basic math

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

Posted on 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.

 

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

Posted on by Dr. Don

 

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.)

Posted on by Dr. Don

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.
Students work with together to learn how to write their numbers.

Fact Families (+ & -) for 1st and 2nd grade

Posted on by Dr. Don

Learn Fact Families to fluency with Rocket Math!

Fact Families Part Two  11 to 18 (add & subtract).  A fact family includes both addition and subtraction facts. This program is Part 2 of Fact Families, coming after Fact Families 1 to 10. You can see to the left the 18 examples of fact families taught in this program starting with Set A; 11-2, 11-9, 9+2, & 2+9.  The sheet shows the sequence of learning facts in the new Rocket Math  program Fact Families Part Two 11 to 18 (+, -).  Each set that students learn from A to R adds just one fact family to be learned, so it isn’t too hard to remember.  (That’s the Rocket Math secret ingredient!) 

Learning math facts in families, is gaining in popularity these days.  Logic suggests that this would be an easier way to learn.  However, the research is not definitive that this is easier or a faster way to learn facts than separating the operations and learning all addition facts first and then learning all subtraction facts.  But learning in fact families is a viable option, and I wanted to have it available for Rocket Math customers.

Part Two is a Best fit for second grade.  These facts come after the facts in 1 to 10, typically learned in first grade, so these are best for second grade.  The 25 fact families in 1s through 10s facts are just enough for one Rocket Math program.  It is a good and sufficient accomplishment for first grade.  With the 11 to 18 in Par Two for second grade there will be a lot of review.  In fact sets S through Z are all review. I have heard that some first grades prefer to keep the numbers small but to learn both addition and subtraction–so this program accomplishes that.

I added Fact Families Part Two 11 to 18 (+, -) to the Universal subscription in August of 2018 bringing the total number of programs in the Universal subscription to 19 (the basic four operations and 15 more!).  As always, new programs are added to the Universal subscription without additional cost as soon as they are available.

I most sincerely want students to be successful and to enjoy (as much as possible) the necessary chore of learning math facts to automaticity. Please give me feedback when you use this new program, Fact Families 11 to 18 (+, -),  as to how it goes for the students.

Intervention Tip: Have students practice test

Posted on by Dr. Don

Sometimes students need to review test problems also.

You know that there is a difference between the test problems and the practice problems, right?  The problems practiced around the outside are the recently introduced facts.  The problems inside the test box are an even mix of all the problems taught so far.  Sometimes students have forgotten some of the older facts.  For example, if there has been a break for a week or more, or if the student has been stuck for a couple of weeks, the student may have forgotten some of the facts from earlier and may need a review of the test problems.

How you could diagnose for this problem.  Have the student practice orally on the test problems inside the box with you.  If the student hesitates on several of the problems that aren’t on the outside practice, then the student needs to review the test items.

Solution. If you have this problem with quite a few students (for example after summer break or after Christmas break) then have the whole class do this solution.  For the next week, after practicing around the outside, instead of taking the 1 minute test in writing, have students practice the test problems orally with each other.  Use the same procedures as during the practice—two or three minutes with answer keys for the test, saying the problem and the answer aloud, correction procedures for hesitations, correct by saying the problem and answer three times, then going back—then switch roles.   Do this for a week and then give the one-minute test.   Just about everyone should pass at that point.

Solution.  If you have this problem with a handful of students, find a time during the day for them to practice the test problems orally in pairs.  If the practice occurs before doing Rocket Math so much the better, but it will work if done after as well.  They should keep doing this until they pass a couple of levels within six days.

If neither the first or the second solutions seem to work, write to me again and I’ll give you some more ideas.

Dr. Don from Rocket Math watches the first Rocket Math app user complete math facts test

Four star rating for Rocket Math Apps

Posted on by Dr. Don

Rocket Math App received 4 Stars!

App Names: Rocket Math Add at Home, Add at School, Multiply at Home, and Multiply at School

Developer’s name: Rocket Math, LLC

App Link :

https://itunes.apple.com/us/app/rocket-math-multiply-at-home/id1048024368?mt=8

Primary School Apps (5-7 Years)

Educational App Store Review

Rocket Math is an offshoot of an existing programme for schools designed to increase children’s speed and fluency in answering simple arithmetic. This app encourages frequent short sessions and is supported by plenty of information explaining its purpose and methods.

The purpose of Rocket Math is to build what its developer terms “automaticity” in arithmetic. A fluent reader does not need to decode simple and frequently encountered words letter by letter. The same can be true for frequently encountered arithmetic.

When automaticity is achieved in arithmetic the answers are available in an instant. The advantages of this, beyond speed, are that it leaves more of the person’s mental processes available for other aspects of the problem. If a person does not have to think about achieving simple arithmetic answers, he or she can concentrate on the more complex and lengthier aspects of a problem.

Rocket Math the app follows on from a well-established programme of the same name based on traditional written resources. Repeat practice and a steady increase in the breadth of the covered arithmetic are at the heart of its methods.

