KTM User Pages
20 Jan 2006 - 16:22
I LOL'd when I read one of Carolyn's patented dry observations on the follies of 21st century math instruction:
Apparently you can understand the heck out of pan balances, and still have trouble with understanding and manipulating equations.
I distinctly recall being charmed the first time I saw a pan balance in Algebra to Go. And of course I loved Carolyn's pan balance drawings:
I also had a lot of fun playing with the pan balance problems in the National Library of Virtual Manipulatives.
But I continued to experience a disconnect between pan balances and 'isolate the variable,' or 'do the same thing to both sides,' until I finally did Investigation 7 in Saxon 8/7: "Balanced Equations."
back from fun-filled Con Ed hiatus We had an ice storm Saturday night, then a wind storm Wednesday morning, and there are so many trees down all over Westchester it's like Hurricane Katrina without the water. Also without the trilliions of dollars in property damage, the loss of life, the breakdown of civil order, the helicopters, Wolf Blitzer, and the international expressions of shock and opprobrium. Apart from that, it's exactly like Hurricane Katrina. Anyway, the electricity went off at noon; the garage door is electric; the car was in the garage; the road to town was blocked; the side roads were blocked; when the electricity went back on the internet connection didn't; and so on. All in all, about what you'd expect.
where was I? Something about a pan balance. Right. I have no idea what I was planning to say about pan balances....apart from the fact that -- it's coming back to me now -- John Saxon can write a Pan Balance lesson like nobody's business. The reason John Saxon can write a pan balance lesson like nobody's business is that he doesn't just slap down a drawing of a pan balance and expect the student to see the light. Instead, he carefully develops his pan balance analogy, presenting the student with a sequence of 3 or 4 drawings of pan balances, one after the other, each one representing a step in the solution of an equation. And he explains the whole thing in words. Words, pictures, numbers, and variables. Kit and caboodle. Here he is:
Equations are sometimes called balanced equations [ed.: wonderful!] because the two sides of the equation "balance" each other. A balance scale can be used as a model of an equation. We replace the equal sign with a balanced scale. The left and right sides of the equation are placed on the left and right trays of the balance. For example, x +12 = 33 becomes
Click. This is perfect. Instead of plopping a pan balance down in the middle of the page and expecting the student to discover its meaning, Saxon explains what the image means, and why it works. Then he takes you through the steps which can only be implicit in a static drawing of a lone pan balance. Then he has you draw your own pan balances. I'm sick Christopher isn't using Saxon. I'm so sick he isn't using Saxon, that I may try to squeeze Saxon back into our 'schedule.'
Saxon - Prentice-Hall smackdown Part 2 I've mentioned Christopher seems to be not only not gaining new knowledge, but to be losing the knowledge he already had. Here's why.
The Saxon pan balance 'Investigation' opens with addition & subtraction equations. Then the same Saxon Investigation proceeds to multiplication and division equations, reminding students in passing that multiplication and addition are related. Prentice-Hall splits all of this up into separate lessons, and never the twain shall meet. Addition and subtraction go together. Multiplication and division go together. Integers go together. Decimals go together. Fractions go together. They're all in their separate lesson-boxes. If the student doesn't make the connection, the connection doesn't get made. I see why David Klein says all American textbooks are constructivist. Technically, Prentice-Hall is a traditional book. But nothing is explained, beyond the bare minimum. It's like a website with a lot of info to sort through (David has made that observation before), or a reference book with problem sets. I don't know why they don't just buy these kids a Dictionary of Mathematics and let it go at that. There's a bunch of them out there.
