My son and I were playing around with an Arduino robot today, which is a small wheeled robot controlled by a small micro controller based computer named ?Arduino?. Part of this involved setting up some simple sensors and LEDs requiring some resistor, so as not to burn stuff up. Since resistors are two small to write the resistance on, they instead use color bands that act as a code which indicates the resistance?
It actually turns out to be a mathematical activity to find a resistor, in a bag of resistors, of a specific resistance, or to determine the resistance of a resistor from its bands. And this all went well. What didn?t go to well was when I applied a multimeter to the resistors to show him that the actual resistance will not be exactly what the bands say because there is a tolerance aspect. And this was where things got dicey. Multimeters automatically change their scale according to the magnitude of the measurement. For example, for a resistor of 50 ohms, the multimeter might read 48 ohms. But for a resistor of 1000 ohms, the multimeter will automatically switch to kilo ohms and might read 0.977 kilo ohms. I asked him how many ohms is that? He couldn?t answer.
Now, there is a conflation of many things going on here. Units of measure (especially strange ones, like ohms), decimals, multiplier prefixes (kilo), fractions, and the arithmetical reasoning behind it all. And as I tried to explain it to him, I could see that he was overloading.
So I do what I usually do in such a situation, work backwards until we are on the same page.
Suppose something is 3 meters long, how many centimeters is that? How many millimeters? Suppose something is 20 centimeters long, how many millimeters is that? etc.
In these problems I am making sure that only multiplication is involved.
When I reversed things, sure enough, things started falling apart. But he was seeing starting to see the issue with his reasoning.
Suppose something is 30 cm long, how many meters is that? 3000 you say? Is 30 cm more than a meter or less? How many kilometers is it? If something is 500 meters long, how many kilometers is that? Is it more or less than a kilometer? etc.
And then the decimal arithmetic, and the meaning of decimals period.
What is 0.977? Ok, well then what is 123? 1x100 + 2x10 + 3x1. Ok, that?s right, so what is 0.977? 977/1000. True, but can you answer like you did for 123? So I wrote, 9x1/10 + 7*1/100 + 7*1/1000. Is this the same as 977/1000? I didn?t expect him to answer this, so I worked it out. 9x1/10 is the same as 900x1/1000, and 7*1/100 is the same as 70/100, and so on. He knows how to add two fractions and find a common denominator, but that seems to go out the window when he gets overloaded, which I understand.
In any event, after a lengthy few hours, he was going back and forth between measurements much better (we didn?t get to actually play with the robot). And he was handling decimals much better. But there is work left to do.
What I got out of all this is just how fast all these *concepts* begin to conflate into something rather complicated. And there are many such conflations happening or on the verge of happening in 5th and 6th grade. I am jumping the gun? To be honest, I don?t recall at what point in school I would breeze through these problems involving all of these pieces at once. I only know that by 8th grade, I was. But I do know that this curriculum stops way short of putting this stuff together. I complained about this earlier with fractions. They teach about fractions but never follow through with a bunch of math involving fractions. They teach about decimals, but don?t follow through with a bunch of math involving decimals. And so on. They also waste too much time on things like tessellation and nets, which at this level goes without a hitch.
These topics that conflate need to be addressed continually or they won?t conflate. They just don?t work if you only treat them sporadically and conceptually. You cannot combine fractions, decimals, prefixes and units of measure, and the arithmetical reasoning behind it all, effectively, unless you have been banging at it continuously for several grades. From introduction right up to algebra?s door. They really have this conceptual thing wrong. It is far too superficial.
Fortunately I have plenty of old books (100 years) that ask these types of questions in every arrangement possible. Maybe they should add arithmetic primers like those to these classes. Forget the picture/concept workbooks. Leave all that jazz in the textbook if you must. Just augment the text with a real arithmetic primer that contains a steady stream of problems that conflates these topics over time.