Comments

"It's the basic trick of algebra; you solve for something by undoing what's been done to it, ..."

I don't like "undo". I perfer telling the student that he/she has to get the variable all by itself on one side of the equals sign.

What about something like

3X/5 = 2X + 1

There is no one mechanical approach to this problem. (Do I sound like a fuzzy?) I would first multiply through by 5 so that I don't have to deal with fractions. This may be inconvenient for some equations. This would give me

3X = 10X + 5

Next, I would collect the X terms together on one side of the equal sign.

7X = -5

Oops. Perhaps I did too many steps at once here. I could have

-7X = 5

Now, divide through by -7 to get

X = -5/7

Students should try to do it different ways to make sure that they can come up with the same answer.

The big picture is the basic rules of algebraic manipulation and the idea that you have to isolate the variable all by itself on one side of the equation.

What about

10X = 3*(5-X) + 5

First, I need to get multiply out to get the X term on the right out of the parentheses, otherwise, I cannot combine the X terms.

10X = 15 - 3X + 5

Now I can combine the X terms on the left and the constant terms on the right.

13X = 20

X = 20/13

If the X is in the denominator, then you have to multiply through by X. X has to be in the numerator and everything has to be multiplied out. Then, you have to collect all of the simple X terms on one side of the equals sign.

Practice makes perfect. EM gives my son "Study Link" pages that contain only 3/4 problems to do a night ... actually, it's not every night!

-- SteveH - 06 Jan 2006

Maybe make him write out the full steps a lot of the time.

E.g.

d/21 = 6/7

d x 21/21 = (6/7) x 21

d x 1 = (6 x 21)/7

d = 6 x 3

d = 18

If he does a lot of those steps, what he is doing will stay in his mind.

Another way might be to give him problems like:

21/d = 6/7

-- TracyW - 06 Jan 2006

I require students to know how to solve these problems formally—as you are suggesting—but also have them look for quicker, informal ways, especially when the numbers are compatible as they are in the examples you present.

I agree with you that it’s important for Ben to be able to isolate the variable using inverse operations. I call this “proof style” solving: How do we isolate w (second example)? We have to multiply, because (w/2.1) x 2.1 = w. We need to multiply the other side by 2.1 to keep the equation balanced, as I’m sure Ben understands from his pan balance work.

You can show him how he is justifying, or proving, the solution this way when he demonstrates the operations on both sides of the equation, and help him understand the importance of being able to do this. But I would recommend encouraging him to develop his informal strategies too, as this can result in a deeper understanding of what’s going on mathematically.

In the first example: For the two fractions to be equal, the numerators and denominators need to have the same “scale factor,” which in this case is a whole number. Ben seems to recognize this, though it might be worthwhile to have him articulate why multiplying 6 x 3 gives d.

Even in the second example, this approach can be used, and Ben might enjoy applying it in this more complex problem. First, reduce 6.4/2.4 to 8/3 (e.g. by multiplying both by 10, then dividing by 8, mentally). Now he can use the same reasoning he used in the first example: 3 goes into 2.1 how many times? (3 x 7 is 21, and you need one decimal place, so use .7) Now, 8 times this same number will give w. 8 x 7 is 56, so 8 x .7 is 5.6.

I typically balance these two approaches by having students do several problems formally, then encouraging them to work informally on the remaining problems, at least those that have somewhat compatible numbers. (If there’s a problem Ben has done informally and is unsure of, or one for which you’d like him to justify his answer, you can have him work it again formally.) Even students who prefer to do everything mentally are usually OK with this, as they know they’ll be allowed to do it “their way” after doing it “the proof way” a few times. I do the same thing on quizzes, giving them a few problems for which they must demonstrate the operations on both sides of the equation, and then allowing them to do several problems using their own methods.

Has Ben learned how to use cross products yet?

-- GretaFrohbieter - 06 Jan 2006

Hi Carolyn,

The Ghost of Fourth Grade Past speaking here...

