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A mom congratulates her son at a baseball game.

Positive Praise: Building Effective Teaching Habits

Posted on July 3, 2019July 8, 2022 by Dr. Don

Positive praise is one of the most effective ways to encourage wanted behaviors from students. Because building habits is not an easy task, here are a few things you can do to start easily incorporating positive praise in the classroom.

Be prepared with positive phrases
Develop the most effective wording
Start Small with two areas you would like to see improved behavior
Practice in the Classroom and watch the effect it has on your students
Grow and expand your positive phrases over time as you master the habit

Be Prepared with Positive Praise Phrases

I distinctly remember trying to help pre-service teachers build the teaching habit of positive praise. I would make suggestions and then observe. Trying to implement my suggestions wasn’t as easy as you would imagine – these teachers would glance in my direction and start the sentence “I like the way you’re . . .” and then trail off without knowing what to say.

Teachers want to use positivity and affirmation with their students, however, in my experience, they don’t always have the appropriate words ready to praise good behavior. Building the teaching habit of positive praise starts with getting the right words ready.

Recently I was reminded of this key component of building the new habit of making more positive statements. I wanted to personally develop this positive statement habit, but for some reason was not making the progress I had hoped for.

I quickly realized that I was making the same mistake I had watched the pre-service teachers make. I was unable to make more positive statements because I did not have any in mind that were ready-to-use.

To build the habit of making more positive statements, I would have to start memorizing some key phrases to keep on standby, ready to use when I needed them.

Positive Praise Example Phrases: How to Develop the Right Wording

The first step in positive praise is learning and developing the most effective wording. Using effective wording means you are getting through to your student, and clearly communicating that you appreciate the good behavior they are exhibiting.

Praise is most effective when it is prompt – when you deliver the praise in the moment. Can you picture a specific scenario in your classroom when many of the students are not doing as you asked, while a few students are dutifully following instructions?

This is the perfect scenario to use positive praise not only in rewarding students with good behavior but also encouraging other students to follow suit. Don’t be afraid to praise good behavior loud and proud for the rest of the classroom to hear!

Here are some examples of positive praise:

Look at Alan so smart sitting in his seat and showing me he is ready to learn. Way to go, Alan.
I see Beto is tracking with his finger while Claudio is saying the facts. That’s the way to help your partner!
Julia, you are so sharp having your eyes on the teacher, so you can learn! I am impressed.
Stacy and Sophia know just what to do, they have their books open to page XX. They are so on top of it!
Fantastic, Justin! You put your pencil down and are waiting for directions. I can tell you’re going to college.
Stephanie is being such a great on-task student by working quietly and not talking.

Start Small: Pick Two Key Behaviors You Would Like to See More Of

Start out by choosing wanted behaviors from the two most annoying or frustrating scenarios you face as a teacher. Stating small will help you build a consistent habit of giving positive praise.

Take these two wanted behaviors and build two praise statements you can easily use in-the-moment. Make sure the statement names the behavior specifically. Always include the student’s name, and keep it simple and affirmative.

Now, take a note card or piece of paper and write down these two statements. Don’t wait! Write them down now and keep this note in front of you while you teach. It will serve as a reminder throughout your day to incorporate positive praise as much as possible.

Practice saying these phrases aloud until you have them memorized and can recall them without having to think about it. The most important step in building this habit? Actually practicing positive statements in the classroom.

With these key components and diligent practice in the classroom, you will quickly build the habit of positively praising your students.

Positive Praise in The Classroom: Will it Make a Difference?

Fortunately, positive praise is free and can be implemented at any time throughout the school year. Start using positive praise now, and watch how your students respond.

Prepare yourself for giving positive praise when you are about to begin those frustrating scenarios. When the activity begins, look for opportunities to praise the behavior you are looking for when you notice students who are off-task.

You will see results when you use positive praise genuinely and with enthusiasm. You will know it is working if you watch for those distracted students taking notice of who is being praised. If you notice this happening, keep it up. The more praise you give for wanted behavior, the more that behavior will occur.

Grow and Expand Your Positive Praise Habit

Now that you know how to promote a specific behavior with positive praise, you can systematically develop statements for all your troublesome areas. Every time students are not doing what you want, think of what you want them to do instead. Behavior analysts call those replacement behaviors.

Positive praise can also be used creatively alongside other motivational tools in the classroom. When I began my teaching career I was in the habit of scolding behaviors I did not want. Early in my career, I learned the effectiveness of positive praise and began incorporating it into my daily routine.

