kitchen table math, the sequel: help desk - 'distributing the negative'

Wednesday, November 30, 2011

help desk - 'distributing the negative'

I'm working with a boy in a neighboring town who can solve equations with positive values like the following:
3 + 2(x + y) = 9
He is having difficulty solving equations that require him to distribute a negative:
3 - 2(x + y) = -3
I remember C. having trouble distributing a negative, and I remember stumbling over minus signs myself when I was a kid. At some point, I solved my problems by deciding to treat minus signs as either a -1 or the addition of a negative, depending on the expression I was dealing with.

Thus -x became (-1)(x) and x-8 became x + (-8).

I don't think anyone ever told me to translate expressions in this manner. Quite the contrary; I have vague memories of reasoning it out for myself on more than one occasion.


Here's the way a sheet I have from Glencoe says to teach distribution of the negative:
Use the Distributive Property to write each expression as an equivalent algebraic
expression.

a. 3(w – 7)
= 3[w + (-7)] Rewrite w – 7 as w + (-7).
= 3w + 3(-7) Distributive Property
= 3w + (-21) Simplify.
= 3w – 21 Definition of subtraction
Unfortunately, this sequence doesn't solve the problem. My student can simplify 3(w-7); what he can't do is simplify 3–2(x+y).

Today I tried having him draw huge brackets around 2(x+y), then simplify the 2(x+y), and then simplify the remaining expression:
3–2(x+y)
3–[2(x+y)]
3–[2x+2y]
3-2x-2y
In effect, I was turning the problem into two distributions: first the 2, then the negative sign.

This approach always worked for me, but the logic of it wasn't obvious to my student.

One more thing: this student probably had Everyday Math in elementary school, and his current class seems to be intensely procedural. The only textbook his teachers are using seems to be a NY state test prep book.

I'm eager to hear any thoughts you have both about procedural teaching (including mnemonics) and about how I might help this student make some sense of the math he's learning. Moreover, and I hate to say this, but if I'm going to help him make some sense of it, I have to do it on the fly. Our time together is extremely limited.

If anyone knows of a good set of "instructional worksheets," that would be fantastic. I'm combing through my own collection.

Last but not least, what do you think of this video?

57 comments:

pckeller said...

The video seems ok except for the last examples which are not helpful. No one would bother applying the rule here -- the natural thing to do would be to plug the given numbers directly into the expressions and perform the arithmetic according to the usual order of operations, never needing to "distribute the minus sign".

This is a rule that certainly has to be memorized and internalized so that you can do it on autopilot. But while you are learning it, it can be puzzling. Like many algebra rules, it can help to apply the rules with concrete numbers to see that the rule is true and that the common "first intuition" is not. So try this:

4 - 2(x -5) =?

The issue is whether it should be

4 - 2x - 10

or 4 - 2x + 10

Why debate? Make up a number for x and see what happens:

Say x = 12
4 - 2(12-5) = 4 - 2(7) = 4 - 14 =

-10

4 - 2x -10 = 4 - 2(12) -10 = -30...nope

4 - 2(12) + 10 = -10...yep

Also, I just remembered how I explained this to my son: give it some context. Say you are calculating your profit at the end of a day of selling stuff. Say you took in $100 but you sold 3 items, each which had a labor cost and a material cost. You could write your days profit as

$100 - 3(L + M)

where you are adding the costs, multiplying by the number of units and then taking that away from your income. But you may also feel like starting with the income and then subtracting the labor costs and then the material costs. That expression would be

100 - 3L - 3M.

This was the approach that clicked for my son.

Hope some of this helps...

p.s. the same guy who chooses the word for the word verification also names furniture for IKEA. I'm sitting on a croligg right now...very comfy

Hainish said...

Maybe turn it into 3 + 2(-x - y)? That way, he is distributing the negative sign first, then the 2.

Anonymous said...

Can your student simplify/expand:

-2(x + y)

??

-Mark Roulo

Catherine Johnson said...

Phillip - Like many algebra rules, it can help to apply the rules with concrete numbers to see that the rule is true and that the common "first intuition" is not.

Absolutely -- BUT (and I'd like to get your take on this) --

I was a bit hesitant to mention this, but this student is wrestling with an attention issue, too, a pretty serious one.

He seems super-bright and intensely verbal.

Sooooo....I've tried showing him - or having him hold a pen in his hand and write down what I say as I say it - the same principle used with numbers as opposed to variables.

It doesn't help IN THIS CONTEXT.

What I desperately need is a real curriculum --- actually, what I desperately need is for the school to adopt a real curriculum.

I believe I've seen kids pick up pretty quickly on an impromptu demonstration of a concept using numbers and then working up to variables -- BUT so far that hasn't worked here.

Any thoughts about that?

Catherine Johnson said...

btw, I 'understand' the problem he's having; it's a problem I've seen other kids have & it's the same problem I've had myself.

(Do 'math brain' types not have this problem? I'm getting the sense that they don't --- ?)

The minus sign is confusing because to a novice it seems like two different things: a minus sign AND an 'operator.' (Is that the word?)

