tag:blogger.com,1999:blog-8231784566931768362.post5256897693821128223..comments2014-04-20T14:44:15.114-04:00Comments on MathNotations: There are twice as many girls as boys: 2G = B or G = 2B?Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comBlogger14125tag:blogger.com,1999:blog-8231784566931768362.post-64070395670694424802013-08-02T13:57:40.143-04:002013-08-02T13:57:40.143-04:00Amazing! Almost 5 yes after publishing this and it...Amazing! Almost 5 yes after publishing this and it still generates a reply.<br /><br />I feel your pain but the truth is that everyone experiences confusion with these phrases regardless of whether or not English is your first language. You're not alone!Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-89686932815457270552013-07-30T19:26:47.923-04:002013-07-30T19:26:47.923-04:00I am a math PhD in Canada and my mother tongue is ...I am a math PhD in Canada and my mother tongue is Chinese. So far my only trouble understanding English expressions about mathematics is this "twice as many" stuff - have to look up the internet every time. Also I know that many of my friends have this same problem.P&Phttp://www.blogger.com/profile/16157601830907796168noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-71099193939588703482009-09-24T12:08:27.709-04:002009-09-24T12:08:27.709-04:00This was a helpful reminder for my students that s...This was a helpful reminder for my students that struggle with writing equations based on the wording...especially the subracting phrases...they tend to write them backwards.<br />Thanks!kmopcolehttp://www.blogger.com/profile/09597428367554417990noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-42736342389814684472009-03-25T09:24:00.000-04:002009-03-25T09:24:00.000-04:00Eric--Congratulations on getting through the first...Eric--<BR/>Congratulations on getting through the first 75% of your 1st year!<BR/><BR/>I'm retired from the classroom at this time, although I still teach SAT classes. My first year teaching was back in '70-71 at Queens College, I went on to teach secondary math for the next 37 years. Experience helps so definitely connect with veteran teachers and ask lots of questions -- I'm sure you already do! I'm not an expert on NY state curriculum but I know there are websites which provide samples of released Regents. My compatriot, Jonathan, who blogs over at jd2718 (just Google that!) is far more knowledgeable about NYS and the integrated algebra curriculum.<BR/><BR/>I would also recommend going to the Achieve/ADP websites. The links are in the sidebar of my blog. There will soon be a link to a full practice test for first year algebra which will provide additional practice for your students. I'm guessing "integrated" means a sprinkling of discrete math topics in the algebra, perhaps with some geometry added in, or am I way off?Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-10014159345909533302009-03-25T07:51:00.000-04:002009-03-25T07:51:00.000-04:00Thank you for your reply and help. I definitely h...Thank you for your reply and help. I definitely have to go over some of those other phrases with my students and the mnemonic will help. You seem like you have a lot of experience. This is my first year as a teacher and I am teaching 9th grade Integrated Algebra in NYC. Do you have any pointers for me? I will be starting to review for the regents exam in about a month and a half... any suggestions. What do you teach? Thanks for all your help.Eric Pflughttp://www.blogger.com/profile/04418307215619640547noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-9616866338277447922009-03-21T13:34:00.000-04:002009-03-21T13:34:00.000-04:00Eric--Strange that this post from some time ago st...Eric--<BR/>Strange that this post from some time ago still generates many visits and occasional comments. I think it's because those of us who have experienced the student confusion resulting from these kinds of convoluted or inverted verbal expressions empathize with each other! Sometimes we simply teach students survival skills as you have done: "When you see this phrase, always ....." <BR/><BR/>Subtraction and division definitely create problems because of the variety of phrases like:<BR/>(1) "M less than N": N-M<BR/>(2) "M less N": M-N<BR/>(3) "M is less than N": M < N<BR/>(4) "The difference of M and N": M-N<BR/>(5) "M is three less than twice N": M = 2N-3<BR/>etc...