Why not start with subtraction in 3rd grade?

Posted on by Dr. Don

Julie asks:

Hi Don, My staff has a question about which operation to start with. 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!

Dr. Don answers:

Dear Julie,
Your teachers are right that a lot of students may not be fluent with subtraction facts. There are several reasons for that. And yes, it would be possible to start with subtraction in 3rd grade and then switch to multiplication as students finish, or by mid-year at the latest. But I would not recommend it because you will then have a problem with not every child getting through multiplication in 3rd grade, which results in a similar problem in fourth grade. What would be better would be to get every second grader fluent in subtraction facts before 3rd grade.

Why? It is important to understand the problem before specifying the solution. Students have trouble learning subtraction facts primarily because they have not achieved automaticity in addition facts first. And why aren’t they automatic in addition facts? Usually because they didn’t start early enough and work on addition facts long enough in first grade to get to automaticity with addition facts.

A second reason students don’t master subtraction during grade 2, happens when the school doesn’t keep track of folders from first grade. If students have to start completely over with addition in second grade, they don’t have enough time (if they are a child who needs a bunch more time to learn facts) to get through both addition and subtraction. They go slowly through addition again, and don’t get into subtraction until well after the middle of the second grade. So the first push is to try to get everyone passing subtraction in 2nd grade.

What you don’t want to do is start over again in subtraction in third grade and struggle through that all year and then not have enough time to master multiplication in third grade. Because multiplication facts are so important, it would be better to do the reverse. Start with multiplication in third grade–because it has priority–and then for those who finish multiplication allow them to “go on” to subtraction. It is much better to start fourth grade strong in multiplication facts (even if you still count on your fingers for subtraction) than to be a fourth grader who is strong in subtraction, but unable to answer multiplication facts!

What does CCSS mean by “know from memory?”

Posted on by Dr. Don

Knowing from memory means not having to think about it.

Two of the best standards from the Common Core State Standards are on our home page:

By end of Grade 2, know from memory all sums of two one-digit numbers and

By the end of Grade 3, know from memory all products of two one-digit numbers.

These standards name the most important elementary math skills of all, because they are the foundation of all further work in mathematics.  But what does it mean to say students know math facts “from memory?”  It means that students don’t have to stop to figure it out.  Say for example a student is adding nine plus seven. A student can figure that out by thinking that because 9 is one more than 8 and 7 is one less than 8, the answer to 9+7 would be the same as 8+8, which is 16.  This is a smart strategy for figuring out the answer, but knowing it from memory means the student simply remembers the answer is 16.

So if second grade students know from memory the sums of all single digit numbers, they can answer any of those problems without hesitation, without having to stop and think about them.  That takes practice, to build up the neural connections, so that students remember the answers instantly without some intervening thought process.  That’s what Rocket Math is specifically designed to do.  Practicing figuring out the answer to facts is NOT the same thing as recalling them from memory.  So any practice procedure that allows students a long time to answer facts, allows hesitations, will not be very helpful in achieving that status of “knowing from memory.”

The peer practice procedures in Rocket Math require the “checker” to follow a “correction procedure” whenever there is a hesitation.  If the student has to stop even for a second to “think about it” they need more practice on that fact to commit it fully to memory.  The “correction procedure” provides that extra needed practice.  Having students complete worksheets on their own will NEVER eliminate that “stopping to figure it out.”  That is why the oral peer practice in Rocket Math is essential.  And that is why Rocket Math really will help students come to “know from memory” all sums of two one-digit numbers.

How do students correct in Skip Counting?

Posted on by Dr. Don

Principal Luebke writes:
Dr. Don,
We have some fast Rocket Math students at our school. We want them to keep working and improving all the time. I want some students to start the skip counting function. What is the correction procedure while practicing? Do the checkers say the next number is ___, start over? Thank you,

Dr. Don answers:
Great question! There should have been some special directions in the Skip Counting Drawer for teachers. I fixed that this morning. Here’s what I posted there.