Maybe it would help to revisit comparing two fractions. As Greta said, For the two fractions to be equal, the numerators and denominators need to have the same “scale factor”... For a short review of common denominators, erase the equals sign. Check if he knows how to compare two fractions when

1. the numerators are equal and denominators aren't

2. the denominators are equal and the numerators aren't

3. the numerators and denominators are all different and nothing is an obvious multiple of anything else.

Then talk about comparing two fractions where one of the four parts is unknown.

Good luck!

-- BeckyC - 07 Jan 2006

Carolyn, Carolyn Morgan here.

TracyW^?'s idea of writing out the problems over and over is really a good idea. I am not the "expert" you're asking for -- I feel inadequate suggesting to you what to do because you're the algebra "brain" and I'm just a 5th gr. math teacher, but I've found that 5-8 grade students really do learn from repetition of the same types of problems over and over. As they are beginning their journey into abstract thinking and unknowns, they will hit "snags" that hang them up along the way. These "snags" are really just gaps in their reasoning. The gaps need to be filled in before they can go on.

OK, I'm a "drill and kill" person, I know that, but repetition will provide students with a method of getting the correct answer while also giving them a chance for the steps to "sink in". Their brains just need to grow a little more in thinking and reasoning about what they are actually doing. The repetition makes the students feel "safe" until they get the concept. I've seen it work over and over again with these developing young minds, some catch it quickly and some hit "snags", but one will suddenly say, two, three, four weeks later, "Oh, I get it!" Suddenly, the reasoning gels and the new concept takes root. But in the meantime, they've been able to solve the problem and be successful.

Math really does need to be taught in small, incremental concepts or steps. Often we skip a step -- it's such a small one that we don't realize that we've skipped it because the reasoning seems so logical to us. Each new step need to be hooked onto a previous concept that is securely planted in the mind. If the previous concept isn't well anchored, the new one won't have anything to catch hold of.

Some students can move through these concepts, just skipping along, seeming to hop over some of the steps. These students can "see" the steps in the problem by just thinking through it.

Others need to plod along, pausing at each concept along the way, mulling it over. They don't "see" the steps by thinking. There may be gaps in their reasoning that need to be filled in. Repetition helps these students "see" the steps and helps them fill in the gaps because they only "see" the steps by doing each one. (But for both types of students, the steps are still there and have to be done in order.)

Well, I've gotten off the subject.

And "cross products", as Greta mentioned, aren't always taught anymore, but it's a great tool. I wish it were used more often. I use it when I'm trainine my Math Olympic students. They love it.

(And by the way, back to my "drill and kill" practicing. When I know that my "brainy" kids have caught on and I'm sure that they really get it, I tell them, OK, I see that you've got this, I'm going to give you permission to do some of it mentally, and as long as I can see that you're getting right answers, I'll let you take shortcuts." I'll even allow these students a time to demonstrate for others, at the board, their method and explain what's going on in their brains.)

-- CarolynMorgan - 07 Jan 2006

Problems like this aren't any different from equivalent fractions:

1\2 = x\4

They can be seen as equivalent fractions, equations, or as proportions. The numerator and denominator are multiplied or divided by the same number to produce each term in the other fraction or ratio.

With w\2.1 = 6.4\2.4, if you don't simplify 6.4\2.4 first, you may be creating more work for yourself.

But if one knows to set up problems like this, like this:

w\2.1 * ?\? = 6.4\2.4

than you can use any of several methods to solve it--intuition, inverse operations, or cross-multiplication.

-- JdFisher - 07 Jan 2006

An example I would use to try to move from 4th grade math to motivate the reasonableness of cross-multiplication:

1. 2/5 compared to 2/7

2. 2/5 compared to 3/5

3. 2/5 compared to 3/7...

wherein we create a new common unit of measurement to convert 2 fifths into 14 thirty-fifths by chopping each fifth into seven parts, and we convert 3 sevenths into 15 thirty-fifths by chopping each sevenths into five parts.

