Friday, December 11, 2015

Math Test Answers

Last week I posed 12 questions related to teaching elementary school math. The test was specifically designed for those who believe a traditional, "back to basics" approach to math is superior to the type of math we are trying to teach through the Common Core in 2015.

So here are the test questions and answers with the  first answer, the traditional math answer (TMA) and second answer, the 2015 Common Core (CCSSM) answer.

1. What is counting?
TMA - "one, two, three four five etc" with no sense of what the numbers actually mean.
CCSSM -  "one, two, three, four, five etc" understanding that three is one more than two and four is one more than three and that when you count five objects the word "five" refers to all five objects not just the last one counted. This is the idea of cardinality and is key to understanding number,

2. What is addition?
       TMA:  34
                + 48
       CCSSM The above is only an algorithm, a piece of arithmetic. Knowing just this is useless in
       problem solving. To problem solve you need to know the concepts of joining, separating,
       comparing and part-part-whole if you want to use addition or subtraction in anything useful. .
3. What is multiplication?
       TMA  112
                x    5
       CCSSM Just like addition and subtraction the above is just an algorithm, a piece of procedural   
       knowledge. It doesn't help you decide which operation to use in problem solving. You'll need to
      know the repeated addition, multiplicative comparison concepts as well as a few more to be know
      whether to divide or multiply
4. What is division?
       TMA:  5/115
       CCSSM See above
5. What is Pi?
        TMA: 3.14 etc
        CCSSM Yes, but this doesn't help a whole lot. Pi is a ratio between the diameter and   
        circumference of a circle. The circumference is always just over 3 times the diameter; or, if the
        diameter is 7 the circumference is 22.
6. What does the 0 mean in 308?
       TMA: A place holder
        CSSMIt means there are 0 tens. Place holder is a meaningless term than has no conceptual  
         value. Children need to learn initially that each place in place value has a value.
7. What is 1/2 divided by 1/4?
        TMA:   "Change the sign and invert the second fraction"
        CCSSM  This is virtual nonsense and doesn't help at all. Knowing that the problem is asking   
        how many 1/4s there are in 1/2 is of far more value especially in a world of calculators.
8. Make up a word problem for # 7 above.
        TMA: I can't
        CCSSM: How many 1/4s in the first half of a football game 
9. What do  you get when you reduce 4/8?
        TMA: 1/2  Unfortunately, most children being taught to "reduce" a fraction  think that 1/2
 smaller than 4/8 because the word "reduce" means to make smaller and the two numbers.1
         and 2 are  smaller than 4 and 8. I joke about reducing 4/8 to 4/8    
        CCSSM; You don't use the word  "reduce" fractions because children think 1/2 is smaller than
         4/8. You  rename or regroup 4/8.

10. Why do you  put a comma after every three digits in a large number?
         TMA: Because that's what you do
          CCSSM Because it marks the repetition of  100, 10, 1 in each place value referent 
11. What is area?
          TMA: length times width
          CCSSM this only allows you to measure the area of regular shaped objects. Area is the two
         dimensional concept of coverage. You can find the area of your hand by tracing it onto squared
         paper and counting the squares.
12. What does a degree measure in geometry?
         TMA: It measures angles
         CCSSM It measures the rotation of an angle about a point. This is the only way you can explain
         why an angle is 45 degrees 4 inches from its origin and still 45 degrees 4 feet from it;s origin.

Children must understand what they learn at the developmentally appropriate level; this is why the common core is so good.

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