In terms of grammar (which will sound nutty to a lot of you, I know), the negative sign in -x is like an adjective, while the negative sign in x-y is like a verb. They are two different things with two different functions.

That's why it always worked for me to turn 8-x into 8+(-x). In rewriting the expression, the addition sign becomes the 'verb,' and the minus sign becomes the adjective.

Catherine Johnson said...

Mark wrote: -2(x + y)

Yes, BUT I'd say he's inconsistent on this.

He absolutely hits a brick wall when you make this expression part of a subtraction problem.

Actually, before you weighed in I was going to ask whether it makes sense just to have him deal with the constant first ---- which he doesn't want to do because his teachers haven't told him to do that.

Still, that's not the right approach because if all he's asked to do is simplify the expression, he won't be able to do it.

Except...what if I asked him to use the commutative property to switch the terms around?

I didn't think of that this afternoon; I kept trying to come up with a sequence that would 'naturally' lead to his being able to simplify this expression.

So, long story short, yes: he can simplify/expand -2(x+y).

Catherine Johnson said...

Maybe turn it into 3 + 2(-x - y)?

boy

I think I can pretty much guarantee I'LL hit the brick wall if I try that!

But I'm going to put it in my notes and see if I can at least point it out.

Catherine Johnson said...

Say you are calculating your profit at the end of a day of selling stuff. Say you took in $100 but you sold 3 items, each which had a labor cost and a material cost. You could write your days profit as

I'm writing all this down, BUT I don't think this is going to work. His strength is that he has a phenomenal memory (it seems to me). He memorizes procedures and rules rapidly, and that's pretty much getting him through.

It is he-double-hockey-sticks for him to add in explanations like yours -- trying to get him through sales tax problems was very, very difficult.

That's another thing: I need CHARTS. Don't know if you all remember Exo saying that in the Soviet Union (or was it Ukraine?) math was always taught with charts.

I have become a believer.

The point of charts, as far as I can tell, is to reduce the load on working memory --- (because charts make it easier to swap content in and out of WM) --- and this student needs to have the load on WM reduced as far as it is humanly possible to reduce it.

That's what motivated my stab at the brackets.

That's why I need mnemonic devices, too.

He's so verbal that I actually got him to try to figure out a mnemonic for "distribute the negative" or "distribute the minus" --- but then we didn't come up with anything memorable!

Meanwhile, throughout all this, this kid is super-bright.

He is sui generis.

Catherine Johnson said...

Why debate? Make up a number for x and see what happens:

Say x = 12
4 - 2(12-5) = 4 - 2(7) = 4 - 14 =


Right!

But then the problem with showing him things -- or with having him write things down as I tell him what to write -- is that .... it's 'too much.'

He doesn't make the jump from the numbers to the next case (although that's not **entirely** true, and I'd say this is the major technique I've used so far).

pckeller said...

Not an easy situation. I run against a similar problem teaching physics to some of my lower level classes: for many students, teaching by analogy is effective. But for others, it doesn't work -- you have to invest so much effort to get to an understanding of the analog and then you find that the understanding doesn't transfer or even make the journey easier the "second" time.

So here's another suggestion that will sound cynical but I don't mean it to be: provide models of a half dozen examples, work them repeatedly until memorized and don't worry about "why" for now. Just say that a minus sign outside can be turned into a plus sign if you then go through and flip the sign of everything inside.

It's OK to use algorithms that you do not understand yet. Sometimes the understanding clicks later. Other times (and I bet this is the most common one) the understanding decays leaving only the memory of the algorithm. For example, of the people who still know that to divide by a fraction you invert and multiply, how many can tell you why?

p.s. ...now, my comfy chair is an edali.

gasstationwithoutpumps said...

I can't offer much help here—I generally only work with kids at the top end of the math distribution, not ones who are struggling with the distributive law.

If it helps any, unary negation and binary subtraction ARE different operators, and it is sometimes a nuisance that we use the same glyph for both.

If the student can handle simpler forms of distributive law, I would recommend one of the transformations you suggested:

a-b(c+d) => a+ (-1)b(c+d)

It is a mathematically sound transformation that eliminates the subtraction that seems to be causing the mental difficulties.

Allison said...

Catherine, part of the reason you think of the - in -x as an adjective and in (x -y) as a verb is because, I bet, you say "minus".

"Minus x" and "x minus y".

Yes, the first *is* an adjective, the second *is* a verb.

A better thing to do is what GSW/OP is saying, and maybe this will help.

Call -x "negative of x" or better still "opposite of x". Keep the "of" in there, too: unary negation means you're negating some single thing, you're taking the negation of that thing. You're always taking the "opposite" in that you're reflecting about the y axis.


Binary subtraction is x minus y, still, or even x "take away" y. But now those two operations do have different words associated with them, even if the symbol is the same.

The advantage of this change in word choice might be to dislodge the confusing and incorrect ideas your student has about something becoming "more" or "less" negative--opposing things switch signs, regardless of which sign they started on.

(cont)

Allison said...

Singapore's Primary Math teaches the distributive property intuitively first by teaching number bonds, and then as practiced in Singapore, the emphasis on mental math makes small calculations like these happen all the time, so the leap to the abstract isn't so big.