Then move on to d/5 compared to 3/7. Try to get Ben to tell you what to do and why.

Then, as Carolyn Morgan said, repetition makes the students feel "safe" until they get the concept. Agreed! But don't forget to demo "wrong turns".

-- BeckyC - 07 Jan 2006

Carolyn Morgan! Long time no see, and we've missed you!

Thanks for all the good suggestions. There is a lot of food for deep thought here. More is to come on this topic.

-- CarolynJohnston - 07 Jan 2006

Or maybe all Ben needs to see is that (using the numbers from my example) 3/7 = "f", and d/5 = f is asking the same question as d = 5 x f. Is Ben always focusing on the details of the specific numbers of each problem? Give him the liberty to call the balancing fraction, "f". <-- voice of unconscious pedagogical incompetence ;)

-- BeckyC - 07 Jan 2006

Carolyn Morgan —

Hi!

Good to hear from you!

WE NEED YOU!

WE ARE DESPERATE!

(um.....I mean, I am desperate)

-- CatherineJohnson - 07 Jan 2006

I'm so blasted at the moment I can't think my way through this....but I think it's the same issue with Christopher....

One thing I'm convinced of now, and this is something Steve mentioned one of his teachers did:

If I had to teach the Phase 4 math class, I would require the students to do as many problems as possible 'formally,' showing each step, AND WRITING DOWN THE PROPERTY THAT ALLOWS YOU TO TAKE THAT STEP.

Christopher's 'knowledge' is beyond fragmented. He has ZERO idea that anything he does is related to anything else, or why he can do it.

Today I wanted Ed to show him that 'isolating the variable' by 'doing the same thing to both sides' is the same thing as 'using inverse operations'......

I don't think Ed got to it, and Ed was reluctant to do it because he thought it would confuse Christopher.

We're now at a point where showing any connection amongst the various tricks he's been taught at all will make him more confused, not less.

I don't know whether writing down the property or principle alongside each step would help him make connections, but at a minimum it would help him remember the properties.....

-- CatherineJohnson - 07 Jan 2006

I have an idea, but I don't know if it's any good. It kinda follows along a thread from Carolyn Morgan, above.

Carolyn mentioned that she sometimes lets her students demonstrate their shortcuts to their classmates. Many of us have experienced how much better we learn a topic once we begin to teach it to others. Perhaps it might be useful to ask Christopher to try to teach you the lesson he learned in math class today. I wouldn't expect him to be able to do it well the first time you asked him, but he might get better at it over time. It might slightly alter how he pays attention in class if he is trying to perpare himself to re-teach the material.

If you can make any progress in this direction, I can imagine a few advantages:

• It would get away from you explaining something, then asking, "Do you understand that?" which is often answered affirmatively even when that's not quite true.

• Christopher would not be starting with a new problem to solve. He may not know how to attack a new problem from his homework. Instead, he would be explaining using a problem that he has already seen worked.

• Once he gets started, he would be expected to stumble at a point where he is really confused. This might tell you more what is missing than trying to work a problem he doesn't know how to start or working on a problem that isn't exactly like what he was supposed to learn today.

• You can couch some of your corrections as pointers to improve his teaching technique, rather than pointing out what he doesn't know. This might be more constructive.

Like I said, I don't know if this is a good suggestion. I haven't tried it, so I don't even have anecdotal results. It's just that it might give Christopher a different point of view and change your dynamic in working on his math with him.

My advice is generally worth what you've paid for it, but comes with a full money back guarantee.

-- DanK - 09 Jan 2006

I would not teach cross multiplication. In fact, I once told my students that "cross multiplication is not one of the operations" we're allowed to use in solving linear equations. I was probably a bit too, uh, animated about it, too. O:-)

If he's like a lot of other students I've seen, he'll add "cross multiplication" to his list of "mathematical operations that occasionally gives the right answer" when dealing with equations having fractions. See http://www.math.umd.edu/~jnd/Fractions.html .