When I saw the behavior I wanted I would give loud and proud praise for all to hear. I decided to couple this by adding marbles to a jar every time I gave praise, as an added motivational tool – so students could see how well they have been doing. It worked wonders on increasing wanted behavior.

Building new habits is never easy, but I can personally say that as a teacher, learning to incorporate positive praise into your teaching routine will not only help students learn, but it will save you a lot of frustration!

If you are currently looking for a job as a math teacher abroad, check out this link on Jooble.

Rocket Math Adds Beginning Numerals & Counting Program

Posted on June 13, 2019April 14, 2021 by Dr. Don

Beginning Numerals and Counting

Dr. Don has created another math program and put it into the Universal level virtual filing cabinet at Rocket Math. This is a beginning program for kindergarten students and is to help them learn counting and numerals. That means they can’t learn on their own, the teacher must provide instruction. Teachers can use the counting objects kindergarten worksheets to effectively teach students to count objects aloud and then match the word with the numeral. You can see the top half of Worksheet A above.

If you’re already a Rocket Math Universal Level subscriber, you can find the worksheet in your virtual filing cabinet. Not a subscriber yet? Get the counting worksheets.

I Do: Demonstration of Counting

Each worksheet begins with a demonstration of counting objects and circling the numeral that matches. On Worksheet A, there are only the numerals two and three to learn. The teacher demonstrates (best with a document camera so all students can see) how she counts the objects and then points out that the answer is circled. Suggested teaching language is something like this,

“I can do these. Watch me count the frogs. One, two, three.. There are three frogs in this box. So they circled the three. Everybody, touch here where the three is circled. Good.

How many frogs were in this box, everybody? Yes, three.

Now watch me do the next box.”

We Do: Counting Together

In the “We Do” portion of the worksheet, the teacher counts the stars first as a demo and then with the students. Worksheet A you all just count three stars. Suggested teaching language is something like this:

“Our ‘We Do’ says to touch and count. Start at zero and count each star.

We are going to touch and count the stars. Put your counting finger on zero,

everybody. We are going to start at zero and count each star. Let’s count.

One, two, three. We counted three stars. That was great!

Let’s do it again! Fingers on zero, everybody. Let’s count. One…”

By Worksheet S the teacher and the students are counting 12 stars together.

The program has a page of teacher directions with suggested language for teaching the worksheets.

You Do: Independent Counting

In the “You do” portion of the worksheet (after learning the numerals with the teacher), the students are asked to count the items in each box and circle the correct number. They are not asked to form the numerals–that’s numeral writing skill. They just identify the numeral and circle it. Besides cute items, there are also dice to count, fingers to count, and hash marks to count–so students can learn multiple ways of keeping track of numbers.

Passing a level requires 100% accuracy. Students who make any errors should be worked with until they can complete the worksheet independently and get all the items correct.

Rocket Math’s Counting objects worksheets for Kindergarten

This Beginning numerals program will build strong beginning math skills for kindergarten students learning the meaning of numerals. Combined with Rocket Writing for Numerals it will set students up for success in elementary math.

If you’re already a Rocket Math Universal Level subscriber, you can find the worksheet in your virtual filing cabinet [use your link]. Not a subscriber yet? Get the counting worksheets.

In What Order Should Students Learn Fast Math Facts?

Posted on April 15, 2019May 8, 2022 by Dr. Don

Basic, Optional, and Alternative—there are a lot of different Rocket Math programs. But which program should you use first? And in what order should you teach fast math facts? Well, it all depends on the grade you teach and the fast math facts your students have already memorized.

An overview of Rocket Math’s fast math fact programs

Rocket Math offers multiple programs because their are several ways to teach fast math facts. Here’s a link to a printable version of the different Rocket Math programs shown here.

The Basic Program

Rocket Math’s basic program includes Addition, Subtraction, Multiplication, and Division (1s-9s). The basic program must be mastered by all students.

The Alternative Program: Fact Families

There is another way to learn facts, which is called Fact Family math. Instead of learning all Addition facts, students can learn Addition and Subtraction facts at the same time. A fact family consists of four related facts, for example: 3+2 = 5, 2 + 3 = 5, 5 – 3 = 2, 5 – 2 = 3. As an alternative to using the Basic Program, students can learn fact families up to 10 in first grade. Then students can move on to the upper fact families 11 to 18 in second grade. There is no clear evidence that this way is better or the separate operations way is better. That’s why we offer both options.