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I really have to write up my Rubik's cube observation -- which pretty much comes to the same conclusion. You may think conceptual understanding at an abstract level is enough to solve something. It's not. You actually have to DO the actual solving solving part to be able to actually solve it. This is the essence of procedural fluency and it is critical. Conceptual understanding is not procedural fluency and is a por substitute for it. I'm coming around to the conclusion that it's nice to have both (ideally) but if I had to sacrifice one ofr teh other I'd sacrifice conceptual understanding -- at least in its directly taught incarnation. Once you have procedural fluency, you will (finally) get (the right kind) of conceptual understanding. And, this conceptiual understanding is not necessarily the same as the conceptual understanding that you get when it is taught directly. For example, doing pan balancing problems may gainyou some conceptual understanding about solving equations. Do enough of them and you'll gain procedural fluency -- solving pan balancing problems, not procedural fluency solving algebraic equations. To gain procedural fluency solving algebraic equations you need to (wait for it) solve a whole lot of algebraic equations. The conceptual understanding you get from solving pan abalancing problems may speed this process along, but it is no substitute. Plus, you've lost a lot of time teaching pan balancing which could have been spent solving algebraic equations. There is an opportunity cost associated with learning pan balancing. I acme to this raalization while trying to solve my son's Rubik's cube he received for Christmas based on the simple seven step insert. All teh conceptual knowledge demonstrated in that insert is virtually useless until you've developed the procedural fluency you gain by manipulating the cube. You can only get this by, well, manipulating the cube. At this point you've fred up your working memory (hello magic number seven) to be able to use that conceptual knowledge you learned. Until, it's as good as useless. After, you've gained procedural fluency, you start gaining insight on how the cube works. It is this conceptual understanding that is the ultimate goal and it is the same conceptual undestanding you get after solving a bazillion math problems. It's the click -- the a-ha or eureka moment -- so many commentors have written about. -- KDeRosa - 20 Jan 2006
Showing the pan balance that matches an equation??? That's backwards. Pan balances are not the important thing here. What if the equation is non-linear? Do the kids think that pan balances work for all equations? Do they think that pan balances represent algebra? UGH! Pan balances are very simplistic. They should be dropped as soon as possible. Maybe you can contort the analogy to fit almost any problem, but you have to have a good reason to do so. -- SteveH - 20 Jan 2006
I agree. Forget the pan balance analogy. What about the following approach? I'll apply it to 2x=132.
Oops, I meant to number the steps 1., 2., 3., 4. -- SusanJ - 20 Jan 2006
Hey, I like that! -- CarolynJohnston - 20 Jan 2006
" ... my son's Rubik's cube ..." My son got a Rubik's cube too, but I was the one who read the instructions to figure it out. It doesn't tell you exactly how to solve it; just some of the basic moves. You have to put them together to solve it. I already knew that there were certain predictable ways to move some of the sides around while keeping the rest the same. I felt like I cheated because I always thought that the idea was to figure out on your own what the moves were. Since I don't like puzzles like Rubik's cube, but I am curious, I read the instructions. Solving the cube was then tedious, but didn't mean much. If I then continued playing with the moves and studying the different patterns (I won't), I might develop a more conceptual understanding, but that is not my goal. I feel the same way about Sudoku. Tedious, but not very meaningful, especially for math. -- SteveH - 20 Jan 2006
"I agree. Forget the pan balance analogy." I agree. Forget the analogy. I guess that is my point. The analogy is not what is important. Another comment: "You may multiply each side by the same number." I like to be more specific and say that you can multiply each term on each side of the equals sign by the same number. This is because I have had students multiply only one term. They don't "see" the imaginary parentheses around each side of the equals sign. -- SteveH - 20 Jan 2006
One of the problems in learning algebra is the move from simple explanations and rules to more complex ones. One of the things I did was to have students break apart complicated equations. The first thing I would do is to have them circle all of the terms (I used to call them "chunks"). These are the (perhaps complex) rational terms that are separated by the plus and minus signs. (Not the plusses and minuses that are in parentheses.) 