Numbers, numbers, numbers.

This is my four hundred and first entry in my Mostly Math Blog making my last one my 400th. By blogging standards this is probably quite insignificant but for me it is quite the milestone. As a math geek fascinated by numbers and the relationships between them there is always something special about a nice round number; something to be celebrated as one does one's 21st birthday or one's 50th wedding anniversary.  Numbers, by themselves, are the markers of our life stories, the moments of significant achievement, the tale of the tape measure so to speak. They bring specificity to life as well as precise comparisons. The also bring a sense of accomplishment or failure, success and challenge. Imagine a life without numbers where we lived in a land of 'ish" in which we only used words like some, many, few, lots to quantify our lives. I am told there are some indigenous peoples in the Amazon jungle who have three counting words; 'one', 'two', and 'many'.

This is all fine and good but when it comes to problem solving in he elementary school the numbers really are just the details. It is what is going on with all the words that surround the numbers that is the most important thing to be thinking about. Does it really matter that much if one train is going 40mph, and not 45mph, and the other 60mph, and not 70mph, when we have to solve the problem involving when they will crash into each other, or how far apart they will be in half an hour. The numbers are, of course, important for calculating the specific answer but they don't tell us what we have to do. It's all the words around the numbers that we have to decipher in  order to work out how to solve the problem.

If Mary has 16 Hotwheel cars and Michael has four more Hotwheel cars than Mary has, how many Hotwheel cars does Michael have? To solve this problem the numbers themselves are pretty useless. The most important thing is what the words around the numbers are telling us is happening. The understanding of the situation we can construct from deciphering the words is what tells us what we have to do in order to solve the problem. Recognition of the numbers themselves only provides us with the accuracy for our solution. Our understanding of the linguistic structure will tell us if this is a joining, separating, comparing or part-part-whole problem. 

When is Three a Half of Eight?

I spent the past two days at a meeting at the Vermont Agency of Education discussing the new Elementary Education Licensure requirements that are due out in a couple of years. The two days of meetings went by quickly and although I don't think we achieved all our goals there was some great discussion and some meeting up with old friends. It was particularly good to see Casey Murrow again after so many years and to know that he is still publishing his wonderful Connect magazine at Synergy Learning. 

It was also good to meet up with some folks from the Vermont Math Institute as we always have really good mathematical discussions. One interesting question that caught my ear that I hadn't  heard before was "When is three half of eight?". Always up to the challenge I thought about the question from every angle I could. I thought about different bases and how the 8-ball is significant in pool. I tried fractions and finally realized that I was getting nowhere. If using 8 as a cardinal number doesn't work is there something else about it that I could use? Perhaps the answer lay in the physical quality of the numeral 8. Bingo! If you slice the numeral 8 in half vertically the half on the right side is 3 and the half on the left side is a backward 3.

This is quite e neat mathematical problem because a fraction only really has value when you know what the one is to which it refers. As long as I kept thinking about 8 as a cardinal number, the identification of a group of eight, I was getting nowhere in solving the problem. If, however, I found a different 'one' to which the 'half' referred perhaps my problem would be solved. Once I had selected the numeral 8 itself it was then only a short step to see the 3 as the right side of the 8.

Now would you rather have half the money in my left hand or a quarter of the money in my right hand?    

Got a Problem?

We used to teach children to find key words in word problems that would tell them which operation to use to solve tehm. We also used to tell them to look at the numbers; they'll "tell" you which operation to use;  "altogther" means add and the numbers 30 and 5 probably means divide since there will be no remainder. We know now that such misguided advice does nothing to help children know which algorithmic operation to use, or, more importantly these days, which key to press on the cell phone calculator feature; add, sub, mult. or div. For example, simple one step addition/subtraction  problems of the type found in the elementary school can usually be classified  as joining, separating, part-part-whole or comprison. The key to solving such problems is to work out what is happening, find the question, select the operation to use to asnwer the question, then check and decide if it seems a reasonable answer. To help children work out what good problems look like we should share examples of  poor problems, or un-problems. In addition to being humorus such problems help us teach children what constitutes a solvable, meaningful problem.

Here are some examples of un-problems;

1. If it rains for 3 hours on Monday how much will it rain for the next 4 days?
2. If it takes ½ an hour for 3 friends to walk home how long will it take 5 friends? 
3. If 2 students have 5 pet gerbils how many gerbils does each student have? 
4. If 2 girls have 3 brothers how many brothers do 4 girls have? 
5. What is the area of your hand if your thumb is 3 inches long and your middle finger is 4 inches long? 
6. If the temperature is 62F today what will it be for the next 3 days? 
7. If 2 squares have 8 sides how many sides do 3 triangles have? 
8. If it’s 60F in Vermont and 72F in Maine what is the temperature  in New Hampshire? 
9. If one train is traveling at 35 mph (clearly an AMTRAC train) and another similar train is going at 45mph what time will they pass each other? 
10. If you eat 2 slices of your birthday cake today and ½ of what’s left tomorrow how much will you eat the next day? 
11. If 140 students are going on a field trip and a school bus will hold 60 students how many school buses will you need?
12. If 2 friends have 5 pets and one of them has 2 cats how many dogs does her friend have?