I’ve discussed my math history in past posts and explained the importance of math sense . Many of us adults can do steps without ever knowing how or why we do them. This can make it extremely difficult to explain math to our children, and even worse be unable to do elementary school math because we are unfamiliar with our own skills. It is perhaps the most tragic thing about American education where there are so many in the adult population who think they don’t know math. That’s not true. They know it but don’t realize it.

I come across Youtube videos and memes that get under my skin because people think “that the kids today” are being showed the “hard way” and the “long way” for no reason.

There is a good reason to introduce concepts rather than steps. Knowing the hows, the whys, the whens are tools a person can use to better use efficient steps, tackle new problems, and learn new methods.

Today I saw one such video that tried to prove that stacking and carrying for addition was faster than models and thus modeling was a waste of time and stupid.

Here is the problem in question:

Sally got 1568 tickets at Chuckie Cheese. Her cousin won 1423 tickets and gave them to Sally. The next day Sally wen back and won 680 tickets. How many tickets does she have in all?

Let’s assume most adults can either add those numbers up from left-to-right from highest place value to lowest, or break out the pencil and paper, stack the numbers on top of each other, and work right-to-left and carrying as they go.

It took the eight-year-old about ten minutes to do the problem with the models (and she made an error), and it took about two to three minutes to do the standard American carrying method. She explained what she was doing with the models, but did not explain the method of the left-to-right carrying method.

This speed of the standard algorithm plus the slow, messiness of the scratch work was given as proof that the models were a waste of time.

*However, it’s not about speed; it’s about understanding.*

**Here is the model:**

The girl in the video used cubes for the 1000 (thousands), squares for 100 (hundreds), single lines for 10 (tens), and circles/dots for 1 (ones). So I don’t have to use special characters in html, I will use # for thousands, @ for hundreds, * for tens, and / for ones (grouped in fives for ease of count).

**Note: there is a reasonable visible reason for using a cube for thousands, square for hundreds, and single lines for ten:
10^3 — ten cubed is 1000,
10^2 — ten squared is 100,
10^1 — ten to the first power is 10, and
10^0 — ten to the power of zero is 1.
**

1568 –> # @@@@@ ***** * ///// ///

1423 –> # @@@@ ** ///

680 –> @@@@@@ ***** ***

We could add up the symbols right now ( 2000 + 1500 + 160 + 11). But the point of modeling is not just to show place value, but also patterns and how numbers can be reorganized/regrouped/deconstructed/manipulated to make calculations more efficient. It allows for logical thinking, discussions, and allows students to explore their options so they can use the right skill for the right job in more complex problems.

In a standard American algorithm we carry over to the next place value when we have “extra” and make a note by “carrying 1”. How can we show that to a child who doesn’t have a strong grasp of place value? What does it even mean to regroup or carry? Why are we even carrying *one* (they might ask?)

It is not a good idea to teach* only* the steps without the reasons why they are done.

Although we can add up each place value now, this set up can be used to model “carrying” by reorganizing some symbols that are in excess of their place value (ten ones, ten tens, ten hundreds, etc.).

1568 — # @@@@@ ***** * **///// ///**

1423 — # @@@@ ** **// **/

680 — @@@@@ @ ******* *****

Ten ones (in blue) become TEN that we can move to the tens place.

Ten tens (in red) become ONE HUNDRED that we can move to the hundreds place.

Ten hundreds (in orange) become ONE THOUSAND we can move to the hundreds place.

It doesn’t matter where in the model we place them as long as they are in the correct place value.

–> # @@@@@ ***** *

–> # **@** ***** /

–> **#**

–> 3000 600 70 1

**We have demonstrated that:**

1568 + 1423 + 680 =

= 2000 + 1500 + 160 + 11

= 2000 + 1000 + 500 + 100 + 60 + 11

= 3000 + 600 + 70 +1

= 3671

With modeling a student can make sense of the shortcuts inherent in the Standard American Algorithm for adding right-to-left. They can see that they aren’t carrying “1”, but are really carrying the excess of 10, 100, 1000 etc. and moving them to their proper place value. They can also see that they can add the numbers up in any configuration desired or break them apart and the value remains consistent. Also, it is clear that, numbers can be added left-to-right or right-to-left.