3(x-5)^3/4xyz + 3a^3/x The plus sign separates the two terms or chunks. I would tell them that these terms are just like a simple number. you can "move" them to the other side of the equals sign if you change their signs. This is the same thing as adding or subtracting the same thing from both sides. Some students feel comfortable moving constants or single variables from one side to another, but not for larger terms. I would tell them that you can move any sort of complex term to the other side of the equation just like a simple constant. Next, I would have them take each rational term and circle each multiplying or dividing factor. I tell them that there are things you can do with each factor without changing the term. For this term, the circled parts (in parentheses) 3(x-5)^3/4xyz becomes [(3)((x-5)^3)]/[(4)(x)(y)(z)] The brackets enclose the numerator and denominator. One of the things you can do is change the order of any of the factors in the numerator or denominator, like (x-5)^3(3)/z4yx (A times B = B times A) You just might want to do that. A term is not some immutable chunk that is glued together. You can also split the factors in the term into two or more separate rational terms that are multiplied together. 3/4 * (x-5)^3 * 1/xyz The '*' is to emphasize the multiplication. Students have to be comfortable with these kinds of manipulations because they all follow from the basic rules of algebra. Many students have trouble moving from the simple idea or understanding of a rule to a more complex one. That is why I felt it took me until Algebra II or Trig as a Junior before I felt that I really could do algebra. The other thing I told students was that you could take any complex term and turn it into a rational term by putting a 1 in the denominator. This idea might seem simple, but it has been a big help to me over the years. Along with this is the idea that all factors (what I call factors of a rational term) have an exponent. If you don't see an exponent, it is 1. Then, I tell them that you can take any of these factors and move it above or below the dividing line. The (x-5)^3 factor could become 1/(x-5)^-3 All you have to do is to change the sign of the exponent. If you don't see a dividing line or an exponent, use the assumed ones. Therefore 5 is the same as 5/1 = 5^1/1 = 1/5^-1 Why do you want to do that? I don't know. A lot of algebra is about manipulating equations one way or another. I remember so many homework assignments that said something like "simplify this" and of course you don't have much of a clue, but you start moving things factoring things, doing anything (!) to see what you get. (Actually, they were quite annoying.) I'll bet others remember the "simplify" problems. You never quite knew whether to expand and consolidate terms or to factor. Of course, the goal was to master algebra at a non-trivial level. If you know about terms and factors, you are less likely to make the following mistake (x-5)/x equals -5 because the two x variables cancel. Actually, I don't like "cancel". Advanced students might "see" the common factors in a rational expression and quickly cancel them, but beginning students need to be much more careful. 3x(x-5)^3/2y(x-5) simplify. First, what are the factors: [(3) (x) ((x-5)^3)] / [(2) (y) (x-5)] Note that in the numerator, the factor is (x-5)^3. It includes the exponent. The (x-5) in the denominator has the assumed exponent of 1. You cannot cancel the (x-5) in the numerator and denominator. You have to apply the exponent rules. They "combine" to leave (x-5)^2 in the numerator. Ideally, you should separate the term to get: 3x/2y * (x-5)^3/(x-5)^1 After a little practice "seeing" the factors, you can do this in your head. There is a long road to proficiency in algebra and there is no other way to get there than with a good book (curriculum), a teacher who knows the material, and lots of practice. -- SteveH - 20 Jan 2006
Hey Susan -- At the top right of the page there's a button called "Edit this page" -- I use it to fix my comments whenever I hit "add comment" without enough proofreading... -- StephanieO - 20 Jan 2006
Stephanie, thanks for the tip. Unfortunately in this case I didn't actually number the items wrong but the wiki-bliki-wacko formatting changed my 1, 2, 3, 4 all to ones. I'd rather use HTML! -- SusanJ - 20 Jan 2006
See TestOfBlikiFeatures -- you can't put a blank line between them. -- GoogleMaster - 20 Jan 2006
Hi SusanJ -- you can use HTML. See the text formatting rules page. -- CarolynJohnston - 20 Jan 2006
2x=132 When teaching how to isolate the variable, I found that not all students recognize that 2x means 2 times x. Understanding this is necessary to come up with the opposite operation. -- CharlesH - 20 Jan 2006
Charles makes a good point. MathML? has terminology for this: "invisible times." Proper terminology is useful and I like what Steve wrote. However, it may be hard for students to realize that ordinary nouns like "term" and "factor" are actual technical terms. Also, "factor" can be used as a verb. -- SusanJ - 20 Jan 2006
I started using the Gambill Method this year. Unfortunately, the textbook I use does not have the answers in the back of the book. I have created solution pages and I put them on-line for the kids to look up. I have attached the one of the files. I am looking for suggestions. -- SmartestTractor - 21 Jan 2006
Smartest, do you teach math? Or are you homeschooling? How is it I never realized this? I'll get back to you with feedback. -- CarolynJohnston - 21 Jan 2006
I teach grade eight English, math, history, geography, physical education, health, visual arts, and science in a JK to 8 school. -- SmartestTractor - 21 Jan 2006