Number bonds are where you see the part-whole relationships around a number. 7 is made of 2 and 5; so 2+5 = 7 and 5 = 7 - 2 and all the other relationships as well. You practice these concretely with counters or rods or coins and then learn the pictorial abstraction of writing 7 as a tree with 2 and 5 as leaves.

As you build up mastering these number bonds, you learn to decompose numbers in ways that make mental math easy, e.g. 123 - 21 doesn't require a written algorithm because 123 is made up of 100 and 23, which is made up of 20 and 3; 21 is made up of 21 and 1, and you subtract the 20 from the 20 and the 1 from the 3, and you easily see you've got 100 and 2 left over:

it would be written with little marks I can't do well here, sadly, but
123 =

123
/ \
100 23
/ \
20 3

is the idea. So then you see you're taking away a 20 and a 1 from the 20 and the 3.

So you're learning to think through that you're subtracting off all of those pieces --i.e. distributing through a negative sign--long before you EVER write ANYTHING that involves actual symbolic manipulation of the negative sign.

I would practice mental math exercises involving subtraction until the cows come home, basically, until the intuitive feel for distributing a negative is present.

start with small things like 45 - 43, where you know he knows the answer, but you still make him do it with the breaking up into the mental math ways, and then go on to bigger numbers like 135 - 38 where you need to find new whole-part relationships to help you
135 is 100 and 35; 38 is 35 and 3, etc.

Michael Weiss said...

I usually try to explain it like this:

Say you have $40 in your pocket, in a combination of bills of different sizes, and I ask you to give me $15. You reach into my pocket and give me a $10 bill and a $5 bill. How would we write the amount you have left?

40 - (10 + 5)
represents "Start with 40, then take away a 10 and a 5".

But you could also write
40 - 10 - 5
which represents "Start with 40, then take away 10, then take away 5 more."

The point is that you are subtracting two things. It doesn't matter whether you think of that as "subtract the total of two things" or "subtract the two things one at a time".

ChemProf said...

I'd actually suggest trying the commutative property and seeing what he does. Does he move over the two or does he move it as negative two? That might help him clarify in his own mind where the confusion is.

I do also agree with Allison about the hazards of "minus" -- try to use the terms "subtract" as the verb and "negative" as the adjective and see if that clarifies things.

When I have worked with students who have these kinds of deep wholes that they can't articulate, sometimes you just have to try things. Sometimes when you do something like have them swap two terms it is revelatory -- "you can do that?"

Hainish said...

"In terms of grammar (which will sound nutty to a lot of you, I know), the negative sign in -x is like an adjective, while the negative sign in x-y is like a verb. They are two different things with two different functions."

To a novice, yes. I was taught (explicitly!) that I could think of the minus sign (the operation/verb) as being a plus sign adding a negative number. So, x - y = x + (-y), every time.

(Actually, maybe I was taught that the negative number//adjective sign is the equivalent of subtraction...but I found that it was helpful used the other way around, too.)

rocky said...

3 - 2(x + y) = -3

My show-off self wants to just move the constants to the left and the variables to the right:

3 + 3 = 2(x + y)
6 = 2x + 2y

But my teacher self wants to help this kid... The first thing we do is explain that only seventh-graders "minus". As algebra students we only add (or combine) positive and negative terms. So

1 - 2 + 3 - 4 + 5 - 6
= 1 + (-2) + 3 + (-4) + 5 + (-6)

Now we can use our commutative property to move them all around. Maybe we want all the postive numbers first and the negative numbers last. We can do that. etc.

Now let's look at the distributive property:

3 - 2(x + y) = -3

3 + (-2)(x + y) = (-3)

Let's multiply out the (-2)(x + y) term. etc.

SteveH said...

I remember figuring out many things that were not directly taught me. Often, you learn the identities and rules in only one way, like (-1)*(-1) = +1. You never see what they mean in other situations. It's inflexible knowledge. These things compound when you get to combined tasks, like:

3 - 2(x + y) = -3

It's like debugging a program. There are too many errors and they interact.

However, you can't go back and start all over again. You have to find and fix something concrete.

Does the student have a problem with any type of +/- number multiplication, or does the problem seem to happen just during distribution. It's a classic mistake to screw up this problem. When it happens, can the student find the mistake?

Everyone makes this mistake over and over and over. I still make it.

I guess my advice is to try and separate the basic skills from the combined skills, understanding versus inflexible knowledge, and mistakes versus practice.


I used to tell students a way to look at equations and expressions. I would tell them to start by circling all terms of the expression. I told them to include the leading sign in with the term. Next, they should change the sign to a +1 or -1 factor. Put parentheses around the +1 or -1. Next, put a plus sign in-between the terms.

Next, I would have them change all terms into rational expressions. If they didn't see a dividing line, I would have them put one in and put a 1 on the bottom. Each term would be written as a fraction with everything in the denominator or the numerator. You would never leave a (-1) factor dangling in front at the same height as the main dividing line. It had to be written in the numerator or the denominator. I had them explain why the (-1) factor could be written in either location.

