tag:blogger.com,1999:blog-7691251033406320222.post8303013233205124853..comments2024-03-26T04:19:38.862-07:00Comments on kitchen table math, the sequel: add this problem to the curriculumCatherine Johnsonhttp://www.blogger.com/profile/03347093496361370174noreply@blogger.comBlogger7125tag:blogger.com,1999:blog-7691251033406320222.post-52363617293562149882011-10-18T19:15:05.869-07:002011-10-18T19:15:05.869-07:00A note on Singapore -- they introduce multiplicati...A note on Singapore -- they introduce multiplication in second grade, but do it intelligently. They introduce multiplication by 2, 3, 5 and 10, then when students really know what multiplication is, they introduce the "ugly" ones. So students don't wind up blindly memorizing the times table, but do wind up mastering it!<br /><br />I will say, the r^2 problem didn't seem too bad to me, but I really liked factoring when we first learned it, and would do it for fun (yes, I am a weirdo).ChemProfnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-4916089058461671432011-10-18T16:55:02.213-07:002011-10-18T16:55:02.213-07:00The triggering question
"r^2 is a multiple of...The triggering question<br />"r^2 is a multiple of 24 and 10. What is the smallest value?"<br />is supposed to invoke the prime factorization:<br /><br />24 = 2^3 * 3<br />10 = 2 * 5<br /><br />Squares must have even exponents on all prime factors, so the smallest<br />r^2 is 2^4 * 3^2 * 5^2<br />and the smallest r is<br />2^2 * 3 * 5<br />That is, r=60.<br /><br />One could get at this more slowly by doing guess and check on the exponents, but recognizing that 2, 3, and 5 must be factors of r seems essential. Note that only easy multiplication facts (2,3,5) are needed for this problem. They could have made it much harder by using 7 or 13, without testing mathematical reasoning nearly as well (more opportunity for careless error).Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-65930039304952689292011-10-18T14:28:57.268-07:002011-10-18T14:28:57.268-07:00Oy, such a good post. Nunes and Bryant have done w...Oy, such a good post. Nunes and Bryant have done work in this area. The short story is that children often exit the 3rd grade (thereabouts) with misconceptions about what multiplication is (and hence what division is) and this can affect virtually all future mathematical learning. (Of course it would.)<br /><br />If I could hope for any idea to take hold in education, it would be this one: The problems that your students are having with math right now in your class might not be due to lack of motivation, the complexity of the topic, bad parenting, crummy resources, video games, or the administration. It might be that we didn't teach them the right math from the beginning. Mastery is not a unidimensional concept. There are masteries that can hobble you for life. "Mastering" multiplication the wrong way is an example that is widespread, I think.<br /><br />--JDAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-53762992311070277072011-10-17T05:05:25.817-07:002011-10-17T05:05:25.817-07:00It may look like a cliff at certain filter points,...It may look like a cliff at certain filter points, like the 7th grade tracking point and the high school math class that you finally flunk. However, the process starts the first day of school. <br /><br />Kids climb the failure cliff year-by-year because schools don't ensure proper grade-level mastery of skills. Our lower school (K-4) talks about (finally!) trying to make sure kids know their adds and subtracts to 20 by the middle of 3rd grade. While they are doing that, proper 3rd grade material is not being mastered. At some point (7th grade) the delayed and weak attempt at mastery stops and kids are tracked with all sorts of left-over gaps and lack of skills. Those gaps lead to a final failure or low peak in math.<br /><br />They talk about balance, but they don't ensure a proper level of mastery even for the basics. Talk of the complexity of math or the difficulty of learning multiplication or fractions lets them off the hook. Schools teach math for an hour each day. Of course learning can be difficult, but they have plenty of time.<br /><br />I call it low expectations, ignorance, and incompetence.SteveHhttps://www.blogger.com/profile/03956560674752399562noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-80765585973321485142011-10-16T21:09:07.654-07:002011-10-16T21:09:07.654-07:00I think that all the complicated names (addends, s...I think that all the complicated names (addends, subtrahends, divisors, dividends, …) make simple ideas complicated. I never could remember which name went with which thing. In fact, I couldn't tell you now what "dividend" refers to, but I can certainly manipulate fractions and rational functions.<br /><br />I think that the naming actually interferes with the understanding, as too much working memory is taken up with the names.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-7530860627514338132011-10-16T18:59:55.665-07:002011-10-16T18:59:55.665-07:00One of the things I liked about Singapore math was...<i>One of the things I liked about Singapore math was that they introduce multiplication right along with addition.</i><br /><br />Interesting.<br /><br />I think multiplication is an 'easy' topic that turns out to be hard later on....<br /><br />I suspect a lot of children who had no trouble learning multiplication have trouble with the concept of factors and factorization down the line.<br /><br />I wonder if introducing multiplication along with addition helps prevent confusion between addends & factors??<br /><br />It might ----Catherine Johnsonhttps://www.blogger.com/profile/03347093496361370174noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-87247728355850009912011-10-16T10:45:14.712-07:002011-10-16T10:45:14.712-07:00"I'm also thinking more attention should ..."I'm also thinking more attention should be paid to teaching young children the terminology of arithmetic: addends, subtrahends, factors, and the like."<br /><br />That sounds like the New Math program that I went through as a kid. Which actually was a fine program - I learned from it - but evidently others disagreed.<br /><br />Learning the times tables is hard because it is a skill completely unrelated to understanding multiplication. I don't think the concept of multiplication is hard, nor division. One of the things I liked about Singapore math was that they introduce multiplication right along with addition.Bonniehttps://www.blogger.com/profile/08364766877630085946noreply@blogger.com