First, I apologize for monopolizing the Daily yesterday and for frightening those who dislike the topic.

Now. Thank you to Slo_:

Cross multiplication is just the short cut of the actual proof. I don't know if 3rd graders would understand the proof or not since it does require algebra. It would take a long time to solve each problem if you had to go thru the entire sequence.

And to Enke:

Buehrle - I think it's because of what you say, when they learn the trick, they stop thinking about the actual numbers and what they represent. and other ways to manipulate them.

And to tselbs:

RCG, as a retired math teacher, I found your "comparing fractions" situation interesting.  Even though the picture approach is neat, it is time consuming.  I don't know what the students have done at that level, but if they have found common denominators, that could lead to an understanding of cross multiplying.  Even though it may not be the least common denominator, the product of the denominators will always yield a common denominator. After changing both fractions so they have this denominator, the one with the larger numerator will be the larger.  Looking at the pattern in several examples can show the new numerators are the product of the original numerator and the other denominator.  I wouldn't call this proof or algebra but it leads to the generalization that cross multiplying gives the desired result.

And to twocat:

Cross multiplication is a trick? On what planet? How the #$%#$%# else do you figure out which of two fractions is bigger? Why teach awkward ways to do something? It just discourages people. buehrle my vote is for you, the teacher, the school and the rest of the educational system to teach cross multiplication. That and to stop using really boring problems to teach math with. The latter is a major personal gripe I have with mathematics education in general and subject for a future rant.

And to Spinach:

I am a mathematician and my concern about the use of cross multiplication in the third grade is that the students may not/probably not understand what they are doing.  Too many of my current college students fear math because they were taught these magical tricks that they could use to do one computation or another that they don't understand and appear to them to be magic.  They work for one use but not for another.  It would be great if the teachers could explain why the trick works and repeat again why it works and repeat again why it works and on and on so that the students will eventually understand it is just logic and not magic.  the problem I have is that many students just try to use tricks that they sort of learned but don't really understand what they are supposed to do  to perform the trick and what the steps are.

It is great for those who do understand the tricks but too many don't.  One of my goals in teaching the math phobes is to try to do some unusual math topics that  are far from the math they have seen before and they may be able to understand and to give them some success in the subject.  It is tough

I don't have the answer but, all of your input has helped me come to terms with fractions and 3rd and 4th graders. As Spinach sees in his college students, they skipped grasping the rationale and view the tricks as magic. Perhaps I should not have used that term. My friend, Rose, the elementary school math specialist wants to prevent what Spinach sees on his end of this topic. That is why she promotes NOT teaching cross multiplication. Mrs." Egg" is the 3rd grade teacher who allowed me to help Imani with her comparing fractions worksheet. Mrs. Egg has taught many different levels of math. She discovered that some students will not grasp the concepts of the "proof" as to why cross multiplication works until late in high school and sometimes not until their early 20's.

This brings me to tselbs comment. The 4th graders have the common denominator concept down. 3rd graders, I am unsure. But I can work with what Tom gave me in his paragraph.

Twocat. You nailed it, too. If we are expecting these kids to comprehend these concepts, we have to give them fun and applicable projects and not just repetitive worksheets.  Is this what you mean?

Mrs. Egg has the class with the widest gap between students I have ever seen. She has kids that will probably never be able to live on their own let alone compare fractions all the way to students who could probably grasp algebra now. And everyone in between. This is why I go to the classroom. Mrs. Egg gives me the struggling students and I work every problem with them. But I rarely see them ever being able to complete a standardized test adequately.

With Imani, I see a glimmer of understanding. What I would like to do with her is flashcards!  I would like to have her learn her basic addition, subtraction, multiplication, division facts!  With her, we did each problem both ways. We drew the pictures and we "checked" the answers with cross multiplication. I asked her if she had learned about common denominators and she looked very puzzled.

Her problem is she has to use her fingers to try to figure basic multiplication and addition!

So. Now. Spinach. I am curious, developmentally, do you think the average 3rd or 4th grader can grasp the logic, even if it is repeated every time?

I shall continue to go to Matoaka Elementary and help the kids however the teacher in charge wants me to. I will reinforce their instruction. Because I really do not want to confuse them. I think that might be all this one old lady can do.

