Breaking Down the Question Barrier

Children need to be exposed to the forms of mathematical thought as well as to the details. What is missing in most instruction is the timely unfolding of these basic forms of mathematical thought. The use of consistent, familiar forms clearly depicts the underlying structure and helps eliminate confusion and uncertainty.

 

Without the structure provided by the Question Forms, classroom instruction often disintegrates into the usual collection of details—facts, operations, and procedures—that have characterized unsuccessful mathematics education for decades. All students (“at–risk,” “at–level,” and “talented” alike) benefit from having new material in mathematics presented as reoccurring manifestations of the same set of basic ideas with which they have already become familiar.

Let’s start with questions in the “single-form” format. Each is a breakdown of the format of questions based on one single concept; this allows your student to lay down a base to build a foundation on.

Counting, Grouping, Intervals: 
Count from _____ to _____ by _____s 

     Count from 2 to 16 by 2s...from 1 to 15 backward by 2s.

Denomination and SAMEness:
 What is another name for _________? 

     Name 2 numbers that add–up to 15...to 25...to 75...to 100.

Subtraction and Addition:
 How far is it from _____ to _____?

     How far is it from 5 to 11?...from 13 to 20?

Order: 
_____ is greater than/less than/equal to _____ .

     Name 3 numbers that are between 7 and 10.

Multiplication:
 How much are _____ groups of _____ ? 

     If you see 8 eyes, how many people are there?...16 eyes?...100 eyes?

Division:
 How many _____ are there in _____?

     How much is 1 + 1?...2 + 2?...4 + 4?...8 + 8?...16 + 16?...(continue as far as you students can go mentally.)

Fractions and Fractional Parts:
 How much is _____ (fractional part) of _____? 

     Half of what number is 7?...is 1?...is 12?
How much is half of 8?...of 12?... of 18?... of 200?...of 1,000?

“The Whole is equal to the sum of its Parts”: _____ (whole) = _____ (part) + _____ (part) 

     A cook bought two bags of apples. One bag of apples weights 15 pounds and another weighs 10 pounds. How much is left after the cook uses 17 pounds of the apples for pies?

Percent: 
_____% of _____ is _____. 

     How much is 50% of 6?...of 20?

Ratio:
 _____ is what part of _____? 

     Is 7 half of 14, or is 7 twice as much as 14?

Change and Variation:
 As _____ gets bigger/smaller, _____ gets bigger/smaller. 

     Does it cost more to buy 5 apples or 10 apples? Why?

Proportion: 
_____ is to _____ as _____ is to _____ . 
     Does it cost more to buy 5 apples or 10 apples? Why?

- Larry Martinek

Back to Basics: The Essence of the Mathnasium Difference

Principles of Math can come across confusing to students, especially when they are bombarded with multiple rules and confusing practices. Mathnasium focuses on introducing new subject matter with a consistent and knowable approach, allowing students to build upon a foundation of knowledge while understanding and adding new principles.

To begin Early in a child’s learning, values are expressed in groups of 10s:

• 10 pennies make a dime.

• 10 dimes make a dollar.

•   10 “one-dollar” bills make 1 ten dollars, and 1 ten–dollar breaks down to 10 “one–dollar” bills

• 100 pennies make a dollar.

• 100 dimes make ten dollars.

• 10 hundreds make a thousand.

• 1,000 thousands make a million.

Some things ”make sense” to cut in half: a candy bar, a piece of wood, numbers... Other things don’t: people, pennies, cars... To diagnose the correct approach, assess the subject matter. Use the examples below as guidance.

• If twice as many people as you expected come on a picnic, then you will need twice as much food.

But, if you are baking bread, twice the regular heat will not get the job done twice as fast.

• When you double the number of pieces, the size of each piece is half as much as it was. (This is the inverse relationship: when one thing goes up, the other goes down.)

Contrary to its reputation, Math is not all numbers, in fact, word prefixes play a large role in setting the numerical value. Take these examples:

• “mono–”means 1: The monorail at Disneyland runs on 1 rail.

