Wednesday, August 26, 2009

Original Thinking about Geometry

Warning: You are about to enter the Land of Original Thinking. If you like the security of following exactly what a text book says and marking your own trail makes you feel dizzy, find another math guide!

Last June, I presented the topic of how Charlotte Mason taught math. Inspired by the soundness of her ideas, I will attempt to implement them this year with Pamela, who is taking Pre-Algebra. Judging by the reaction of people in my audience, the most controversial recommendation was the following:
Pronounce a sum wrong, or right––it cannot be something between the two. That which is wrong must remain wrong: the child must not be let run away with the notion that wrong can be mended into right. (Volume 1, Pages 260-261)

What! No partial credit? But . . . but . . . but . . .

I know!

I even admitted to my class I had never really attempted this. Someone suggested a brilliant way to do it, so I went home and immediately adopted her idea with David, who was working his way through the final third of MUS Algebra II. We loved it!

What she suggested is to assign only half of the problems and mark what is right right and what is wrong wrong. Then, assign an extra problem for each wrong one, keeping in mind Charlotte's admonition that "he may get the next sum right, and the wise teacher will make it her business to see that he does" (Volume 1, page 261). How? Assess what went wrong and resolve this issue before trying more problems, whether it be careless execution, misunderstanding one little thing, or completely missing the boat.

Charlotte believed that arithmetic trained children to be accurate and clear-thinking. We foster slipshod habits of mind by "the copying, prompting, telling, helping over difficulties, working with an eye to the answer which he knows" Volume 1, page 260.

While Charlotte Mason's books do not go into great detail on math, clues are scattered online. The latest complete PNEU schedule for the highest form available online is for 1922: Charlotte Mason broke math up into three short lessons (Arithmetic, Geometry, and Algebra), using different books for each. I modeled our plan around this idea and spread Math-U-See's thirty-lesson schedule for Pre-Algebra over 36 weeks. The following is my plan for Geometry (roughly one-third of the book) in 20 minutes every school day, five times a week:

Week 1 - Assess and fill in gaps for the area of a square/rectangle/triangle/circle.
Week 2 - Introduce surface area of cubes, rectangular solids, and rectangular and triangular pyramids by using concrete objects. Transition to pictures of them by making shapes from paper and unfolding them to figure out surface area.
Week 3 - Transition from unfolded shapes to labeled pictures to drawing surface area for word problems.
Week 4 - Do selected surface area problems from Lesson 15 of MUS Pre-Algebra.
Week 5 - Introduce the volume of rectangular solids by slicing up blocks of cheese or using building blocks. Transition from blocks to slices and transition to labeled pictures.
Week 6 - Transition from drawing pictures for rectangular solid volume to word problems and working through the equations.
Week 7 - Introduce Pythagorean theorem by measuring by hand, squaring each side, and looking for a pattern. Transition to figuring it out from a labeled picture.
Week 8 - Transition from solving the Pythagorean theorem for labeled pictures to drawing right triangles for word problems.
Week 9 - Do selected Pythagorean problems from Lesson 10 of MUS Pre-Algebra.
Week 10 - Introduce volume of cylinders by finding the volume of stacks of coasters, pineapple, etc. Transition to labeled pictures.
Week 11 - Transition from labeled pictures to drawing cylinders for word problems.
Week 12 - Do selected volume of a cylinder problems from Lesson 24 of MUS Pre-Algebra.
Week 13 - Introduce volume of pyramids with a square base and cones by filling different-sized objects of these shapes with rice and pouring them into rectangular solids and cylinders with the same dimensions. The goal is demonstrate that one is one-third of the other. Transition to labeled pictures.
Week 14 - Transition from labeled pictures to drawing pyramids and cones for word problems.
Week 15 - Do selected volume of a pyramid and cone problems from Lesson 27 of MUS Pre-Algebra.
Week 16 - Assess memory of ratios and proportions. Using concrete activities like cooking and whole/part pictures and thinking to reintroduce ratios and proportions.
Week 17 - Transition from whole/part pictures to drawing them for word problems.
Week 18 - Do selected ratio and proportion problems from Lesson 19 of MUS Pre-Algebra.
Week 19 - Measure the sides of similar polygons (while practicing knowing their names) and look for a pattern. Transition to doing the same for labeled pictures.
Week 20 - Transition from labeled pictures to similar polygon word problems.
Week 21 - Do selected similar polygon problems from Lesson 20 of MUS Pre-Algebra.
Week 22 - Practice carrying and borrowing in inches, feet, and yards using strips representing each. Transition to labeled pictures, drawing, and equations.
Week 23 - Transition to labeled pictures, drawing, and equations. Then, try it with adding and subtracting time.
Week 24 - Do selected adding and subtracting time problems from Lesson 26 of MUS Pre-Algebra.
Week 25 - Read temperatures of various liquids in Fahrenheit and Celsius. Measure freezing and boiling temperatures of water too. Show the logic of how the conversions are made using two strips of paper with these two temperatures in both units.
Week 26 - Transition to drawing and equations.
Week 27 - Do selected Celsius to Fahrenheit problems from Lesson 16 of MUS Pre-Algebra.
Week 28 - Show the logic of how the conversions are made using two strips of paper with these two temperatures in both units. Transition to drawing and equations.
Week 29 - Do selected Celsius to Fahrenheit problems from Lesson 17 of MUS Pre-Algebra.
Week 30 - Show the same civilian time and military time until she sees the pattern. Develop a formula for converting time.
Week 31 - Try adding and subtracting multary time as was done with measurements and civilian time previously.
Week 32 - Do selected adding and subtracting military time problems from Lesson 28 of MUS Pre-Algebra.
Week 33 - Add and subtract measurements through concrete activities: feet and inches, yards and feet, pounds and ounces.
Week 34 - Add and subtract measurements through pictures: feet and inches, yards and feet, pounds and ounces.
Week 35 - Add and subtract measurements by drawing for word problems: feet and inches, yards and feet, pounds and ounces.
Week 36 - Do selected adding and subtracting measurement problems from Lesson 29 of MUS Pre-Algebra.

In the following posts, I will include my plans for Algebra and Number Theory and what we covered in our first week of each.

4 comments:

Penny said...

You never cease to amaze me. I want to be like you when I grow up. lol

JEMD1966 said...

In last year's math I gave Timothy incentive for accuracy by assigning him three rows of problems, and if he was perfect through the first two rows he did not have to do the third. It really helped him to be careful!

The Glasers said...

Penny, it helps to know really smart people and I have met so many online and at conferences . . .

sarah in the woods said...

I am new to living math, and this really gives me motivation to work on making our math better.