Optional Programs

The rest of the fast math facts programs like Rocket Writing for Numerals or Skip Counting are optional. You should only offer these programs to students once they have memorized the fast math facts through the Basic Program or the Alternative Program.

The only exception would be in a school where Kindergarten students did not get a chance to learn how to quickly and easily write numerals. In that case, you might take the first two months of the first grade year to run students through Rocket Writing for Numerals before beginning Addition (1s-9s).

Let’s take a closer look at how to implement each program in different grade levels.

First grade math facts: Learn Addition

Rocket Math fast math facts programs for first graders include:

The Basic Program
- (1s-9s) Addition
The Alternative Program
- Fact Families (1-10) Add & Subtract
Optional Programs
- Rocket Writing for Numerals
- Add to 20

If first grade students are taking all year to get through sets A-Z in Addition in the Basic Program, they need some extra help. You should intervene to help students who take more than a week to pass a level. Often they need to practice better or practice with a better partner. Some may need to practice a second time during the day or at home in the evening. First grade students who finish the 1s-9s can move on to the Add to 20 Optional Program for the remainder of the year.

Likewise, if you choose to teach Fact Families (1-10) Add & Subtract from the Alternative Program instead of using the Basic Program, your students can use the Optional Programs for supplemental learning purposes.

Second grade math facts: Learn Addition and Subtraction

Rocket Math fast math facts programs for second graders include:

The Basic Program
- (1s-9s) Addition
- (1s-9s) Subtraction
The Alternative Program
- Fact Families (1-10) Add & Subtract
- Fact Families Part Two (11-18) Add & Subtract
Optional Programs
- Subtract from 20
- Skip Counting

Second grade students must have completed Addition before starting on Subtraction (1s-9s). They can also test out of Addition through the Placement Probes. Second graders who cannot test out of Addition in first grade or didn’t complete it in first grade must focus on Addition. Only after getting through Set Z of Addition should they move into Subtraction.

You can substitute the Basic Program’s (1s-9s) Addition and (1s-9s) Subtraction for the Alternative Program’s Fact Families (1-10) Add & Subtract and Fact Families Part Two (11-18) Add & Subtract.

Second grade students who complete Addition and Subtraction 1s-9s (or the Alternative Program) can move on to Subtract from 20. Students who finish Subtract from 20 can do Skip Counting, which does a great job of preparing students to learn Multiplication facts.

Third grade math facts: Learn Multiplication

There aren’t any Alternative Programs available for third graders from Rocket Math. There are only Basic and Optional Programs. These include:

The Basic Program
- (1s-9s) Multiplication (priority)
- (1s-9s) Addition
- (1s-9s) Subtraction
Optional Programs
- 10s, 11s, 12s Multiplication
- Factors

In third grade, Multiplication has priority—even if students have not mastered Addition and Subtraction. Multiplication facts are so integral to the rest of higher math that students are even more crippled without Multiplication facts than they are having to count Addition and Subtraction problems on their fingers. So do Multiplication first. Then, if there’s time, students who need to do so can go back and master Addition and Subtraction. Once all three of these basic operations are under their belts, students can go on to 10s, 11s, 12s in Multiplication (one of the Optional Programs). If students successfully progress through each program and there is enough time left in the school year, introduce the Factors program next.

Fourth grade math facts: Learn Multiplication and Division

Like the programs for third graders, there aren’t any Alternative Programs available for fourth graders. There are only Basic and Optional Programs, which include:

The Basic Program
- (1s-9s) Multiplication (priority)
- (1s-9s) Division (second priority)
Optional Programs
- 10s, 11s, 12s Multiplication
- Factors

In fourth grade, students need to have completed Multiplication before going on to Division. If they complete Division, they can go on to 10s, 11s, 12s Division, followed by Factors, and then equivalent fractions (shown in the fifth grade section below).

Fifth grade math facts: Learn all basic operations first, then they can branch out

By fifth grade, students should have completed all four basic operations (1s-9s) within the Basic Program (or the Alternative Program for grades one and two). If students have not completed these basics (and cannot test out of them with the Placement Probes) then the sequence they should follow is Multiplication, followed by Division, then go back and complete Addition followed by Subtraction. The same recommendations hold for students in any grade after fifth.

Once students have mastered the basics (1s-9s add, subtract, multiply, divide), the supplemental pre-algebra programs are recommended. These will help more than learning the 10s, 11s, 12s facts. I would recommend this order:

Factors
Equivalent Fractions
Learning to Add Integers
Learning to Subtract Integers or Mixed Integers

Math Fact Apps: Do they help – Or Waste Time?

