Ambition, Distraction, Uglification and Derision

In my math ed class this week I talked about ambition and distraction and uglification and derision but no-one knew what they were so I asked who had read Alice in Wonderland and could remember the Mock Turtle. Perhaps three students out of the eighteen raised their hands. I was initially somewhat alarmed by this but then I started thinking that what is termed a classic piece of literature  probably changes over time. But I'm not an expert in this area so I'll just wonder about it.

Something else we did pursue in class through a really interesting discussion is  whether the incredible amount of time we spend teaching algorithmic procedures  such as 54   is still
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worth it given the culture in which we live. Is the use of endless tedious practice exercises really worth it given the extent of the work we now do on numeracy skills as well as the technology-based tools available for computing. Shouldn't we be spending our valuable time teaching children to compute simple problems mentally and more complex problems using some sort of calculator? Shouldn't we be spending our time teaching children the meanings of the operations such as recognizing joining, separating, comparing, part-part-whole, groupings, Cartesian products and multiplicative comparisons?  The essence of solving a math problem is deciding which operation to use. Once this has been done the problem becomes simple arithmetic which can be solved either mentally of through the use of a calculator. In the discussion with the students I was playing devil's advocate; well maybe!

There are very few places in more advanced mathematics where the use of paper and pencil algorithms is required over mental or technology based computations.

Worried Problems - Whirred Problems!

Word problems! the one thing guaranteed to turn just about every student off mathematics. Remember the one about two trains approaching each other?  And then there are the remarkably insane strategies that are used to help children solve simple word problems like the one in the image to the left. There are some amazingly daft suggestions in this strategy such as find the "key words" as if they would really give you a magic way to solve the problem.

 "Do the math" is equally ridiculous. Why is "math" limited to the word "compute". Isn't the whole thing math?

Several years ago Thomas Carpenter came up with the idea that all simple addition/subtraction "problems" could be identified as involving part-part-whole, combining, separating or comparing. He

also identified another set of concepts related to multiplication and division; things such as the equal groups, area and the multiplicative comparison concepts. The main task in solving a quantitative problem is to work out what is happening in the problem in the context of one, or more, of these concepts and what is being asked. Once this is done, the solution is easily found by using one of the four operations using one's head, paper and pencil or a calculator.

To show the significance of understanding this, consider these non-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 students have?

  4. If 2 girls have 3 brothers how many brothers do 4 girls have?

  5. If the temperature is 62F today what will it be for the next 3 days?

  6. 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?

  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 you eat 2 slices of your birthday cake today and ½ of what’s left 

     tomorrow how much will you eat the next day?

10. I 140 students are going on a field trip and a school bus will hold 60 
     students how many school buses will you need? 

You could get some really interesting answers by using the "Attacking Word Problem" strategy. e.g. (Each student would have 2½  gerbils)