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Yesterday, my 5th grader came home with three multiplication quizzes that her class had done in class.
The first quiz was the three’s tables. 3 x 1 = ___, 3 x 2 = ___, 3 x 3 = ___, etc. up to 12. The next two quizzes covered the four’s and the five’s.
I wonder why these fifth graders are still being quizzed on the multiplication tables. Are these just quick exercises to practice math facts? Is it an assessment to see which students haven’t learned them yet? If so, why would they test the entire class if only two or three students were lagging in this area?
We are located in an affluent NYC suburb where public school spending per pupil is about $19,000. If our students aren’t fluent in their math facts by fifth grade, we’re not getting our money’s worth. However, at least our students probably possess “deep conceptual understanding” and can apply lots of “higher order thinking skills”.
I really don’t know what to think. However, I have given up on believing class time is being used efficiently to teach the most important fundamental academic skills.
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25 comments:
Tex,
At my sons' elementary school, there is a big push to get parents to volunteer to help kids with basic math facts...even in the fifth grade. Have you noticed this at your school?
No, not that I know of, Paula.
Our school has told us that math facts are important and are being taught. However, the reform curriculum they’re using doesn’t seem to support mastery. Some teachers do supplementary drilling.
I just heard something amazing today! My daughter’s special ed teacher told me he was so glad that she had mastered her math facts, because that allowed her to focus on learning concepts and using her brain power to work on strategies to solve word problems.
That’s the first time I’ve ever heard something like that from any of her teachers. Maybe he forgot to drink his kool aid today. :-)
Congrats Tex on hearing something amazing today! I don't think I've ever heard a teacher say why learning math facts is crucial. They simply say your kid must know them.
Doesn't your school use TERC? Our school uses some math investigations mixed in with a more traditional math program. However, without attending Kumon, I am not so sure my kid would have done so well this year.
Is your daughter using TERC? If so, this matches the 3rd grade TERC books I have. 3rd graders using TERC are still working on single digit addition. They are *also* working on other (more advanced) stuff, so it isn't like they are stalled at 3+5, but it is quite clear that they are *not* expected to have mastered single digit addition by 3rd grade. I can't tell if they are expected to have mastered 3+5 by the *end* of 3rd grade, but I don't think so ...
So ... if your daughter is using TERC, then this is at least consistent with the 3rd grade books. I can't tell you what the expected outcome is either.
-Mark Roulo
Nope, my daughter’s using Think Math, new this year.
They started to learn multiplication back in 3rd grade. At the time, my daughter was having difficulties with mastering basic math facts. Her teacher told me not to worry, because it was more important for her to learn the concept of multiplication.
However, I intervened with Saxon and Kumon. And, my daughter and other parents tell me how some of these 5th graders are still struggling with basic math facts.
And I learned something depressing this week. I have not verified, but I was told by one teacher that my school dropped any 5th grade lessons on reducing fractions or finding common denominators. They’re still spending a lot of time shading in diagrams to depict fractions. Lots of work with denominators of tens. Dumbing down, if you ask me. That’s fine, because it seems to be in line with NY standards. Sigh.
Actually, I think the dumbing down is occurring everywhere, and not just with NY standards.
Ugh. I really object to overreliance on diagrams for fractions. I know too many kids who can ONLY think of fractions as pieces of pizza or chocolate bars. It is terribly limiting, and it appears to me that the longer the kids rely on pictures, the harder it is for them to move to the abstract.
This reflects a larger problem of too much time spent with manipulatives and visuals. In fact, there have been many times my kids have been encouraged by teachers to "concretize" their abstract understanding, when I think the goal in math education should be just the opposite: move away from the concrete into the abstract.
In fact, there have been many times my kids have been encouraged by teachers to "concretize" their abstract understanding, when I think the goal in math education should be just the opposite: move away from the concrete into the abstract.
Exactly. And as much as I like the bar diagram approach in Singapore Math, it should be viewed as a means to an end, with the end being abstraction. In fact, Singapore's national syllabus for math has as a goal: concrete-pictorial-abstract. But the hangers on who are promoting Singapore in the US without fully understanding what it is about glom on to bar modeling as if it's the only thing Singapore Math is about, with books about how to do bar modeling. That's not the name of the game. The intent is to use bar modeling as the middle ground to get to abstraction via algebra, not to teach an algorithmic approach to how to use it. My daughter didn't like bar modeling because she couldn't draw the bars "pretty" so she solved the problems without it, which was fine with me.
my school dropped any 5th grade lessons on reducing fractions or finding common denominators
They probably did.
TRAILBLAZERS ends with the 5th grade book and never teachers division of a fraction by a fraction.
I should check to see whether it teaches common denominators.
It may.
But think about it. Unless the school supplements, these kids go into 6th grade never having divided a fraction.
concrete-pictorial-abstract
When I first started reteaching myself math, I had a TERRIBLE time transitioning from pictorial to abstract.
I felt -- it really was a "feeling" -- that if I couldn't visualize something I didn't understand it.
