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

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

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

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

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.

 

Math teaching strategies #6: Teaching a new concept using a common sense name

 

Here’s an example of the problem.   A brachistochrone (pictured above) is a curve between two points along which a body can move under gravity in a shorter time than for any other curve. It is the same curve as a cycloid, but just hanging downward.   A cycloid is the path traced by a point on a wheel as the wheel rolls, without slipping, along a flat surface. The standard parametrization is x = a(t – sin t),y = a(1 – cos t), where a is the radius of the wheel.

Introducing this concept and the term brachistochrone at the same time would be designed so the teacher can use brachistochrone and its concept in instruction.  However, because the term is new and the concept is also new, when the teacher uses the new term during later instruction, the students will have difficulty bringing the concept into their minds. “So a brachistochrone has some other cool properties. What’s the primary thing we know about the brachistochrone?”

Instruction not working.  If you watch instruction where the term and the concept have been taught simultaneously, confusion ensues when the teacher uses the term.  You’ll see students looking away as they try to bring up the explanation of that weird new term in their memory (see the pictured example?).  Sometimes the teacher will notice this and give a thumbnail definition or example of the term, and the students will then remember.  However, the teacher should then realize that the concept was not connected to the new vocabulary term.

Math teaching strategies: Teach new concepts using a common sense term at first.  

When a new vocabulary term is used to introduce a new concept, students will need a lot of practice with recalling the term and the definition before it can be used in instruction.  On the other hand, students can quickly understand and use new concepts and ideas in math if they don’t have to learn a new word for it.  Using a common sense term, the idea or concept can be worked with, the implications studied and it can be applied to real world problems almost immediately.  Students can later quite easily learn proper vocabulary terms for concepts they understand and recognize.  Here’s an example. 

The “shortest time curve.”  It is more efficient and effective to teach the concept first and use a common sense term for it.  I would call a brachistochrone the “shortest time curve.”**  Instruction would proceed with the, “Do you remember that ‘shortest time curve’ we talked about last week?”  Students would easily be able to remember it.  Instruction would go like this: “So ‘the shortest time curve’ has some other cool properties. What’s the primary thing we know about the ‘shortest time curve’?”  Students would easily be able to answer this question.

Then after students have worked with the concept of “the shortest time curve” for a couple of weeks, you can add the vocabulary term to it. “By the way, the proper mathematical name for “the shortest time curve” is called a brachistochrone. Isn’t that cool?”  Students will want to learn its proper name as a point of pride about knowing this fancy term for a concept they already “own,” rather than a point of confusion.

**Actually that’s what brachistochrone means in Greek: brakhistos, meaning shortest and khronos meaning time.

 

Rocket Math: Can students really learn this way? (It seems too easy.)

At first, it may seem like the way Rocket Math presents the same simple facts over and over, is so easy it must be a waste of time.

       But like anything you learn, you have to start where it seems easy and then build up to where it is hard.  Rocket Math has been effective helping students learn their math facts for over 20 years.  It is designed according to scientifically designed learning principles, which is why it works, if students will work it.  Rocket Math carefully and slowly introduces facts to learn in such a way that students can achieve fluency with each set of facts as they progress through the alphabet A through Z.  Let me explain.
         Set A begins with two facts and their reverses, e.g., 2+1, 1+2, 3+1 and 1+3.  Dead simple, huh?  But in answering those the student learns what it is like to instantly “know” an answer rather than having to figure it out.  The student says to himself or herself, “Well, I know that one.”  The student learns he or she can answer a fact instantly with no hesitation every time based on recall and not figuring it out.  The game requires the student to answer the problems at a fast rate, proving that he or she knows those facts.  Once that level is passed the game adds two more facts and their reverses,.  The same process of answering them (and still remembering Set A) instantly with no hesitation every time.  When that is achieved, the game moves the student on to Set C, two more facts and their reverses.  Eventually, every student gets to a fact on which they hesitate (maybe one they have to count on their fingers), meaning they can’t answer within the 3 seconds allowed.  Mission Control then says the problem and the correct answer, has the student answer that problem, then gives two different facts to answer and goes back to check on the fact the student hesitated on again.  If the student answers within 3 seconds then the game moves on.
 
     In the Take-Off phase the student is introduced to the two new facts and their reverses.  That’s all the student has to answer.  But the student has to answer each one instantly.  If the student is hesitant on any of those facts (or makes an error) then they have to Start Over and do the Take-Off phase over again.  They have to do 12 in a row without an error or a hesitation.  Once the Take-Off phase is passed the student goes into the Orbit phase, where there is a mix of recently introduced facts along with the new facts.  The student has to answer up to 30 facts, and is allowed only two errors or hesitations. After the third error or hesitation the student has to Start Over on the Orbit Phase.  Once the Orbit phase is passed, the student goes on to the Universe phase, which mixes up all the facts learned so far and presents them randomly.  Again the student has to do up to 30 problems and can only hesitate on 2 or them or he or she has to start over.  But once the student proves that all of those facts can be answered without hesitation, the game moves on to the next level, introducing two more facts and their reverses.
      In the Worksheet Program, students practice with a partner.  In the Online Game the student practices with the computer.  In both versions of Rocket Math the students follow the same careful sequence and slowly, but successfully build mastery of all of the facts in an operation.  It’s hard work and takes a while, but we try to make it fun along the way.  It will work for everybody, but not everybody is willing to do the work.  At least, now you understand how Rocket Math is designed so it can teach mastery of math facts.

Dictating Sentences, do students need automaticity in spelling too?