Next, I would have them circle all factors in every term. This was tricky and instructive for many students. After that, I would have them write an exponent for all factors, even if they didn't see an exponent.

It's one thing to say that a*b = b*a, but quite another to see it in action with a complex rational term.

Also, we have:

1/a = a^-1

No big deal unless you see it in action with a complex rational term. It seems so wrong.

There are many ways to approach math, but I wanted students to "see" expressions and equations as terms and factors with exponents. I was never explicitly taught to do that, but that's why I've said that it took me until my high school trig class to feel like I had completely mastered algebra.

However, rather than get really weird about the rational terms and the exponents, you could do a limited breakdown for

3 - 2(x + y) = -3


Term 1 Term 2 Term 3
[+3] + [(-1)*(2)*(x+y)] = [(-1)*(3)]

This knowledge won't help if the problem is just practice with remembering the minus sign in the distribution. I know of no way to do the distribution and then add in the minus sign somehow. They just have to practice this until it becomes automatic. In most cases I've seen, this problem is not understanding, but practice.

lgm said...

The students I have helped in the past usually can't 'see' the negative sign after they start distributing. They'll write:
3-2(x+y)=3-2x+2y. It is helpful to them to use their fourcolored pen, switch colors and underline the -2, then switch back and continue. They then can make themselves 'see' the negative sign as they continue to distribute the terms. It also helps some to work at the whiteboard and explain each step - many have a procedure from class, but they weren't able to process all of it in class.

Catherine Johnson said...

Allison wrote:

Catherine, part of the reason you think of the - in -x as an adjective and in (x -y) as a verb is because, I bet, you say "minus".

Interesting!

Of course that makes perfect sense, doesn't it?

I've been musing about the (apparent) difference between 'word brains' and 'math brains' (I WILL find that study & post the abstract)....I guess all the intensive SAT prep, which has involved intensive thinking-about-SAT-prep, caused me to start 'rubbing' those two ideas up against each other.

I'm wondering how often words get in the way for me (and for others) -- and also how often I don't have the words I actually **need.**

I'm still trying to think this through....but I believe there's something there.

In particular, I've been thinking about the fact that I was taught virtually no grammatical vocabulary whatsoever. I don't think that was good, for reasons that I think go beyond the fact that "knowledge is good" and "more is more."

Very interest.

It's true: I use 'minus' primarily as a verb -- although I do say sometimes "minus one" meaning "negative one."

That doesn't matter, however. If for me 'minus' is primarily a verb, which it is, that's going to be tripping me up.

lgm said...

It seems to me that math brains have both the words and the graphics, and in addition have the generalization to the abstract. My objection to my district is that the majority of teachers are leaving out the graphics and the opportunity to use the abstract notation. Now that we have no math club, I have to grab Dolciani or use AoPS to fill in the gap. In my day, everyone that took college prep math had to be proficient in all three to score the 'A'. 'Taint so now.

Jen said...

I'd go with two of the things here:

**So here's another suggestion that will sound cynical but I don't mean it to be: provide models of a half dozen examples, work them repeatedly until memorized and don't worry about "why" for now. Just say that a minus sign outside can be turned into a plus sign if you then go through and flip the sign of everything inside.

It's OK to use algorithms that you do not understand yet. Sometimes the understanding clicks later. Other times (and I bet this is the most common one) the understanding decays leaving only the memory of the algorithm. For example, of the people who still know that to divide by a fraction you invert and multiply, how many can tell you why?**

This is the main thing. Do a ton of problems like this in a very procedural way (see below). Honestly, it doesn't matter whether or not he "gets the concept" yet. Better he can DO the problem first, then after layers of practice it may turn into a more abstract understanding. Or not, but he can at least do it!

If you talk to special ed teachers, the number of repetitions it requires for some kids (and we're not talking very impaired) to get something is easily in the 50x or more range for something that might take another kid 2 practice problems at most to get, commit to memory and understand.

So Two: I'd go with a procedure like several have mentioned above, one that works for every situation:

3 - 2(x + y) = -3

TO

3 + (- 2)(x + y) = -3

TO drawing a little curve line from above the (-2) to the x term and another from the (-2) to the y term

(in my mind's eye, that first term is like a basket going first to the first term and then to the second term and dropping off a negative 2 each time. I'm weird, what can I say!)

to rewriting it:

3 + -2x + -2y = -3

And then do it about 20 more times in that set-up (with the rewriting and the lines EVERY TIME)

Then do the same with something like this:

3 - 2(-x + y) = -3

another big bunch of times -- way past easy point -- do it until he can do it without thinking, without asking you, without looking at you each time)

And then again like this:

3 - 2 (x - y) = -3

and then with the "hard part" starting the equation:

-4 (x + y + z) - 6 = -9

and so on and so on.

Honestly, you have to do it so many times over, in so many seemingly similar but somewhat different situations that it becomes second nature. Not exciting, surely, but 100s (possibly 1000s =:-0) of reps will do it!

Jen said...

Uh, rereading that, I'm not sure I made it clear not to do the exact same problem a jillion times -- switch up the numbers or make other small changes!

