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A third grade girl attempts, unsuccessfully, to add several large numbers using an Investigations Math strategy. She then adds them successfully using traditional "stacking" (disallowed at school) in a fraction of the time the Investigations method took her:
Filmed and edited by a fellow concerned parent who is a specialist in math remediation.
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14 comments:
Hi everyone -
I'm copying Steve H's comments from the previous posst into this window:
SteveH:
They need lots of paper and a big eraser. If they skip the erasing part, maybe they could track down errors. My first reaction was that the school must have taken a LONG TIME to teach that method. Wouldn't partial sums be more effective? If this is for visual learners, do they separate kids by learning type? What is the downside of teaching non-visual kids with a visual technique? If all of that was just to learn the concepts before using the calculator, then what do they do with percents and fractions? When do they drop the pictures of pie and teach symbolic math?
This is more about low expectations than understanding. You can always trade time for more understanding, but it this case, they don't seem to get that right.
At the very least, they could define a more compact picture for 1000 and 100. The picture for 1000 could consist of a vertical line followed by three O's. The picture for 100 could be a vertical line followed by two O's.
February 20, 2011 5:12 AM
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Blogger SteveH said...
It's one thing to try for deep understanding, but quite another to define it and decide when and if it happens. Go ahead and use boxes and other symbols. Does it work? How can you tell? Everyday Math uses stacking and partial sums to teach understanding. Is that not good enough for TERC? Some kids understand perfectly well with stacking and carrying. Does TERC believe that that can't happen by definition? Should math be reduced to some sort of lowest common denominator of understanding? Do TERC students EVER understand what lowest common denominator means? How do they test for that? If they get the wrong answer, can teachers figure out whether students really understand the concept? How many wrong answers can you get and still receive an "A". Oops, the school probably uses authentic rubrics.
They live in an eduland of fairy-thought. How many TERC students can dance on the head of a pin?
How many TERC students ever get to algebra in 8th grade without outside help? Of course, they will say that it's just a matter of engagement and motivation. It couldn't be TERC. Blame the kids. Blame the teachers. Blame the parents and society. However, they did seem to figure out that in the old days the problem was "traditional math". As the SNL Church Lady would way: "How con-VEEN-ient!".
February 20, 2011 6:12 AM
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Anonymous Anonymous said...
SteveH, your idea about representing numbers with marks (a line and three circles for a thousand) is very much like an idea that ancient Hindu mathematicians used! You have constructed a very good representation for numbers, and if it works for you, then go ahead and use it!
February 20, 2011 6:23 AM
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Blogger SteveH said...
On top of that, there is additional understanding in the symbols and positions. You can talk about the numbers of zeros. You can talk about how the columns match up. Imagine. You can teach understanding with that picture! You can then teach partial sums and follow that up with an understanding of why carrying works.
With their picture tecnique, a line is 10, a square is 100, and a cube is 1000, but then what happens to their symbolism? What is a 1? What shape is 10000? What pictures do they use when they try to teach scientific notation?
February 20, 2011 6:37 AM
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Blogger SteveH said...
I wrote my first letter to the editor of our state newspaper over 10 years ago about TERC. At the time, our school was using the even worse MathLand, but was looking at moving to TERC. It ended up choosing Everyday Math. Our schools talk about how their math scores are better than before ... with MathLand.
February 20, 2011 6:47 AM
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What drawing would they use to represent 10,000? A hypercube?
Thank god mom is teaching her at home.
SusanS
I want to know how long it would take her to get the answer right.
That should be the sequel.
She draws the cubes & sheets & etc. and gets the answer wrong....then she draws them again and...gets the answer wrong again?
What are the odds?
Wow. That was absurd.
Thanks for stopping by my blog Terri :)
Mary
I can see the value of using this method for a couple of days as a segue into place value and using the "columns" of the stacking method to keep track of the different powers of ten.
The kids should work several examples before they need to use composition (carrying). It might even be wise to refer to the pictures again to explain how composition works for an arbitrary column and the next higher column.
