Comment on anything that you like, but here are some prompts:

• Do you like my questioning? (Because I didn’t. Notice that moment when I pause and try to unask my question?)
• Describe, as best you can, the way that this kid thinks about the Distributive Property.
• What would you say next, if you were me?
• Does this video have implications about the way that you’d teach this topic? (And, come on, don’t give me that “they need lots of practice stuff.” Of course they do. But what else?)
• Do you prefer having videos over images on this site? (Because I have about a half hour more footage from my work with this kid…)

Looking forward to a great bunch of comments here. Don’t let me down?

Update:

Here’s how I responded:

• I’m intrigued by where this conversation might have gone. I like the student’s last remark that they will still equal the same thing. What is interesting to me here is that he is more intent on moving constants rather than collecting the variable. I understand the desire to eliminate negatives, I have that desire still myself, but I would have guessed that students would quickly latch on to the idea that moving variables has a higher priority than moving constants. I don’t dislike your question at all, but I would think that delaying that question would be more productive in terms of having the student explain his thinking here.

• Andy

I think this student does not fully understand how operations affect terms. I would like to see how he uses the distributive property in something like this: 8 – 5 (2n – 4).

• 1) I don’t dislike your questioning. Believe I’ve said similar things – and also had times I wish I’d waited 5 seconds to see where something was going. One possible alteration might be, instead of “why can you do that” ask “why would you want to do that”. You can add 4 to both sides, you can add 9 to both sides if you want! Whatever keeps the balance. But only the 4 is actually useful.
2) He thinks about the distributive property as a multiplication of numerical/variable values, which include separators. Like a plus, or a comma. Andy makes an interesting point (in the prior comment) about when the negatives start coming into play.
3) “Let’s see where this goes then.” Because honestly, I do not see a math mistake in this video. Once he has 2x+10=x, hopefully he’ll realize he needs to collect x’s. This will either lead to x+10=0, assuming he recognizes the need for zero, or 10=-x, which means dealing with a negative…. but no matter how you look at it, he hasn’t messed up. Yet. (I grant he HAS potentially made his work harder, but that’s not wrong.)
4) Not really, but then distributive property is nothing new to me.
5) I wouldn’t say a preference, but it does add an extra dynamic in that you can follow the thinking process. On the negative side, I felt I’d need extra time to watch rather than just glancing at an image, which is partly why the comment is so late. (I also rewatched the video twice in the course of commenting.) Maybe a balance?

• I’ve been burned by assuming that when a student uses the terms ‘it’ and ‘thing’ in an explanation, I know what the student is referring to and that the student is correct. The video stopped too early to know how you responded, but I’d push the student on specifically _what_ is equal.

• I like your prompts! I’ll respond to a bunch of them.

1. Do you like my questioning? (Because I didn’t. Notice that moment when I pause and try to unask my question?)

I think it’s a good question but I would have saved it until after the solution was reached.

2. Describe, as best you can, the way that this kid thinks about the Distributive Property.

Very much as an algorithm to follow rather than as a fact about the behavior of operations. I’d want to ask later “Can you draw a picture that illustrates the first step you did there, showing that 2(x+3) = 2x + 6?” But then I love my area models.

3. What would you say next, if you were me?

Having gone down that path, I’d want to keep probing about what’s going on here. I might aim toward “what is the relationship between the equation you’re writing at this step and the equation you had in the previous step”, aiming toward the idea that we have a sequence of equivalent equations.

4. Does this video have implications about the way that you’d teach this topic? (And, come on, don’t give me that “they need lots of practice stuff.” Of course they do. But what else?)

It makes me want to spend more time with these problems broken down into tiny pieces (like, first solving x + 3 = 10, then solving 2x + 3 = 10, then later solving 2(x+3) = 10, then …) but instead of practicing the tiny pieces with 30 examples, get some serious explanation-writing of what to do, why to do it, and why it gives correct answers. Maybe 3 examples so they don’t have to do all that explaining on the same one. I’ve recently given a lot of thought to good explanation prompts and am hoping I can explain to kids the distinction I’m aiming for with those three questions, to get the procedure, the explanation of how to select it as a proper procedure for this situation, and then the explanation of why the procedure is mathematically valid. I’d love to learn more about other people’s successes in that kind of direction.

5. Do you prefer having videos over images on this site? (Because I have about a half hour more footage from my work with this kid…)

No, I find videos a pain in general, but sometimes they capture answers to a lot of the questions that we are often following up with here (like, “can you tell us what the kid was thinking when they wrote that line there?”) so sometimes they’re a good thing.

• In the followup I think that using “x” for your pound symbol is potentially pretty hazardous, but otherwise I like the “equal weight” analogy.

The “take away 4 pounds” got a bit confused too, because the student had initially written +4 and then later *said* “take away 4 from over here and put it over here” which got you taking away. So then you went to adding pounds, which was a good way to head, but then it went sideways again because of the subtracting 4 instead of adding 4 or, even more efficiently, subtracting 6. Now we’re headed for a good conversation on checking answers and finding mistakes!

I wonder how many kids out there are confused between “these things are equal to the things in the previous step” and “if the things in the previous step were equal, then these things are equal too.”

• Amanda

I think you should have waited until he was done with the problem and then tackled the “why can you do that” part. Then have him explain the entire problem using boxes and you assisting with the analogy. Then have him do another problem and ask the “why” in the middle of that question.

• Sid Hollander

First, you have a wonderful site. Second, the comments I pass are from a math teacher of 76 years who still loves the classroom. So, if the math has not changed since I left and 2x + 6 = x -4 was the original equation it appears that at the end when you replied, “Beautiful!” that you actually accepted an incorrect answer of x = -2.. My advice to you and your students would be to never terminate a problem (or a video) without checking your answer.

I might have objected/suggested/prompted real quickly to the students ‘picture’ of the equation for several reasons.
1. never an equal sign.
2. As soon as he put 9 pounds in each box, I think a great error was made. The boxes represented the variable. If one was estimating a nine for the value of x then you/he needs to evaluate the .equation to see if x=9. Since that was not the intent (i. checking 9 for a solution then I’d suggest/insist that the 9 (OMG not ‘x’s’) be treated as constants.
3, At this level I have found that an illustration of a see-saw gives a strong visual.analogy.
O
_Box + Box + ^^^^^^_____Box + Balloon |
^
Sorry for the incomplete underline. mean the ^^^^^^ to be triangular weights. The lighter than air balloon on a string does nicely to represent a negative weight. I doubt that this student would ‘mess-up’ with this analogy. .