Allison said...

-- Honestly, it doesn't matter whether or not he "gets the concept" yet. Better he can DO the problem first, then after layers of practice it may turn into a more abstract understanding. Or not, but he can at least do it!

Except for many kids, this doesn't work. They don't understand, and so their procedural knowledge is at the mercy of their brain remembering it right. Under stress, they may not. They may mix up which procedure to use when.

When you teach "just do the procedure" for multiplication of fractions, you teach
"just multiply the tops and then multiply the bottoms."

okay, so when the student who doesn't understand now tries to remember the rule for addition of fractions, they misgrab that rule, and think "just add the tops and add the bottoms".

Worse, they have no understanding to tell them why that's wrong.

Same with the division of fractions: which fraction do you invert? the first or second? does it matter? not to the weak procedural user.

They are lots of kids good at getting the procedure right without knowing why, but it doesn't mean all kids are. A kid who has gotten as far as Catherine's student with these huge gaps in understanding is unlikely to be that student, or they would be getting the procedure right already.

Jen said...

Uh, actually, that kid is likely to be exactly the kid who this helps.

It's like the research on smiling. Just smiling -- being told to smile, so it's not natural, it's not mood based, it's just forced smiling -- improves your mood. It's a two-way system and in many cases the direction we poo-poo is the stronger system. That is, the physical sensations sent to the brain via the smiling muscles are interpreted by the brain in a much bigger/more effective way than someone thinking about being happy or making themselves happy.

Right now? The kid has NO ability to get the problem right. Just seeing the problem likely raises his heart rate and makes all sort of math thoughts fly around in his head.

Doing the problem correctly one on one with a tutor (not as independent work or homework where he's as likely to be doing it wrong as right) will begin to allay some of that. After you've done something 20 or 30 times, it doesn't panic you anymore. You aren't thinking of six different things to do, you aren't trying different things everytime and not knowing what is and isn't working. You are able to begin to see the bigger picture.

It's true in a regular classroom, procedures are unlikely to be well understood -- and to be incompletely learned. This is especially true for kids who need lots of practice to begin to get something. Most classrooms can NOT provide the sort of practice that is needed, correct practice in the right quantity, over several days. I'd recommend that even after the kid "gets" it that she start with a couple problems like this at every tutoring session, too.

Understanding begins to happen AFTER you have done something correctly many, many times and your brain has had the chance to process those experiences and build a conceptual framework for them. If you never get to the point of doing it correctly in some sort of relaxed way, that abstract understanding isn't happening.

Catherine Johnson said...

Just smiling -- being told to smile, so it's not natural, it's not mood based, it's just forced smiling -- improves your mood.

Absolutely.

Catherine Johnson said...

Allison wrote: They don't understand, and so their procedural knowledge is at the mercy of their brain remembering it right.

Right. That is the HUGE problem with purely procedural knowledge.

Actually, "procedural knowledge" is the wrong term here (although I think it's the term most people use).

I think we're really talking about Willingham's "inflexible knowledge."

I wrote a whole long comment last night, sparked by Allison's observation about language, that Blogger rejected because it was too long (!) --- and thinking about it today, I was drawing a distinction between my own procedural knowledge of multiplication, which is strong AND which includes an understanding of when to use multiplication -- and some kind of 'deeper' conceptual understanding.

As I've retaught myself arithmetic and beginning algebra, I've found that there are many procedures I'm completely fluent in AND that I certainly understand in the sense that I comprehend when to use them and when not to use them.

What I lack is Liping Ma's "PUFM": "profound understanding of fundamental mathematics."

I wish we had agreed-upon terms for the various modes of 'knowing,' 'remembering,' and 'understanding.'

("We" meaning everyone who thinks about these issues, not ktm readers & writers specifically.)

Catherine Johnson said...

Very interest.

Uh, I meant "Very interesting."

Catherine Johnson said...

lgm wrote: The students I have helped in the past usually can't 'see' the negative sign after they start distributing. They'll write:
3-2(x+y)=3-2x+2y. It is helpful to them to use their fourcolored pen


WOW.

THANK YOU.

I am getting a four colored pen TODAY.

The other problem is that this student's handwriting is pretty bad, and I get the sense that it's somewhat difficult - and 'resource-consuming' for him to be writing at all.

I don't know WHAT is going on with our public schools.

This student's mom tells me he has low working memory, so what we have here is a **very** bright kid who in order to develop MUST have as many ways of taking loads off WM as humanly possible --- and no one taught him to write fluently.

Our schools desperately need precision teaching.

Catherine Johnson said...

chemprof wrote: I'd actually suggest trying the commutative property and seeing what he does.

That is EXACTLY what I'm thinking this morning.

Catherine Johnson said...

rocky wrote: As algebra students we only add (or combine) positive and negative terms

I'm gonna tell him you said so!

Catherine Johnson said...

Seriously....I just may do that.

I may say: I TALKED TO ROCKY.

Catherine Johnson said...

Jen said: provide models of a half dozen examples, work them repeatedly until memorized and don't worry about "why" for now

Absolutely. I try to do that every time I see him --- WHICH IS WHY I DESPERATELY NEED "INSTRUCTIONAL PRACTICE" MATERIALS.

