Q&A

Questions and Answers

Some people have found it difficult to understand this page.  Please do not feel bad if you find it difficult to follow and understand this page at first read.  Some people have a degree in math and they have a difficult time understanding these concepts.  The following questions were asked by a person who  "claimed"  to be an expert in math.   Keep this in mind as you read this and you will see that he does not understand the basic concept of groups and counters or even what a number is.   If you take the time to read this page and think about it you will be able to see math from a different point of view.  You will understand that you are smart and can understand math  if it is explained to you correctly.

Question

Is a piece of wood a shelf, or a table leg?

It can be used for both. The wood can be used for either a shelf or a leg or both. When the wood is used as a table leg, it is a wooden table leg.   We have partly defined it by its use and its location. The location and use are part of what defines it. The same is true for single line symbols.   We define them by their use and location.   

To define the term "wood" you must describe its physical and tangible traits, use and location. You must describe it in a way that other can understand. You can describe it as a plant that has been processed into a building material. A more detailed description would be " a hard fibrous substance beneath the bark of the stems and branches of trees and shrubs that are cut and prepared for use in making things."

The same is true of a number. You must define it by its tangible traits, its use and location.

A number is a name given to a symbol, that represents a group(class), that represents an address in a three dimensional space, that has a sequential list of order incremented by one plus one (counting), that locates an object (thing or unit) in that space.

A number is a collection (group) of things (units)

(A)means one    (is) means equal to    (of) means times.

(A) number         (is) one group            (of) one thing (unit)

(1) number         (=)   1 group                 (x)    1 dot

           N                (=)        g                       (x)        d

therefore   N  =  g  x  d      ( a number has groups and things or it in not a number.)

Question

Aren't numbers just numbers?

Answer (by Owen)

No, numbers aren’t just numbers. They are a complex combination of two single line symbols. A number has two parts. One part is a single line symbol called a group symbol and the other part is called a unit symbol or counter (because we can count them). Single line symbols have no value unless the use is given. This is why we must specify what the units are. We must specify the units and that defines how the single line symbol is to be used. The use and location of the single line symbol define what the symbol is. You must have something to add or you cannot add. If you have no units, your answer is always zero.

Are you saying that the statement

3 + 5 = 8

is meaningless without units?

3 + 5 = 8 is meaningless because it has no units. It means nothing because you did not say what it is that you have that you would like to add.

Your statement is: 3 groups ( 0 units) + 5 groups ( 0 units) = 8 (zero units)

The student is told the solution is "8". The solution is zero {0} because anything multiplied by zero (0) is zero (0).

(8 groups) times (0 units) is { 0 units}

(8 X 0 = 0 units)

The 3, 5 and 8 are group symbols. That means without units the answer is zero not eight. The formula for a number is:

 (groups) X (counters) = Number.

A Single line symbol can be used as a group symbol or a counter symbol. This is one thing that is causing confusion and is one of the major factors of math anxiety.  Children are told to do calculations that do not mean anything.

In the question solve 2 + 2= 4

isn’t the answer just 4 without any units?

In the problem "solve" there are no units given (zero units). That means we have two groups of zero we are going to combine with two other groups of zero.

This is the math for solving 2+2=N (zero units)

Formula:    Group (counter ) + Group (counter ) = N (counter)

step1:       2 groups (0 units) + 2 groups (0 units) = N

step2:   1 group (2 x 0 units) + 1 group (2 x 0 units) = N

step3:                        (0 units) + (0 units) = {0 units}

step4:      The solution is zero

Formula: Group (counter ) + Group (counter ) = N (counter)

step1:    2 groups (1 dot) + 2 groups (1 dot) = N (dots)

step2:                (2 x 1 dot) + ( 2 x 1 dot) = N (dots)

step3:  1 group ( 2 dots) + 1 group ( 2 dots ) = 1 group {4 dots}

step4:      The answer is we have four dots.

I believe that I can add 2 and 2, without having any units involved:

2 * (1 dot) + 2 * (1 dot)

= (2 + 2) * (1 dot) (Distributive property)

= (4) * (1 dot)

Your statement shows that you do not understand the difference between group symbols and counter symbols. Your work is incomplete, Vague and misleading because you use information out of context and nothing is defined.

First:.........you did not write down the formula for a number.

                Group (counter ) + Group (counter ) = N (counter)

Second:.... you did not explain why you used the distributive property

Third:........ you do have units because you have dots in your problem.