Let's not laugh at pictures that try to explain place value. They are a means to an end of full understanding, but I think most kids would be ready to leave the pictures behind after about three weeks.
The pictures will be useful again when the kids learn about decomposition (borrowing) in subtraction.
Kids take a long time and many problems to learn arithmetic. We want to rush them through it because it is tedious to us; but to them it is new. Adding 1995 + 7 is a big deal for a kid, because it applies the principle of composition over and over.
All.the.tedious.drawing. Argh! Maybe some place value cards to segue into it for a day or even two but this....I did make dd the future teacher watch the video. We're planning to have her infiltrate and double cross.
Good grief - even though that was the edited version, it was still painful to watch.
Speaking of which, I asked students in my high school to figure out the annual cost of the thrifty food plan for a three person family with two 25-year old adults and a 7 year old. Students were getting stuck on just beginning this problem, which only involved arithmetic. Seniors in my Economics class could not begin the problem and by the end of my presentation, there was only one student who even started. This was after seven minutes. Note that this school was 73nd in the U.S. News high school ratings.
"Let's not laugh at pictures that try to explain place value. They are a means to an end of full understanding, but I think most kids would be ready to leave the pictures behind after about three weeks."
They should be well past learning about the basics of place value at this point. It might be useful to reuse some sort of pictures when they start learning about adding numbers greater than 10, but not 3 weeks worth; not even one week's worth. The video was about using pictures to add 4 digit numbers, and they were told not to use "stacking". At the very least, they should be at the level of using partial sums. Even Everyday Math uses those. If TERC can't progress past pictures at this stage, what do they do when they get to fractions?
Also, what exactly is "full understanding" and how do you know when kids have achieved it? What determines the point when you can drop the picture crutches? Surely it has to be before adding 4 digit numbers.
"Adding 1995 + 7 is a big deal for a kid, because it applies the principle of composition over and over."
It shouldn't be when they get to problems like that. That's the whole point. They shouldn't have to go back to some sort of "full understanding" or pictures to figure it out.
"Kids take a long time and many problems to learn arithmetic. "
"Many problems?"
Since when do curricula like TERC or EM push practice? They are against drill and kill. They think that some vague sort of understanding or picture is enough to solve any problem.
The video is laughable because it shows the rote silliness of educational thought.
I like the Right Start Math abacus for teaching addition with carrying. It reinforces place value but eliminates the need for drawing pictures. All the student needs to do is to push the beads in the proper columns (ones, tens, hundreds, etc.) It also is easy for the student to visualize during the transition from using the manipulative to doing pencil & paper calculation.
The beauty of using pictures to teach composition and decomposition is that when the children are confronted with adding pounds and ounces, they will not simply assume that you can "carry" the tens across columns, but understand that pounds are bundles of 16 ounces. The old New Math tried to use different number bases to teach the same thing, but it was more than the kids were ready for.
How many times have you seen kids trying to carry 100 minutes into the next hour?
Our place value system is a wonder of design and was developed over many years. In some ways it makes things too easy. Kids need to see below the surface simplicity of "carrying" so they can deal with weights and measures, time, fractions, days and weeks, etc.
The Montessori math materials for place value worked well for my child, at home. They use a similar geometric construct to the TERC drawings -- single beads, rods of ten beads, squares of a hundred beads, cubes of a thousand beads -- but handling the manipulatives is way more efficient than drawing the darned things. The activities done with these beads build on each other until the child has the skills to construct and deconstruct numbers, do addition with carrying, and do subtraction with borrowing.
But then the beads can be tucked away and the kid can keep on going with written numbers, using the standard algorithm. My son is 4 and we've already reached that stage. (For multidigit addition with carrying, anyway; not yet with subtraction.)
The point is, we used the manipulatives merely as a bridge to the abstractions of place value and carrying. That's all. I can't imagine being forced to always grind out the arithmetic with pictures, like in this video!
The whole thing looked like a huge load on working memory to me. The little girl is obviously bright and clearly understands what she is supposed to do (how many children is that true of?), and still gets the wrong answer. I imagine for many children the whole thing is just a muddled mess. Sad.
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