How Students Should Practice SKIP COUNTING

Students should practice by saying the skip counting series in order from memory. They learn the harder series in parts, so they only have to say the part they are learning at that point. For example, in Set G students are to learn the first four numbers of the count by 9s which are 9, 18, 27, 36. When practicing in Set G, the checker says: “Count by 9s to 36.” [That’s exactly what it says on the little cloud at the base of the rocket, making it easy for the checker!] The student then says, “Nine, eighteen, twenty-seven, thirty-six.” Of course, in Set H the student says the 9s to 63, and then in Set I all the way to 81.
Saying series in the same order every time is very important as it creates the verbal chain. Eventually, after many repetitions, an amazing thing happens. Whenever the student starts to say the first part of the skip counting series the rest of the series will pop into mind unbidden. (I try to use the word “unbidden” at least once in everything I write – just because I can.) This automatic coming-to-mind is called “automaticity” and is the goal of practice.

The student should say the series in order without any hesitation. I really mean NO hesitation! Now I will say that a few different ways to prove that I am really serious. I want students to practice these series until they are as automatic as saying their name. If even a slight pause is needed to think of the answer, I want them to practice until it comes to mind without any effort at all. This will enable them (after these series are learned) to easily learn multiplication facts and go on to concentrate on the higher functions of math.

CORRECTION: Each time an error or hesitation is made, the helper/checker should follow the following correction procedure. It is really important to do this correction procedure. The correction procedure is part of that “secret important stuff” that makes Rocket Math work.
1. Helper states the whole series up to that number, for example: “Nine, eighteen, twenty-seven, thirty six.” (If the student has said the right answer but hesitated somewhere in the middle, the helper can confirm it by saying, “Yes, that was right, but you hesitated, so let’s practice that some more. Nine, eighteen, twenty-seven, thirty-six.”
2. After the helper says the series once, the helper and the student should say the series together twice. “Say it with me: Nine, eighteen, twenty-seven, thirty-six. And again, nine, eighteen, twenty-seven, thirty-six.” Then have the student repeat the skip counting series three times.
3. Go back and do the previous series [just one, not three!], which is enough so this series comes up again before the student forgets it. (Rinse and repeat as necessary.)

Note that this same correction procedure is to be used each time there is an error or hesitation. If the student hesitates again after they went back one series and started again, just repeat the correction procedure. Say it together twice, then three times without help, go back one series and start again. Repeat this practice until there is no hesitation. Extra practice on a series, to lock it into memory, is important work and should not be considered a bad sign. THAT is what we are doing here—LEARNING!!

Getting stuck in Rocket Math Worksheet program–A solution

Posted on by Dr. Don

A teacher asks:
We have several students that are highly skilled in the math area but are also on the borders of perfectionism. They are having difficulty passing the writing “40” goal even though they all did more than that on their “Writing Speed Test.” They easily pass when they say them orally. What would be your recommendation to do with these students or tell their parents? The classroom teacher is willing to listen to each one of them to see if they can pass all levels on just one oral try but really doesn’t want all students or parents to start expecting this. He says these students are truly some of his top students (95th % on state standards).

Dr. Don answers:
Thanks for letting me know about their results in the Writing Speed test, and the fact that they can answer over 40 problems in one minute orally. This is clearly an example of my nice clean theory meeting the mess of reality. There is no logical reason why students who know the facts well enough to pass orally AND who have the handwriting skills to write the answers are not able to pass the written tests. It doesn’t make sense to hold them back from moving along and learning more facts if they are automatic with the facts–as demonstrated by the oral test on each level. The point of the Daily One Minute tests is to find out if the students know the facts without hesitation. If you know that is the case, you want to move them along to learn more facts, BUT…..
You want a policy that encourages students to pass the written test if they can at all, because that is more efficient and more fair. On the other hand you don’t want to hold students back completely if they really know the facts without hesitation.

