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.

Choose a Learning Track for Online Game

Choose from ten Learning Tracks

In the Rocket Math Online Game every student needs to be started in one of the ten Learning Tracks.  A student’s Learning Track can be changed at any time**, but one must be chosen to begin with.

If you are entering the Student Login individually, you can use the pull down menu to select a learning track, as illustrated to the right.

The ten learning tracks are numbered as follows. If you are using the csv method of entry you’ll need to enter the number for the track.

  1. Addition 1s through 9s
  2. Subtraction 1s through 9s
  3. Multiplication 1s through 9s
  4. Division 1s through 9s
  5. Fact Familes (1 to 10) add and subtract, ex.4+5, 5+4, 9-4, 9-5
  6. Fact Families (11 to 18) add and subtract, ex. 8+7, 7+8, 15+7, 15-8
  7. Add to 20, example 13+4, 4+13,
  8. Subtract from 20, example 15-3, 15-12,
  9. Multiplication 10s-11s-12s,
  10. Division 10s-11s-12s.

You can click below to see a google document showing all the problems learned in each of the Learning Tracks.

Click to see the problems in the tracks.

Considerations, or what to choose when?

Begin with the basics.  The four basic operations are most important and typical expectations is one of those per grade level, so Addition in first, Addition then Subtraction in second, Multiplication, then go back to Addition and Subtraction in third, and Multiplication then Division in fourth grade, and then going back to get Addition and Subtraction if those haven’t been learned.  Make sure your student have worked through the expected basic operations for their grade level BEFORE doing any of the other optional Learning Tracks.

Another way to learn basic Addition and Subtraction Facts.  Learning in Fact Families is another order to learn. Fact Familes (1 to 10) add and subtract would be chosen in first grade.   Fact Families (11 to 18) add and subtract would be mastered in second grade.  You can choose this sequence instead of the basic addition and basic subtraction fact Learning Tracks.   Optionally, Fact Families is also a good way to review for students who have already learned the basic addition and subtraction facts in first or second grade.

Optional Learning Tracks. Add to 20 and Subtract from 20 are additional problems that the Common Core feels should be committed to memory.  They are composed of facts you can figure out if you know the basic 1s through 9s facts, but can be learned AFTER the basics are learned, if there is time in first or second grade.  They should not be assigned until after the student has mastered the basic 1s through 9s addition and subtraction facts.

After students learn the basic 1s through 9s multiplication facts, if there is time, they can move on to 10s, 11s, 12s.  After basic 1s through 9s  division facts are learned (and all the other basic operations are learned) then the 10s, 11s, and 12s are a good use of time.

 

**See “How to change Learning Tracks” in the FAQs and Directions document.

How to purchase Online Game subscriptions

Here’s information (that may not be apparent) about how to purchase Online Game subscriptions.  First you register for a free account at https://admin.rocketmath.com for the Rocket Math Online Game.  The next step is to to purchase game subscriptions with our No risk 30 day trial. 

Non-credit card options

 

If you wish to buy subscriptions by sending in a Purchase Order here’s a link to our order form. Or, if you wish to order online with either PayPal or a PO number click this link to get to that page

Either in PayPal or with a PO we will give you 13 months for the one year price, and if you tell us you no longer want your subscriptions during the first month, we’ll cancel your subscriptions and cancel the invoice. With PayPal we’ll give you a full refund if you don’t want to keep your subscription.    

If you ask, I can also manually give you a 30 day free trial–without you having to enter a payment method.  I’ll give you access to all the subscriptions you’d need for your free trial period.  Then, if you wish to continue and purchase we can send an invoice.  Just contact [email protected], with the number of subscriptions you would like to use during your free trial. 

Credit card procedure

Go to the “My Profile” page to order subscriptions.  There you click on + Add Subscriptions on the “My Profile” page of your account.  It looks like this picture. 

No gotcha here–See how the auto-renew is turned off by default?  

After you click on + Add Subscriptions this dialog box will pop up.

This person in this picture has payments set to yearly.  So the price for one subscription is $3.89 for the year.

If you leave it set to monthly, the price will be $.50 (50 cents/month).
To order more than one subscription hover in the box and you’ll see arrows to increase the number of subscriptions.  This box will automatically discount to $2 for quantities of 20 or more and down to $1 for quantities of 100 or more.
Hit the green payment data to pay with a credit card.  You’ll get this Stripe dialog box.  You fill out your credit card info and hit the pay button, but remember, you will not be charged a thing for 30 days.

Monthly, non auto-renewal expires if you don’t act.

Note: As long as you leave the renewal period set to monthly, and leave auto renew set to OFF in your profile, then your subscription will simply end after 30 days.  No matter how many subscriptions you order, your credit card won’t be charged until you login and renew.  So you can try the game for free to see if it’s worth paying for with no risk of being charged for it.  When you decide it is worthwhile, come back into “My Profile” switch the renewal period to yearly, and make sure you have as many subscriptions as you want, and then change to “Auto Renew.”   You can switch it back  to non renew after you renew, but there’s no other way to renew ahead of time with the credit card.  But if your subscription has expired, you will see a green “Renew Subscription” button in “My Profile” and you can click on that to renew.

Yearly renewal gives you lower prices (and still no charge for the first month).

If you are pretty certain, go ahead and set the renewal to yearly and then order your subscriptions.  You’ll get the best price and you’ll automatically get the discounts for quantity.  Your credit card will not be charged until the end of your 30 day trial, so if you cancel before then you do not pay a thing.