Simple Adding and Subtracting of large numbers.

Tonight I reviewed adding and subtracting with the kids. As I was explaining things, I observed the following principle..

When your student is unable to answer a question, instead of telling them the answer, show them how to figure it out.

When dealing with mathematics, this often involves a lot of counting.

To make it more meaningful and interesting to your student, get the numbers from them. Have them say a number, or use some significant number like their age.. It makes it more meaningful to them, and as a result, they will remember it easier.

Now on to the basic exercise..

I started with a quick review of number houses.. starting with writing down the numbers up to 10.. as in

1 2 3 4 5 6 7 8 9 10

Then pointing out that when you get to 10, the numbers can't fit in the house any more, so they move next door. In the process they change from ten ones, to one ten.

Next I write a large number down, and ask them, while pointing to each number "how many ones", "how many tens", "how many hundreds"... You could go as high as thousands, but that might stretch the vocabulary of the four and five year olds. To review, ask them the same questions without pointing.

Next I got a couple of numbers from them and wrote them down on the board in the vertical form.. as in..

9

+ 8

I then talked about adding steps, as an introduction to the number line.. Which is a great way to show the answers as well as negatives, being opposite..

I used little arcs to represent each step along the number line starting from zero.. so..
After showing this exercise, I return back to the addition diagram and fill in the 17.. like so..

9

+ 8

17

This gives them a concrete observation to associate with the abstract representation of how we write the addition problem down to solve it.

Generally, the more concrete observables a student has to associate with a principle, the easier it will be for them to understand and recall it. If a student is having difficulty with a principle, see how many ways you can help them to experience it. For counting, addition and subtraction, use beans, fingers, toes, blocks, etc.. I use steps for the number line specifically for the utility it has when considering negatives or subtraction which I will illustrate in a moment.

Now that I have illustrated addition, I add another 8 below the 17, and add in the subtraction sign, and then I update the step diagram, starting at the right end, and looping upside down and backwards, count 8 steps back..

9

+ 8

17

- 8

then I count the remaining 9 steps again from the left, and I can update the diagram again like so.

9

+ 8

17

- 8

9

At this point, you can point out that the top and bottom numbers are the same. This is important, as this will allow them to check their own work later, once they get the hang of things.

Building a scale

Got a new video-camera, and last night I went through a practice run for filming construction of a balance scale from foam core project board, and dowels.

I'm rather pleased with the results, the beam that I made, balances well, and the project seemed to go together well. Though a balance made from foam project board would certainly function, I do have some concerns about long term durability. Still, it functions, with a minimum of tools and supplies, and there is a certain benefit from being able to experience the building of it.

For anything longer lasting however, better tools are called for.

More to come as I get things figured out.

Observation, the Master key to the Sciences

Whenever we set ourselves out to discover truth, or to learn, it is important to consider how we can know something.

For the purpose of this book, I am going to state that in order to know something, we must have personal experience with it. Otherwise, regardless of how much confidence we may have that something is true, if we do not have experience with it ourselves, it is a belief.

As a result, when we are striving to know, to learn, it is important to experience as much as we can for ourselves. Now, this of course has it's limits. It is difficult for us to experience walking on the moon, and likewise, other experience me may wish to avoid, as they can have harmful and lasting consequences to ourselves and others. As a result, a good scientist will strive to make sure their experiments and observation is safe, both for themselves, and others.

However, I did not set out to talk about safety, but rather observation and experience.

When we consider that all knowledge, all experience, is gained through the senses. Therefore, I would like to take a moment, to talk about what our senses are. Often in school, we learn about our five senses, those being touch, taste, hearing, sight, and smell. However, I would like to use a much broader definition for our senses, as anything which we experience, or are aware of.

Therefore, our senses not only include the five senses, but also include our thoughts, our emotions, our sense of humor, and that niggling sense that something is missing, or is not quite right.

As an exercise, I would like to ask that you spend a moment and write down your own list of 'senses' or in other words, things that you can be aware of. Seriously, stop right now, and write down your own list..

---

Now, as I consider my own list of things I can be aware of, the first thing that comes to mind is cold and it's opposite of heat, and then pain, pleasure, humor, hunger, thirst, texture, pressure, rhythm, pitch ( frequency ), harmony, dissonance, volume, direction, sweet, salty, sour, bitter, smells ( long list there ), light, darkness, colors, shapes, spacial relationships, fear, peace, love, lust, longing, loneliness, happiness, joy, fatigue, energy, alertness.

As you can see from my list, we can be aware of a great number of things. In short, we are packed with senses, if we take the time to pay attention to them.

Sources for Ballance Scales

It has been my experience that it is critical to have a balance scale to make some critical observations of the natural laws that make mathematics work. These observations work best with a two pan balance. I am working on developing a cheap yet accurate balance for this purpose, but in the mean time, I thought yall might appreciate some alternate sources.

Ohaus^® has a small selection of two pan balance scales, and while I am not satisfied with their accuracy, they are available currently on the market.

Primer^® Balance
Max Capacity: 2,000g
Readability: 1g
Starting At: $15.75
from: ScalesOnline

Unfortunately, this seems to be little more than a toy. The fact that it's Readability is only 1g concerns me.

School Balance
Max Capacity: 2,000g
Readability: 0.5g
Starting At: $24.00
from: ScalesOnline

This would by my recommended minimum. While made of plastic, the readability of 0.5g is somewhat better, and should be sufficient for the purposes of studying the science of mathematics, until the student is ready for a more precise scale.