Posted on March 25, 2019February 20, 2023 by Dr. Don

Just doing math is not enough to develop math fact fluency.

If you want to help your kid succeed in golf, you must first help them develop a good drive. Before they can successfully go out and play a game of golf, they will have to spend some time on the driving range. Right? The same thing goes for math. To help your kid succeed in math, you must first help them develop math fact fluency. Math facts are single-digit problems like 5 +9 or 6 x 8. The times tables are math facts in multiplication. Math facts are the fundamental units of all math problems. They are the drivers, if you will, of doing math. Children who struggle to remember basic math facts will find learning math and doing math difficult. We know what that looks like, right? On the other hand, students who can easily answer all math facts, who can answer math facts with fluency, find math a breeze and are successful and confident in math. Math fact apps can be a great way to help with learning math facts and improving math fact fluency.

Not all math apps build math fact fluency.

There are a lot of apps out there that look like they would help your child learn math fact fluency. If they have to answer math facts, won’t that work? Not really. Just playing a game that asks you to answer facts won’t help you learn new facts. In fact, most apps for practicing facts discourage students who don’t know their facts well. Why? Because most of the people designing the app don’t have any experience teaching or knowledge of instructional design. A teacher with expertise in instructional design, like the Rocket Math Online Game App creator, knows how to effectively teach a student new math facts (or any facts) and knows an effective math app from an ineffective one.

Three essential features of an effective math fact app.

There are plenty of ineffective math apps. Some apps don’t give the answers when a student doesn’t know them. Some apps just fill in the answer for the student and then move on. When the student doesn’t know the answer, the app has to teach it. To teach a math fact that the student doesn’t know, the app has to do these three things:

Tell the problem and the answer to the student, if needed.
Ask the student to give the correct answer to the problem from memory.
After a short delay, ask the problem again a couple of times, to be sure the student can remember the answer.

The key job in developing math fact fluency is learning new facts. By doing these three things, the app will be able to teach the student new facts, and the student will develop fluency beyond what they already have.

An effective math app will only teach a few math facts at a time.

Math facts are all very similar, which is what makes them hard to remember. No one can learn a bunch of new and similar things all at the same time. A person can only learn two, three, or four facts at a time, and you cannot expect to learn more at one time. That’s enough for one session. The student has to practice those facts a lot of times to commit them to memory. Once or twice is not enough. It also won’t help to practice the same fact over and over. Proper math fact fluency practice intermingles new math facts along with facts the learner has already memorized. No more than two to four facts should be introduced at a time. If a student has to answer a lot of random untaught math facts, you will have a very frustrated learner.

How to build math fact fluency by practicing

Some students learn to solve addition problems by counting on their fingers. That’s a good beginner strategy, but students need to get past that stage. They need to be able to simply and quickly recall the answers to math facts. An app is good for developing recall. But the app has to ask students to answer the facts quickly, faster than they can count on their fingers. The app has to distinguish when a student recalls the fact (which is quick) from figuring out the fact (which is slow). Second, the app must repeatedly ask the learned facts in a random order, so students recall them. But the app should only throw in new facts once all the facts are mastered and can be answered quickly.

Introduce new facts only when old facts are mastered!

The trick to effectively teaching math facts is to introduce new math facts at an appropriate pace. If you wait too long to introduce math facts, it gets boring and wastes time. If you go too fast, students become confused. Before introducing new facts, students need to master everything you’ve given them. An effective app will test whether students have mastered the current batch of math facts before introducing more facts. And it will also introduce math facts at a pace based on student mastery. That’s the final piece of the puzzle to ensure students learn math facts from an app.

Rocket Math Online Game App–try it for 30 days for free

You probably guessed that we know of an app that meets all these criteria, the Rocket Math Online Game. It is like a driving range for math–it will develop math fact fluency. But don’t take our word for it. Try it for free for 30 days. If your child or students use it daily for 30 days, you’ll know they are learning and becoming more fluent with math facts. Keep it going, and they will become confident and successful in math. And that’s the point, isn’t it?

*The Rocket Math Online Game App focuses on two facts and their reverses at a time, such as 3+4=7, 4+3=7, 3+5=8 and 5+3=8.