With the model we can show the convenience of regrouping and the student can visualize the method behind the shortcuts, and will learn and understand the algorithm better and use it faster. The model is a temporary, transitional method.

In terms of the video, the question shouldn’t have been, “Which method is faster?”

Although, the child could follow the steps of adding with carrying right-to-left did she:

- understand that she is not really carrying one?
- know she can add the same numbers left to right? Can she do so?
- know if she did he numbers mentally she doesn’t have to carry right-to-left? Instead she can add effectively left-to-right?
- have the ability to check her work by adding top to bottom?
- add the number columns only in order top to bottom, or can she reorganize them into friendlier configurations?
- does she have the ability to add the numbers using the model concept (no drawing), but also without using carrying notations (in other words can she construct: (1000+ 500 + 60 + 8) + (1000 + 400 + 20 + 3) + (600 + 80), stack them, and add left-to-right)?
- have the concept that she can use any symbol in the model she wants as long as they represent the face value she desires?

(Recall: there is reason the model introduces cubes for thousands, squares for hundreds, and single lines for ten. –>

10^3 — ten cubed is 1000,

10^2 — ten squared is 100; and

10^1 — ten to the first power is 10

10^0 — ten to the zero power is 1)

- not understand the model, why?

The model method is not the end of he journey. It is a tool used to explore Number Sense and to introduce more complex and efficient methods of calculation. This is no different than using hand held “base 10 blocks” to demonstrate mathematical concepts to children. They aren’t meant to use the blocks forever.

In no other subject is there is so much undermining and apprehension of the unfamiliar.

If a teacher uses a concept to teach letters and reading, there is not an uproar over the method even if they are strange. Songs, models, art, deconstruction and reconstruction are all part of the process of learning to read. It is not just memorizing words. Yes, some words don’t follow the rules and must be learned by sight, but for every other word the student must learn patterns and analyze new ones based on the skills they’ve accumulated. Children are encouraged to explore words, discuss construction, and *make mistakes*.

The same should hold true for mathematics.

It can be intimidating to look at a paper and not know what is going on. However, if we approach the unfamiliar math-gobbeldy-gook with the right questions, we can lend support and encouragement even when we are unsure. And imagine this: When the teacher first saw “new old math” they had a similar reaction! How do you explain *what you do automatically* to children who are starting from scratch?

Math is not something that comes as easy to me as words and that’s okay. I don’t need to be a rocket scientist; I just have think longer, that’s all. It makes me more aware when i teach adult learners to tackle their math fears. What makes all the difference in the world is an attitude shift. My husband is dyslexic and I can only imagine how frustrating it can be for him to help our kids with reading or to read to them when it is so easy for me (although, I tell him that I wouldn’t know he had dyslexia if he hadn’t told me).

I am not saying that there aren’t issues with curriculum and the school systems (don’t get me started). I think the Terc Mathematics book and Everyday Mathematics are poorly written on both the student and teacher end, however, the *concepts* are sound and important foundations to hone the inherent math skills present in all of us.

If a child comes home struggling, perhaps the concept needs to be broken down further. Likewise, if a child comes home and is unchallenged then parents can present and encourage advanced skills that support the basic practice (if basics have been mastered).

This is what is naturally done, for example, when a Kindergartener knows their Alphabet before school starts but others don’t. *People don’t oppose the practice of the mastered alphabet! * Instead, the teacher and parents offer additional challenges. Situations of frustration and boredom can be addressed to support rather than oppose curriculum.

At the end of the day, working together is what will improve education.

*What gobbeldy-gook has your child come home with that made you go, “Say what”?
*

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Aspiring writer, wife, mother of two, owner two cats. Teacher, lover of science, books, science fiction, fantasy, and video games.

J.M. Ames

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