<BR/><BR/>Division Phrases:<BR/>(1) "M divided by N": M/N<BR/>(2) "The quotient of M and N": M/N<BR/>(3) "The ratio of M to N": M/N<BR/>(4) "M is what fractional part of N?": M/N<BR/>(5) "M of N" (where M is a fraction): MN<BR/><BR/>These tend to cause issues because subtraction and division are not commutative so order is critical.<BR/><BR/>Then you have terms like factor and multiple. I tend to keep it concrete:<BR/>3 is a factor of 18<BR/>18 is a multiple of 3<BR/><BR/>I've occasionally used thhe islly mnemonic:<BR/> ___<BR/>F)M<BR/><BR/>Most students listen to the FM band so maybe they will remember that the "factor" (F) is on the outside of the division and the "multiple" (M) is on the inside. Then again, transferring from division form to fraction form can be an obstacle for students.<BR/><BR/>If I think of others, I'll add them. If others have their favorites, pls share them!!Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-22710505810138621792009-03-21T11:16:00.000-04:002009-03-21T11:16:00.000-04:00Thanks Dave... I'll be using this one in my 9th gr...Thanks Dave... I'll be using this one in my 9th grade Integrated Algebra class and I'll let you know how they do. A similar problem that most of my students have trouble with is something like "seven less than ten". They always want to write 7-10 rather than 10-7. It is one of those backwards ones. Do you know of any other backwards phrases to be aware of?Eric Pflughttp://www.blogger.com/profile/04418307215619640547noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-23891314751494508512008-09-10T18:10:00.000-04:002008-09-10T18:10:00.000-04:00Thank you Denise. It's very gratifying to see how ...Thank you Denise. It's very gratifying to see how these play out with real students. I'm not surprised by the results at all. Your daughter will eventually make sense out of the convoluted wording which is the real nub here.<BR/><BR/>Now I want to check out her post...Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-9731597632182956382008-09-10T18:03:00.000-04:002008-09-10T18:03:00.000-04:00Here are our results from last Friday:middle schoo...Here are our results from last Friday:<BR/>middle school--right:wrong = 3:2.<BR/>high school--right:wrong = 3:5, with one person caging his or her bet by writing both A and B on top of each other.<BR/><BR/>I gave the middle school students the "Which group is bigger?" and about 10 seconds to respond. I wanted to push the high school students to give an instinctive response, so I allowed them only 3-5 seconds.<BR/><BR/>When we went over it in class, I think everyone understood how testing a few numbers can be a useful check to see if you have the equation correct. Whether they will remember to do that in future ratio problems remains to be seen.<BR/><BR/>I also tried this with my 4th grader, who naturally gave the wrong answer. Then I went over the scaffolding questions and several numerical examples with her, and she could answer all the questions correctly. But when we went back to the original, she again gave 2G=B as her answer. <BR/><BR/>She definitely understood the situation, and she was confident that the 2G=B equation expressed exactly what was going on. I think she must have been interpreting the equation as if it were a ratio: "2 girls for every boy." <BR/><BR/>By the way, the puzzle inspired her to write a blog post: <A HREF="http://kittenspurring.wordpress.com/2008/09/08/tons-of-girls/" REL="nofollow">Tons of girls</A>.letsplaymathhttp://letsplaymath.wordpress.com/noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-71985561723789763142008-08-31T09:12:00.000-04:002008-08-31T09:12:00.000-04:00Denise--I agree that the order of the words or sym...Denise--<BR/>I agree that the order of the words or symbols as in Jonathan's example is a critical factor. I've been thinking that it may even go beyond this to the essential core of problem-solving. We need to train our students to think more deeply about what they're doing. The natural tendency for most students (and adults too for that matter) is to respond quickly, almost automatically, rather than take the time to read more critically and THINK about what they are doing. Pretty obvious, huh! As teachers we say these things to our students, but it often falls on deaf ears. We tend to therefore fall into the trap of 'survival mode' for our students, giving them a list of keywords or phrases to look out for. You know what I mean, Denise. "If the problem uses this word, children, then you should..." <BR/><BR/>Now, I do feel that some of this is absolutely necessary to help our students avoid common traps in the phrasing or in the math. However, I suggested in my post