I'm writing problems on the fly, too, and I'm trying to write a logical, coherent sequence of problems while also keeping my student in the same room with me --!

Catherine Johnson said...

Michael - I owe you an email about WM, etc. --- !

Catherine Johnson said...

chemprof wrote: I do also agree with Allison about the hazards of "minus" -- try to use the terms "subtract" as the verb and "negative" as the adjective and see if that clarifies things.

Absolutely.

Very good advice.

rocky said...

LOL! Yes, we're big kids now. :)

And remember the rule for adding positive and negative numbers: if they're the same sign you add them, and if they are opposite signs you subtract them, and the biggest magnitude wins!

Allison said...

--Doing the problem correctly one on one with a tutor (not as independent work or homework where he's as likely to be doing it wrong as right) will begin to allay some of that. After you've done something 20 or 30 times, it doesn't panic you anymore. You aren't thinking of six different things to do, you aren't trying different things everytime and not knowing what is and isn't working. You are able to begin to see the bigger picture.

YOU may be able to see the big picture, but it doesn't mean this kid or most kids can. Your own personal experience and intuition drive this belief--experience with your kids, too, maybe. But that's a bias. The idea that mimicking experts will make us experts is cargo cult education. It's essentially the idea here, but it's not borne out by the data.

Experts see these connections, and because we are experts, we go back are "rewrite" our historical understanding of our old experiences--we now see the connections that we didn't as children. We forget how little as children we made those connections naturally. It's not 30, 40 times. It's 30,000 times that we've done those problems.

Even in tutoring situations, generally, procedural knowledge doesn't just with practice turn into understanding. For the bright math naturals, yes, but not for the rest. Instead, you're practicing getting the procedure right when the tutor's there, and often, the student tries to game the tutor "just tell me the answer" becomes more sophisticated "is it a 2 here? a -2?" The reasoning has not been learned, and pull away the scaffold of the tutor, and it falls apart.

But more, it isn't difficult to teach WHY the distributive property is true. So there's no excuse for not doing the two parts together, seamlessly. Singapore does it from the beginning this way. Even in a remediation environment, it can be done.

Jen said...

"we now see the connections that we didn't as children. We forget how little as children we made those connections naturally. It's not 30, 40 times. It's 30,000 times that we've done those problems."


Exactly -- that's my argument, not yours! I absolutely did NOT make these connections as a child, I was just lucky enough to have two things. One was the ability to follow a procedure and to see when it was needed. The other was someone "giving me the words" that developed the concepts many years later, after I had years more experience manipulating numbers.

I didn't say 30 or 40 times through would make this child get it -- I said 30-40 times through is barely one day's work toward getting it.

I didn't say don't explain what is being doing as you PRACTICE repeatedly. In fact, the shortest, most succinct yet mathematically correct way of describing what you're doing should be repeated frequently while doing it. A mantra even.

I didn't say don't try to use visuals or little stories that, again, illustrate or illuminate the concept while doing it (in fact, I gave an example of a simplistic image my brain still uses in these situations).

I just said that doing those things is NOT ENOUGH without lots and lots of concentrated practice to go along with it.

And my experience has been with kids 2-5 grade levels behind, with very little basic skill knowledge (for instance, a 7th grader who couldn't count up or down accurately to add and subtract her basic facts, and 6th and 7th graders who couldn't verbalize whether breaking a big pile into groups would mean the resulting smaller piles would be bigger or smaller than the original pile to be divided.) Some kids have had virtually no experience that counts toward their 10,000 times even after years of schooling.

I tried to windsurf once long ago, with someone who knew how standing by. I understood how sails and wind work, I understood totally what the person teaching me was saying about how to loosen certain muscles and grip with others, when to lean and when to bend my knees, etc.

However, seeing and hearing all of those things did diddly-squat for me when I stood up on that board. My brain was happily screaming reminders about all of the pieces, but I had to just DO it, over and over, with some commentary on which of those rules I had or hadn't followed.

Certainly even 30-40 practices wasn't going to make me even the tiniest bit proficient. However, 30-40 practices daily for several weeks, certainly would have improved my skills, whether or not I was given any more instruction.

But hours of having the concept explained to me? Wouldn't have had any effect after the initial learning, would it? It's the doing that counts.

kcab said...

I'm late to this, but your student sounds enough like my daughter that I thought I'd just get her opinion and share that. She's a HS freshman now & has finally internalized the distributive property, thank goodness.

Her opinion was basically the same as several above - rewrite the equation to make it clear that the expression in parentheses is multiplied by a negative number. I believe that she has used the technique of drawing arcs between the inner terms and that on the outside. I don't think anyone told her to do that, she just did to keep track.

She had trouble with the distributive property even this last summer. What helped was doing a lot of problems and getting rapid feedback. (Not from me! She won't listen to me!) She used Alcumus (in the mode where she was working on specific objectives) and was taking an AOPS algebra course to refresh her memory of the subject.