• “bi–” means 2: A bicycle has 2 wheels.

• “tri–” means 3: a Triceratops has 3 horns.

• “qua–” means 4: a quartet has 4 players

• “dec–” means 10: a decade is 10 years

• “cent–” means 100: percent means “for each 100”

• “mil–” means 1,000: a millennium is 1,000 years—a mile is “1,000 paces”


Finally, I touch on a few points of knowledge for your student (if this is for parents, “parents have kids… teachers have student) that will assist him or her build the base for understanding the previously dissected, whole and parts method.

• A quarter of an hour is 15 minutes (not 25).

• There are 4 quarts in one gallon. A “quart” is a quarter of a gallon.

• Whole basketball and football games have 4 quarters.

• Four quarters make a whole dollar.

• One half dollar is the same as 2 quarters.

• Half-time in a basketball game comes after 2 quarters.

These basic understandings set the cornerstone for laying the foundation of knowledge to continually build and evolve your student’s mind, the Mathnasium way.

The Whole Equals the Sum of its Parts

Word problems can be tough even for the math-minded. The challenge lies in correctly converting words to the numbers and symbols of an equation. One method that helps is the concept that “the whole is equal to the sum of its parts.” Start with these three questions:

• What is the whole in this question? Is its value known or unknown?

• What are the parts in this question? Are their values known or unknown?

• What is the relationship between the whole and its parts? Which remains constant in the question? Which changes? How does it change?

By figuring out what are the parts and what is the whole, we can decide whether we need to perform a synthesis (“building up”) or an analysis (“breaking down”) to solve the problem at hand.

Synthesis:
If the whole is unknown, then the task is to build it up from its known parts:

• If the parts are equal, we multiply.

• If the parts are not equal, we add.

In other words, affirming that “the whole is equal to the sum (total) of its parts.”

Analysis:
If the whole and one or more of its parts are known, then the task is to find the remaining part(s) by breaking down the whole, using the known part(s):

• If the known parts are not equal, we subtract.

• If the known parts are equal, we divide.

Basically, “Each individual part is equal to the whole minus all of the other parts.”

By identifying which category the problem falls under, we can designate a relationship and determine the plan of attack; this is called the whole-part method. Below are a few examples of the method in action.

Ex. #1:

A box contains some marbles. 6 of the marbles are red, 5 are green, and 14 are orange. How many marbles are in the box?

In this question, the whole (the total number of marbles) is unknown. Since the parts (the number of red, green and orange marbles) are known, we can use the Key Synthesis Concept to find the whole:

total # of marbles = (# of red) + (# of green)+ (# of orange) = 5 + 6 + 14

= 25 marbles.

Ex. #2:

A train traveled 200 miles at an average speed of 50 miles per hour. How long did the trip take?

In this question, the distance traveled is the whole and is known. In each hour the train traveled an average of 50 miles, so:

time of trip = (distance traveled) ÷ (average rate of speed)

= 200 miles ÷ 50 miles per hour = 4 hours.

Ex. #3:

Another sandbox contains 100 pounds of mixed sand, 15% of which is brown sand. The rest is white. How much white sand must be added to make the mixture only 5% brown.

In this question, there are 15 pounds (15% of 100) of brown sand. As we add white sand, the whole (the total amount of sand) changes, but the amount of brown sand remains constant (at 15 pounds).

What we want is “15 out of the (new) total” to equal “5 out of 100” (5%, the desired outcome).

We can ask the following equivalent questions,

“15 is to what number as 5 is to 100?” or
“5 out of 100 = 15 out of what number?” or “5/100 = 15/what number.”

All of these methods yield the same answer: 300 pounds. Since the box already has 100 pounds of sand, 200 pounds (300 – 100) must be added.

The whole-part method allows us to identify the first step of a word problem bringing us one step closer to the solution and making math make sense.

- Larry Martinek 

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’ts

• 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.

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 2^nd through 5^th 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 2^nd through 5^th 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.

ADDITION TIPS

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 

SUBTRACTION TIPS

1)    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.