You may think conceptual understanding at an abstract level is enough to solve something. It's not. You actually have to DO the actual solving solving part to be able to actually solve it. This is the essence of procedural fluency and it is critical. Conceptual understanding is not procedural fluency and is a por substitute for it. Plus I'm finding that conceptual understanding is MUCH MUCH easier than procedural competency, let alone procedural fluency. -- CatherineJohnson - 21 Jan 2006
Smartest Tractor You MUST let us know how it goes with Gambill. I'm COMPLETELY sold on choral response now (due to attentional issues) — are you using the scripted question-and-answer approach, too? -- CatherineJohnson - 21 Jan 2006
Once you have procedural fluency, you will (finally) get (the right kind) of conceptual understanding. This, I don't agree with. I've had procedural fluency in all kinds of arithmetic, without conceptual understanding. I've come to think that the proper 'manipulative' is the word problem. Without those - without applications - you can do procedures til the cows come in. You don't get the conceptual understanding. -- CatherineJohnson - 21 Jan 2006
Pan balances are very simplistic. They should be dropped as soon as possible. Maybe you can contort the analogy to fit almost any problem, but you have to have a good reason to do so. Saxon uses the pan balance lesson brilliantly, I think; then he drops it. I think when you're first learning something that has the problem Wickelgren talks about - it looks exactly like all the other stuff you've learned - a visual image like a pan balance may function as a mnemonic device. I now have, in my head, the image of an equation becoming 'unbalanced' - one pan dropping down & the other flying up. It underlines the concept. I'll teach this lesson to Christopher. -- CatherineJohnson - 21 Jan 2006
I threw our Rubik's cube out. -- CatherineJohnson - 21 Jan 2006
oh, btw, I think I realized why Carolyn may be having trouble with Ben doing problems like X/5 = 20/30. Asking him to write this out in steps is getting way ahead of the game. You have to start with equations like this pan balance equation first. The whole idea of 'doing the same thing to both sides' and 'isolating the variable' seems to be extremely hard for kids - especially if they've been taught NOTHING to mastery. Christopher doesn't see that 'doing the same thing to both sides' is the same thing as writing: 5 = 5 2 x 5 = 2 x 5 He would definitely solve a ratio problem by seeing it as an equivalent fraction problem & then just 'knowing' the factor in his head. Factors seems almost to come 'naturally' to him. -- CatherineJohnson - 21 Jan 2006
Smartest Tractor has your writing program turned up??? and: what do you think of Killgallon's book (if you've had a chance to look at it - )
-- CatherineJohnson - 21 Jan 2006
Catherine, Step Up to Writing should arrive by the end of January. I ordered it on December 5. I called the Canadian dealer earlier this week and, to make a long story short, it is on its way from Sopris West, to them, and then repackaged and sent to me. I ordered Killgallon's book the other day. It hasn't arrived in the mailbox. The Gambill Method has been rather interesting, and effective, strategy in my classroom. Needless to say, the kids have never been exposed to the idea. I really messed up the other day when I tried to combine two ideas, like terms and the distributive property, into one day. We went over Thursday's lesson again (Lesson Nine - Solving Equations in More Than One Step). The current results for the unit. -- SmartestTractor - 21 Jan 2006
Hey -- those are box-and-whisker plots! What do the dots mean -- are those outliers? -- CarolynJohnston - 21 Jan 2006
outliers Yes. -- SmartestTractor - 22 Jan 2006
Charles & everyone When teaching how to isolate the variable, I found that not all students recognize that 2x means 2 times x. Understanding this is necessary to come up with the opposite operation. I have a question about this. I'm pretty sure Steve has mentioned the same 'issue' (Carolyn's favorite word).... Do you have a special approach to making sure kids understand this? btw, someone (Tracy maybe?) advised putting all terms inside parens, and I noticed just this week that Saxon does that. Saxon not only puts all the terms inside parens, he writes a + sign for the positive numbers, like so: (+2) (-3) It's very helpful. I wish I'd had Christopher doing that from the start. -- CatherineJohnson - 22 Jan 2006
Or he'll write: (+2) + (-3) + (-6) + (+2) = X -- CatherineJohnson - 22 Jan 2006
Steve One of the problems in learning algebra is the move from simple explanations and rules to more complex ones. One of the things I did was to have students break apart complicated equations. wow! I like that idea. This is like sentence diagramming (a bit), only for math. That's a terrific idea; I don't think I've seen it done. -- CatherineJohnson - 22 Jan 2006
Steve I'm going to try to STEAL the time to put this in Equation Editor, so when I post it I hope you'll copy edit to make sure I transcribed correctly -- -- CatherineJohnson - 22 Jan 2006
Smartest Tractor I am looking for suggestions. meaning? -- CatherineJohnson - 22 Jan 2006
I don't like all those parentheses. The whole idea of math notation is that it is compact. It says what it means and means what it says. I'd rather have the kids practice turning math notation into grammatical sentences to ensure that they understand the standard notation. 2x = x - 3 Two times x equals x minus three. Or, two multiplied by x equals x minus three. Of course, now I'm worrying if the kids can tell the difference between a times sign that resembles an x and the letter x that stands for an unknown. -- SusanJ - 22 Jan 2006