$Students and teacher playing multiplication games with dice sitting in a circle in a classroom$

Why Multiplication Games Are Awful & What to Do About It

Posted on March 13, 2019March 10, 2019 by Dr. Don

As a university supervisor of pre-service teachers, I’ve seen my share of bad lessons. Among the most painful were when student teachers would try to liven up their lessons to impress me by having the students do a math game. My student teachers wanted their students to learn math facts and to do so in a fun way. The picture above is typical of what I would see. Here are the reasons that most multiplication games that the student teachers implemented were awful.

(For multiplication games that work in and out of the classroom, check out Rocket Math’s Worksheet Program and Online Game.)

Waiting for your turn at a multiplication game is not learning!

As you can see in the picture above, all but one of the students are just waiting for their turn. They aren’t doing math. The students are just watching the student who is playing. No one likes waiting, and your students are no exception. Any game that has turn-taking among more than two students wastes time.

Make sure your multiplication games are structured so all or most students are engaged and playing all the time. You want students to have as much engaging practice as possible while practicing math facts at speed. If everyone can be doing that at the same time, that’s optimal. No more than two students should be taking turns at a time.

A multiplication game that allows using a known strategy to figure out facts (like finger counting) is not learning!

Learning math facts involves memorizing these facts so that students know them by memory, by recall. Committing facts to memory is why there is a need for lots of practice. If the game allows time for students to count on their fingers or use some other strategy for figuring out the answer to facts, then there is no incentive for students to get better.

In the lower left corner of the picture you can see one student counting on their fingers—which is better than just watching—but is not learning the facts, it is just figuring them out. The most able students in an elementary school are able to memorize facts on their own when they tire of figuring them out day in and day out. But the rest of the students will just do their work patiently year after year without memorizing if you don’t create the conditions for them to memorize facts.

Make sure that your multiplication games reward remembering facts quickly rather than just figuring them out. Speed should be the main factor after accuracy. Fast-paced games are more fun and the point should be that the more facts you learn the better you’ll do.

Multiplication games that randomly present ALL the facts make learning impossible.

It is a basic fact of learning that no one can memorize a bunch of similar things all at once. To memorize information, like math facts, the learner must work on a few, two to four facts, at a time. With sufficient practice, every learner can memorize a small number of math facts. Once learners master a set of math facts, they can learn another batch. But if a whole lot are presented all at once, it is impossible for the learner to memorize them.

Make sure your multiplication games are structured so that each student is presented with only facts they know. A good game presents only a few facts at a time. As students learn some of the math facts, more can be added, but at a pace that allows the learner to keep up. The optimal learning conditions are for the learner to have a few things to learn in a sea of already mastered material.

Rocket Math Multiplication Games

We designed Rocket Math games to help kids gradually (and successfully) master math skills. Students use Rocket Math’s Worksheet Program to practice with partners, then take timings. Students can also individually develop math fact fluency—from any device, anywhere, any time of day—with Rocket Math’s Online Game.

How to Grade 1-Minute Math Fluency Practice Tests

Posted on February 7, 2019May 8, 2019 by Dr. Don

Katy L from Wilson Elementary asks: How can I keep up with everyday Rocket Math grading? Do you teach students to grade their own 1-minute math fluency practice tests?

Dr. Don answers:

Only grade 1-minute math fluency practice tests if students pass

An integral part of the Rocket Math Worksheet Program is the 1-minute math fluency practice test. One-minute fluency practice tests are administered every day, to the whole class, and only after students practice in pairs for two to three minutes each. Check out the FAQs page to learn more about conducting 1-minute math fluency practice tests in class.

Teachers do NOT need to grade, score, or check daily Rocket Math 1-minute math fluency practice tests unless the student has met their goal. Students do NOT need to grade their own daily Rocket Math fact fluency tests either.

Why grading each math test is not important

The important part of math fluency practice is the oral practice with the partner before the test–what’s going on in this picture. Because the students are orally practicing every day and getting corrections from their partners, there should be VERY FEW errors on the 1-minute math fluency written tests.

Correcting written tests doesn’t help students learn anyway. Corrections are only helpful if they are immediate, the student has to acknowledge the correct answer, and remember it for a few seconds–all of which is part of the oral correction procedure. “Correcting” what’s on the paper takes a lot of time and does not help students learn more, so it shouldn’t be done. But you have to check them before declaring that the student has passed a level.

How do you know if a student passes?

Students should have a packet of 6 sheets math fact fluency sheets at their level. Each Rocket Math student has an individual goal. For example, if a student has a goal of 32 (based on their Writing Speed Test) and they only do 31, they know they did not pass. If the student does 32 or more, they pass!