Fourth:...... you failed to complete the last step!

last step is:   (4 groups) times (1 dot) = 1 group {4dots}

The final answer is a sentence:   You have four dots.

It is ok to add (2 groups) to ( 2 groups) to get (4 groups) but your answer is not 4 groups  (your four is a "single line symbol " and not a number)!!    It is not a number until you define the groups.   You must finish the task and multiply by one dot to get {4dots}. Now you have a number - one group of four dots. (Use brackets or a circle to indicate your answer).

I was taught that I can't add dots and that I can only add numbers.

Answer (by Owen)

I’m sorry this happened to you.  The truth is you can only add dots or things. The Single line symbols only represent the dots and the dots represent things so you cannot add symbols unless they are defined. If you do not define the symbols, you cannot add them.

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Here is another example. You have two baskets. Each basket has one egg. Your friend has two baskets and each basket has one egg. How many eggs do you have all together?

Formula: Group ( counter ) + Group ( counter ) = N

                                           You + Your friend = total eggs?

step1:      2 baskets ( 1 egg) + 2 baskets (1 egg) = N (eggs)

step2:   ( 2 baskets x 1 egg) + (2 baskets x 1 egg) = N

step3:                    1 (2 eggs) + 1 ( 2 eggs) = 1group{4 eggs}

step4:     The answer is you have a total of four eggs .

(answer should be in brackets or circled)

How can you multiply pi times the square root of three with your system? How many dots would that be?

The DotMath for kids system is for children in K-4. The units you give to them must be within the curriculum for those grades. Square roots and pi do not apply to these students.

They'll apply to the students later. What If the students have 'learned' about fractions the wrong way by placing too much emphasis on models?

Most students have problems with fractions because they are told to memorize facts they do not understand. The problem is that not enough models are used that explain fractions properly and children do not have a true need for fractions. We must create a need for them to motivate them. The thing that limits students is the confusing way fractions have been taught in the past. To learn fractions there must be motivation, comprehension and practice.

We can use fractions to _model_ certain situations, e.g., cutting pies into pieces; but that's a model. It's not what's going on with the numbers themselves. It certainly doesn't place any constraints on how the numbers or operations are defined

Answer (by Owen)

The model does define the number symbols for that model.  The single line symbols and operations are defined in that model because we defined them.  The numbers can be redefined for a different model because the group number symbols have no value on their own (0 value) and that is why we must redefine the symbols for other models.  Teachers need to give the units when they ask math questions. This will help the students understand what they are being asked and how to do the math.

I was taught different than this. I think this sets the students up for failure later on. Don't millions of students learn their addition and multiplication tables without involving dots, or units of any kind.

There are many systems that use dots to teach math.  There are number lines,  charts, graphs and many other ways that dots are used and students taught this way do well.  Many students are told to memorize facts and are not taught with dots.   Most of these students do have trouble learning basic math.  There is a major problem with math anxiety because of the way math is taught. There are people doing studies on math anxiety,  full time,  because so many people are affected by it.   Canada and the United States have some of the worst math scores in the world but spend the most money on education.   Many students have trouble learning the basic math functions because they cannot prove any of their answers.

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Math is a short hand language to help us solve real world problems. We translate from language (text) into math symbols to speed up the process of calculating and to reduce the space, paper and time needed to calculate. This is why it is important to know what the units are. If you have no units, your answer will always be zero because anything multiplied by zero is zero. We also need to know what the reason for the calculation is. You can calculate forever and never stop if that is your goal ( boring and pointless).   It is important to get a solution to a real problem to make it worth the effort.

To solve a math problem we should:

Start with the correct formula and end with the solution to the problem { in brackets}, or in a circle, to show the final answer (with units)

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Math is made easy when it is fun. The child learns faster and remembers facts longer when they have fun. "The Fun Book" is called that because it is fun. In fact, it was a pre-kindergarten girl that named the book "The Fun book" because she had so much fun with it.

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About pi

Owen’s Pie

If pies are round and pi (r) squared and two pies are what I have.

How much pie would I have, if they are eight inches round?

Would I use pi(r) squared, to find what two pies are?

If I use pi (r) squared to find what two pies are;

Would I use the distance, or would I use the (r)?

If I used the distance, twice that of a chocolate bar

Would I use the distance, of that candy bar?

If I use the distance to find what two pies are;

Do I halve the distance, or do I use the (r)?

Or did I eat the pies and the chocolate bar?

I must confess I ate them and the bar with glee.

The answer to this question is as simple as can be.