Here is a possible policy that will balance the two. If a student feels he or she really knows a level, after two tries in class that student can choose to stay after school (or in at recess or come in early) and take another test with the teacher. [This allows the teacher to watch the student take the written test to see if there are any maladaptive behaviors such as erasing answers or skipping around that are causing the problem.] Then if the student doesn’t pass on the written test, the teacher can then listen to the student orally say the answers and if the student answers more than 40 problems in a minute, award the level as passed. One level at a time.
The students still have to try in class two days and try once more after class, but then can move on if they really know the facts. This puts some of the burden on the students.  We want to be sure they aren’t just doing it orally because it is easier and gives them more attention. This two tries policy gives them an opportunity to save themselves some time if they can pass in writing, but ensures that they move along as they need to academically. It also allows the teacher to watch the written testing to see what is going on there.  I think it will seem fair to the other students (who are passing in writing) because these students are not getting a special pass–instead they are having to come in on their own time to do this.
As I noted in another post it is extremely important to preserve the value of doing the work to pass the levels in Rocket Math. The work and the level playing field makes the whole exercise of Rocket Math meaningful and valuable. Don’t let anyone pass levels without doing the practice and taking the tests on each level. Otherwise you make the other students feel like dopes for having to work at it when others get it for “free.”

Teaching the value of hard work

Posted on by Dr. Don

A teacher asks:     

Our teachers just had parent/teacher conferences and had a few parents concerned about their student “not passing” levels in Rocket Math. The students AND parents of these students are having a hard time with their child struggling on Rocket Math when it is apparent that they “know” their facts. Their parents don’t know why they should have to have the speed when they clearly know their facts and these students are truly some of the top students (95th%ile on state standards). Although it has given those students some perspective on what it feels like and how you handle not accomplishing something with ease.  If they score 60 or above on their two minute timings consistently, should they be required to pass all levels?  What would be your recommendation to do with these students or tell their parents?

Dr. Don answers:

One of the most important benefits of Rocket Math is that it teaches students the value of hard work.  By practicing orally with their partner each day, and doing the correction procedure properly, students find they can learn math facts to the level of automaticity–to where they can answer them instantly without thinking and without hesitation.  That takes some practice and work, even for gifted students.  But everyone can do it with enough practice.  Although it is only ten minutes a day, the work of Rocket Math is very important in teaching students the value of their own efforts.  Students learn that even if they can’t pass initially, if they practice every day (and maybe some more at home with a parent or sibling), they get to the point that they can answer those problems as fast as they can write.  When they achieve this they are justly proud of themselves, because they know they earned the achievement through their own efforts.  Learning this lesson is quite possibly even more important than the math facts themselves.  This is an important lesson for life–that you benefit from working hard at something even if it doesn’t come easily.

The only way you could take that away from those students is by rewarding some of your brightest students with the same accomplishment without having to work through the levels.  You can use the placement probes to determine if students even need an operation–they can “test-out” of the operation in the beginning of the year.  But once you have determined that students need to work through the operation, the worst thing you could do to the class would be to suddenly announce that some students have “passed” without doing the work.  That would make everyone else feel like a dummy for having to work at it.

I will write a separate post on the things you can do for students who get stuck and can’t pass in six tries.  However, I want to stress that a key outcome of Rocket Math is learning the value of hard work in school.  Don’t do anything to undermine that.

How do you complete the Individual Student graph?

Posted on by Dr. Don

Here are four examples of how to complete the vertical axis on the Individual Student Graph.

Amy writes:
I have a question about the Individual Student graph form. Can you send me example of a completed graph? I understand marking 10 points lower but the 0…5…..0…5….0…5 axis confused me.

Dr. Don answers:

Amy,
Here are some examples of how you would fill out the vertical axis of the Individual Student Graph depending on what the student’s starting score was on the Two-Minute Timings. The form says, “Set starting point of vertical axis at the nearest ten below the student’s first 2-minute timing (e.g., if first timing is 37, begin graph at 30, etc.).”

If a picture is worth a thousand words, then these four examples should make the procedure clearer. Thanks for asking for examples–which is often the best way to explain/teach something!