What to do when a student beats their goal (passes)

If a student meets or beats their goal, then have them stand up, take a bow, and then turn their folder into a place where you check to see that all problems were answered correctly. When YOU check (after school?), make sure all of the completed problems were correct and the student met their goal. If so, then you put the unused sheets in that packet back into the filing crate and re-fill the student’s folder with a packet of 6 worksheets at the next level and hand the folder back the next day.

When students receive the new packet of worksheets, they know to color in another letter on the Rocket Chart (and maybe put a star on the Wall Chart).

What to do if a student doesn’t pass?

Students who don’t meet their goals, don’t pass. These students should put the non-passing sheet into their backpacks and take the sheet home for more practice.

The next day they will use the next sheet in their packet of 6. If you want to give them points, do that the next day after they bring back their worksheet where they did a session at home (signature of helper should be there) and all items on the test are completed. If that’s done, they get full points.

Sometimes you’ll catch errors on sheets that students turn in as “passes.” If you see an error, the student doesn’t pass. As a result, the student keeps the old packet and has to continue with that same level worksheet.

For more information about conducting 1-minute math fluency practice tests in class and how to implement the Rocket Math worksheet program, visit the FAQs page.

Math Teaching Strategies #2: Ensure math facts are mastered before starting computation

Posted on January 17, 2019July 4, 2023 by Dr. Don

Rocket Math can make learning math facts easy. But even more important it can make teaching computation easy too! One of the first teachers to field test Rocket Math was able to teach addition facts to her first grade class, and then loop with them into second grade, where she helped them master subtraction facts as well. She told me that because her second graders were fluent with their subtraction facts, they were ALL able to master regrouping (or borrowing) in subtraction in three days. What had previously been a three week long painful unit was over in less than a week. All of them had it down, because all they had to think about was the rule for when to regroup. None of them were distracted by trying to figure out subtraction facts.

Math teaching strategy: Get single-digit math facts memorized before trying to teach computation.

When math facts aren’t memorized, computation will hard to learn, hard to do, and full of errors.

When math facts aren’t memorized, computation will be hard to learn. I used to think computation was intrinsically hard for children to learn. Because it was certainly hard for all of my students with learning disabilities. But none of them had memorized the basic math facts to the point where they could answer them instantly. They always had to count on their fingers for math.

When I learned more about the process of learning, I found out that weak tool skills, such as not knowing math facts, interferes with learning the algorithms of math. When the teacher is explaining the process, the student who hasn’t memorized math facts is forced to stop listening to the instruction to figure out the fact. When the student tunes back into instruction they’ve missed some essential steps. Every step of computation involves recalling a math fact, and if every time the learner has to turn his/her attention to deriving the math fact they are constantly distracted. That interferes with the learning process.

When math facts aren’t memorized, computation will be hard to do. Having to stop in the middle of the process of a multi-digit computation problem to “figure out” a fact slows students down and distracts them from the process. It is easy to lose your place, or forget a step when you are distracted by the difficulty of deriving a math fact or counting on your fingers. It is hard to keep track of what you’re doing when you are constantly being distracted by those pesky math facts. And of course, having to figure out facts slows everything down.

I once stood behind a student in a math class who was doing multiplication computation and when he hesitated I simply gave him the answer to the math fact (as if he actually knew them). He loved it and he was done with the small set of problems in less than half the time of anyone else in his class. Children hate going slow and slogging through computation. Conversely, when they know their facts to the level of automaticity (where the answers pop unbidden into their minds) they can go fast and they love it. That’s why “Because going fast is more fun!” is the Rocket Math tag line.

When math facts aren’t memorized, computation will be full of errors. When I learned more about basic learning, I found out that the frequent student errors in computation were not simply “careless errors.” I thought they were because when I pointed out simple things like, “Look you carried the 3 in 63 instead of the 6.” my students would always go “Oh, yeah.” and immediately correct the error. If I asked them they knew that they were supposed to carry the number in the tens column, but they didn’t.

I thought it was carelessness until I learned that such errors were the result of being distracted. Not by the pretty girl next to you, but by having to figure out what 7 times 9 was in the first place. After going through the long thinking process of figuring out it was 63 they were so distracted that they carried the wrong digit. Not carelessness but distraction. Once students instantly know math facts without having to think about it, they can pay full attention to the process. They make far fewer errors.

In short, don’t be cruel. If you have any autonomy available to you, first help your students memorize math facts and then teach them how to do computation in that operation. In other words, teach subtraction facts before subtraction computation. If you help them get to the point where math fact answers in the operation come to them without effort, you’ll be amazed at how much easier it is to teach computation, for them to do it and at the accuracy with which they work.

