Door-2-Math Blog

We start by lining up 2 single digit numbers vertically. The reason is that we need to set up the habit of writing vertically from the get-go. When math progressively gets more complicated and more involved, it’s a tremendous advantage if each step of the solution can be checked quickly. Writing vertically makes things easier.

Take a look at the following two side-by-side comparisons. Imagine if you are the teacher who is grading the paper at the end of a long working day. Ask yourself, which one are you more likely to give partial credit if the final result was wrong.

Find 1/7 + 3/8 + 3/4

Case 1:

(1/7) + (3/8) + (3/4) = 1/7 x 8/8 + 3/8 x7/7 + 3/4×14/14

=8/56 + 21/56 + 42/56 = (8+21+42)/56

Case 2:

1/7 + 3/8 + 3/4

= 1/7 x 8/8 + 3/8 x 7/7 + 3/4 x 14/14

= 8/56 + 21/56 + 42/56

=(8 + 21 +42)/56

You get the point. Help your child to develop the awareness of math “performance” early because a good grade/score has two components: good understanding of the material tested and clear communication of that understanding to the grader, aka the teacher.

Now that you understand why stacking vertically is important, let’s get started.

9
+7
__
16

(not 9 + 7 = 16)

Notice a couple of things. On the vertical stack, there are four numbers and they are lined up neatly. Going back to our chapter earlier, the number 16 has two “chairs” where 1 is in the “mommy chair”. This number is called “carry”

Addition by regrouping happens like this:

9                                          (3 + 6)

+7                                         7

__                                          _____

10 + 6

To make this addition even more efficient, we place carry of 1 as a “dot”. This will greatly speed up the addition when the number of items goes up to, say, ten single-digit numbers.

(1) 9

(2) 7 * 3 aka 13

(3) 8 * 1 aka 21

(4) 6 * 7 aka 27

(5) 5 * 2 aka 32

(6) 4 * 6 aka 36

(7) 3 * 9 aka 39

(8) 9 * 8 aka 48

(9) 7 * 5 aka 55

(10) 6 * 1 aka 61

_____________

61 – 6 dots of carry 1 each

Let’s slow down and take a closer look at what happened in the above column:

(calculation will be posted here)

This may look ridiculously slow, but when we are faced with a large column of numbers, we are only adding two single digit numbers at a time. By assembly-line our task of adding multiple numbers into small repetitive tasks, we greatly improve our speed.

The second, more important advantage is that we can now use this isolated, deliberate exercise learn a fact about math that many kids (and adults alike) are faced to reconcile: In math, close enough just doesn’t do. Math is an exact science where result trumps effort.

Give a child a column of ten numbers to add, there are exactly nine steps before a final answer appears. If any of the nine steps is wrong, then the effort put in getting the answer is immaterial. It’s one of the hardest lessons a youngster must learn or his/her performance will always lag behind his/her conceptual understanding. As this gap gets larger, a child will naturally assume that the cause is his/her ability to learn. Very few, if any, come to question the ability to deliver, aka test-taking skills is amiss. Some give up altogether while others compensate by studying harder. When studying harder only produces marginal improvement doubts set in.

All this can be avoided if at an early stage, parents can create an environment where a child has to face the unpleasant task of deliberate consistency backed by double-checking.

Addition by regrouping fits nicely toward building that awareness. Progress slowly from adding 2 numbers to adding 10, 12, 20 numbers. A child will not know the luxury of getting done without checking each step. If you are up to the task, I’d even recommend what I call a doubling method.

The doubling method goes like this: In exchange for the freedom of naming the number of addition problems to be done on a given day themselves, I tell my students that they’d have to add twice the amount of problems they got wrong to the next day’s assignment. In the years that I’ve used it, the record was 32 additional problems (2,4,8,16,32). Did that student get the importance of double-checking? You bet!

Depending on your patience level, I would recommend first explaining why you’re doing what you’re doing before implementing any double/triple policy. As important as the test taking skill to any student, it is not worth trading a child’s learning curiosity nor is it worth damping a child/parent relationship.

Having established such a firm foundation in number sense and visual memory, we can now turn our attention to the next task at hand. Double checking.

A word of caution: As with establishing any good habit, repetition plus patience is the key. Whenever you feel you’re losing it, walk away. There’s nothing worse than trading your child’s innate (often irreplaceable) curiosity with a temporary gain in result.

Some of the tricks that worked well for me (and other parents) are:

- set a timer ahead of time on how long you both agree to work.

- get an engaging book to rad while working with your child side by side, so that you don’t breathe down his/her neck.

- have a plan B ahead of time if you know you’re short on patience.

- Write down a trigger word on a note card and ask your child to “stick it in your face” when she is getting stressed out. Some of the effective words are:

“Love me”

“I’m only a child”

“I need a hug”

“remember my curiosity is more important”

“I’m tired”

I don’t recommend using rewards because I didn’t want my kids to associate “learning” to “chores”. To me, learning itself is rewarding. Think about it, if learning is equated with reward, then “no reward” is equivalent to “no learning”. There have been quite a few excellent books on intrinsic (like pride) reward vs. external reward (like candy and money) and how in the long run the extrinsic reward system slows down learning.

Learning to regroup 10 is the last of the building blocks you need to foster. Luckily, we can turn to our two hands for this one.

The first combination to make 10 is obvious: 5 + 5

Next, nudge one finger closer to the opposite hand, you can see the next combination pretty easily: 6 + 4

Continue doing so, other combinations are “jumping out”: 7 +3, 8 + 2 and 9 + 1.

A word of caution here is to slow down. We invented this game of “what’s my cousin” to accomplish this. “I’m a number. Me making 10 cousin is 8. What number am I?”

“2″ they’d yell from the back of the car.

What I was trying to get them to see in their mind’s eyes, is that 8 + 2 are “buddies”, complimentary numbers, in making 10.

Practice this “cousins” as much as your child will let you. Remember dripping water? It will put a hole in a boulder.

Congrats! You’ve done one of the most tedious jobs there is!

While dropping the kids off to school, I’d ask, “