I used to keep a overhead slide of a Far Side Cartoon By Gary Larsen with me as I introduced my two hour lecture on Problem Solving. It showed a picture of the devil and a bunch of books; on closer inspection all the books were story problems and the caption read "Libraries in Hell".
It was worth a chuckle. The point of learning the language of mathematics is for problem solving. Counting and computation as I have said many times are for doing math but the point of being able to count quickly (using basic operations) is to solve problems. Story problems show some of the uses for algebra and computation, build up problem solving skills. Problem solving skills include logic and reasoning, which together help form critical thinking skills. Some of the skills learned along the way include using mathematics to express reality, that is, translating real world situations into mathematical symbols and then knowing what those symbols mean. Turns out the same symbols can be used to express lots of different things...
Of course the vid gets cut off at the very end. If you watch it we will see that Spaceman Spiff catches him after 3 minutes. How far away from the moon base are they when he does? An astounding 90+% of students can NOT do two stage problems, that is solve a problem to get the answer to another problem. Once I find X, I can evaluate the equations and see if they are same. In mathematics once we find one variable we can usually find other variables easily. If we understand the concepts.
In calculus we study one and several variables. I used to start my pre-calc lecture where we just did linear algebra by asking the people in the room especially college graduates and teachers to give me a short definition of calculus anybody could understand. 99.99999% of the time none of them could. They might come up with some convoluted definition that even they were somewhat unsure of or come up with a very complex definition that they understood but when asked the rest of the room was not able to make sense of it...and I we would all laugh. This happened in cities from Maine to Maui and everywhere in between. The stress test was always could a little kid "get it." I have a simple definition. Look for it in an upcoming blog post meantime think of one for yourself if you've studied calculus.
In linear algebra we just begin to delve into calculus concepts and the most basic concept of all: the idea of a variable and how two variables interact. For now we are just fooling around with X. Later, as we see at the bottom of the page we can get into Y. (Make up your own joke.)
"The most powerful single idea in mathematics is the notion of a variable." ~K. Dewdney
So we start off with X.
The basic concepts in use here are Hero Zero, No Fun Get Back To One and of course the rectangle.
They can then use those concepts and algorithms to DO math, rather than just memorizing rules and process...which we have seen DOES NOT WORK. But who is in charge of math education? The 5% who easily memorized rules and process...see the problem? Good, because they don't.
The problem solving page is starting to grow. There is also a Password Protected Problem Solving Page now with more stuff on it. There are tons of common, perennial classic, story problems that students get exposed to during their journey through mathematics. Rate and Distance, Boat and Stream, Percents, Mixture and Solution, (I have some 5% solution and some 12% solution, how much 12% solution do I have to mix with 5% solution to get 8% solution)...there also some like in the case of ordered pairs that they don't get exposed to and never get to understand. I know a ton of students that could not tell you that an ordered pair could be a story about a water tank or a doughnut factory and the resulting equation could be a graph of the production or amount of water as it fills the tank. All they "know" is slope intercept form, y = mx + b, and points on a graph. Half the time they can't remember which is the x and y axis. How this relates to a story about a tank of water or factory or anything else.
In that short 60 second clip I don't tell the story that went with the symbols but in the longer video OF COURSE I do. Here is a story:
I have a doughnut factory. After 3 hours, there were 16 cases of doughnuts, and after 8 hours, there were 31 cases of doughnuts how many cases where there to begin with and how many cases do they make per hour? That's the most basic story. You can do the same with a tank of water, after 3 hours 16 feet of water after 8 hours, 31 feet of water in the tank. How fast is the tank filling up and and how much water must have been in there to start?
Then you can throw extra bells and whistles like the tank holds 40 feet of water how long does it take to fill? Go negative and drain the tank...use 39 and make them do fractions...but make sure the concepts are understood before you do any of that.
You can find many amazing and easy ways to explain these in the Series A Manuals from Mortensen Math. Eventually, I will make videos of all of these kinds of problems but it takes time because I take the many steps in the degrees of difficulty...and don't just dive in and do the problems. Parents and Home schoolers seem to appreciate this, but it's driving some of the math teachers crazy.
This method is slower at first but pretty soon it goes faster and faster until it surpasses traditional methods by far and you see little ten year old kids that seem like geniuses...it all starts with a game for 4 year olds called "what's under the cup?"
There is also some problem solving going up on Sarah's Page.