If the starting score was 9, you would set the starting point of the vertical axis at zero.

If the starting score of the two-minute timing was 18, you would set the starting point of the vertical axis at ten.

If the starting score of the two-minute timing was 25 you would set the starting point of the vertical axis at twenty.

If the starting score of the two-minute timing was 34 you would set the starting point of the vertical axis at thirty.

Must students say math facts in a certain order?

Posted on by Dr. Don

It is actually more important than you might think, that students practice by reading facts in a consistent way.

Rachel asks:
Hi Don,
After using Rocket Math for a week, I have a question. My daughter often reverses the order of the numbers when reading off the facts (i.e. 1+5 when it’s really 5+1). Of course, this doesn’t affect the answer in addition, but I wondered if I should correct her? She sometimes does it upwards of 50% of the time. Anyway, I just wondered if I should be concerned about her reversing the numbers, and if so what I should do about it.   Thanks, Rachel

Dr. Don answers:

If I were still running a school I would be offering you a teaching job right now! What a good question! So your daughter is doing something that most people do, which is trying to simplify the task and ignore the difference in the order. Because 5+2 and 2+5 are both 7 why not just think of them as the same thing?***

However, there is a risk. If a student always says “five plus two is seven” and never says it the other way around they will not have the jingle-like memory of “Two plus five is…seven” in their brain. Then when they encounter 2 + 5 and read it aloud to themselves the answer won’t pop into mind automatically. They would probably puzzle a second, realize it is the same as 5+2 and then know the answer is seven, but it won’t be automatic. [That by the way is what I’m trying to illustrate in the picture above, which isn’t Rachel’s daughter!] We want that automatic answer to pop into mind, unbidden, without having to think about it. In other words, we want it so that when your daughter says to herself, “Two plus five is…” the answer “seven” pops into her mind without having to think about it.

Whew, this is a lot of rationale, but I know you can follow me. This means that you want to treat reading the problem in the wrong order as an error. When she reads the problem in the wrong order (says “Two plus five is seven” when the problem reads 5+2) correct by saying the problem in the correct order with the answer. You say, “Five plus two is seven.” This, by the way is why our correction procedure is for the checker to say the whole problem and the answer, so the checker can correct the order of reading the problem without causing confusion. Then have her repeat it three times and go back three problems.

She of course, will tell you, “But, it’s the same!” Just reply with, “You have to say it the way it is written.” You can tell her I said so!

PS. When you get to multiplication, this gets even more tricky, because there’s a good case to be made for reading multiplication fact problems up, because that’s how we say them when we are doing multi-digit multiplication problems. But that is a whole other blog!

***Interestingly, when doing the Rocket Math app, the learner/player is presented with both facts in the same set–mixed between the two as they are being learning. When I am playing the app, I find I can’t remember if I got both of those to answer or just one. Although the app gives both, I just put them together in my mind to make it easier, and don’t even notice the order.

How can you improve writing speed?

Posted on by Dr. Don

Tina asks:
Hello Don,
Do you have any recommendations for improving writing speed? My son’s school does not use Rocket Math, but we use it at home. He knows his addition facts rather quickly orally but is stuck at a much lower level at school because he cannot write them fast enough.
Thanks, Tina

Dr. Don answers:
Tina,
That is a very good question. Yes, you can improve writing speed. Increasing writing speed will come with practice, but a special kind of practice. The biggest problem slow writers have is that they “draw” the numerals. That is to say, they decide how to make the numerals look like they should and then draw them, rather than having a set way of doing the numbers. Step 1 is for them to learn how to most efficiently write the numerals using strokes that consistently go down and from left to write. Students need to learn the right way to form the numerals and then practice it exactly the same way over and over until it becomes habit. In Step 2 the students need to practice writing the numerals small enough to fit on the line, while still forming them the right way. In Step 3 and 4 students need to practice writing the numerals until they are fluent (speedy and still form them correctly and legibly).

