The truth is, subtraction is challenging. I was glad to hear Dr. Moldavan confirm as much in class. I have the procedural knowledge down pat (if I didn't at my age, I'd have bigger problems), but now that I'm thinking in terms of understanding instead of performing, I'm seeing that I don't have the comfort level with subtraction that I'd like if I'm going to teach this fundamental concept.
I don't think I'd find it useful to practice a bunch of subtraction problems. But I would like to explore in more depth what it means to say that "division is repeated subtraction."
Maybe I should see what other resources are out there besides Khan Academy. But on the other hand . . . why fix what isn't broken, am I right?
So let's start at the beginning--with their video The idea of division:
Well, that WAS interesting. This video explains that you can think of 24 ÷ 3 as either 3 equal groups within 24, or as groups of 3 within 24.(This concept reminds me of Bonnie Jeanne's post on arrays and how you can flip them on their sides).
So on to repeated subtraction.
Wait . . .
The Khan Academy search bar doesn't bring up anything for repeated subtraction?!
After taking a moment to pick up the shattered pieces of my math life, I'm ready to move on. . . .
One of the things I've learned in this course is that unlike most other subjects, I learn math best when someone explains it to me. Videos are much more useful to me than textbooks are in this context. So I'll go straight to the source for All Things Video: YouTube.
Whoa--here's a link to a video combining two of the core concepts I'm exploring on my PLP: Division as Repeated Subtraction Using a Number Line!
Oooh. I really like her description of division using the example of 6 ÷ 2:
I don't think I'd find it useful to practice a bunch of subtraction problems. But I would like to explore in more depth what it means to say that "division is repeated subtraction."
Maybe I should see what other resources are out there besides Khan Academy. But on the other hand . . . why fix what isn't broken, am I right?
 |
| What can I say? I'm a loyalist. |
So let's start at the beginning--with their video The idea of division:
Well, that WAS interesting. This video explains that you can think of 24 ÷ 3 as either 3 equal groups within 24, or as groups of 3 within 24.(This concept reminds me of Bonnie Jeanne's post on arrays and how you can flip them on their sides).
So on to repeated subtraction.
Wait . . .
The Khan Academy search bar doesn't bring up anything for repeated subtraction?!
 |
| You betrayed my loyalty, Khan Academy! |
After taking a moment to pick up the shattered pieces of my math life, I'm ready to move on. . . .
One of the things I've learned in this course is that unlike most other subjects, I learn math best when someone explains it to me. Videos are much more useful to me than textbooks are in this context. So I'll go straight to the source for All Things Video: YouTube.
Whoa--here's a link to a video combining two of the core concepts I'm exploring on my PLP: Division as Repeated Subtraction Using a Number Line!
Oooh. I really like her description of division using the example of 6 ÷ 2:
"This question is asking you, 'How many times does this number [2] fit into this number [6]?' "
I might reword her question as:
"How many groups of 2 can you divide 6 into?" Because that wording includes the word divide--and we're teaching division.
Okay, so I see in the video that the presenter is subtracting 2s to get to 0 . But I still don't get why division is considered repeated subtraction; after all, couldn't I just as easily start at 0 and see how many 2s I have to add to get to 6?
Let's see what else we can find. . . . Check out this video--Interpret division using repeated subtraction:
This video is only three minutes long. But it has two key phrases that break this concept open for me.
Phrase 1 is at 2:15: "We had ten objects. We wanted to divide by two. Meaning we wanted to make groups of two, or give two to each group."
- When I give two to each group, I am subtracting 2 each time. I am giving those twos away. Giving something away and subtracting it are the same to me conceptually. And now I understand why we would say "division is repeated subtraction."
Phrase 2 is at 2:49: "The quotient is how many times we subtracted the divisor."
- This is one of those simple phrases that one person might gloss right over, but that could shift someone else's understanding (namely mine). Basically, the answer to any division problem (quotient) is how many times the second number (divisor) fits into the first number (dividend).
But perhaps most importantly, now I understand something else: "multiplication is repeated addition" is very different from "division is repeated subtraction":
- Multiplication actually is repeated addition: 3 x 3 is simply a shorter way of saying 3 + 3 + 3. But
- Division is not necessarily repeated subtraction: For example, you can also start from 0 and add your divisor until you hit your dividend to get to the quotient--which does not use subtraction at all.
In other words: "multiplication is repeated addition" is a mathematical fact. "Division is repeated subtraction" is a mathematical process.
How cool is that?!
Hi Erin,
ReplyDeleteI love all of the videos, and memes you threw into this! I have learned that the way I learned math back in my day, to how it is being taught now is way different. But I say that in a good way. I think that I would look at division the way you stated: "This video explains that you can think of 24 ÷ 3 as either 3 equal groups within 24, or as groups of 3 within 24.(This concept reminds me of Bonnie Jeanne's post on arrays and how you can flip them on their sides)." I love arrays! I find that using them made me develop a better relationship with math. I don't know about you, but I am loving this idea of the multiple approaches to learning how to solve math problems, in general. And hey, it helps out when I am helping my daughter do her homework, winning!!
I also like the different approaches to looking at division.Two groups of two, giving you each two meaning subtracting two each time. Now I understand how this is repeated 9/3 is 9 minus 3 minus 3. But how do you know to just subtract 3 two times? What if you kept going and got to 0? I am still not totally clear on that so I think I have to rematch the videos.. It makes sense with addition, but as erin stated and I agree, mutiplication is repeated addition FACT, division is process.
ReplyDeleteI do however like the idea of repeated groups, 24/3 is 24 split into 3 equal groups so what are those groupings? That makes sense to me.
Hi Erin,
ReplyDeleteFirst I'd like to say that your gifs are hilarious. Thank you for that. Secondly, I completely agree that subtraction is so simple and comes autopilot to me at my age, and I don't really remember what it was like to learn it as a child in elementary school. I could see it being a slightly difficult concept to fully understand by a child, and I can see that it would be a challenge to teach as well. I am similar when it comes to learning - I prefer videos, or hands on learning rather than teaching myself how to solve problems through a text book by following instructions. It always seems like it is more efficient to learn that way, which is why I believe it is important to attain the skills to teach children how to learn subtraction verbally and allow them to visualize what is being done in order to fully grasp the concept. After children fully understand the concept of subtraction, then they will be able to move on to division which is essentially repeated subtraction, like you said early in your post.
By the way, this is Sylvia (not sure why my name hasn't been popping up)
ReplyDeleteErin,
ReplyDeleteAgain, love the approach of making this exploratory learning and proofing is really engaging. It makes hard to tackle math, suddenly nothing more than a puzzle to solve. Partial quotients is the latest method for teaching division. I am still a little partial to old school approach. It seems the new approach, although more visible in understanding place values, the processing time for kids to grasp it seems to put them at a disadvantage with time constraints. It seems like long division is being pushed to the curb for these new methods of "chunking" and "gridding" or even for bigger numbers, the use of a calculator! Never thought of in my day--unless you were so advanced it didn't matter. While I feel these alternative methods are helpful for clarification of the division process, they are time consuming when kids are made to "show their work" with them. I'm picturing the teaching that goes on in a public school math class, where 3-4 approaches for how students solve this problem and are then expected to remember all those different ways/steps, and show their work! It boggles the mind--I do wish we would go back to the standard requirements for the division process(as well as other standards for showing work) as you so clearly and eloquently explained it. Thanks for sharing!
Great feedback everyone. Erin, I love the videos. Your posts are always engaging!
ReplyDelete