Children are taken through a series of stages in which they are faced with a rapid succession of arithmetic questions. Remember, the purpose of this app is to build fluency in frequently encountered arithmetic problems, not complex ones. As such, the questions will be simple ones and, at first, until the breadth expands, there will be little variation in them. Only three seconds is allowed per question so, for some children, developing enough fluency to progress will be difficult but others will thrive on the challenge.

Answers are given by typing them onto a built-in number pad. The app is simple to use and looks attractive. Its space-travel styling and theme add a game-like feel although it is not a game. Speech provides a response to incorrect answers and provides encouragement between levels. It all works very well and provides the exact type of practice that it promises.

An unusual but useful feature is that the app enforces its little-and-often recommendations by insisting on a thirty-minute break after 5 minutes of play. As multiple sessions are likely to yield better results than a single, marathon session, this is an excellent feature that will prevent children from relying on a last-minute catch-up rather than a steady engagement with the app. This, combined with a useful breakdown of each child’s performance in the student report screen, provides reassurance to adults that their children are making the best possible use of the app.

A family of apps is available and potential buyers should think about which they need. Two of the apps cover addition and subtraction and two cover multiplication and division. Your choice here is obviously dependent on what aspect you would like to cover.

The remaining choice is between a school and a home version. They are identical in functionality except that the home version is free to download with a lengthy trial period. The school version has a flat, one-off, fee. Prospective teachers would still be wise to download the home version first so that they can appraise the app’s suitability.

If they choose to utilise the app within their school then buying the school version will be a simpler process than the in-app purchase of the home version. It will also allow schools to utilise the volume purchasing programme whereby they can receive a discount for buying twenty or more of the same app.

Parents will be pleased to see that the app caters for up to three children. As each child engages with the app, parents can check to see how they are performing and offer help, encouragement or rewards as they see fit.   Some useful background information on the app’s purposes and usage are provided within the app itself and a more comprehensive overview of the Rocket Math ethos is available on the developer’s website.

All of the Rocket Math apps provide a learning opportunity that is tightly focused on realising their goal of improving children’s arithmetic fluency. As such, if this is a goal that you also share, you will find them good value and useful apps.

Foolproof method for finding factors

Posted on by Dr. Don

Knowing when you’ve found ALL the factors is the hard part.

Students have to learn how to find the factors of a number because several tasks in working with fractions require students to find the factors of numbers. Thinking of some of the factors of a number is not hard. What is hard is knowing when you have thought of ALL the factors. Here is a foolproof, systematic method I recommend: starting from 1 and working your way up the numbers. This is what student practice in the Worksheet Program Factors Learning Track.  Students also learn the pairs of factors in this sequence in the Online Game.

Dr Don has a white board type video lesson that explains this in 6 minutes.

https://www.educreations.com/lesson/view/how-to-find-all-the-factors-of-a-number/46790401/

Bookmark this link so you can show it to your students.

How to find all the factors of numbers
Always begin with 1 and the number itself-those are the first two factors. You write 1 x the number.  Then go on to 2. Write that under the 1. If the number you are finding factors for is an even number then 2 will be a factor. Think to yourself “2 times what equals the number we are factoring?” The answer will be the other factor.
However, if the number you are finding factors for is an odd number, then 2 will not be a factor and so you cross it out and go on to 3. Think to yourself “3 times what equals the number we are factoring?” There’s no easy rule for 3s like there is for 2s. But if you know the multiplication facts you will know if there is something. Then you go on to four—and so on.

The numbers on the left start at 1 and go up in value.  The numbers on the right go down in value.  You know you are done when you come to a number on the left that you already have on the right.  Let’s try an example.

Factors Answers d

Let’s find the factors of 18.  (To the left you see a part of a page from the Rocket Math factoring program.)
We start with the first two factors, 1 and 18. We know that one times any number equals itself. We write those down.
Next we go to 2. 18 is an even number, so we know that 2 is a factor. We say to ourselves, “2 times what number equals 18?” The answer is 9. Two times 9 is 18, so 2 and 9 are factors of 18.
Next we go to 3. We say to ourselves, “3 times what number equals 18?” The answer is 6. Three times 6 is 18, so 3 and 6 are factors of 18.
Next we go to 4. We say to ourselves, “4 times what number equals 18?” There isn’t a number. We know that 4 times 4 is 16 and 4 times 5 is 20, so we have skipped over 18. We cross out the 4 because it is not a factor of 18.
Next we go to 5. We might say to ourselves, “5 times what number equals 18?” But we know that 5 is not a factor of 18 because 18 does not end in 5 or 0 and only numbers that end in 5 and 0 have 5 as a factor. So we cross out the five.
We would next go to 6, but we don’t have to. If we look up here on the right side we see that 6 is already identified as a factor. So we have identified all the factors there are for 18. Any more factors that are higher we have already found. So we are done.

Now let’s do another number.  Let’s find the factors of 48. 

We start with the first two factors, 1 and 48.  We know that one times any number equals itself.

Next we go to 2.  48 is an even number, so we know that 2 is a factor.  We say to ourselves, “2 times what number equals 48?”  We might have to divide 2 into 48 to find the answer is 24.  But yes 2 and 24 are factors of 48.