now I'm worrying if the kids can tell the difference between a times sign that resembles an x and the letter x that stands for an unknown you can count on them not knowing it! The whole idea of math notation is that it is compact. It says what it means and means what it says. Those two ideas aren't the same..... When you use extra parentheses, the expressions and equations are less compact. But they say the same thing. I'm now insisting that Christopher use parens around negative numbers. Otherwise the minus signs disappear. If this course were set up so that there was practicing to mastery on any skills ever I might feel differently. But we have zero practice. -- CatherineJohnson - 22 Jan 2006
What about using a colored highlighter for the minus sign or the negative number? I understand your goal but my concern is there are going to be too many parentheses if you continue to add extra, non-essential parentheses when he gets to problems where parentheses are essential such as expanding (a-b)^2 or factoring a^2-ab-2b^2. -- SusanJ - 23 Jan 2006
Meaning? Is it clear? Easy to follow? Too cluttered? Should the solutions have a written explanation beside each line of the equation? As a parent, would this be useful or useless when trying to help your child? -- SmartestTractor - 23 Jan 2006
"This is like sentence diagramming (a bit), only for math." Exactly. Since one of the courses I taught was college algebra (for credit, although algebra is, by definition, remedial in college), I was (hopefully) not starting from square one. Square one is where you worry about whether students know that 2x is 2 times x. (I mentioned before that in eighth grade algebra, I thought that maybe X could be two different numbers at the same time.) That's why I could focus on the things I talked about in my previous post. I wanted the students to look at an expression or equation and "see" all of the things that could be done, whether it was expanding or factoring. They would have to look at an expression and break it into it's component pieces - circle the terms and isolate the factors. I taught them that these are the basic building and manipulating blocks of algebra. One of the purposes of my post was to look at how the (necessarily) simple techniques used to teach introductory courses affect a student's progress towards a deeper, more complex understanding. That is why I think it is important to emphasize the basic rules right away, like A/1 = A and its converse A = A/1 and A/A = 1 or 1 = A/A and A*B = B*A I remember seeing these early on and thinking how trivial they were. It was only much later that I found out how they justified all sorts of algebraic manipulations. It's easy to see that 3*5 = 5*3 but, perhaps not so easy to see that 3*x^7*y^-4 = y^-4*3*x^7 = (y^-4)*(3)*(x^7) This is very easy to see if you can separate the factors and you really understand the basic rules. If students are forced to justify each step of an algebraic manipulation with a basic rule, then they will really understand why everything works, rather than just think of simple ideas like "canceling". And, 3x^2(x-5) = 3x^3 - 15x^2 should be as easy as 3(x-5) = 3x-15 -- SteveH - 23 Jan 2006
Susan What about using a colored highlighter for the minus sign or the negative number? Hey! That's a fantastic idea! I have NEVER seen it used..... One thing I've noticed about math-teaching since I was a kid is that they've come up with all kinds of useful 'tools' like that....for instance, they have a much more precise way of teaching children to write out borrowing, carrying, etc. But I haven't seen anyone use color to make the negative sign 'pop.' Christopher desperately needs that. He has a HORRIBLE time seeing the negative sign. Great idea. -- CatherineJohnson - 23 Jan 2006
Smartest Tractor. Is it clear? Easy to follow? Too cluttered? Should the solutions have a written explanation beside each line of the equation? As a parent, would this be useful or useless when trying to help your child? oh, great! I'll take a look again — -- CatherineJohnson - 23 Jan 2006
I think it's FANTASTIC. This is EXACTLY what I need as a parent — and what Christopher needs as a student. He needs to see the solution as well as the answer, at least at this stage. It's beautiful! -- CatherineJohnson - 23 Jan 2006
Since one of the courses I taught was college algebra (for credit, although algebra is, by definition, remedial in college), I was (hopefully) not starting from square one. Square one is where you worry about whether students know that 2x is 2 times x. WIT AND WISDOM (I'm using key words so I can find these things again & get them into WIT AND WISDOM.) WIT AND WISDOM is shaping up to be my favorite page in the blooki.... -- CatherineJohnson - 23 Jan 2006
Steve your posts are fantastic I'm going to find time to rewrite them in Equation Editor & get them 'up front.' -- CatherineJohnson - 23 Jan 2006
!SmartestTractor Warning -- In the attached pdf there is an error in problem 32 -- kids need to multiply by 2, not -2. The format with the red symbols is very cool! How is it working -- does it work well for the kids? -- CarolynJohnston - 20 Feb 2006