-- PaulMiller - 09 Jan 2006

Paul, Although I only have a student in Singapore Math 3B, I am going to print that html out and work through the concepts presented.

I am finally learning fractions beyond the "tricks" stage that I "learned" in public school. At the youthful age of 38, I have fallen in love with math.

-- NicksMama - 09 Jan 2006

I would not teach cross multiplication. In fact, I once told my students that "cross multiplication is not one of the operations" we're allowed to use in solving linear equations. I was probably a bit too, uh, animated about it, too. O:-)

I still haven't read this thread closely, but I'm with you on this.

I think both Anne & Steve said the same thing on another thread — and Ed told me he had an excellent h.s. math teacher who told them NO CROSS MULTIPLICATION.

-- CatherineJohnson - 09 Jan 2006

Problems like this aren't any different from equivalent fractions:

1\2 = x\4

I still haven't read this thread carefully (!) but this caught my eye.

Christopher always solves these problems as equivalent fractions.

To me, that seems OK; it seems reasonably 'conceptual' — conceptual meaning he has some idea of why his solution works, as opposed to just having memorized it in class.

Of course, it starts getting difficult with less 'friendly' numbers.....and I'm thinking that may be a nice, logical teaching sequence.

When the problems grow more difficult, he sees the need for setting up equations, using the properties, etc.

At least, I think he does.

-- CatherineJohnson - 10 Jan 2006

Nick's Mama

I am finally learning fractions beyond the "tricks" stage that I "learned" in public school. At the youthful age of 38, I have fallen in love with math.

Me, too!

Me, three!

-- CatherineJohnson - 10 Jan 2006

This topic has actually reminded me of a discussion I had with one of the math ed grad students before break. She was trying to argue that a student needs some sort of non-mathematical metaphor for mathematical concepts in order to display "real" understanding.

I merely asked what sort of non-mathematical metaphor can describe something called a "linear functional." (Briefly, a linear functional is a function from a vector space into its underlying field.)

I claimed that I understood fractions because I have seen and can carry out and explain the construction of the rational numbers from nothing but the axioms of set theory. I said I see no difference between fractions and equivalence classes of pairs of integers and that I more or less manipulate fractions algorithmically, and that was good enough for me. At this point she became rather disgusted with me. :-)

-- PaulMiller - 11 Jan 2006

"She was trying to argue that a student needs some sort of non-mathematical metaphor for mathematical concepts in order to display "real" understanding."

This is neither necessary or sufficient. It might be "nice" for some kids to relate mathematical concepts to non-mathematical metaphors, but it depends on the metaphor. It wouldn't be a rigorous relationship and, by definition, would not imply "real" understanding. Math is math. Axioms are axioms. Metaphors are approximations.

Perhaps you can ask her what she means by "real" understanding. One could argue that being able to come up with a non-mathematical metaphor is one criteria of showing understanding, but that does not seem to be her proposition. Perhaps she needs a course in mathematical logic.

-- SteveH - 11 Jan 2006

I agree. In fact, there is some good reseach showing that conceptual understanding can come either before or after procedural understanding. What isn't clear to me so far is which is the optimal way to go about it, if there is such a thing.

The thing that really burnt me up about our little discussion was that if I can tear apart and put back together a mechanical device, I can generally claim that I understand how it works. But, it seemed like she was saying that the same is not valid for something like the real number system.

-- PaulMiller - 11 Jan 2006

"This is neither necessary or sufficient. It might be "nice" for some kids to relate mathematical concepts to non-mathematical metaphors, but it depends on the metaphor. It wouldn't be a rigorous relationship and, by definition, would not imply "real" understanding. "

I agree. I don't know about you guys but I've tutored people who have been able to state beautiful non-mathematical metaphors for a given mathematical problem and have shown no signs of conceptual understanding. Or procedural knowledge. They've just memorised the metaphor.

