tag:blogger.com,1999:blog-7691251033406320222.post8952330160450550658..comments2024-03-26T04:19:38.862-07:00Comments on kitchen table math, the sequel: multiplying fractionsCatherine Johnsonhttp://www.blogger.com/profile/03347093496361370174noreply@blogger.comBlogger38125tag:blogger.com,1999:blog-7691251033406320222.post-55134162190209652062007-01-18T14:16:00.000-08:002007-01-18T14:16:00.000-08:00"The point is, she needs practice in skills beyond..."The point is, she needs practice in skills beyond whatever level she's on..."<br /><br />Exactly the problem we ran into after about 9 months in KUMON Math. Meg made it from 3A to halfway through C, but we really wanted it for help in fractions. That's still a level and a half away, and it seems they've really slowed her down though she knows mult facts very well. The cost was also increasing. So we'll try to go back to Saxon 54 on our own, though KUMON did provide the motivation. KUMON has helped Meg in math this year, but she's in a special ed class now and they're covering stuff she knows easily. Next year in Jr High, sped uses SRA's CMC and can make the instruction much more individualized as to where Meg's gaps are.KathyIggyhttps://www.blogger.com/profile/11699911427332930599noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-63503435020026054752007-01-18T11:51:00.001-08:002007-01-18T11:51:00.001-08:00I may have to suck it up and haul Christopher back...I may have to suck it up and haul Christopher back to KUMON.<br /><br />He doesn't want to do it, and I don't really blame him; it's overkill <I>for him</I>.<br /><br />KUMON is an example of educational rigidity.<br /><br />The program is great, but they don't allow individuals to pay for it and then use it as they see fit.<br /><br />I have a friend who's been taking her child to KUMON for over a year. Her child never passes the end-of-level test because she does the problems too slowly. <br /><br />She's probably just being methodical; who knows.<br /><br />The point is, she needs practice in skills beyond whatever level she's on (subtraction? - I think she's having trouble moving from vertical problems to horizontal problems or vice versa).<br /><br />But she's (probably) stuck.<br /><br />Nevertheless, I'm probably going to have to tell Christopher we're doing KUMON over the summer. <br /><br />He's just getting nothing at school.Catherine Johnsonhttps://www.blogger.com/profile/03347093496361370174noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-65196227875456051042007-01-18T11:51:00.000-08:002007-01-18T11:51:00.000-08:00I may have to suck it up and haul Christopher back...I may have to suck it up and haul Christopher back to KUMON.<br /><br />He doesn't want to do it, and I don't really blame him; it's overkill <I>for him</I>.<br /><br />KUMON is an example of educational rigidity.<br /><br />The program is great, but they don't allow individuals to pay for it and then use it as they see fit.<br /><br />I have a friend who's been taking her child to KUMON for over a year. Her child never passes the end-of-level test because she does the problems too slowly. <br /><br />She's probably just being methodical; who knows.<br /><br />The point is, she needs practice in skills beyond whatever level she's on (subtraction? - I think she's having trouble moving from vertical problems to horizontal problems or vice versa).<br /><br />But she's (probably) stuck.<br /><br />Nevertheless, I'm probably going to have to tell Christopher we're doing KUMON over the summer. <br /><br />He's just getting nothing at school.Catherine Johnsonhttps://www.blogger.com/profile/03347093496361370174noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-32510796318555160832007-01-17T15:18:00.000-08:002007-01-17T15:18:00.000-08:00Barry--
I absolutely agree with that. My daughte...Barry--<br /><br />I absolutely agree with that. My daughter's third grade teacher fit Catherine's description of being young and fresh out of ed school. She "didn't know what she didn't know." After several requests for more practice, more repetitions, etc., we finally gave up and took our kid to KUMON. We felt that we didn't have the mental discipline to consistently "after school" her; we needed the external motivation of KUMON.<br /><br />However, I did continue to "preach my message" to that teacher; I think at some later point, some of what I was trying to convey to her finally sunk in. Of course, it was too late for my daughter's class, but I am optimistic that it helped future classes.Karen Ahttps://www.blogger.com/profile/06132085963945807124noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-35757145052517328202007-01-17T14:43:00.000-08:002007-01-17T14:43:00.000-08:00Karen,