The point where you can't visualize comes pretty quickly once you hit fractions, I think. I couldn't visualize a fraction divided by a fraction, nor could I visualize a fraction multiplied by a fraction, either, in spite of all those groovy fraction multiplication squares they draw in math books.
Heck.
Can't find a good example right now.
This one from New Zealand is close.
wow!
New Zealand has a MUCH BETTER WAY of teaching fractions of fractions.
fraction of fractions tool
"TRAILBLAZERS ends with the 5th grade book and never teachers division of a fraction by a fraction."
Constructivists have this neurotic fear of teaching efficient algorithms like invert and multiply. They'd rather not teach division of fractions at all than to defy constructivist dogma if alternatives are not readily available.
"The point where you can't visualize comes pretty quickly once you hit fractions, I think. I couldn't visualize a fraction divided by a fraction, nor could I visualize a fraction multiplied by a fraction, either, in spite of all those groovy fraction multiplication squares they draw in math books."
I've tried desperately to visualize division by fractions until it hit me one day. The trick was to visualize how many times a piece fits into another piece. For example, if you have half a pizza you can fit two quarters of a pizza. Of course, visualization gets dicey when the divisor is bigger than the dividend, e.g. 3/4 divided into 1/4 or the numbers are unfriendly or even hostile.
For multiplication, the shaded squares work remarkably well for me in the visualization department, keeping in mind that "of" means multiply. So 3/4 of 1/2 would necessarily have to be less than 1/2 as logic and the picture indicates.
My daughter just worked on her first Saxon 8/7 Investigation. Visualizing a fraction of a fraction is precisely what was happening in that lesson. She had to cut a number of pre-printed circles out into halves, fourths, on so on through twelveths and then was asked to come up with various configurations using the "manipulatives".
1. What fraction is half of 1/2?
2. What fraction is half of 1/4?
3. What fraction is half of 1/3?
4. What fraction is half of 1/6?
5. What fraction is 1/3 of 1/2?
6. What fraction is 1/3 of 1/4?
16. Two 1/4 pieces form half a circle. Which two different manipulative pieces also form half of a circle?
Now that's what I call great use of "manipulatives".
Singapore 5A uses the rectangles and shaded portions thereof to illustrate fraction of a fraction. So does "Arithmetic We Need" by Brownell,Buswell and Sauble (1955). And--I hate to say it--so does Everyday Math, in fifth grade. Then in sixth grade they do it by using a number line. NZ's method is very nice and elaborate, and is an animated version of the rectangle shading.
Singapore's can be supplemented in that fashion; as you know the SM books are very spare in instruction and its up to the teacher to know how to conduct the lesson.
They'd rather not teach division of fractions at all than to defy constructivist dogma if alternatives are not readily available.
That's the only reason I can think of for omitting this topic.
It really is amazing.
I love the Saxon Investigations. I did every one of them in 8/7.
I was actually sorry to see that the high school books don't have them!
keeping in mind that "of" means multiply.
That was a huge moment for me, figuring out the meaning of "of" back when I was first relearning arithmetic.
I remember bugging Ed, in bed one night, about why "times" would be the same as "of." I could see it with "1/2 of 2"; I couldn't see it with 2x2.
I was beside myself.
Ed didn't exactly know, either, but he'd taught arithmetic to G.E.D. students in Newark, so he had a much better grasp of arithmetic than I did.
Finally he said, "2x2 is 2 units of 2."
The light went on. What a relief.
That's a funny thing about those "early days"...I was much more obsessive and distressed trying to relearn the material. I just didn't get it at all, and I really did struggle trying to develop some conceptual understanding. I would obsess over basic principles of fraction arithmetic.
That almost never happens now. For one thing, I tend to understand new concepts when I encounter them in Saxon; when I don't I can usually figure them out.
It all makes so much more sense now.
The trick was to visualize how many times a piece fits into another piece.
You know, that is a much better place to start than with the "fraction multiplication square."
I now get those squares, but I had NO CLUE what was going on with them when I first encountered one.
I felt about those things the same way I felt about lattice multiplication.
Overwhelmed with stuff to look at and mystified.
The "cognitive load" on both is extremely high.
That's why starting with just one "piece" and dividing it into smaller pieces is a good idea.
Interestingly enough, in addition to Saxon 8/7 my daugher is also working on Singpore 5A right now and about to move on to 5B. She began Singapore 4A/B last year in about January when I realized Everday Math was leaving gaping holes in her understanding and the chaotic way it moved around was making her nuts. Anyway, what was interesting about watching her do the Saxon Investigation was that she began the first two questions explicitly following directions and using the manipulatives, but wanting to get through the lesson just started answering the questions without them from about the third one. She didn't really need the manipulatives to understand the lesson. As far as I can tell, she's able to understand (abstract) fractions of fractions without manipulatives (concrete). I'm not sure if its the Singapore or that it's something she just "gets". Her spatial abilities are extremely developed and this may play a role.
I think I've mentioned before that a couple of the "math brain" boys in my little Singapore Math class had no patience for the bar models.
What I failed to mention is that the Saxon Investigation was wonderful for me. I am certainly not a "math brain".
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