Saturday, March 19, 2011

Clear Thinking about Math Part 1

Sometimes, the best plans fall apart. Pamela seemed to track the review of fraction addition I did early last month. When we went back to area and perimeter problems, something got lost in translation. Fortunately, a friend posted about how she made tiles with numbers and symbols to use on a magnetic board with her grandson. Having to write added enough challenge to chip away at his working memory and make it difficult to a new concept to gel. Pamela writes and draws very well, but I thought it worthwhile to remove any potential memory drain while she was trying to visualize adding fractions. I suspect Pamela lost her thread of thought every time she had to stop and draw her thinking.

I also heeded Mary Boole's advice in an old Parents' Review article,
"Beware of writing, in play-lessons, anything which does not represent some process actually going on in the child's mind."
I created a set of pie charts in a spreadsheet, some representing wholes, some representing wholes split into fractions, and fractions. I stayed simple by limiting it to halves, thirds, fourths, and sixths. I cut out all the shapes, covered them with clear contact paper, and cut them again to make them more durable. Before we worked on a problem, we sorted between wholes and fractions to help Pamela familiarize herself with these homemade manipulatives. You can see the first step in our first lesson in the video below. Since Pamela understands fractions, I am using very declarative language as we collaborate.

Then, we started working on her problem, adding 4 3/4 and 4 1/2. Before writing, she set up a model for each addend so that she could represent her mental process visually and spotlight what adding fractions and simplifying meant. You can watch how we worked through the problem together: first, she made both denominators alike. Because I didn't build any models for eighths, she had to think through another option: fourths.

Then, she added them and ended up with an improper fraction 5/4.

Using the models helped her see what she was doing when converting to a mixed fraction and adding the wholes again.

The video below shows how we collaborated step by step. We wanted to show what we were doing physically and write it on paper.
Working together like this cleared up other glitches. Pamela had a habit of forgetting to write the wholes until she needed them again. While she usually remembered to pick the wholes back up when she needed it, that mathematically incorrect habit could lead to disaster in algebra. When finding a common denominator, she tended to multiply the denominators (2 x 4 = 8) rather than going for the least common multiple (4). The lack of eighths forced her to think of a smaller denominator, which turned out to be the LCM. We worked on similar problems together for about a week. I made a set of twelfths for more challenging ones. Then, I faded myself out of the picture and she did well flying solo without anymore issues.
"Let his arithmetic lesson be to the child a daily exercise in clear thinking and rapid, careful execution, and his mental growth will be as obvious as the sprouting of seedlings in the spring." ~ Charlotte Mason (page 261)


Bonnie said...

Thank you for the links. I think I've read CM's words but need to refresh myself with my last one at home. When you have a child whose strongest bent is to numbers, it is most delightful! That was my oldest except he came up with ways to do math different than I taught it! He's the math guy in accounting now.
I need to hear you next week when you come to bless us.

AutismOnABudget said...

I'm saving this post for my granddaughter later. We did make the manipulatives like you described to teach her recognition of fractions. We need to practice for a little more with that. Thanks for the reminder.

Phyllis said...

This is a great lesson. You have a real talent at breaking down learning to its smallest components. Bravo, Mom!