Ohaus Harvard Junior Balance
Max Capacity: 2,000g
Readability: 0.5g
Starting At: $59.29
Includes Weight set.
from: ScalesOnline

Having the critical parts of this scale made of metal, should make it more resistant to rough treatment that kids can put stuff through.
The addition of a standard weight set is also nice, though most of the time I use pennies and nickels as they are a precise weight ( 2.500g and 5.000g respectively ) and are common place. Especially in the beginning where the student needs to observe the properties of mass and mathematics, and having a large quantity of items that are the same weight is useful.

Harvard Trip Balances
Max Capacity: 2,000g
Readability: 0.1g
Starting At: $146.25
from: ScalesOnline

The improved readability of this scale, and metal construction means that this scale will take longer for the student to out-grow. This can be used for a number of scientific experiments in other sciences as well, as studying basic mathematics.

Ballance Scale Continued.

As I have been reviewing my scale design, to work out a potential manufacturer. AS a result, I have been reminded of a couple of other design elements. both have to do with adjusting the center of gravity for the beam.

Before I get into the adjustments, I think it will help to cover why a balance beam scale works in the first place.

For an experiment, cut a circle out of cardboard ( no it doesn't have to be perfect ) and shove a pencil through the center. then shove another pencil through near the edge..

If you then will hold the center pencil by putting your fingers under the pencil, so the wheel can turn, you will find that the outer pencil is beneath the center pencil. As long as you are holding the center pencil, The force of gravity will always try to push the outer pencil to be under the center pencil.

Now, let's add in another pencil on the edge, say 90 degrees from the other one. As the wheel hangs, both outer pencils will be trying to reach the spot under the center point, and you will end up with both getting half way there. The ratio between how close each one is to the center line, will be determined by the ratio between their lengths.

As the angle between the outer two pencils increases, it becomes more sensitive to differences in weight between the two pencils, with the most sensitive being where the pivot points are exactly opposite each other.

Making a Ballance Scale

Once a student has gotten the hang of making a measurement, and recording that measurement on paper, they are ready to move on to using a simple balance scale. In order to be use a balance scale, it is important that they have one. Care should be taken in obtaining one, as I have seen a-lot of very pretty decorative balance scales that are anything but functional.

As I was figuring how to have the students experience mass, this I designed my own balance scale, in the process I discovered a few simple rules.
The beam of a balance scale, is the heart of the machine. The three pivot points illustrated as the red triangles above, should be exactly the same distance apart, and all on the same line down the center of the beam.
If the pivot points are past the center line line, as illustrated above, the beam will never balance, and will tend to act like a teeter-totter moving to one side, or the other.
If on the other hand, the pivot points are not to the center line, the balance will be lacking in sensitivity, and will not function very well. Of the two problems, the lack of sensitivity, is by far the less critical of the two, so if you are not able to place the pivot points exactly, make sure your margin of error puts you on the lack of sensitivity side. The closer to the center line you can get the pivot points, without going over, the more sensitive your scale will be.

to be continued.

Teaching the Science of Mathematics

I am working on a book, that will go over the science of mathematics, from the perspective of how to learn / teach it.

I have been having some trouble writing the introduction to my book.. so I think I'm just going to start writing blurbs and bits of it, and see where it gets me..

The method I use for teaching mathematics is essentially guiding the student through the discoveries of the principles of mathematics via the scientific method. The student should observe a principle in action in the natural / physical world around them, then come up with an explanation, and then see if that explanation holds through experimentation and further observation. And that makes it sound a lot more complicated than it is.

This method does however dig to the very core of what Science is, and how it applies to our every day lives..

As a good friend of mine says.. "Principles Govern". It is through our observations that we discover those principles..

As it turns out, we naturally learn by applying the scientific method, and we do so instinctively as soon as we are able to start observing the world around us. So this should not be overly complicated or difficult. It can however, become a lot of fun.

Discovering the principles for ourselves, also has a few other benefits.
1) We don't have to take anybody else's word for it. We know of ourselves. This enables us to have faith in those principles, and confidence in our ability to think and reason. I'm not sure I can
emphasize this benefit enough.
2) Because the excitement of discovering the principle for ourselves, is inherently full of meaning and feeling, it sticks in our memories far more firmly, than rote repetition of someone else's answers will. Thus we are able to learn faster, and in more depth.
3) The skills developed by going through the discovery process extend into all other aspects of our lives. How do we know if God lives? Through our experience. This will also help the student have confidence to try new things, and be unafraid of having the experiment go badly, as they will be focused on what they can learn from the results.

so.. where do we start?

Well, the first step, is to observe. When working with the science of mathematics, we are often going to be observing how many, and of what. One of the first excercises, is to be able to record our observations by writing how many of something, and what that something is.
Or, in-other words.. Counting..

As one of the first exercises I would have a student do, is to count things, and write down how many things, and what those things are. I would encourage them to write it down in the form of..

10 fingers, 2 eyes, 1 nose, etc..

This becomes especially meaningful if the student can count something meaningful to them. For example, how many parts are in a lego kit, or anything else that the student may be interested in.

As I work with younger ( and older ) students, one of the things that I find really helps them to feel at-ease, is by explaining the behavior of grown-ups in terms that they understand. As a result, I will explain that when describing the "Of What" part of the measurement, that "Mathematicians are Lazy", since Lazy is something they can associate with. And since mathematicians are so lazy, they often write down only one or two letters for the name of what it is they are counting.

So, from our previous example...

10 f, 2 e, 1 n...

This introduces them early to the appearance of having letters mixed with their numbers, and will make the transition to algebra much easier.

Well, that's the time I've given myself to write for the moment.. so
more soon..