Author: Dr. Don

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.

Dictating Sentences, do students need automaticity in spelling too?

Posted on by Dr. Don

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.

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.

What if teachers won’t do Rocket Math?

Posted on by Dr. Don

Don’t argue, just prove it works! 

Joyce asks: 

How can we encourage the teacher who refuses rocket math and administration does not reinforce (or enforce) the program’s use?

Dr. Don’s response:

  Joyce,

     This is a great question.  Frankly, one of the most annoying things I found during my time as a teacher were the constant “new” fads.  I got sick and tired of being told to do things I knew would not work.  I don’t blame people for being skeptical or an administration that won’t go to bat for a new curriculum.  I think it is the responsible thing to do. Which is why schools should test everything for themselves, which isn’t that hard to do.  Prove to yourself it works with your students in your school with your staff.  Then you know it is worth doing.  Only then do you have a responsibility to reinforce the program’s use, only after it is proven.
In one of the first schools to use Rocket Math we had a veteran teacher who said she did not think Rocket Math would be any better than the things she had been doing to help her students learn math facts for years.  The principal wisely allowed as how that might be possible, but asked if she would be willing to test her assertion.  Rocket Math has 2-minute timings of all the facts which the students take every couple of weeks.  The principal asked if she would give that test to her students at the beginning and the end of the year and compare her results with that of other classes.  She agreed.  At the end of year the Rocket Math students were far higher in their fluency than her students, even though at the beginning of the year her students had been more fluent than the other students.  At that point she said, “Well this proves it to me.  I’ll be using Rocket Math next year.”
   Just use those 2-minute timings as pre and post tests and see if there is anything that will beat Rocket Math.  Any teacher worth their salt should want to use a curriculum that is effective and helps students learn.
I have the following standing offer on my website.  If any school will conduct research comparing Rocket Math to some other method of practicing math facts and share your results–I will refund half of the purchase price of the curriculum.  If a school finds some other method is more effective, I will refund 100% of your purchase price.

Webinar with Dr. Don: How to Prepare students for math success.

Posted on by Dr. Don

On Thursday May 3rd, the Educational App Store is hosting a seminar with Dr. Don, “How to prepare students for math success.”   Pacific time will be 8:30 AM, Eastern time 12:30 PM and London time will be 4:30 PM .

This 30-minute webinar focuses on the importance for future math success of developing fluency and automaticity with math facts and how to help students achieve it.

Dr. Don Crawford, the author of Rocket Math and Justin Smith, CEO of the Educational App Store will discuss

  1. What are math facts and why are they important for future math success.
  2. What happens when students haven’t memorized math facts.
  3. How can you best help students learn math facts.

Here is the link to register for the webinar.  https://www.educationalappstore.com/webinar/how-to-prepare-students-for-math-success

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.

Why give the Two-Minute timings in Rocket Math?

Posted on by Dr. Don

To prove whether students are making progress in learning math facts.

First of all, understand that the two-minute timings are NOT a teaching tool.  They are an assessment tool only.  Giving a two-minute timing of all the facts in an operation every week or two allows you to graph student performance.  You graph student performance to see if it is improving.  If the graph is going up, as in the picture above, then the student is learning.   If the graph is flat, then the student is not really learning.

The individual graphs should be colored in by students allowing them to savor the evidence of their learning.  The graphs should be shared with parents at conference time to prove that students are learning.  

Progress monitoring with two-minute tests are a curriculum-free method of evaluating a curriculum.  If you use the same tests you can compare two methods of learning facts to see which one causes faster growth.  This makes for a valid research study.

This kind of progress monitoring over time is also used in IEPs.  You can draw an aimline from the starting performance on the two-minute timing to the level you expect the student to achieve by the end of the year.  (Note the writing speed test gave you goals for the two-minute timing which you could use for your end-of-year goal.) The aimline on the graph, when it crosses the ending date of each quarter, will provide quarterly objectives that will enable quarterly evaluation of progress–required for an IEP.