Beating a dead horse is sometimes necessar...Karen,<br /><br />Beating a dead horse is sometimes necessary to get one's point across, I'm very sorry to say.Barry Garelickhttps://www.blogger.com/profile/01281266848110087415noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-40401456212304690292007-01-17T14:10:00.000-08:002007-01-17T14:10:00.000-08:00"I'm an example of "learned the principle/concept ..."I'm an example of "learned the principle/concept later"."<br /><br />My 12th grader made a similar comment the other day to the effect that she thought that procedural fluency with math helped her to eventually grasp the concepts. <br /><br />My 8th grader catches on to the concepts much quicker; she always has. But (am I beating a dead horse here?), we took her to KUMON for 2 1/2 years (3rd through 5th grade) so that she could become procedurally fluent.Karen Ahttps://www.blogger.com/profile/06132085963945807124noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-42964342318001246752007-01-17T13:35:00.000-08:002007-01-17T13:35:00.000-08:00I used to be quite haughty about the sanctity of m...<I>I used to be quite haughty about the sanctity of mathematical concepts until having to deal with a daughter who is ADD.</I><br /><br />Well this is the CRUX OF THE PROBLEM having only teachers in their 20s.<br /><br />None of them has been mugged by reality yet.Catherine Johnsonhttps://www.blogger.com/profile/03347093496361370174noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-2431218051016410902007-01-17T13:34:00.000-08:002007-01-17T13:34:00.000-08:00Saxon often does a simple word-description of what...Saxon often does a simple word-description of what's been done.<br /><br />Here's an example from Lesson 113 in Algebra 1.<br /><br /><B>The distance traversed by a car traveling at a constant speed is directly proportional to the time spent traveling. If the car goes 75 kilometers in 5 hours, how far will it go in 7 hours?</B>.<br /><br /><I>solution</I> We will use four steps.<br /><br />Step 1: <I>D</I> = <I>kT</I> write the equation<br /><br />Step 2: 75 = <I>k</I>(5) <I>k</I> = 5 solve for <I>k</I><br /><br />Step 3: <I>D</I> = 15<I>T</I> put <I>k</I> in the equation<br /><br />Step 4: <I>D</I> D = (15)(7) <B><I>T</I> = 200 <I>K</I></B> solve for <I>D</I><br /><br />I bet it would help Christopher just to write these word labels. <br /><br />I think it might help him keep track of where he is in the computation.<br /><br />I'm just this year starting to be able to get him to write each step directly below the preceding step.Catherine Johnsonhttps://www.blogger.com/profile/03347093496361370174noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-84834688053792710162007-01-17T13:24:00.000-08:002007-01-17T13:24:00.000-08:00Steve -
wow!
I'm so word-oriented myself I hadn...<B>Steve</B> - <br /><br />wow!<br /><br />I'm so word-oriented myself I hadn't even thought of having Christopher write a/a = 1<br /><br />I was going to have him write "multiplicative identity."<br /><br />In fact, at this point, it would be MUCH more useful for him, in terms of procedural <I>and</I> conceptual learning, to write a/a=1.<br /><br />One possible problem is the fact that I don't think he's been shown many a/a = 1 type principles. I'm afraid he's mainly learned these things, when he's learned them, in words.<br /><br />More learning sacrificed to SPEED SPIRALING.<br /><br />You're absolutely right that constructivists don't want kids writing a/a = 1.<br /><br />They don't even want kids writing multiplicative identity, I don't think.Catherine Johnsonhttps://www.blogger.com/profile/03347093496361370174noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-30351722226159632742007-01-17T12:09:00.000-08:002007-01-17T12:09:00.000-08:00"Would constructivist educators say that writing o..."Would constructivist educators say that writing out your explanation is the same as justifying each step?"<br /><br />This is not the same as writing down the mathematical rule, like a/a = 1. They want students to write it down with words that somehow convey a common-sense, or gut-level understanding of what is going on. Of course, there is nothing more common-sensical and accurate than a/a = 1, but kids don't discover that. They would prefer something like:<br /><br />5*6/5 = 6<br /><br />"Well, like you can separate the 5's so that, like, you get 5 divided by 5. If 5 is the whole and you have five of the thing, then that is like having the whole pie and that is equal to one. Then, like, if you have one of the 6, then that is just 6."SteveHhttps://www.blogger.com/profile/03956560674752399562noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-56384421877543293022007-01-17T12:00:00.000-08:002007-01-17T12:00:00.000-08:00I don't remember being given an explanation of why...I don't remember being given an explanation of why one could cancel when multiplying fractions (in 6th grade) but I liked doing it, and that was all that mattered then.<br /><br />When I had algebra, teachers got down on us for calling it cancelling, telling us it was "dividing out", which I suppose is more descriptive. In any event, algebra made a lot of things about arithmetic a lot more obvious. I'm an example of "learned the principle/concept later". <br /><br />Perhaps this is frowned upon, but if learning the concept interferes with learning the procedure, then something's gotta give. That's sort of Saxon's philosophy. I used to be quite haughty about the sanctity of mathematical concepts until having to deal with a daughter who is ADD.Barry Garelickhttps://www.blogger.com/profile/01281266848110087415noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-42870899558751972802007-01-17T11:54:00.000-08:002007-01-17T11:54:00.000-08:00"...unless the child just so happens also to inven..."...unless the child just so happens also to invent the phrase "multiplicative identity."