Letting Life Lead
I guess this video got under my skin more than usual (see part 1), so I decided to explain why it is important to be able to understand why this adding-up skill is useful.
As an individual who struggled with math because I was part of the experimental group who were taught steps and not concepts, I think I am in a position to advocate for these basic skills. Demonstrate basic addition and subtraction skills right and you are setting up the necessary foundation for multiplication, division, and algebra.
43 – 13 = ?
If you want to get the answer (especially in your head), you can add-up starting with 13. Though, you don’t have to. An adult needs to remember that even though they know the answer easily, children are new learners. They don’t have the experience with two digit factors to be able to work it out quickly. Especially in this horizontal presentation. What some adults might be doing (because they are not used to adding up) is to visualize 43 and 13 in the vertical format and then mentally subtract the 3’s and then 4 minus one going backwards. Which will get you the answer without a pencil and paper.
It’s automatic. If this is how you learned how to subtract and the only way you can do it, it can be very difficult to visualize another way. Try that in your head, though with a three digit number, then a four digit number. An adult also understands place value and that they aren’t actually subtracting 4 minus 1. They know they are doing 4 tens and 1 ten. Forty minus ten. That is also a concept new to children.
Number sense is more than knowing the steps.
How can you show that subtraction and addition are related? If you can add in any order, why can’t you subtract in any order? Why can we take shortcuts like saying 4-1 rather than 40-10 and arrive at the same answer of 30?
Let’s look at the video demonstration Common Core Video:
43 – 13 = ?
Let’s say that we have introduced the students to the concept of “addition and subtraction are related” and we tell them that you can find the answer to a subtraction problem by starting with the smaller number and adding up until we get the bigger value (adults might remember this concept as checking your subtraction by adding your result to the lowest number).
Let’s say that the students are also already familiar with adding with 5’s and 10’s and can recognize number pairs that give us those values such as 2 + 3 (friends of five) and 8 + 2 (friends of ten). Fives and tens are very easy to add.
We can demonstrate the power of fives, tens and zeros when adding like this:
We say: Start with the lowest value and add up to make fives and tens until you get to 43.
13 + 2 = 15 (ends in five)
15 + 5 = 20 (use five to make a value ending in 0)
20 + 10 = 30 (use ten to make a value ending in 0)
30 + 10 = 40 (use ten to make a value ending in 0)
40 + 3 = 43 (add in what’s left without going over)
10 + 10 + 5 + 2 + 3 = 30 (add up your chosen values)
Is 43 – 13 = 30?
At the same time we’ve done this adding-up concept, we’ve shown not only the relationship between addition and subtraction, but also useful number pairs that add up to five (3 + 2) and the concept that you can add numbers up in any order. AND we’ve shown how using fives and tens can make adding go faster. A teacher might ask the students if using 10’s and 5’s are the only way to add-up? Do we need to do them in that order? Does it work with any subtraction? How can we use this skill to check our subtraction or to make subtraction easier. A teacher has a choice to reinforce this concept in a number of ways through games, art, and hands-on activities.
We could think this:
13 + 2 = 15
15 +20 = 35
35 + 8 = 43
20 + 8 + 2 = 30 (add up your chosen values)
We can then ask the students if they found it easier, the same, or harder to use 8 in the list? That’s right, teachers can ask a student’s opinion. Some kid might have discovered that you can do the following and get the answer and shares it with the class:
13 + 2 = 15 (that ends in five)
15 +20 = 35 (can I do two tens at the same time?)
35 + 10 = 45 (oops I went over!)
45 – 2 = 43 (wait, can I subtract the extra 2?)
20 + 10 + 2 = 32 minus 2 is 30!
Another student can play off of what their classmate discovered and could think this:
13 + 30 = 43 (if you can add two tens, can I add 3 tens at once?)
thus 43 – 13 = 30
And someone else could suggest this:
13 + 7 = 20
20 +20 = 40
40 + 3 = 43
20 + 7 + 3 = 30
This foundation will actually make it easier for students to learn the standard borrowing (regrouping) method that most parents know how to do.
Some people might think this is a lot of trouble when you can just have the students memorize the steps for the standard algorithm that has you find the answer going from right to left. However, doing it that way without explanation doesn’t open up the doors to discovery of number relationships or why that method even works in the first place. How many adults who learned that method in school even understand that they are doing something similar to this:
43 – 13 = ?
3 ones minus 3 ones = 0 ones (that will be the right most digit in the answer)
40 tens minus 1 ten = 3 tens (the answer is thirty)
How would you explain to a child what you are doing when you do it that way? Can you explain why you go backwards? Why don’t you do the tens first and the ones second? Is there a reason? Why do you say 4 minus 1 rather than 40 minus 10? How do you know the answer is 30 and not 3? Did you do 30 plus 0 ones? Why or why not?
My grandparents were confused when I was doing math the “new way” (the way a lot of parents today know how to do) which was to them confusing. Why are you adding those columns of numbers backwards? Why don’t you add up the numbers left to right? Why are you doing the subtraction that way, why aren’t you doing it this way or this way? They thought regrouping by writing on the top was messy and confusing. Why do you always do your division the long way, they’d ask? I was thinking, “What other way is there to do it!?”
Well, I didn’t understand it then, but I understand now.
I could only do the math one way so that I could get the answer by following the steps I memorized. I was unable to use other methods because I didn’t even understand why the method I was using worked! I just knew that if I did the steps I’d get the answer. Though I could get the result, I lacked understanding. Now, it wasn’t that my grandparents didn’t have some issues. They, too, couldn’t understand the “new” way as well either because they also had some holes in their understanding. Not as big as mine, but there. They came from a generation who memorized math facts and multiplication tables by rote, but did they truly get the concept? Maybe some.
And I’ll tell you this. Before I went to kindergarten I could do arithmetic the way my grandparents taught me. I remember the old math workbooks I could do quite vividly. A lot of it mentally. But when I got to first grade, the experiments began and I got horribly confused because I didn’t understand what the steps they were teaching were meant to do. Not that the steps are bad. The method is sound. But teaching the steps without the concepts is a bad idea.
This is why I am happy that teachers are now using ten frames and other visual methods to show concepts. That foundation is so important! I am pleased that they are showing students why things work and not having them simply memorize steps (I am not so pleased about grading them on the learning; that’s just stupid). I bought math games, an simple 100-bead abacus, and a Soroban abacus app for my children in order to supplement those concepts before they even got to school. Math needs to be around them every day just like words are.
Let the teachers teach. They finally have a giant toolbox to use in order to teach concepts in math in the same manner they can teach writing and reading. If you want to fight about something, get the government off their asses about the standardized tests.
Watch this video from the UK and see what a US classroom using these methods could be like with support (and lucky schools are). Thinking, collaboration, and enthusiasm. Embracing common errors as learning opportunities.
I leave you with this: Why are nearly all the memes focused on math education but not about reading or writing?
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