Thursday, October 18, 2012

Larry’s Math Do’s and Don’ts

In December 2004, the Wall Street Journal reported that American kids are “Economic Time-bombs” because they are not learning enough math to be the good problem solvers the nation demands. With this in mind, how are your children performing in math? 

Most American children lose ground to students all over the world who are preparing themselves for success in the future, which includes, like it or not, a lot of math. And why is this happening?

Unfortunately, this is because year-after-year students are put in classes for which they do not have the prerequisite knowledge necessary for success.  Under these conditions, it is difficult, if not impossible, for teachers to ensure that students are acquiring the number sense in elementary school and the solid pre-algebra skills in middle school needed to be successful in Algebra and the higher math classes required in high school and college, regardless of their choice of majors.   To ensure your child is on track, I have created some simple Do’s and Don’ts. Do’s:
  • Do “math” with your children just as you read with them
  • Do make sure your children get their math homework done in a timely fashion
  • Do meet with your child’s math teacher from time to time so you know what is going on beyond the report card. Work with your child’s teacher to set realistic goals for current school year and for the future.
  • Do check to see if your child (one major tip per grade level):
    • Second Grade: Is “fluid” with single-digit addition and subtraction.
    • Third Grade: Can find half of even and odd numbers.
    • Fourth Grade: Know times tables “by heart.”
    • Fifth Grade: Can order fractions using benchmark numbers.
    • Sixth Grade: Is able to mentally calculate percents using “friendly” numbers.
      • 7% of 300 = ? are “friendly” number, whereas 7.132% of 321.097 = ? ain’t so friendly!
  • Seventh Grade: Is able to convert fractions to decimals to percents.
  • Pre-Algebra: Can effortlessly add and subtract positive and negative numbers.
  • Algebra: Is able to solve simple equations “by inspection.”
  • Don’t let your negative experiences in the math classroom influence your child’s education.
  • Don’t let your child use a calculator until he has developed genuine Number Sense.
  • Don’t let your child be put in a math class that he is not ready for, that is, a class where he does not have the prerequisite knowledge necessary for success in the class.
  Following these Do’s and Don’ts will not solve everything, as students need to continuously nurture their math skills and Number Sense with practice. However, use these as guidelines that will help steer your child in a strong direction.

Back to School Checklist

Imagine this scenario.

One day a student arrives at the college or university of his choice for the first time as a freshman. He is so excited to start his new adventure as a college student, inevitably facing new challenges and meeting new people. Knowing that things would not come easy, but ready for all that was ahead of him, this student is confronted with the Mathematics Placement Exam to see what level of college math the student is qualified to take.

The results of the exam make it clear that the student is nowhere near ready for the course that will help him achieve college credit, an unfortunate situation that occurs far too often with incoming college freshman. Before taking the math course of his choice, he will have to revisit previous math concepts by taking a non–credit course, delaying his progress toward college graduation.

This is not an uncommon situation. Nor is it a situation without a solution, as we will see in a minute.  This is a reality that many students go through when entering college, causing them to give up on potential careers in fields that relate to math. We want to stop this from happening early on! The problem can be fixed in grade school.

With that in mind, I have created a Back to School Checklist consisting of three simple questions:
  1. Are your child’s math skills ready for the coming school year?
  2. Is your child being placed in the right math class?
  3. Do you have resources identified in case your child needs extra help with math?

To properly gauge your child’s math capabilities, ask them the appropriate question for the corresponding grade-level they just finished:
  • Second Grade: Can you efficiently compute 7+8+9-10?
  • Third Grade: How much is 99+99+99? This should be done mentally!
  • Fourth Grade: Count from 0 to 7 by 1 3/4?
  • Fifth Grade: Which is greater: 9/10 or 18/19? Explain.
  • Sixth Grade: Half way through the second quarter, how much of the game is left?
  • Seventh Grade: How much is 6 1/2 % of 250? Explain.
  • Pre-Algebra: On a certain map, 6 inches represent 25 miles. How many miles does 15 inches represent? Explain.
  • Algebra: Solve 4x+3=0. This should be done mentally!

