Thinking About Elementary Math

Recently I've had some questions come my way about what the younger kids need to be able to do in order to be successful in the future years. I am no expert, but I am very interested in getting a discussion started about this from a Middle School perspective.

Specifically:

• I think in grades 1 - 4, the most important skills to develop are obviously basic (up to two-digit) addition, subtraction, and "nice" multiplication and division using the times table. For the young kids, manipulatives are very important in order for them to understand the meaning of these operations. 
• In Germany (and probably other places as well), they use a triangular diagram to teach the idea of inverse operations in elementary school. For example, the diagram below reinforces that 60, the total, can be divided by 5 to get 12, or divided by 12 to get 5. The two bottom operations multiply to form 60. The neat thing about this triangle is that it extends into algebraic relationships such as D = rt, D/r = t, or D/t = r. My lawyer friend who grew up in Israel told me that this is how they learned basic operations in school.
  
• I think that by grade 5, kids should be able to do addition from left to right, in order to build up their estimation skills. For example, adding 638 + 290, yo can look from left to right to get 800... And then when you look just one digit ahead, you can already estimate that 9+3 is bigger than 10, so the result is actually 900 something. 928, to be exact, if you keep adding from left to right, peeking just one digit ahead each time. The nice about this is that even if kids only quickly looked at two numbers being added, they can already estimate the sum reasonably.
• I think that by grade 5, kids should be able to multiplication of two-digit by one-digit numbers in their heads. Teaching kids to break down (82 times 6) into 80 times 6, plus 2 times 6, reinforces two things: placement values and the idea of distributive property within arithmetic. It makes introducing the algebraic distributive property in middle-school a breeze, if kids already have seen it in action.
• Sometime in grade 5 or grade 6, when kids start to learn conversion from decimals into fractions, this should be done using their proper naming of numbers. 5.6 is read properly as "five and six-tenths", and the way we write that in fractions is immediately 5   6/10. Going backwards, they should be able to do the same, at least for base-ten fractions. 3   9/100, is read as "three and nine-hundredths", which writes as 3.09 in decimals.
• There are a lot of resources out there for fractions already, but I think that the most important representation is the number line and the comparison of numbers. To find a fraction of any number, the kids need to know that 1/n is one out of n equal pieces, so k/n just means that size, multiplied by k. I think the concept behind fractions is so so so SO important, so it should always be done in context.
• Dividing by simple fractions can be done similarly using reasoning. I teach my middle-schoolers how to intuitively divide 5 by 1/3 by first asking them what is 1 divided by 1/3. We draw diagrams until everyone can see why it is 3 (and I use language like, "how many times does 1/3 fit into 1?"). And then I ask them what is 5 divided by 1/3. ("How many times does 1/3 fit into 5?") The language that you use with fractions, I think, has an immediate impact on the children's understanding of the operations. Of course, this does not bypass the need to show them the manipulation of fractions in division, but it helps to add meaning to the otherwise rote/abstract operations.
• By the way, the triangle (shown above) can be used to reinforce why 5 divided by 1/3 is 15, and why 5 divided by 15 is 1/3. One of Geoff's friends has a good analogy to cutting potatoes in order to illustrate this. (You can cut 5 potatoes into 15 groups of 1/3 potatoes each, or if you already knew that you wanted to make 1/3 potatoes the size of each group, you can make 15 groups.)
• I recently wrote a short email outlining my recommendations for decimal division in Grade 6, so I'll just paste it here. "I think for decimal division, kids should be able to reason through step-by-step, starting with normal division. For example, to teach 7.2/6, I’d start first with 72/6 = 12, and then ask kids what they think 7.2/6 will be, and then ask them what 7.2/0.6 will be, and then 7.2/0.06, etc. You can use it to introduce this idea of ratios between numbers. Have them practice this on other decimal pairs instead of teaching the rote “moving it over this many times” trick.
  
  
  Also, this is a useful trick: 700000/35000 = 700/35 = 20. Or 840/120 = 84/12 = 42/6 = 7. They should always reduce before division, if they can. It’ll make their lives much, much easier down the road to not have to divide with so many digits involved.
  
  
  Another thing is that, I don’t know how familiar the Grade 6 kids already are with fractions, but I think that if the division is very messy, the kids should stop after the unit digit and just write the rest as a fraction. The most important thing from Grade 7 on is that they can estimate decimals, such as to know that 11/7 is between 1 and 2, just past 1.5 because 11/7 = 1   4/7. They don’t need to really get that it is 1.57142857142857142857142857142857 since we all have calculators…."
   
  What do you think? Disagreements? Something I missed? I would love to hear what all the MS teachers have to say about what makes a child successful coming into MS.