Math teaching strategies #3: Teach computation procedures using consistent language

Posted on January 17, 2019January 17, 2019 by Dr. Don

Improv can be entertaining, but it will frustrate students trying learn a procedure.

Much of math, and especially computation, is about learning a process or a set of procedures. [I am assuming you are practical enough to know that we cannot expect elementary aged children to re-discover all of mathematics on their own, as some people recommend.]

Learning a procedure means knowing “What’s next?” If you ever learned a procedure (for example a recipe) you know that it is between steps, when you ask yourself, “What’s next?” that you need help from the written recipe. Students are no different. Just showing them what to do is usually not enough for them to be able to follow in your footsteps. You need to teach them the steps of the procedure. As with anything you teach, you are going to confuse your students if you do things in a different order, or with different words, or different steps. What you call things, and some of how you explain yourself, and some of the sequence of doing the procedure is arbitrary. If you are improvising you will do things differently each time and your students will be confused. At a minimum you need it written down.

Math teaching strategy: Use a script or a process chart to keep the instructions consistent.

We know a lot about how to help students learn a procedure. We know we need to consistently follow the same set of steps in the same order, until students have learned it. We know we need to explicitly tell students the decisions they must make while working so they know what to do and when, in other words, we have to make our thinking process overt. We know we need to be consistent in our language of instruction so that students benefit from repetition of examples. And finally, we know we need to careful in our selection of problems so that we demonstrate with appropriate examples how the new process works and where it does not work.

Guess what? You can’t do all of that when you are improvising your instruction and making up the directions on the fly. To be able to do all that, you need a script and pre-selected examples. Many teachers have been taught to use a chart of steps, posted in their classroom, to which they refer as they model a procedure. The same effect can be achieved with a script, so that the teacher uses the same wording along with the same steps in the same order. If you improvise, it won’t always be the same, which will confuse your students.

You have to learn when and how to make decisions. Every math procedure involves looking at the situation and making decisions about what and how to do what needs to be done. You have to know what operation to use, when to borrow, when to carry, where to write each digit and so on. Because you as the teacher already know how to do the procedure, it is tricky to remember to explain your thinking.

Math teaching strategy: Teach a consistent rule for every decision students must make.

Good teaching involves first explaining your decision-making and then giving your students practice in making the right decision in the given circumstances and finally to make them explain why–using the rule you used in the first place. First, you teach something like, “Bigger bottom borrows” to help students decide when to borrow. Then you prompt them to explain how they know whether or not to borrow. All of that should be asked and answered in the right place and at the right time. A script or a posted process chart will help you remember all the decisions that have to be made, and what to look at to make the right one. Without a script it is very unlikely that you will remember the exact wording each time. You need a script to be able to deliver consistent language of instruction.

Math teaching strategy: Plan ahead to carefully choose the right examples.

With some math procedures it is quite hard to choose the right examples. The fine points can be obscured when the examples the teacher happens to come up with, are not quite right. The examples may be an exception or handled differently in a way the procedure has not taught. So for example borrowing across a zero is different than across other numerals so the numbers in a minuend must be chosen carefully rather than off the cuff.

Also, when teaching a procedure it is essential to teach when to use the procedure and when not to use that procedure. It is important that the teacher present “non-examples,” that is, problems in which you don’t follow that procedure. I have seen students who are taught, for example, borrowing, using only examples that need borrowing. Then they turn around and borrow in every problem–because that is what they were taught. They should have been taught with a few non-examples mixed in, that is, problems where borrowing wasn’t necessary so they learned correctly when to borrow as well as how to borrow. Choosing teaching examples on the fly will often end up with more confusion rather than less.

If it bothers you to see students as frustrated as the one above, then find* or write out a script for teaching computation so that you can be consistent and effective. Trust me, your students will love you for it.

* You may want to look at the “Learning Computation” programs within the Rocket Math Universal subscription. Here are links to blogs on them: Addition, Subtraction, and Multiplication. These are sensible, small steps, clearly and consistently scripted so each skill builds on the next.

Math Teaching strategy #4: Teach only one procedure at a time

Posted on January 17, 2019January 17, 2019 by Dr. Don

It’s far better to know only one way to get there, than to get lost every time!

There are educational gurus out there promoting the idea that by giving students multiple solution paths it will give them a deeper understanding of math. Generally these experts know this from teaching pre-service teachers in college, some of whom come to have some insights by learning multiple paths to the same goal. Sorry folks. What works for pre-service teachers in college, does not [and never will] apply to most children.