Make sure these questions are solved as efficiently as possible because if the appropriate measures are taken, higher grades will only be the beginning of your child’s math experience. Your child will have an improved attitude towards math, they will be better prepared for future math classes, and they will be prepared for college level work when the time comes.

And from college… Who knows where math can take your child?

Finger Count Beware!

I want to introduce the importance of “numerical fluency” and learning basic “number facts.”

Picture this scenario.

A teacher asks a classroom,
“If you spend 70 cents, 80 cents, and 90 cents, how much did you spend altogether?”
The teacher is thinking,
“7 + 8 + 9 = 24. With a zero at the end, the answer would be 240 cents.”
However our “finger counting” students, which is sadly too many of them, are thinking,
“7 + 8 = 7…8…9…10…11…12…13…14…15,” then “15 + 9 = 15…16…17…18…19…20…21…22…23…24…25 (oops).”
Many times finger counters get the wrong answer because they either count too many or too few.

Now, since the process of “getting it wrong” is so uninspiring and time–consuming, not surprisingly, many students report being “bored” in math class. In addition, the process has taken so long that the student is no longer in the flow of the lesson, which in this case, is learning about how to “add a 0 at the end.”

The term “number facts” includes all addition, subtraction, multiplication, and division problems resulting in single–digit and double–digit numbers (up to 24 for addition and subtraction, and 144 for multiplication and division). Examples of number facts include:

 3 + 7 = 10 13 – 5 = 8 5 x 9 = 45 120 ÷ 10 = 12. 

In school, great emphasis is put on rote memorization of “number facts.” This emphasis is misguided.

“Numerical Fluency” is the ability to “effortlessly recall—to “know by heart.” Students should be able to tap into their reliable, quick, and knowable ways to answer “number facts” questions.

Many students in 2nd through 5th grades and higher have a limited grasp of numerical fluency. Hence, their ability to stay in the flow of new lessons is extremely limited. This makes mathematics a frustrating and painful process for everyone involved—the kids, the teachers, and the parents!

Memorization seems to be the more understandable route initially, but it does not promote the mathematical thinking and problem solving skills that are required for long–term success in math. Eventually, most students will forget what they memorized.
I suggest that it is fairly easy to forget that which you have memorized, and nearly impossible to forget that which you have learned.
What students need to do is to build mental structures, frameworks for learning, so that they will know the basic number facts in a matter of a second. Then they won’t have to worry about “forgetting.”

In my next Blog posting, I will detail a process for teaching virtually any child how to “effortlessly recall” the number facts, paving the way for future success in the mathematics classroom. 

Numerical Fluency Tips – The Visual Element

In the last blog posting I discussed the importance of developing numerical fluency (the ability to effortlessly recall and use basic number facts). Unfortunately, many students in 2nd through 5th grades, and at times even higher, have a limited grasp of numerical fluency. To avoid making mathematics a frustrating process for all those involved in the teaching and learning process, today I will go over some tips, designed for parents with children in elementary school, to effortlessly recall number facts.

Before I begin…

Think about the last time you attended an excellent presentation. It could be a classroom lecture, a keynote speaker discussing the importance of social media, or even a YouTube clip explaining how to tie a tie. The common quality in any excellent presentation is a facilitator explaining goals and objectives right from the beginning to ensure the audience is on the same page. After that, a great follow up tool is the visual element.

Visual elements add impact and interest to a lesson. Pictures are useful in reinforcing many concepts. Let’s look at this image for example. 

 Possible questions you can ask are:
  • How many circles are there in the picture?
  • If each circle is a penny, how much money is shown in the picture?
  • If each circle is a dime (a nickel, a quarter, etc.), how much money is shown in the picture?
  • Shade in half of the circles. How many are not shaded in?
  • Shade in half of the circles that are not shaded in. Now how many circles are not shaded in?
  • Again, shade in half of the circles that are not shaded in. Now how many circles are not shaded in?
When a child has a visual element to look at, concepts as simple as counting, or slightly more complex like fractions, become easier to understand.