I *think* that it would be useful to do a lot of this type of problem with numbers assigned to x and y (and the constant outside too). Use values s.t. the correct answer should be obvious from basic multiplication facts. No doubt Alison is correct that Singapore would be helpful, but I'm guessing this student is old enough to be insulted if it is obvious you are using elementary school materials.

lgm said...

"Under stress we regress" - procedures without understanding tend not to be performed perfectly when stressed. Wasn't that in Willingham's book?

Here's an algebra tile tutorial that might be helpful to you and your student:


He should start out understanding the distributive property with whole numbers, using pictorial or manipulatives to group. He should pratice the basics 2x8=2(5+3),6x9 = 6(4+5)etc. Then go on to negative numbers and so forth up the strand. He's basically missed several years of this strand, but he will have lightbulbs go on while he brushes up on his math facts, which will help him as algebra goes on. There's no royal road here, he has some good work ahead of him.

On the handwriting, he may do better if he slows down and makes an effort to write big. Sometimes a flair marker helps because of the friction and the contrast, sometimes a bigger body pen (the 4 color pen is great). I usually have them work on marker board with me.

If his school district has a contract with castle learning or study island he might find practice problems with solutions there.

lgm said...

Retyping link:
UTexas Distributive Prop w/ Tiles Tutorial

Jen said...

""Under stress we regress" - procedures without understanding tend not to be performed perfectly when stressed. Wasn't that in Willingham's book?"

Page number?

One thing to remember though is that this kid isn't forgetting something he previously learned, he's not gotten to that point yet, for whatever reason.

Also, most procedures don't work as well under stress, regardless of understanding. Understanding helps if you can then realize the stress, slow down, calm down and work it through. Back to the SAT examples earlier, *that's* what you don't have time to do. That's why they call it stress. ;-D

Jen said...

I could at least get some problems that you could use as ideas for fixing on the spot from here:

http://www.math.com/students/worksheet/algebra_sp.htm

I put in 20 for "distributive property" and got a worksheet that had all problems like this:

-3(2x-9) = 27

You could spice them up adding in little extras? At least it's not horrible!

Jen said...

And because I can't seem to comment enough...just wanted to agree with the marker board comment above. All ages seem to love their marker boards and it just is more fun to erase it all away than to end up with pages of messy pencil pages to toss.

At school, we'd use a sock as the eraser (clean ones, bought just for that purpose at dollar stores).

Catherine Johnson said...

There can be no excess of commenting!

Catherine Johnson said...

No luck with the white board thus far (tough nut to crack, I know!) BUT we have been working on the sofa in the living room, and I finally put my foot down and moved the two of us to an actual table, which is SO MUCH BETTER.

Gotta go find the other comment about the white board.

Allison said...

--But hours of having the concept explained to me? Wouldn't have had any effect after the initial learning, would it?


I never once suggested "explaining" the concept. Explaining is not the same as teaching someone to the point of conceptual understanding. You break a difficult concept into smaller parts and make sure those are understood to mastery.

Nor did I suggest not practicing problems.

Maybe you and I are in more agreement than disagreement, but I can't tell. I disagreed not with the idea that you shouldn't practice, or that you shouldn't practice with lots of variability.

I disagreed strongly with this
"Honestly, it doesn't matter whether or not he "gets the concept" yet. Better he can DO the problem first, then after layers of practice it may turn into a more abstract understanding. Or not, but he can at least do it!"

I'm saying someone helps a student properly distribute by making sure they get what distribution means. They conceptually master the distributive property symbolically by working with whole numbers concretely or mentally, and performing a variety of grouped operations--so they understand that's how numbers work, first and foremost. No symbols at all. No -2s to distribute. First, you learn about part-whole relationships, and operations on those parts and whole.

Look, does the student even understand why you can make an "x" into a "-2x"? What that means? Does he understand X IS A NUMBER, so the symbol -2*x is another number? That may be obvious, you think, but it isn't obvious to lots of students, who don't seem to get when you get to play arithmetic on undefined symbols and when you can't.

I'm not suggesting someone give the student a Primary Mathematics workbook or textbook either. I'm suggesting studying the teacher's guides to understand how to teach part-whole relationships, number bonds, mental math strategies, and work up from there. These are interactive sessions, where you start to uncover what other holes and misconceptions are in this boat, because while bailing is necessary, you're going to need to cover the holes or bailing first isn't going to work.

SteveH said...

Is it ever possible for a tutor (one-on-one) to provide full understanding at any level? Isn't this only accomplished when the student does a set of problems on his/her own? What I'm saying is that mastery and understanding are linked and that mastery is ultimately a solo task. It's OK to try to achieve this goal while tutoring, but don't expect it to happen. Worse yet, does the student expect that you are going to do that? This doesn't absolve bad teaching, but you have to see that there are limits even with the smartest, most well-prepared kids.

Catherine Johnson said...

Allison wrote: Call -x "negative of x" or better still "opposite of x". Keep the "of" in there, too

That is fantastic advice. (I'm just now settling in to digest all of these comments.)

I solved another problem with this student by figuring out, for myself, that the pronoun "of" was critically important.