True, there are multiple ways to solve most arithmetic problems. They have been discovered over centuries across multiple civilizations. While one might dream of knowing all the ways to do long division, it’s far better to have one reliable method learned than to simply be confused and to get lost each time. Just as in directions to go someplace, it is hard to remember all the steps in the directions. When you’re new to a destination, the lefts and the rights are all arbitrary. If you get two or more sets of directions, you are going to mix the steps from one way with the steps from the other method and you will not arrive at your destination.

Math teaching strategy: Teach one solution method and stick to that until everyone has it mastered.

In real life, as in math, once you learn one reliable method of getting to your destination, you are then free to learn additional ways, or to try short cuts. But please don’t confuse a beginning learner with short cuts or alternative methods. It adds to the memory load and there are additional things to think about when trying alternatives. Before the learner is solid in one method, the new information is likely going to get mixed up with the not-yet-learned material, leading to missteps and getting lost. Teach the long way every time and leave them to finding the short cuts on their own time.

But teachers say, “I want them to have a holistic understanding of what they are doing!” Which is laudable, but that understanding has to come AFTER a reliable set of procedures is mastered. There’s no reason that additional learning can’t be added to the student’s knowledge base, but it can’t come before or in place of learning a simple, basic, reliable procedure. These admirable goals of getting a deeper understanding of math are fine, but they require MORE teaching than what used to be done, back when we were only taught algorithms for arithmetic. There is time to learn more than the algorithms, if we teach effectively and efficiently. Unfortunately, the deeper and more profound understandings in math can’t precede or be substituted for teaching the algorithms.

If you don’t believe me, ask a typical middle school student to do some arithmetic for you these days. Few of them have mastered any reliable procedures for doing long division or converting mixed numbers or adding unlike fractions, etc. It’s time to accept that teaching one way of doing things is better than none.

Math Teaching strategies #5: Separate the introduction of similar concepts

Posted on January 17, 2019December 23, 2019 by Dr. Don

Teaching two similar concepts and their vocabulary terms at the same time creates confusion.

The classic example is teaching parallel and perpendicular on the same day. The two concepts have to do with orientation of lines and the new vocabulary terms for them are similar. So teaching them at the same time means some or many students will have the two terms confused for a long time. That is known as a chronic confusion–possibly permanent. They will know that orientation of lines is one of those two terms, possibly, but will be confused about which is which. The predictable conversation with the teacher goes thusly:

Teacher: See these two lines lines. Are they parallel or perpendicular?

Student: I think they’re perpendicular.

Teacher: Well…

Student: No, wait! I know. They’re parallel.

Teacher: Yes, you’re right. I’m glad you’ve learned that.

In case you missed it, the above student did NOT know the correct term. The student just knew it was one of two terms, but unsure as to which one. So the student picked one and as soon as there wasn’t confirmation by the teacher, switched to the other term. When you’re busy teaching it is easy to get fooled by that kind of response into assuming the students really did learn it.

Other examples abound in math. Teaching numerator and denominator in the same lesson is common. Teaching the terms proper fractions and improper fractions on the same day is another example. Acute and obtuse angles are yet another pair of chronically confused concepts that are introduced simultaneously.

Math teaching strategies: Separate the introduction of similar concepts in time.

If you teach one concept and only one of the pair, there’s no cause for confusion. It will still take a lot of repetition and practice for it to be cemented into memory, but students will be clear. If they use the term, for example parallel a lot of times in conjunction with examples they will soon (in a couple of weeks) be able to recall. However, you should pair the concept with non-examples of the concept. Not the opposite necessarily but just not an example of the concept.

For example: For the picture to the right you would ask the students.

A “Are the two lines in item A parallel or not parallel?” Ans: Not parallel.

B “Are the two lines in item B parallel or not parallel?” Ans: Parallel.

C “Are the two lines in item C parallel or not parallel?” Ans: Not parallel.

Then after a couple of weeks you could introduce perpendicular. Again teach it on its own and then contrasted with non-examples until the vocabulary was clear. Probably for a couple of weeks. Only then can you combine both terms in the same lesson.

Another advantage of this approach is that you have avoided the other common mistake. Students sometimes come to the conclusion that all pairs of lines are either parallel or perpendicular. This method of introducing the concepts helps them realize that there are non-examples of each concept, parallel lines and ones that aren’t as well as perpendicular lines and ones that aren’t as well.