<br /><br />Double LOL!Instructivisthttps://www.blogger.com/profile/01652458042291988959noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-19613182488284272362007-01-17T11:45:00.000-08:002007-01-17T11:45:00.000-08:00lynng is right. The rule could be: "The sooner you...lynng is right. The rule could be: "The sooner you can simplify or reduce a fraction, the better.", although I could probably come up with an example where that wasn't true.<br /><br />How about:<br /><br />3 4/7 * 2 7/10 = ?<br /><br />25/7 * 27/10 = ?<br /><br />each can't be simplified, but<br /><br />25*27/7*10 = ?<br /><br />can now be factored to get<br /><br />(5*5*3*9)/(7*5*2) = ?<br /><br />or<br /><br />My high school teacher made a point that we really should write it as<br /><br />5/5 * (5*3*9)/(7*2) = ?<br /><br />then<br /><br />1 * (5*3*9)/(7*2) = <br /><br />to get<br /><br />(5*3*9)/(7*2) = ?<br /><br />135/14<br /><br />In any case, kids should know that simplifying isn't necessary. Do it at the end or don't do it at all, unless the problem states that it has to be in its simplest form.<br /><br />That reminds me of all of the algebra problems that said: "Simplify this".<br /><br />Sometimes it wasn't clear what that meant. Do we expand or do we factor or do we ask another student what they are doing?SteveHhttps://www.blogger.com/profile/03956560674752399562noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-41198320249012671192007-01-17T11:43:00.000-08:002007-01-17T11:43:00.000-08:00smackdown!
I have to see this!
but first I have ...smackdown!<br /><br />I have to see this!<br /><br />but first I have to take a walk in freezing-cold 20-degree weather!Catherine Johnsonhttps://www.blogger.com/profile/03347093496361370174noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-91124826165440856542007-01-17T11:41:00.001-08:002007-01-17T11:41:00.001-08:00Actually I just made up the problem because I was ...Actually I just made up the problem because I was at work. My goal was to just come up with some numbers that could be factored and cancelled out.<br /><br />I should of posted last night when I had the work in front of me, but I was too busy performing a smackdown of one of my commentors over at parentalcation.TurbineGuyhttps://www.blogger.com/profile/09781298806992944235noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-31996129034106919682007-01-17T11:41:00.000-08:002007-01-17T11:41:00.000-08:00"Shortcut Math" is, if you don't mind my saying so..."Shortcut Math" is, if you don't mind my saying so, probably the right term for the kind of math education to which both the NCTM <I>and</I> mathematicians object.Catherine Johnsonhttps://www.blogger.com/profile/03347093496361370174noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-9366157377385924562007-01-17T11:39:00.001-08:002007-01-17T11:39:00.001-08:00RUSSIAN MATH also asks students, fairly often, to ...RUSSIAN MATH also asks students, fairly often, to write out each step in a computation and write the property that justifies the step on the same line.Catherine Johnsonhttps://www.blogger.com/profile/03347093496361370174noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-55487503518558653792007-01-17T11:39:00.000-08:002007-01-17T11:39:00.000-08:00Would constructivist educators say that writing ou...<I>Would constructivist educators say that writing out your explanation is the same as justifying each step? Are you saying they focus too much on the writing part?</I><br /><br />My sense is that constructivist educators - I could be wrong here - want a fair amount of what they think of as "originality."<br /><br />They really do seem to seek "invented" or "discovered" strategies that, of course, won't have names like "multiplicative identity" unless the child just so happens also to invent the phrase "multiplicative identity."<br /><br />When I look at examples of children's explanations of how they solved a problem on the web, I don't see simple lists of properties used to justify each step.<br /><br />Does EM teach the commutative property as the commutative property?<br /><br />I should look to see if TRAILBLAZERS does this.<br /><br />Saxon Math teaches the properties and their names very early on.Catherine Johnsonhttps://www.blogger.com/profile/03347093496361370174noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-88757560537740430772007-01-17T11:33:00.000-08:002007-01-17T11:33:00.000-08:00While Singapore stresses the ability to quickly do...<I>While Singapore stresses the ability to quickly do computations, in general this doesn't happen until the child is proficient at the justification which he must do the long way.<br /><br />The heart of the matter is the multiplicative identity and it's far more important that the kid understand the reasoning behind canceling out AND be able to apply it rather than whether he does it sooner or later in a particular computation.</I><br /><br /><br />Perfect.<br /><br />This is the heart of the matter for Christopher's class.<br /><br />I shouldn't even call it "procedural" teaching.<br /><br />It's really shortcuts.<br /><br />3 years of Accelerated ShortcutsCatherine Johnsonhttps://www.blogger.com/profile/03347093496361370174noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-75283953615793746912007-01-17T11:31:00.000-08:002007-01-17T11:31:00.000-08:00For which grade level is that?