-- TracyW - 11 Jan 2006

"This is neither necessary or sufficient. It might be "nice" for some kids to relate mathematical concepts to non-mathematical metaphors, but it depends on the metaphor. It wouldn't be a rigorous relationship and, by definition, would not imply "real" understanding. "

I agree. I don't know about you guys but I've tutored people who have been able to state beautiful non-mathematical metaphors for a given mathematical problem and have shown no signs of conceptual understanding. Or procedural knowledge. They've just memorised the metaphor.

-- TracyW - 11 Jan 2006

"This is neither necessary or sufficient. It might be "nice" for some kids to relate mathematical concepts to non-mathematical metaphors, but it depends on the metaphor. It wouldn't be a rigorous relationship and, by definition, would not imply "real" understanding. "

I agree. I don't know about you guys but I've tutored people who have been able to state beautiful non-mathematical metaphors for a given mathematical problem and have shown no signs of conceptual understanding. Or procedural knowledge. They've just memorised the metaphor.

And personally I was on my second year of calculus before I realised why doing the derivation and setting it equal to 0 gave you the maximum or minimum. I knew the graph, and how the rate of change reached 0 at the peaks or lows and had reproduced them neatly on tests. I just hadn't made the conceptual jump yet and the whole setting to 0 regardless of whether you wanted a minimum or a maximum seemed magical.

-- TracyW - 11 Jan 2006

Carolyn

what worked? Inquiring Minds Want To Know! I'm still rooting for calling the balancing fraction "fred" and dividing d by a really weird number. We need some formative assessment done on all of this "backseat driving"!

-- BeckyC - 12 Jan 2006

I actually have not buckled down to this yet... I've been swamped at work (that's why I've been so scarce lately) and have been doing only 'reactive teaching'.

I will do an assessment tonight and give him some of these problems to do.

In response to the comments about cross-multiplication -- I generally don't want to teach just the cross-multiplication trick, at least not without having firmly gotten across the idea that you can do the same thing to both sides of an equation. The reason was that one sort of problem I would occasionally see with older students was a direct result of misunderstanding how cross multiplication worked.

Here's an example of the problem problem:

w/3 - 4/5 = 6/7.

The kids would want to solve for w by applying cross multiplication, so they would multiply w/3 by 5 and 4/5 by 3, to get:

5w - 12 = 6/7.

It's probably errors like these that get constructivists foaming at the mouth about traditionalists and their bag of rote tricks -- because kids don't necessarily understand all the tricks they learn.

Of course, simply refusing to teach them anything isn't a viable option...

-- CarolynJohnston - 12 Jan 2006

update

We had a long session this evening on exactly this topic. This is one that doesn't seem to stick very well. We spent a long time on the problem

d/1.2 = 15/9

with me emphasizing: isolate the variable by undoing the division operation, make sure you do the same thing to both sides.

And Ben saying: mom, sometimes you can just SEE that the answer is 2. (!)

And me saying, well Ben, I just want you to know what to do when it's the kind of problem you can't do that easily.

Boy, this topic is just not easy for him. I intuit (for what that's worth) that the best way to teach Ben is to give him steps to follow, with reasons for each step, and let him learn by repeating those steps (and having me repeat the words that go with them). It's just turning out to be a big lump for him to swallow. We'll have to cover this ground every night for a while.

-- CarolynJohnston - 12 Jan 2006

However, intuiting that the answer to this problem is 15 * 1.2/9 appears to be much easier for Ben than it is for anyone else.

-- CarolynJohnston - 12 Jan 2006

My feeling is maybe just leave it, if he does happily think in a mathematical way, and instead give him some more difficult problems.

2/x + 5 = 6/1.2

2/(x+5) = 6/1.2

2/(x+5) = 6/1.2 + x

If he's like me, you can make a lot of arguments, but he's not going to listen unless his own experience convinces him otherwise.

-- TracyW - 12 Jan 2006