In my next blog posting, I will go over tips to effortlessly recall addition and subtraction facts. For now, use visual elements to practice concepts with your child!

Numerical Fluency Tips – Addition/Subtraction Facts

The first few times a child looks at addition and subtraction problems can be a confusing experience. It is very important for children to be able to effortlessly recall reliable methods to answer number facts.

One excellent method is Filling in the Gap. Take a look at the following example.

8 + ___ = 13

To find the missing number, we can fill in the gap between 8 and 13 by solving:

8 +___ = 10       and      10 +___ = 13

Once the two easier problems are solved, adding the two answers together will give the child the desired result. Take a look.

Since 8 + 2  = 10 and 10 + 3 = 13, the “gap” is 5 (2 + 3).

Check the result: 8 + 5 = 13.

And in words…

How far is it from 8 up to 10 (2) — how far is it from 10 up to 13 (3)? From 8 up to 13 is 5 (2 + 3) so 8 + 5 = 13.

Now try this:

7 + ___ = 16

7 + ____ = 10 and 10 + ____ = 16

7 + ___ = 16  
Since ____ + ____ = ____, the “gap” is ____.

So, 7 +____ = 16

The ability to use this “up to and over 10” method relies on the student knowing a series of prerequisite skills.  Here is a list of those skills.  When mastered, these skills will enable students to have effortlessly recall of addition and subtraction facts.
For each tip, I will also provide practice problems.


1)    Doubles

5 + 5 =                                             9 + 9 =

2)    Doubles plus/minus 1

5 + 6 = 5 + 5 + 1 =                        8 + 7 = 8 + 8 – 1 =

3)    Counting on (start at x and count up by y)

7 + 2 =                                              8 + 3 =

4)    Breaking down numbers

6 + ___ = 9                                        7 + ___ = 11

5)    How far apart are two numbers? (How far is it from x up to y?)

How far apart are 6 and 10?

How far is it from 9 up to 12?

6)    Combinations that make 10

8 + 2 =                                              6 + 4 =

7)    10 plus a number

10 + 7 =                                           10 + 9 =

8)    10 plus what number?

10 + ___ = 16                                  10 + ___ = 19 


How much is left?
Use the notion of “how much is left” when the numbers are fairly far apart, and count down.
For example, 12 3 is best thought of as “counting down from 12 by 3.”

2)    How far apart are the two numbers (how far is it from the smaller number up to the bigger number)?
Use the notion of “how far apart are the two numbers” when the numbers are fairly close to each other, and count up.

For example, 12 9 is best thought of as “how far is it from 9 up to 12.”

For the following examples, decide which method you would use: How much is left or How far apart

100 – 98 =                       100 – 3 =                          100 – 87 =                       100 – 15 =

Stay tuned for my next Blog posting to learn some more tips to effortlessly recall number facts.

The Name of the Game – Denomination, Quantity, and SAMEness

 When you add two apples with three apples you get five apples. When you add two bananas with three bananas you get five bananas.

When you add two apples and three bananas do you get five banapples?

You don’t get five banapples just as if you add pennies and nickels you don’t get pickles. The reality is that you cannot add two things that have different names.

According to The Law of SAMEness you can only add or subtract things that have the same name.
There are Four Principles I want you to think about:
Principle 1:
A thing can be simple or complex. It could be as simple as a penny in your hand or as complex as the total number of eyes, noses, and fingers of all the people on Earth.

Principle 2:
Every thing has a name.
The name is that thing’s denomination.

Principle 3:
Things have a number associated with them. The number can be as general as singular and plural or as specific as a number (i.e. a person, several people, two apples, a half gallon milk, twelve days).

The number is the quantity of things we have.

Principle 4:
Combining Principles 1, 2, and 3, we learn to think in terms of
Quantity and Denomination—“How many–of what?”

The name of today’s game is Denomination, Quantity, and SAMEness.

Every thing in math has a name. This name gives the denomination of the thing, making it unique within a larger group of things.

For example:
Tuna, shark, and salmon are denominations of fish.
Halves, quarters, and thirds are denominations of fractions.
Pennies, nickels, dimes, and quarters are denominations of coins.

