Issues in implementation

Are students really “friends?”

Posted on by Dr. Don

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

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

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

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

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

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

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

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

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

cheetah running fast

Timed Math Fact Fluency Expectations by Grade Level

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 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).”

Timed Math Fact Fluency Expectations by Grade Level

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 the 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 you expect a timely response 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 take tests 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 35 problems per minute rate seem to be the lowest noted in the literature.

The Correct Math Fact Rates

Other authors noted research that indicated that “students who can 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 seems more likely to continue to accelerate than the lower end at 30.

What is the recommended time to finish problems?

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 can write digits (Stein et al., 1997, p. 87).” For example, a student who writes 100 digits per minute expects to write 67 digits per minute. This 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 you multiply that percentage  by 40 problems to give the expected number of problems per minute. For example, a child who writes 75 digits per minute would expect 75% of 40 or 30 facts per minute.

If measured individually, a response delay of about 1 second would be automatic. In writing, 40 is 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, students can count answers on their 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.

Conclusion

With the right tools, any student can develop math fact fluency and have fun while doing it! Students use Rocket Math’s Subscription Worksheet Program to practice with partners, then take timed tests. Rocket Math also offers math facts practice online through the Rocket Math Online Game. Students can log in and play from any device, anywhere, any time of day! Start a free trial today.

Both the worksheet program and the online game help students master addition, subtraction, multiplication, and division math facts.

 

 

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.

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!

What best honors & motivates achievement?

Posted on by Dr. Don

Recognition for real, tangible accomplishments, that not everyone gets.

What makes for a great award, or great recognition that really motivates?  In the final analysis, recognition, like an olympic gold medal, is not about what you receive–it’s about how hard you worked to get it.  If students worked hard, and accomplished something real and tangible, then the recognition they are given, regardless of its form, will be valuable and meaningful.  A paper certificate given out by an adult that represents weeks or months of effort, an honest accomplishment, will be highly prized.  Those are the certificates that are posted prominently in the bedroom or on the refrigerator at home, because it was hard to get.

Remember that when you want to honor student achievement at the end of the year.  If you give awards to every student, then an award means little or nothing.  If on the other hand, students know they had to work and put forth effort to earn the reward, then it is a real honor.  Rocket Math has many built in landmarks of accomplishment that are great to recognize publicly.  Certainly completing an operation is one of the most commonly celebrated achievements.
Chase

This student’s teacher tweeted this picture of the student and his Rocket Chart, proving his accomplishment.  This is something to be really proud of, because it represents a real, tangible accomplishment.  Another accomplishment is when a student beats his or her individual best in two-minute timings.  Yet another tangible Rocket Math accomplishment is being able to pass two levels in one week or ten levels in a month!

What motivates students to try to achieve is knowing what has to be done and believing they can do it. This is another reason why recognizing real, tangible accomplishments works so well. If the other students can see what their recognized peer did and they understand what has to be done to get there, they are motivated to get some of that glory for themselves. Getting through Level Z of Rocket Math is something students know they can do, if they just keep working at it. It is hard to believe you will become Student of the Month, if you don’t know what the previous recipients did to achieve that honor. But if you know that working hard and practicing your math facts every day can get you there–then you can believe it is possible.

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!

Can all 2nd graders finish subtraction?

Posted on by Dr. Don

Julie writes,

In our district, we have data that shows students are struggling with subtraction. We really want to put emphasis on getting the subtraction facts memorized. What are your thoughts about 3rd grade starting with subtraction in the beginning of the year and switching to multiplication the second half of the year regardless of having completed Z in subtraction?

Thanks, Julie

Dear Julie,

Multiplication facts are VERY important and you want to be sure that all third graders have enough time to master them. The best solution would be to get all your second graders through subtraction during the second grade year. Here are some suggestions to get better results–where more students develop automaticity in subtraction facts in second grade.

1) Start Rocket Writing for Numerals in the second half of Kindergarten, so that students leave K able to write numerals quickly.

2) Start Rocket Math Addition in the first month of first grade (because most all students) are able to write numerals fast enough. If you have a few who are below 18 boxes in a minute–give them extra work on Rocket Writing for Numerals to get them up to speed and into Rocket Math Addition before the end of October.

3) Monitor closely students who don’t pass an operation in six days and make sure they are getting a bonus practice session each day at school or encourage their parents to practice with them at home. Some few students will need extra help to practice twice or three times a day to make progress as fast as others.

4) Save folders from grade to grade (over the summer) so that students don’t have to start an operation over from the beginning (just continue on from where they left off) and so can finish an operation in a timely manner.

5) If students get stuck for more than a week, and definitely when they come back from the summer in the middle of an operation, have them practice the test problems orally as well as the outside problems. If they are only practicing the outside, and they are slow or have forgotten some of the cumulative review problems in the test, they need to practice the test problems (as well as the newer problems on the outside) to bring them up to speed.

I think that continuing a student in subtraction at the beginning of 3rd grade from where they left off in 2nd grade is an OK policy, but it is far better to have as a clear goal to get students through subtraction during 2nd grade. Hope this helps. -Don