HOWEVER. One thing I am emphatic about is discouraging these kids from labeling themselves!  One little girl told me, "I am just not good at math."  I told her she was not allowed to say that. She can say, "I need to work more on math."  And you know what?  She smiled at me. She smiled big and said, "Yes. I need to practice." I told her we can practice together. And I told her she is a very bright little girl.

Rosie - I volunteered to help out with the remedial math students in my daughter's 5th grade class.  That was a few years ago.  I was shocked at the range of abilities.  A few kids were still counting on their fingers to add 7 plus 11 and would get the wrong answer otherwise, actually have no clue how to get the answer without counting on fingers.  Some rote memorization was missing I guess.  Those kids needed extra help, and I don't know if they were getting it.  I imagine the range of abilities in reading and writing was not as dramatic, but I didn't volunteer in that area.  Anyway, I had 5 kids, and they were all very dedicated to trying to learn.  That's what I remember second most.

So. Now. Spinach. I am curious, developmentally, do you think the average 3rd or 4th grader can grasp the logic, even if it is repeated every time?

I really can't answer this since I know nothing about the mind of a third grader.  Its been about 50 years since I was one and I have been in college now for over 40 years.  And I have some questions about what college students can grasp.  But I feel fairly sure that if they were to grasp the logic then we would need to repeat it a lot.  I need to do that to college students.  A good strategy for dealing with major ideas is to tell the class what I am going to tell them.  Then I tell them it, and the tell them was I told them. And maybe I will do it the next class also.  There is no harm in repetition.

I think that over the last 40 years there has been a big change in what is taught to the students.  We are teaching the subjects earlier and earlier, in many cases we a teaching them concepts that are too abstract and difficult for their young brains to understand.  They are learning some algorithms to do things but they don't understand them and so math is becoming more and more difficult and obscure and more students are fearing it.

This may be in large extent because mathematicians  make the subject more mysterious so we can impress everyone else by how smart we are. There have been many times that I hear mathematicians talking among ourselves in plain English but if they go talk to students or the general population they will use a language (jargon) that is unknown to people outside of our secret society with the point to confuse every one else. This makes us feel smarter.

This has nothing to do with running, so i will add that I ran the Wednesday Hump Day tonight at the local running store.  A nice 5k run every week.

Hmm?

I was querying the math teachers because I thought all teachers had to take Developmental Psychology (I had to take it for my BSN). I thought the class was quite valuable and I wish everyone getting a degree in anything should take the course. The course should be mandatory for new parents. <Hyperbole>

My husband has a PhD in Mechanical Engineering and has taught some grad. courses through GWU and Old Dominion. I believe the psychology course he took as an elective was beneficial when he was dealing with some of his students. But that is very different from 3rd and 4th graders. But when I talked to him about cross multiplication and fractions he showed me how he would help the kids with the problems and he did not use cross multiplication. I guess it does take a rocket scientist...

ANYWAY.  Thank you again for participating in this discussion.

Really, who cares how long it takes a frictionless block to make it down a 45 degree ramp that is 5' long on the moon.  But that is what the problems students are given look like. Those fractions you were dealing with, related to what? Nothing I bet. Just a bunch of problems with no useful application in sight. If I had to teach fractions I would start with something like: Joe's sub shop is worth $124 and has 12 shares outstanding. Mike's burgers is worth $84 and has 5 shares outstanding. Is 1 share of Joe's subs worth more or less than Mike's burgers? This is a real problem that they will, at some point have to deal with, assuming they ever invest in a company.

I believe my husband might care about your first example. In fact, I think figuring out that kind of thing IS in his job description.

As for Mike and Joe, how do you set up the problem? What does the term "Share Outstanding" mean? I never took a course in economics. Although DS1 (Joe) has a BA in economics. He will have a BS in chemical engineering in May. He didn't figure out what to do with the economics degree. I think you have to get a masters or PhD with economics, right? He wasn't interested in doing that. Oh well.

I would choose Joe's subs because I like hoagies better than burgers and my son's name is Joe. Is that the right answer?

Hmm?

 

I was querying the math teachers because I thought all teachers had to take Developmental Psychology (I had to take it for my BSN).

No I got a PhD and you don't need to know nothing to teach in college.  If you got those three letters after your name you are qualified to teach in college. The only classes I took as a grad student were math classes, although I almost got the opportunity to get a couple art history courses to qualify for my PhD.  A long story I won't go into.