After knowing the name of some thing, you should also know the quantity of that thing in the given problem.

For example:
Three sharks quantity = 3
Half of a dozen quantity = ½ and 6
Two hours in minutes quantity = 120

After distinguishing between the denomination and the quantity associated with each thing, it is important to understand the functionalities of these things. This calls attention to The Law of SAMEness.

Keep the rules of quantity and denomination in mind will enjoy math more because the rules you have learned will make more sense to you! For more practice, work on these problems:

1) 1 nickel  + 1 dime = 5 cents  + 10 cents = 15 cents (if you change the name to cents and adjust the amount appropriately)

2) 1 dog + 1 cat = 1 dog + 1 cat (or 2 animals, if you change the name), just as in Algebra x + y  = x + y,  (or 2 unknowns, if you change the name)

3) ½ + ¼ = 2/4 + ¼ = ¾ (Just as the names “nickels” and “dimes” had to be changed to the same name, so too the denominators must be changed to the same name (getting a common denominator).

Try these:
1) 5 nickels + 50 cents = ___________  (See Example 1 above.)

2) 2 cats + 3 dogs + a cat + 2 dogs = ________________ (See Example 2 above.)

3) ½ + 1/3  = _____________ (See Example 3 above.)

Remember, there is no such thing as banapples

A Flea's Flea - Proportional Thinking

 When my son Nick was seven years old, he told me, “Hey Dad, if the world was the size of a basketball, our house would be as big as a flea’s flea!”

I was stunned at this incredible display of Proportional Thinking.  While I thought to myself that our house would actually be the size of a flea’s, flea’s, flea… I also thought “Hey, full credit for a seven year old.”

This demonstration of a “sense of scale,” placing our home in proportion to a basketball sized world, was setup by verbal questions I asked Nick that involved proportional thinking.  At the same time, I helped him develop strong mental math skills (we’ll dig deeper on this in another post).  Let’s take a look at some more examples.

Imagine an amusement park filled with all sorts of rides, clowns, arcades, food – the works! Now along with that, place a carrousel, ten times bigger than every other attraction, somewhere into this amusement park. It would completely stand out because it would be out of proportion.

Understanding these concepts involves a working knowledge of Proportional Thinking.

Proportional Thinking is “thinking in accordance to amount.“ The amount of one thing can be adjusted according to the amount of something else. Many kids have a difficult time visualizing proportionally. This is because they need to be able to develop a mental image of a sense of proportion – “a sense of scale. “

Some sample questions you can ask your kids are as follows:

If you put a dime in a machine and get 2 pieces of gum, how much would you have to put in to get 6 pieces?

If 3 tennis balls cost two dollars, how much will 12 balls cost?
A sense of scale can be activated by asking the following:

What part of 12 is 4? What part of 100 is 25? What part of 1,000 is 10?

Number Sense does not happen by accident. It must be carefully instilled and nurtured by teachers and parents.  Parents, a couple of minutes spent on one or two questions a day will pay off in terms of mathematical growth in the long run. Ask questions on the way to and from school, at the market, and in such unlikely places as an amusement park!

Be aware that ‘sense’ is understanding things in context, as in the carrousel example above. Context provides a framework of ideas and information that can be used in a meaningful way.  Now that we understand the components of Number Sense: Counting, Wholes and Parts, and Proportional Thinking, let’s begin using our Number Sense! 

Wholes and Parts

Add 17 + 18 + 19. Do it quick! If you reached for a scratch sheet to get to the solution of 54, you didn’t do it wrong, but here’s a different way to think about this question:

First, round each number to the nearest 10 (20 in this case), and add the 20 three times, making a total of 60.

Now, take away 19 from 20 (1), 18 from 20 (2), and 17 from 20 (3). That gives you a total of 6 (1 + 2 + 3).  This 6 represents how much “extra” you added by adding 20 three times instead of just adding 17, 18, and 19.

Finally, take the 6 from 60 and you get 54 as your solution!

Try this:  99 + 99 + 99.

By doing this mentally, we can help children to develop strong numerical fluency skills.