Catherine Johnson said...

chemprof wrote: try to use the terms "subtract" as the verb and "negative" as the adjective

5 - 2(x+y)

So I would read this as "five subtract two times the value x+y."

or: 'the value of x+y'

or: 'the sum of x + y'

(or 'the difference of x and y' if it's x-y ---- ?)

What word or words do you use specifically to indicate the presence of the parentheses?

rocky said...

For some reason that doesn't sound right. I would say "five minus two times the quantity of x plus y".

"But", (continue patter...), "as algebra students, we convert it in our heads and say: five plus negative-two times the quantity of x plus y".

"If I wrote what I was thinking it would be: 5 + (-2)(x + y)"

I just realized my world is smaller than I thought it was. I thought we all used the word "quantity".

rocky said...

Allison, I really think that Euclid's geometric way of explaining it is the best. Maybe I'm just a "visual learner", but how can you go wrong?

+--x--+-----y-----+
|.....|...........|
a.....|...........|
|.....|...........|
+-----+-----------+

a(x+y)=ax+ay

Ok, there's a little handwaving over the negative number issue, but that is why you have to teach the distributive property before negative numbers. Wu uses the distributive property to prove that a negative times a negative is a positive.

Allison said...

rocky,

:)

I try not to be circular in my reasoning or Wu will catch me :)

I think Euclid's right, but Euclid didn't have negative numbers, did he ? :) It's so amazing to think about the geometry the greeks did. When I was taught "the greeks didn't have negative numbers" I ignorantly took that to mean they didn't understand their need, not just that they lacked symbology. They had the most magnificent ways of proving things to handle them, didn't they?

I'm with Rocky. I say "quantity". I would say "five minus 2 times the quantity of x plus y" OR if I was standing in front of someone, I would indicate the quantity with my voice and hands--do other people not do this?

"five minus (mini pause) two times(hands cupped as parentheses) (pause) (then rushing) x plus y", emphasis on the "x plus y"
which sounds different than

"five minus two times x (pause) plus y" which would be
5 - 2x + y

Allison said...

well, maybe I'd say "the quantity x +y"

it would come out as
"five minus 2 times the quantity x plus y."

ChemProf said...

Allison and rocky are right, it sounds funny to use "subtract" that way. You'd say "subtract x from y" not "y subtract x".

I'd tend to say "five minus two times the sum of x and y", but quantity is also nice to really read the equation from left and right.

However, when you start talking about what to do, then I'd tend to switch to subtract, as in "okay now subtract x from both sides".

rocky said...

Here's another "geometric" idea (so you don't get foiled again).

+-- 3x ---+- -2 --+
| ........|.......|
x . 3x^2 .|. -2x .|
| ........|.......|
+---------+-------+
| ........|.......|
5 . 15x ..|. -10 .|
| ........|.......|
+---------+-------+

(x+5)(3x-2)=3x^2+13x-10

It's not really geometric, it's just an "organizer" for your work. The neat thing is that you can work backwards from that grid to factorize the polynomial. It's the fastest method I know.

SteveH said...

I found that students rarely had a problem with the distributive property. They had a problem with what to do with a minus sign versus a negative number.

As I mentioned in a post above, I liked to tell students that they could always change a minus sign into a separate factor of (-1) or put it with the number to the right (-1)(2)=(-2).

This would give them

5 + (-2)(x + y)


I think this is important in many situations. What if a student ended up with the following after simplifying an expression:

3y/(-2x^2)


What do they do with the minus sign? So many students get stuck here. Change it into a factor (-1).

What about something like:

5 - (x+y)/(x-y)


where (if drawn in math form) the minus sign looks like it's stuck pointing right at the dividing line? It's not so much about what one should or should not do in this situation to "simplify" the expression, it's about what, mathematically, can one do with this? Many kids get stuck thinking that "simplify" is some sort of mandatory one-directional thing and that the expression will not be legal unless that process is done.

My view is that students don't need one understanding (or words) that explains what to do with a minus sign during distribution. They need how to deal or understand what to do with a minus sign in general.

So, the above equation becomes

5 + (-1)* (x+y)/(x-y)

Unfortunately, many draw the (-1) factor dangling right in the middle in front of the dividing line. Where does the factor belong? Students need to know that it's really

5 + (-1) * [(x+y)/(x-y)]

or

a + b*(c/d)

in factor form. I was really big on making students know the factors in complex expressions.

This could then be written as

a/1 + b/1*(c/d)

Students have to see that the dangling (-1) factor is really in the numerator. They could then write the expression as:

5 + [(-1)(x+y)]/(x-y)



I'm assuming that Catherine is not talking about a student who has difficulty with the distribution operation or with multiplying numbers with different signs.

I found that changing the minus sign into a (-1) factor was the best way to provide a general approach to any minus sign issue. It also opened up discussions of many other misunderstandings.


For this expression

3y/(-2x^2)

I would draw it like this:

[(3^1)(y^1)]/[(-1)^1(2^1)(x^2)]

Then, I would change it to

(3^1)(y^1)((-1)^-1)(2^-1)(x^-2)

At least I tried to get them to that point. Students hated it because it was anti-simplify.