Next we go to 3.  We say to ourselves, “3 times what number equals 48?”   The answer is 16.  We might have to divide 3 into 48 to find the answer is 16.  But yes 3 and 16 are factors of 48.

Next we go to 4.  We say to ourselves, “4 times what number equals 48?”  If we know our 12s facts we know that 4 times 12 is 48.  So 4 and 12 are factors of 48.

Next we go to 5.  We might say to ourselves, “5 times what number equals 48?”   But we know that 5 is not a factor of 48 because 48 does not end in 5 or 0 and only numbers that end in 5 and 0 have 5 as a factor. So we cross out the five.

Next we go to 6. We say to ourselves, “6 times what number equals 48?”  If we know our multiplication facts we know that 6 times 8 is 48.  So 6 and 8 are factors of 48.

Next we go to 7.   We say to ourselves, “7 times what number equals 48?”   There isn’t a number.  We know that 7 times 6 is 42 and 7 times 7 is 49, so we have skipped over 48.  We cross out the 7 because it is not a factor of 48.

We would next go to 8, but we don’t have to.  If we look up here on the right side we see that 8 is already identified as a factor.  So we have identified all the factors there are for 48.  Any more factors that are higher we have already found.  So we are done.

Why should you renew your Rocket Math Worksheet subscription?

Posted on by Dr. Don

Reason #1 to renew: So we can be here for you.

Your subscription enables Dr. Don and our team to be available to help you with Rocket Math.  We can be here to answer your questions, help you with your implementation, provide customer service at assistance@rocketmath.com and to continue to expand our offerings.
Every day we correspond with and often get to talk with customers who have questions and need help. We have added programs such as Add to 20, Subtract from 20, Fact Families, Skip Counting and others to the Worksheet Subscription. (Check out the list on this link.) We are quite excited about the Learning Computation programs for addition, subtraction, multiplication and division.  Additionally, we have added tools such as the classroom wide aimlines in Excel to keep track of the progress of your whole class.  Most exciting of all, I have developed the Online Tutor that students will be able to use at home and at school with one log-in. Your subscriptions make all of that possible.  Thank you.

Reason #2 to renew: It will extend your permission to copy. 

The second reason to renew is that your subscription gives you “permission to copy” the materials only for a limited time.  After the subscription expires, permission to copy the materials also expires (it is no longer granted).  As you can see, the footers on our materials now show your name and the expiration date.  Don’t embarrass yourself by continuing to print these materials after your subscription has expired.  In order to continue printing and copying the Rocket Math worksheets, I ask that you please renew your subscription.

Reason #3 to renew: Access the virtual filing cabinet.

The third reason to renew is so that you continue to have access to the Rocket Math filing cabinet on the web.  New and updated materials are constantly being added to the filing cabinet. The materials now number in the thousands of pages!
 You can much more easily print from the website filing cabinet, than you can from a paper master copy.  You can print from home and even print from your phone.
Worksheets are updated in the filing cabinet.  When anyone finds an error in one of the thousands of worksheets, I can change it the same day in this virtual filing cabinet.  When anyone asks for a new form, I can share it with everyone instantly.

“Knowing” means never having to figure it out

Posted on by Dr. Don

Most people, for example, know their name, by memory.

In a previous blog I discussed  What does CCSS mean by “know from memory?”    

A reader asked the following question:

This topic of “know from memory” is something I have been digging into as a special educator. I wonder what your thoughts are about whether certain accommodations from these “know from memory” standards would actually be modifying the curriculum?

For example, if we used “extra time to respond” and the student had to use their fingers or some other method to count, would they then not be doing the standard?

This relates to where I’m at in middle school math, but I think that it’s reflected in the continuum of the common core maths.

Thanks.

Dr. Don’s response: 

Actually, your example is very clear that it is not “knowing from memory.” You are describing “deriving from a strategy” or what I call, “figuring it out.” When you know it from memory, when you recall the answer, then you stop having to “figure it out.”

Knowing from memory and figuring something out are two very different things. I used to ask workshop participants to imagine sitting next to me in a bar and asking me for my name. What if, instead of saying, “Hi, my name is Don,” something different happened?  What if, like the man pictured above, I was puzzled and said, “Wait a second, I have it here on my driver’s license.” Most people would likely turn their attention elsewhere while wondering what kind of traumatic brain injury I had sustained! They would very likely say to themselves, “OMG, that man doesn’t know his own name.”

The purpose of the verbal rehearsal that is a daily part of Rocket Math is to cement these basic facts in memory. Then when a student says to themselves, “8 times 7 is,” the answer pops into their mind with no effort. It takes quite a bit of practice to achieve that. However, the ability to instantly recall the answers to basic math facts makes doing mathematical computation a relative breeze. It make seeing relationships among numbers very obvious. It makes reducing fractions and finding common denominators easy. That’s why the Common Core thinks “knowing from memory” is so worthwhile. It’s why I began promoting Rocket Math in the first place.