I like your explanation a lot. Now I’ve got more to think about. Moreover, I’m wondering how we present ALL of this to students. For better or worse, I know I was not taught to do math the way that I am doing math with my students.

It seems like I might finally get this right just in time for retirement (25 years from now). ]]>

a-(b+c)

1. Algebra deals with the signed number system.

2. So subtraction is “moving to the left” or “moving down”

3. How much am I moving to the left from position a ?

4. b+c

5. Do I have to do this move in one go ?

6. No, you can move b to the left, followed by c to the left

7. So a-(b+c) is the same instruction as a-b-c

8. Done ! ]]>

The number line is great for understanding adding and subtracting when you can get the kids to invest in it. I probably use a number line in my 8th grade class once per day and my Algebra 1 (8th, 9th, 10th) class at least twice a week to demonstrate addition, subtraction, and absolute value one more time. But students often fight drawing a number line for themselves and find it tedious or childish when asked to do it even when it makes the answer clear! Pre-teen and teenage rebellion? A case of sophomoric bravado? I don’t know. I know that with my oldest daughter (age 6), I encourage her to add and subtract on her fingers so that she sees it happening, and I will encourage her to use a number line when we get do more subtraction. (And she’ll do it because she wants to make daddy happy…) We also spend a lot of time talking about number places. My daughter claims to not be able to add two-digit numbers in her head, but when we discuss number places she does just fine.

As for a – (b+c), an ideal lesson would like this:

Me: What operations are in this expression?

S(tudents): Subtraction. Addition.

Me: Anything else? [I know there is multiplication, but most students do not recognize it for this situation.]

S: …

Me: Ok, what do we do with subtraction signs? [I want them to see the distribution on their own. Remember, they're invested a little in the problem because they are pretty sure that they are correct.]

S: Add the opposite.

[Now I write a + -1(b+c). Depending on the group of students, I may need to remind them that the term (b+c) = 1(b+c) which most students will be ok with.]

Me: Oh! Look! Hmm…What operations are in this expression?

S: Addition and Multiplication (Distribution).

Me: Remember, being able to speak clearly and describe ideas well is important. When we multiply a value by -1 we are finding the opposite of the value. So we are really adding “a” by the opposite of this entire sum [point to b+c].

S: Can’t we just distribute?

Me: Yes, but you also should anticipate and try to understand what you’re doing.

[Write a + -1 * b + -1*c.]

Now we have distributed -1 over the sum. What is the opposite of positive b?

S; -b

Me: What is the opposite of positive c?

S; -c

Me: Good, so we have a + -b + -c. Of course, on the ACT they think that subtraction is a simpler form than “+ -” so we have a – b – c.

(Short version: change subtraction to addition of the opposite (need to write -1), distribute the -1, rewrite any “+ -” as subtraction).

I am probably more verbose and possibly even bad in my method of teaching (I keep learning), but at least I am trying to get students to think about what they are doing due to websites like this. 5 years ago, I would have just said:

Me: We have a – (b+c). So we just change the signs on the numbers in the parentheses like this: a – b – c. Remember, if you see a negative outside parentheses it changes the signs on all the terms on the inside. Get it?

S: …

Me: Ok, moving on…

Don’t know if you were expecting such a long response, but you did get me thinking. Thanks, Viv!

Don.

]]>I know part of this is that I have enjoyed math all my life, but I’ve read enough comments below his videos to know that many other people (a significant number not having always appreciated math) feel the same way.

]]>The general problem is simply this. For thousands of years of human culture, we’ve been conditioned to believe that memorization skills were the key to higher learning. And for a long time, that was true, because our technology wasn’t good enough to make memorizable facts widely-accessible. People who could memorize algorithms had a significant advantage *because* they were competing against people who had no way of looking up the answers.

I don’t know whether to be optimistic or pessimistic about the future. The mere fact that you can even write this blog, and have dozens of people read about it and think about mathematics education, means that you’re already far ahead of your grandparents and their grandparents before them. So things are already moving in the right direction. But I guess one way of thinking about it is that our educational system is not nearly moving fast enough to get to a level where we can complement our advances in technology. So is that bad news because our educational system is slower than ideal? Or is it good news because our technology is faster than ideal?

]]>I am not a teacher , but I am intrigued that the use of the number line does not help the kids to understand that a-b = a+-b? I have never explicitly explained to my 6.5 yr old, but we have played games with adding negative numbers to positive numbers and I’m hoping that this will start to lay the foundation for the subtraction rule. In the UK, the kids use the numberline extensively in Years 1 and 2 and I assumed that this would help them understand that subtraction is just like adding negative numbers. Sorry if this is a really naive question, just trying to get my brain around the ‘new maths’ (which by the way I do like and think kids enjoy much more than our ‘old maths’). Out of curiosity, how do you explain a-(b+c)= a-b-c and at what age/grade are your students in?

Kind regards,

Viv

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