7th grade "acceler...<B>For which grade level is that?</B><br /><br />7th grade "accelerated" math<br /><br />The course is so hideously jumbled these kids are going to end up being behind<br /><br />A mom who dropped her older child out of this course, which has apparently always been conceived as a "killer" course, told me the kids who dropped down sometimes ended up with higher scores than the kids who stayed & struggled on<br /><br />The Phase 4 math course in Irvington is probably the worst course I've seen in my life no matter who's teaching itCatherine Johnsonhttps://www.blogger.com/profile/03347093496361370174noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-74818448647912072892007-01-17T11:29:00.000-08:002007-01-17T11:29:00.000-08:00sheesh, just lost my comment
sighsheesh, just lost my comment<br /><br />sighCatherine Johnsonhttps://www.blogger.com/profile/03347093496361370174noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-31237214134143298822007-01-17T11:28:00.000-08:002007-01-17T11:28:00.000-08:00Why was the problem written
1 5/10 x 2 3/6
why ...Why was the problem written <br /><br />1 5/10 x 2 3/6<br /><br />why not reduce first?<br /><br />1 1/2 x 2 1/2 or as an improper fraction:<br /><br />3/2 x 5/2 or 15/4<br /><br />strange that the problem was written that way.LynnGhttps://www.blogger.com/profile/11467061079495021347noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-68132004439154466402007-01-17T11:23:00.000-08:002007-01-17T11:23:00.000-08:00"I think, was from Steve who talked about a teache..."I think, was from Steve who talked about a teacher he had in high school who had students justify every step in every calculation."<br /><br />Yep. He didn't even like the idea of canceling, because students used it in the wrong places. If they saw the same thing in the numerator and denominator, the kids would cancel them, whether or not they were factors of the fraction. Kids would cancel the 5's in the following expression<br /><br />(x+5)/5<br /><br /><br />He drilled into our heads that a/a = 1 is not canceling. Actually, there is no such thing as canceling. I think he was the one who said that there is no such thing as cross-multiplication.<br /><br />The answer to the question is that students have to be able to show that multiplying mixed fractions works out any which way you do it.<br /><br /><br />5 3/10 x 3 5/6 = ?<br /><br />I would recommend that they first change it to improper fractions:<br /><br />53/10 * 23/6 = ?<br /><br />My son likes the shortcut of multiplying the denominator times the whole number and adding in the numerator. All of this goes over the denominator. He doesn't like it when I force him to explain exactly why he can do this.<br /><br />You could have kids show that they get the same answer if they multiply the following:<br /><br />(5 + 3/10)(3 + 5/6)<br /><br /><br />Then, I would look to see if these fractions can be reduced. If you want to factor the numbers in the numerators and denominators, that's fine, but students should practice this enough to reduce or simplify any fraction in their heads.<br /><br />You can't do that in this case, so you could multiply them next, like this:<br /><br />(53*23)/(10*6)<br /><br />Then see if there are any common terms to cancel - oops, I used that word. Intermediate reducing or canceling is always helpful. What works to simplify the problem one time may not work the next time.<br /><br />This is not like long division, where there is one method that is powerful and efficient. If you want kids to be ready for algebra, you have to focus on the basic rules, like a*1 = a, and a/1 = a. I remember thinking that these were stupid and trivial rules until I saw how they worked on complex expressions. If you really understand the rules and how to apply them, then combining and simplifying expressions like this:<br /><br />a*x^3/(x-2) + 55 - 5*(x-2)^3<br /><br />is easy.SteveHhttps://www.blogger.com/profile/03956560674752399562noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-51073881098171854382007-01-17T11:07:00.000-08:002007-01-17T11:07:00.000-08:00I f**ked up. Post corrected. Principle is the sa...I f**ked up. Post corrected. Principle is the same.<br /><br />Rory <-- red facedTurbineGuyhttps://www.blogger.com/profile/09781298806992944235noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-74591351500754029232007-01-17T10:59:00.000-08:002007-01-17T10:59:00.000-08:00Catherine,
About the “justify each step” advice?
...Catherine,<br /><br />About the “justify each step” advice?<br /><br />Would constructivist educators say that writing out your explanation is the same as justifying each step? Are you saying they focus too much on the <b>writing</b> part?Texhttps://www.blogger.com/profile/12015376722906862434noreply@blogger.com