Math Teaching Strategy #1: Help students memorize math facts

Once students know the procedure, they should stop counting and memorize!

When it comes to math facts like 9 plus 7 or 8 times 6 there are only two things to know.  1) A procedure to figure it out, which shows that you understand the “concept.”  2) What’s the answer?

It is important for students to understand the concept and to have a reliable procedure to figure out the answer to a math fact.  But there is no need for them to be required to use the laborious counting process over and over and over again!  Really, if you think about it, even though this student is doing his math “work” he is not learning anything. 

Math teaching strategy:  Go ahead and memorize those facts.

(It won’t hurt them a bit.  They’ll like it actually.)

Once students know the procedure for figuring out a basic fact, then they should stop figuring it out and just memorize the answer.  Unlike capitals and countries in the world, math facts are never going to change.  Once you memorize that 9 plus 7 is 16, it’s good for a lifetime.  Memorizing math facts makes doing arithmetic MUCH easier and faster.  Hence our tagline

Rocket Math: Because going fast is more fun!

Memorizing facts is the lowest level of learning.  It’s as easy as it gets.  But memorizing ALL the facts, which are similar, is kind of a long slog.  Some kids just naturally absorb the facts and memorize them.

Math teaching strategy: Find a systematic way for students to memorize.

A lot of students don’t learn the facts and keep counting them out over and over again.  They just need a systematic way of learning the facts.  Students need to spend as much time as necessary on each small set of facts to get them fully mastered.  If the facts are introduced too fast, they start to get confused, and it all breaks down.  Each student should learn at their own pace and learn each set of facts until it is automatic–answered without hesitation and without having to think about it.  This can be accomplished by everyone, if practice is carefully and systematically set up.  It should be done, because the rest of math is either hard or easy depending on knowing those facts.  And don’t get me started about why equivalent fractions are hard!

 

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.