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 

Friday Afternoon Cartoon

Want to make change? This cartoon illustrates how making change for a dollar can be an exercise in computation. Try it with your kids, but aim for 5 different ways to make change, not 293!

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 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. 

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?

and…

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.