Number Sense is the ability to appreciate the size and scale of numbers, in the context of the question at hand. Three elements establish Number Sense: Counting, Wholes and Parts, and Proportional Thinking. We already discussed Counting. Today, we will focus on Wholes and Parts.

The concept of Wholes and Parts is the backdrop for many mathematical concepts.

A whole that is broken into equal parts creates fractions. 100% of something represents a whole. Many people do not have a clear idea of what a fraction represents nor do they break down the word ‘percent’ for what it is: per-CENT—“for each 100.” Without a solid understanding of Wholes and Parts, solving word problems becomes very difficult.

Wholes: equal to the sum of its’ parts.

Parts: equal to the whole minus the other part(s).

Let’s dig deeper, starting with complements. A complement is the amount needed to make a whole complete.  Problem solving comes to the ability to identify the missing part(s) or the complement.

Children should be asked to visualize and answer a question like:

Together we have 10 pieces of candy. You have 7 pieces. How many pieces do I have?  Here, the whole is 10 and one part is 7.  So, the other part is 10 minus 7 (the whole minus the part, you know).

This will help set up their understanding of complements.

Children also need to be introduced to the fraction “half” as being “2 parts the same.” Before other fractions are introduced (1/3s, 1/4s…) children need to master questions like:

How much is half of 6? 3? 7? 20? ½? 99?


Half of what number is 5? 10? 25?

Wholes and Parts creates a strong understanding of the structure of mathematics, eventually building up a child to understand how to solve complex fractions, equations, and word problems.

When dealing with Wholes and Parts, kids really like examples that deal with cookies, sandwiches, or anything they can eat. Try this example:

You have a box of cookies. You get to eat half of them after lunch and half of what you left over after dinner. After dinner, you have 3 cookies left. How many cookies did you start with? The answer is 12.

When a word problem is set up like this, children can often visualize the situation. If visualization doesn’t work, have a Plan B (drawing a picture), or C (using physical objects) until the child understands.

For every kid, there is a way to explain every topic in a way that makes sense to them. 

Counting, The Basis of Number Sense

I have often heard students coming out of a class saying, “That stuff doesn’t make sense!” This is because many students have not developed a good general sense of the mathematical subject matter that is presented at school. Students are not provided with enough context when they learn material in school. Context provides students with a suitable environment to integrate new ideas and information in a meaningful way. Unfortunately, rather than integrated learning, fragmented learning (learning without a sense of continuity) takes place in today’s schools. To fill in the gap between fragmented and integrated learning, student’s need to establish Number Sense.

So, Number Sense… what is it? Number Sense is the ability to appreciate the size and scale of numbers in the context of the question at hand. There are three elements that fall under Number Sense: counting, wholes and parts, and proportional thinking. Today we will focus on counting.

Counting, simply put, is the ability to count from any number, to any number, by any number, forward and backward. When I ask students to explain what counting is, they will usually respond by counting from 1 (1, 2, 3…). Although this is completely correct, kids need to grasp how to count from other arbitrary numbers, for instance, 28 (28, 29, 30…). How about when counting by 2s? Starting from 2 (2, 4, 6…) is easy to get down. Can our kids do the same when starting from 3 (3, 5, 7…)? After a good deal of practice, an experienced counter will know how to count to 250 by 1s forward and backward; to 300 by 2s, 5s, and 10s; and to 3,000 by 100s.

As children are learning to become experienced counters, they should also be learning how to group the numbers they count. Parents, ask your child questions like “A group of 10 take-away a group of 3 leaves how much?”

Another important idea at this stage is interval: the distance from one number to another—the space between two numbers. From 6 to 7 is 1, and from 7 to 8 is 1, making a total distance from 6 to 8 of 2. Another important aspect of counting is its connection with the basic math operations: addition (counting how much altogether), subtraction (counting how much is left), multiplication (counting in equal groups), and division (counting how many of these are in that). The basis of Number Sense begins with counting.

Remember, children don’t hate math, they hate being confused, frustrated, and embarrassed by math. Once they understand math, the passion will follow naturally.