These two minute timings are a scientifically valid method of proving whether students are learning math facts, in the same way that tests of oral reading fluency prove whether students are learning to read.  They can be used to prove to a principal or a curriculum director, for example, that Rocket Math is working and is worth the time, paper and money it requires.

 

Integers learning tracks are a part of Worksheet subscription

Posted on by Dr. Don

Using a vertical number line can help provide certainty.

Adding and subtracting positive and negative numbers can be confusing for students. You can either start with a positive or a negative number and combine it with a positive or a negative number. That makes for four types or patterns of problems. Then when you consider addition and subtraction, the total is 8 problem types. Rocket Math Worksheet Program has three learning tracks to help students learn how to deal with integers. Mixed Integers include all eight types, whereas Learning to Add Integers and Learning to Subtract Integers each just deal with four types. [Mixed Integers may be too hard for some or all of your students–meaning they can’t pass levels in 6 tries. In that case, put them through the Learning to Add Integers and Learning to Subtract Integers first.]  

Part 1: Using the vertical number line to solve integers problems

The first issue for students is just to be certain of the answer. A vertical number line, where “up” is more and “down” is less helps provide certainty.

I have posted a series of free lessons online (links below) that use a vertical number line and a consistent procedure to take the confusion out of the process. Students can solve all eight types of problems with the same process on the vertical number line. Using the vertical number line there are two rules to learn. Rule 1: When you add a positive or subtract a negative, you go up on the number line. Rule 2: When you subtract a positive or add a negative, you go down on the number line.

So the first thing to figure out is what you are being asked to do (add or subtract a positive or a negative), and then use the rule to tell you whether you’re going up or down. Next step in the procedure is to circle the starting point on the number line. Once you circle the starting point, you show how far you’re being asked to go. You simply make the right number of “bumps” going either up or down from where you start. That gives you the answer without any uncertainty. These online lessons are quick (about 2 minutes) and identify a pattern of whether the answer is like the sum or the difference between the numbers. Once students can recognize the pattern they can begin to answer fluently and without a struggle

(1) Mixed Integers Set A1 Positive add a positive

(2) Mixed Integers Set A2 Positive subtract a positive

(3) Mixed Integers Set D Negative add a positive

(4) Mixed Integers Set G Negative subtract a positive

(5) Mixed Integers Set J Negative subtract a negative

(6) Mixed Integers Set M Positive subtract a negative

(7) Mixed Integers Set P Positive add a negative

(8) Mixed Integers Set S Negative add a negative

 

 

Part 2: Using the Rocket Math Integers learning track(s) to develop fluency in recognizing the type of problem

Here is a part of a page from the Mixed Integers learning track. The paired practice part of the program helps students learn to quickly and easily recognize each pattern.  First, students use the vertical number line to work on a problem. In this example: -6 minus (-4). Then they have a set of problems with the same pattern (a negative subtracting a negative), which they should be able to answer orally without using the number line. Each worksheet includes all the types learned so far in the learning track.

As with all Rocket Math programs, there is a 2 to 3-minute practice session (at this level, I’d recommend 3 minutes), with a partner. Then the two switch roles. The practice is followed by a one-minute test. If the student can answer the problems in the test fluently (essentially without hesitations), the level is passed. As always, the student’s goals are individually determined by a Writing Speed Test. If a given level is still difficult, the student stays with that level a bit longer.

When a new pattern or type of problem is first introduced the one-minute tests will have a whole row of problems that are the same pattern. When the student passes the level, the next test will have two types of problems in each row. The next level has 3 types in a row, culminating in the fifth level, where the problem types are mixed. This way, the student develops fluency in recognizing the type of problem and how to derive the answer quickly. The Learning to Add Integers and Learning to Subtract Integers learning tracks take more time to learn the patterns, while Mixed Integers move more quickly.

Here’s a link to sign up for a Worksheet Subscription.