Not a Math Moron After All

[judifilksign]

 Attended a teacher training today about bridging gaps between how elementary school math is taught, and how middle and high school math is taught.  Very enlightening.

Many high school teachers are cross because they do not think that students have "the basics" any more.  In fact, they do.  But the methods of teaching math in the past ten years have shifted at the elementary level, but the ways that high school math teachers are building are still using the old-school methods, which the upcoming freshmen have never seen before.

Lattice multiplication, partial sums, partial quotients and partial dividend methods, in which you solve problems with place value firmly in mind, and not just setting up math problems by rote, and chunking through a process without understanding it are things the high school teachers haven't seen, so they have to spend time teaching the rote methods, and cannot build on a child's understanding of the distributive property.

We were given worksheets in which we had to show as many different ways to solve a problem as we could think of.  Addition and subtraction, multiplication and division, and finding the area of a trapezoid.

I had scribbles everywhere.  I showed a lot of my thinking in weird ways, but came up with the correct answers.  At least once, I started a process and crossed it out because it wasn't working.  They pounced on that, and asked why I had abandoned that effort, and when I said that it didn't match my estimate, they broke down why, and how I could have "saved" that calculation using the method I'd started.

I had pictograms.  I had groupings.  I had triangles making squares added to rectangles to simplify formulas.  I had a mess!  Dividing 689 by 5 using squares for one hundreds, circles for tens, and hatch marks for ones amused people.  But it showed the concept clearly, and we discussed how I would use it with the math manipulables for my special education students.

We were given student work of the same examples, and another strength I had was figuring out how the students had solved the problems, and being able to reconstruct what it was that they were thinking as they did the problem.  A couple of the other teachers had a hard time with this, and my talk-out-louds to myself in what was going on proved to be helpful for others who could determine right or wrong answers, but not how the student got there.

I had, I was told, a strong numbers sense - the very thing they tried to develop in the students, but so often could not.  I spent a LOT of time asking question after question after question, and none of the math teachers were bothered that I took up so much of their time.  I evidently provided insight as to the thinking processes of their students in ways that hadn't been accessible, because I had the shared adult vocabulary to ask the questions that kids would just say "I don't know" to.  One told me that I made a workshop that she was enduring for the credit points into an interesting one for her.

I found it funny that part of the training for the other teachers was "What questions can you ask Judi to enhance her understanding of this concept?"  And many times, the answer back was "I want to see what she's doing.  I'm getting this better by watching her do it."

I think that if I had been taught in this way when I was in elementary school, I'd've done much better, even though I have dyscalculia (like dyslexia, but reversing numbers instead of letters.)

Comments

[lisa-marli.livejournal.com]

Yeah - Math Genius!
Hey, even Bookkeepers (I use Numbers for a Living) can reverse numbers. We do it a lot. That's why we learn the rule of 9 - If things look off by 9 look for a reversed number.
And Calculators and Spreadsheets are my friends. Not because I can't do the math - I really am good at numbers - but because Large Collections of them can get muddled and reversed and need correcting. It is easier to correct a spreadsheet, or see the error in the Calculator tape...
And especially these days, as the Math teachers learn that there are Many Ways to the Correct Answer, people who have more than one are appreciated.
My Father in Law had a Math Degree, and could do vast sums in his head (and would probably be labels Aspergers, if he was in school today). 3 to get you 1 he rarely used the Rote Methods that were taught him in school. Math Geniuses rarely do.

[tibicina.livejournal.com]

I used to drive my teachers crazy because I would 'skip' or 'combine' steps. Or sometimes come up with more elegant solutions than the one they wanted or whatever. Of course, they drove me crazy because they'd get upset at my doing, say, a whole bunch of associative stuff as one step. I think I really did end up staring at one at one point and saying something like, 'If you can't follow this or don't understand that the associative property means that any string of things connected by plus signs can be rearranged any way you want, perhaps you should be doing something else.' It may not have been the most politic, but it did get across that I knew what I was doing and they were being stupidly pedantic. I don't mind pedantic, but I strongly object to the stupid.

Of course, the other important point sometimes is that math != arithmetic. And many people who are /good/ at higher math really aren't at arithmetic, particularly if they're dyslexic or dyscalculic or even have a bad enough case of dysgraphia. The worse case is when people who /would/ be good at higher math get so discouraged by arithmetic and the over reliance on speed and memorization that they never discover that geometry or calculus or group theory is something that they can just do.

[judifilksign.livejournal.com]

I remember loving Geometry. I also thought it interesting that during the class, I saw one of the teachers solving problems the way my husband does, from left to right.

[mdlbear]

Just so. I _still_ have a lot of trouble with arithmetic -- that's what computers are for. And slide rules before that. But I was a math major in college.

[kyrielle]

And at the opposite end, in Junior High, I was accused of using a calculator on a math achievement test (administered in the guidance counselor's office, and he left me alone with it, and he left his calculator on the desk, and WHY?) because I didn't write out all my steps.

Seriously, for 11 * 17, they figured I cheated because I didn't write out the 170+17, so I must not have really done it. (My mother said to them, "Even *I* can do that one in my head, and I'm not good at math; Laura is.")

Then again, I got in trouble in math class when 315 * 12 was done as 3150 + 630 - it should have had three steps since the top number was three digits. Um. Seriously, people?

[judifilksign.livejournal.com]

Your last example is one of the ways we're being taught to teach the students. If they can use tens, then other digits, it is quicker, and makes use of number sense.

[joecoustic.livejournal.com]

I had a horrible math block most of my life. Add in that my mother was a University Math Professor and it was just a real joke in elementary school. Whenever she taught me tricks that made sense I got in trouble for using them. On the other hand she told me that after seeing what I went through and my issues it changed the way she taught math - many of her students were studying to be elementary school teachers and hated math.

I used to also work for a company were we graded students essays and I usually could do what you could do and deconstruct what they had meant to do. I often did this the season I worked at the IRS too. Unfortunately this was often not an acceptable way of dealing with the forms or papers. *sigh*

[judifilksign.livejournal.com]

The instructors validated all of the mental math problems and counting back change things my mother had me do as a child. Those activities probably built most of my ability to estimate and have a numbers sense.

So, thirty years later, I appreciate it.

Other Ways

[(Anonymous)]

My grandmother worked in the billing department of a department store at the beginning of the 20th century, before adding machines were in use. Everything was calculated by hand. She showed me how she could add up a column of three-digit numbers all at once. She had her own way of adding up a column of single-digit numbers, basically by picking out combinations that added up to ten. She'd gather the 7+3s, the 6+4s, the 8+2s, etc. and keep track of the unmatched ones somehow.

It started me on the path of realizing that there's more than one way to solve a problem.

Re: Other Ways

[judifilksign.livejournal.com]

Yes, this was one of the methods we were told to encourage our students to use.