the Stars Game Center!

Posted on by Dr. Don


How can you entice your students to play the Race for the Stars game in a center?

By letting them post their record/best time for completing the game boards. Race for the Stars is a great game for students to practice their math facts. The game provides 24 problems tiles students can race to put down next to the answers in the game board as fast as they can. There is now a Game Center Kit that give you everything you need to set up a center in your room that students will want to visit. The key is the poster with room for student labels with their names. Then next to their name is a place for them to write their best time at filling the game board. Well, actually their partner with the stop watch should probably write down their time. All the students in the class can rotate through the Game Center to record how fast they can fill either or both of the two game boards. One, for levels A-K are the first facts to be learned in Rocket Math. The second game board, L-Z are the facts learned in the last half of the Rocket Math levels.

But how can you motivate your students to play the game again and again to get the practice they need?

Here is the coolest part of the Race for the Stars Game Center. Students can go back and get someone to time them filling the Race for the Stars gameboard again. If the timer with the stopwatch sees that the student beat their previous best, they get to record the new record time, AND cover the old time up with a star sticker. Getting to put up a star sticker next to your name proves you were able to beat your own previous best–that is impressive! In fact, the savvy teacher will make a daily stop to see who in the class has been able to add a star sticker to the poster next to their name, showing that they beat their previous best time.

And that is how you motivate students to use the Race for the Stars Game Center (item #2112) to practice their math facts in their spare time!

Can you avoid summer losses with Rocket Math?

Posted on by Dr. Don

Take up where they left off before the summer!

Don’t think students have to start over in Rocket Math. They have learned the facts so well that with careful review they can take up where they left off!

There is a way to start off the year on the same set on which students left off at the end of last school year (providing you know where that was). You do need to do a slightly different procedure at the start of the year, however.

Notice on the Set L sheet above that only five of the facts on the One-Minute Test are new–the rest are review from the previous sets. That means that practice around the outside will help with the new facts, but won’t review those older facts from previous set. If you test the students on any set after the summer they might not pass because they need a little review of those older facts.

Here’s the way to beat the summer forgetting:
For the first week of school have the students add another practice session with the One-Minute Test each day. Give them the test answer keys, give them 2 to 3 minutes with their partner to orally practice the test problems with the same correction procedure as usual. Hint: have them take the sheets home and practice with a responsible sibling or a parent as well. A week to ten days of this extra review of the test problems and they will be successfully passing levels in Rocket Math–starting right from where they left off in June!

Five interventions for frustrated students

Posted on by Dr. Don

Erica writes:
My son currently is in second grade and uses Rocket Math in his school. He has been doing addition and is on Level “S”. Most of his class has moved onto subtraction and multiplication.
My concern with him is where does this leave him next year in 3rd grade?? Is he left behind? He knows the addition facts orally but fails to meet his goal on the 1minute drills due to his anxiety and frustration with being timed. He struggles to move forward even though he knows his addition facts!
With this scenario, how does Rocket Math help? How will he ever move on to learn subtraction and multiplication? He’s a smart kid but can’t seem to succeed with this method!
Please help me to see otherwise!

Dr. Don answers:
Erica, I can see why you are frustrated. Students should not take more than three to five days to pass a level in Rocket Math and no more than a year to pass an operation such as addition. The rule is: If you child is frustrated by Rocket Math–it isn’t being done right! The school should not be complacent and should not leave your son to fall way behind his classmates. If a student takes longer than six days to pass a level I recommend that the school or the teacher should intervene. Interventions should happen in a matter of days, rather than allowing students to stall for weeks or months. You are writing in response to my post, so you know you can read the directions, even get the program and work with your son at home to help him. What interventions should be tried with your son?

1. My first intervention would be to practice Level S test (inside the box) orally with your son in the evenings once or twice. If he has been on Level S for a long time, there are facts on the test that need review–through oral practice. Use the same correction procedure we recommend everywhere: if he makes any hesitation, give the correct problem and answer, have your son repeat it three times, then back up three problems and go again. A couple of days of that and he should pass handily at school.

2. If that didn’t help your son pass in two days, my second intervention would be to watch your son take a written test and see what is going on, see if there is evidence of frustration, or anxiety, or if there are behaviors during testing that are interfering with good test results. If you want to send me video of your son taking the test at home, I could probably tell what is the issue that is holding him back. There may be some test-taking behaviors he can learn, such as not stopping during the test, or not erasing sloppy answers, which would improve his test results.

3. The third intervention I would do is give the Level S test orally. There are a number of reasons a student might not be passing and I have blogged and have video clips on YouTube that address what to do with students who are “stuck.” You write that you know that he “knows his facts.” Probably because when you ask him a fact, he can answer it immediately, without having to stop and figure it out. If that is true he should be able to verbally tell you the answer (not read the problem, just say the answer) to 40 facts in a minute at Level S. You could test him at home to find out if he does know the facts at Level S. If he can orally answer 40 facts on the Level S test, he knows the facts well enough and should have passed. If he is not passing the written test even though he can verbally say the answer to 40 facts in a minute, then his writing goal is off for some reason. Another piece of data that would suggest his writing goal is off, is if he has been stalled at some rate of problems and hasn’t improved his rate for a week. That suggests that that number of facts (whatever it is) is all he can actually do, and his writing goals need to be revised down to the number he can write when taking a one-minute written test (assuming he is on a level on which he can verbally answer 40 problems in a minute). In a couple of places in the directions (FAQs), I explain that, so you can share with the teacher.

4. What if, as you suggest, he is freezing up during the written test due to “anxiety and frustration with being timed?” The best way to overcome anxiety is to keep doing the thing that makes you anxious, which is why most students stop being anxious about Rocket Math after a couple of weeks. A fourth intervention would be to practice taking the test in writing–but untimed. If he completed all the items on the test several times at home, untimed, he would stop being so anxious about doing it under timed conditions. Most students also understand why they are being timed (to see if they know the facts without hesitation). He will not get unduly frustrated if you explain to him this is just a race and if he doesn’t give up he will keep getting better until he wins. Of course, if his writing goal is too high and he can’t possibly meet the goal, he will become frustrated.

5. If your son cannot already orally say the answers to 40 problems on the Level S test in one minute, he needs some more practice. My fifth intervention would to practice with your son–be his partner. He may not have a conscientious partner at school and may not be getting the most out of his 2 minute practice time. I routinely find that when I practice with students (the right way with correcting hesitations as well as errors) even once, they suddenly pass or come very close. The quality of the practice is critical to learning to answer these facts without hesitation. If practice allows students to stop and figure out the fact every time they will take a very long time to get to knowing those facts instantly. If that is the case, if you practice orally with your son once or twice an evening at home, the right way, he will begin to pass every few days at school. He will finish addition this school year.

In third grade I recommend that all students start multiplication at the same time even if they have not “finished” subtraction. Multiplication facts are far more important, so subtraction facts can wait. If you closely follow your son’s progress in learning multiplication facts with Rocket Math, you can intervene in time to make sure he does not get frustrated or fall behind. It should take him three to five days to pass a level, but no longer than that. With some extra practice at home you can be sure he will be successful. Knowing basic facts instantly will be very important for him, so don’t give up!