Few teachers realize how similar spelling and math facts are.

Both spelling and math facts are tool skills. Tool skills are things which one needs to do academic work, tools you use to do other things.  The tool skills of spelling and math facts (like decoding) need to become so automatic that students don’t have to think about them.

If students have to think up the answer to math facts, it makes doing computation harder.  The process of figuring out a math fact distracts from the mathematics being done.  Similarly, if you have to stop and think of the spelling of a word (like the boy pictured above) while you are trying to write, it distracts you from thinking about what you are trying to write.  Students are more successful and better able to show what they know and better able to focus on learning when their tool skills have developed to the level of automaticity.

Daily practice develops automaticity.  Developing automaticity with math facts and with spelling requires a lot of practice.  Daily practice is best and a few minutes a day is optimal.  That is why Rocket Math is designed the way it is–to provide that daily practice.  So Dictating Sentences gives each member of the pair ten minutes a day of practice writing sentences composed of words they know how to spell.

Dictating Sentences is spelling with a twist.  Instead of spelling one word at a time, in Dictating Sentences (now part of the Universal Level Rocket Math Worksheet Program) students are asked to write an entire sentence from memory.  They work in pairs and their tutor has the student repeat the sentence until it is learned.  Then the student has to write the whole sentence from memory.  It turns out this is considerably harder than writing words on a spelling test, so it is challenging practice, and does a lot to help students develop automaticity with spelling.

Working in pairs.  As you know from Rocket Math practice, students enjoy working in pairs.  And when one partner has an answer key the practice can be checked and corrected.  Sound research shows that immediate correction and editing of misspelled words is the fastest way to learn the correct spelling, so that’s what we have the student tutor do.  After each sentence is written every word is checked and practiced again until it is correct.

Mastery learning. The program is structure so that all the words are learned to the level of automaticy.  Students keep working on a sentence until it can be written without any errors.  They work on the same lesson for as many days as is needed for them to spelling every word perfectly in all three sentences.  Each sentence persists for two or three lessons, so that the student is required to write it from memory and spell every word perfectly for several days in a row.

Earning points by being correct and going fast.  Students earn two points for every word that is spelling correctly the first time.  Every word on which there is an error is worked on until it too can be spelled correctly, earning one point.  The faster students go during their ten minutes, the more points they can earn.  Students graph the amount of points earned and try to beat their own score from previous days.  Teams can be set up and competition for the glory of being on the winning team can enhance the motivation.

Individual Placement.  There is a placement test.  Students begin at the level where they first make a mistake.  Student partners do not need to be at the same level, so every student can be individually placed at the level of success.

Add Login info for classes with csv file

Assign Subscriptions. The orange box on your dashboard shows the number “Unassigned Subscriptions” you have that can be assigned to students. You can give these subscriptions to students by using the blue + Import Students Logins From CSV button.
That page–the pop-up labeled “Import Student Logins From CSV” looks like this picture to the left.
Begin at #1 and click on “CSV template to fill in” to get a properly formatted starting point.
See the blank csv template to the right. You’ll enter the student’s first and last name, make up a username and a passcode for the student. Enter the code number for the learning track they will start in. You can change it at any time. Add the Teacher Mgr’s email if you wish to connect the student to a different teacher that you set up in your account.
Once you have completed the file, save it to your computer as a CSV file (it’s an excel file now, so you have to choose Save As and find Comma Separated Value -CSV in the list).
Now go back to the pop-up labeled “Import Student Logins From CSV” and do #2 Choose file and browse to the csv file you just saved and select it. Then go to the bottom of the page at #3 and hit the blue button that says “Parse CSV.”
After you hit “Parse CSV” you’ll see a list of your students. Scroll to the bottom and hit the blue button that says “Import Students.” Then they will be set up in the system.
If something goes wrong, you can use the red button on your Dashboard that says “Delete ALL students!” It is extreme, but it will clear out all of your student data, allowing you to start over and re-import.
If you have a bunch of trouble, send me your csv files and I will do the import for you. -Dr.Don

Add Login info individually for Online Game

If you have few enough students, you can simply add their login information individually.  In your dashboard click on the blue button that days + Add Student Login.

Up pops this dialog box.  Enter the student’s first and last name, then create a username and passcode.  It only has to be unique to your school or family, so make it simple and easy to enter.

Then be sure to choose a Learning Track from the pull-down menu.

If you are the owner, you are also the first teacher.  If you have other Teacher Managers, be sure to connect the student with the teacher you want.  After you hit the green Save button your student is ready to play.

Adding Teacher Managers to Online Game

The person who first sets up the account is the owner (probably you).
The owner is automatically the first teacher.
 
Next, if you need help, you can set up additional teachers and give them subscriptions.
Go to the Teacher Mgr page by clicking on the Teacher Managers link in the left hand navigation.
Then click the blue Add Teacher Mgr button in the upper right.
You’ll see this dialog box (below) in which you enter the name and email. Don’t worry if you give them the wrong number of subscriptions.  When you enter the csv file with the student logins, there is a place to enter the teacher for each student.  The system will increase the number of subscriptions given to each teacher if necessary to accommodate what is in the csv file.
When you hit the green “Create” button the system will create a password for that teacher and email it to the email you entered for them.  It’s a hard password, so they might want to change it.
Don’t wait too long to let the teacher know about the incoming information or they’ll miss the email and won’t know how to enter the system.
Repeat as needed to add more Teacher Mgrs to help you monitor students.
Next, you will go on to assign student login information to your “unassigned” subscriptions so the students can login and play.