that we need to encourage some other constructs or frameworks for them. I was interested in your take on the "Which is the larger quantity, which is the smaller quantity?" This metacognitive approach seems to me to be more effective than merely memorizing that you must remember to reverse x and y when encountering the phrase "y less than x." Of course, in the light of day, we will all tell them to reverse it, because it is a survival tactic! <BR/><BR/>The other heuristic I mentioned was concretizing the problem, that is, training our students to replace "y less than x" by "what is 4 less than 7?" It's not enough for us to show them this. We need to train them to do this by themselves, since it may not be natural for most. Of course, some youngsters do not need these strategies. They do these things instinctively.<BR/><BR/>By the way, I use the 'reverse cross-multiply' strategy for handling the "3x = 5y, what is x/y?" type. Students write a proportion x/y = __ / __, then fill in the blanks to insure that the result is 3x = 5y when cross-multiplying. Thus the "3" must be diagonally opposite the x, etc. Yes, this is a tactic, but it does have meaning if the student understands why cross-multiplication is a valid operation. I wonder how many students really know WHY that method works!Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-66769923059256787542008-08-30T14:00:00.000-04:002008-08-30T14:00:00.000-04:00I am definitely using this when our co-op classes ...I am definitely using this when our co-op classes start next Friday. I have two groups: middle school and high school. The youngers can have the "which group is bigger" hint, but I'm giving it to the older kids straight. I wonder how many of them will get it?<BR/><BR/>I think the trouble with these (and many other word problem mix-ups) is that the order of the words in the sentence is different from the equation. Students read <I>twice ... girls ... boys</I> and the "correct" equation seems so obvious that they don't bother to think. Similarly with Jonathan's ratio problem.<BR/><BR/>Deniseletsplaymathhttp://letsplaymath.wordpress.com/noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-18703874681390313632008-08-29T22:28:00.000-04:002008-08-29T22:28:00.000-04:00Hmm. I know you are right. Teach it, they still ge...Hmm. I know you are right. Teach it, they still get it wrong.<BR/><BR/>But I have great success teaching them to find the ratio of <BR/>x:y if 3x = 5y<BR/><BR/>(It's counterintuitive, but easy. Solve the equation for x/y)<BR/><BR/>What if we taught kids to write g/b = 2, and then solve?<BR/><BR/>Just thinking out loud.<BR/><BR/>Jonathanjd2718http://jd2718.wordpress.com/noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-19257593653836714282008-08-29T09:12:00.000-04:002008-08-29T09:12:00.000-04:00Dave,That's the same problem we all face: The corr...Dave,<BR/>That's the same problem we all face: The correct equation seems so obvious to us, how could they not "see" it! Try imagining how you felt the first time someone handed you directions for some new technology and told you it's so easy!!<BR/>I write this blog to share my own experiences and maybe help a few along the way. Thanks for sharing.<BR/><BR/>I'm tempted to compile a list of common English phrases that create confusion. Most Algebra 1 texts provide plenty of practice for translating common verbal expressions but we could always use more! <BR/>Also, standardized tests (SATs, ACT, Algebra I state tests, American Diploma Project End of Course Exam in Algebra, etc.) generally include some of these.<BR/><BR/>What are some other phrases you've come across that confound childrens' brains?Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-44876114713401985112008-08-29T08:53:00.000-04:002008-08-29T08:53:00.000-04:00Agreed. They spend so much time complaining about...Agreed. They spend so much time complaining about "when will we ever use this?" Then you give them a situation and they hate it. Not only the translation of words into math, but then that their "real life" answers are not nice integers! Ahhhhh!<BR/><BR/>This is where I struggle when I teach algebra 2. It has become so second nature to me which is the right equation that I struggle to understand how they can possibly see the other way. Just reading the title here was so confusing to me that I was trying to figure out how the other one might possibly make sense (maybe if there were negative boys and girls?).Calculus Davehttp://www.blogger.com/profile/14039458440867020542noreply@blogger.com