A student can practice each page of Rocket Writing several times. How many times you ask? See my blog on the topic of How much practice is enough in Rocket Writing, because it is interesting to see that you can arrange it so that you trust your son to know how much practice he needs.

Rocket Writing for Numerals is part of the Universal Level of the Worksheet Program. If you only have a basic level subscription, you can upgrade to that.

Are extra practice sessions helpful?

Posted on by Dr. Don

More practice is helpful as long as it is motivating.

Rachel asks:
When doing Rocket Addition with my daughter: I plan to do 2 three-minute practice sessions during our school day with the one-minute timing after the second one. Then, if she doesn’t pass, should I have her work on those facts again in the evening, perhaps with Dad? Or do we just pick up again the next day?

I’m encouraged to hear that with enough time and practice she will be able to memorize math facts to automaticity. This is good news and motivation for me to keep working with her. Memorizing has always come easily for me, and I’ve tried many different techniques while remaining at a loss as to how to help her. After reading the teacher directions, I can see that I was introducing new facts too quickly, before the others were completely memorized. I love how helpful and directive Rocket Math is! Thanks again.

Dr. Don answers:
Rachel,
As for another session with Dad in the evening, it will only be beneficial if it is motivating and that depends on how you structure it and how she perceives it. It could be punishing or it could be a motivating treat. If she has some control over whether or not to do it, but she is “allowed” if she wants to show Dad how well she can do them (and more importantly he reciprocates by being impressed!) then by all means, “let” her do that. Also, if she doesn’t pass her test after the second session, and she wants to, she could have a special bonus chance to try to pass with Dad, but that would mean practice AND a test with him. Either of those scenarios could make the extra practice session in the evening a motivating treat and that would be good. If she perceives it as no fun and extra work when she should be enjoying time with her father, then don’t do it.

Oh, and in that regard do have her fill in her Rocket Chart and color in the levels as she passes them. That’s definitely something to show Dad when he comes home. And take a look at the Achievement Awards and use them from time to time.

By the way, as someone who has struggled with sport-like skills my whole life, it was a huge revelation to me that I could learn to do things if I was willing to work longer and harder and more consistently at it than anyone else. Knowing that I could develop mastery was the prerequisite to be motivated enough to persevere until I got there. If you can help her persevere until she masters these things you can help her develop the perseverance habit itself–which is way more important than math facts. You’ll need to be impressed with her hard work and mightily impressed by her accomplishments but once she sees for herself that she can achieve difficult things if she perseveres, she has learned a most important life lesson.

How much practice is enough in Rocket Writing for Numerals?

Posted on by Dr. Don

Students balance a desire for comfortable mastery against a desire for novelty.

A home-schooling mom asks:
After having read the Rocket Writing for Numerals teacher’s directions, I have a question about implementation: Should I have her do the same page twice in one day (at separate times) to help her get more practice? After re-reading the teacher directions again today, I also think I need to go back and do more demonstration and air writing.

Dr. Don answers:
Regarding Rocket Writing for Numerals, the focus of the air writing and demonstrations is to achieve accuracy and consistency in the way to form the numeral. Once she consistently knows how to form the letter (starting in the right place, making the strokes in the right direction, etc) then the rest is developing the motor skills. More air writing is not needed once formation is consistently correct.

Yes, you can have her do a page twice in a day. How many days in a row is needed before you can move on to a new page is not established by research. It would be different for each student anyway. If you watch her, then you can decide, or you can encourage her to decide.

You want a page to become easy or routine for her. She doesn’t have to do it perfectly, but don’t move on if she still seems to be struggling or having to go very slowly. You should move on if she seems to be unchallenged by the page. You can also engage her in deciding if she feels she is ready to go on to the next sheet or wants to practice on the same page some more. Generally, once children get the idea of what it feels like to master a performance, they want to do so and students balance that desire for comfortable mastery against a desire for novelty. My favorite image is of skateboarders in the park who practice and practice until they have a particular move down–but then move on to try something new when they think they have it.