Fraction A Day - Day 9 - An Extremely Hard Problem

As mentioned yesterday I said I would work on the problem using Etoys (the above images are from the that project).  The image show notes (using "text box" from the connectors category of the supply bin) about questions I have and some design issues.

In doing this I am thinking I may want to switch things around.  The study interview started with what is 1/2 + 1/3, but that was to assess understanding, I am thinking I should create a version where the experiences lead to this.

This problem was based upon the problem on page 40 of: The Development of the Concept of "Fraction" from Grade Two through Grade Twelve. Final Report. Part One, Part Two and Appendix. - Robert B. Davis 


Also saw an article in the NY Times  Brain Calisthenics Help Break Down Abstract Ideas which mentioned  a 2010 study, researchers at UCLA and the University of Pennsylvania had sixth graders in a Philadelphia public school use a perception-training program to practice just this.  (more detailed version here)

Fraction A Day - Day 8 - Mapping the Developmental Evolution of a Child's concept of Fraction

On pg. 40 of The Development of the Concept of "Fraction" from Grade Two through Grade Twelve. Final Report. Part One, Part Two and Appendix. - Robert B. Davis it states:

"In the case of fractions, the developmental evolution of each concept is important. At first, one probably defines a/b by taking a candy bar, or a pizza, or something else, dividing it into b pieces, and taking a of them. ..."
"When one encounters 'improper fractions,' this meaning fails. One cannot use this meaning to speak of (say) 5/4" We must introduce the concept of unit. We divide each unit into b equal pieces, and take a of them. Now, with these new definitions, we can easily deal with 5/4, although we shall need two units in order to do it."

So one step in our process to "map" is the developmental evolution of each concept (okay its a BIG step, but someone must have done this before). The challenge is to understand where the child is and what questions and experiences you can provide to help them progress. It is unrealistic to expect elementary school teachers, who must teach all subjects, to know let alone be masters of the concepts and maps in all subjects. So, how can you do this if the teacher may not understand certain concepts or the map?

How can we help students move from their existing mental models to more powerful and more sophisticated models? (please post answers as comments ;)


Obviously it depends on their models, any mis-conceptions they may have, and other factors.


The following problem is from the interview excerpt starting on pg 45.  The The student in the interview was described as "a generally bright resourceful 5th grade girl" who reported that "in her previous school there was very little instruction in arithmetic". : 


"I want to show you what most people think is a really hard problem: 1/2 + 1/3"


She was stumped and did not know how to solve the problem. Later the interviewer then brought out cuisenaire rods and asked: "If we want to talk about "one half"  and "one fourth" which rods do you supposed we want to call "one"?


Once she had the physical model she was able to answer "three fourths."  She did this rather quickly it seems and without using the rods completely. She had simply identified which rod was "one half" and which was "one fourth" and then "solved the problem just by thinking about it.  The concrete model and "enabled her to build up an appropriate representation in her mind".


Today I will start working on an Etoys version of the problem, perhaps with a pre and post assessment.

Taking Tic-Tac-Toe to the next level

If you give a child an answer,
you solve a problem for the day.
Teach a child to find the answers,
you prepare her for a life.

So, how do you "Teach a child to find the answers?" Let's illustrate by example.
Parent Ground Rule:

You can't tell the child the rules of the game, let them figure it out for themselves, they'll retain more and feel better about themselves

The Game Example

Parent: (Stage Direction: Create a grid of dots on a piece of paper like this)
. . . .
. . . .
. . . .
. . . .
Parent: Lets play a game on this board I have created.
(Stage Direction: Look at your always beautiful, and sometimes frustrating child)
(Say your child's name) Pick any two numbers from 0-10.

Child: okay 2 and 8

Parent: Great, Now lets go to the game board.
(Starting in the lower left corner of the board, count from ZERO to the first number your child said. See diagram below:
. . . .
. . . .
. . . .
0 1 2 .
(next using the second number your child gave you start counting up from the dot you just picked. See diagram below:

. 4
. . 3 .
. . 2 .
. . 1 .
. . 0 .
(when you get to four, ie: you are off the board stop and say) Sorry that's off the board, my turn.

Parent: I pick 1 and 2
(Again count from ZERO, starting with the dot in the lower left corner and pointing to each dot as you count. Now mark an X a shown)
. . . .
. X . .
. . . .
. . . .
Parent: Your turn pick two more numbers between 0 and 10
Child: I pick 2 and 3
Parent: Okay lets check it out.
(Start counting from the dot in the lower left corner of the board using the first number your child gave you. Then start counting up from the the first number using the second number and mark your childs spot with an 0, as shown:
. . O .
. X . .
. . . .
. . . .
Parent: (say it with enthusiasm and pride: state your child's name) Congratulations, you seem to have figured out the rules of the game!
(Teaching Tip: here you are giving specific praise on something your child probably struggled with at first then figured out all by themselves)
Game Teaching Suggestions:

1. Your child most likely won't figure this out on their second guess, it will probably take them a while, but have patience they will figure it out.
   
2. If you have multiple children, teach the game to your younger child first (if they can play tic-tac-toe, they can play this game) and don't let your older child see you playing. Then have your younger child compete against her older brother. Then sit back and watch the sense of pride as your daughter's face beams as she womps her older brother. He will get frustrated, but that frustration will drive him to learn the game. This is good frustration channeling.
3. If they get too frustrated, STOP. Our goal is to help them be success. If you sense your child's frustration level is too high, change the way you ask the question, such as "pick two numbers between 0 and 3", then switch back on the next turn to between 0 and 10.
4. Always start counting from zero It took us humans thousands of years to discover zero, its an important number. More history on Zero in a later posting.
   
5. After a few games have your child explain the rules to you and how the game works. This also helps your child to start reflecting on what they are learning.
6. Its okay to win the first game. Your child will feel a greater sense of accomplishment if they lost the first game and then came back to win.

What your child's learning:

1. Geometry - Cartesian Coordinate System
2. How to communicate what you know (see teaching tip #5)

Now try it with your child and please let us know how it went by clicking on the comments link below. We also appreciate any suggestions and editorial corrections you have in the comments section.
Coming Soon:

1. Books suggestions that go along with this lesson
2. Taking tic-tac-toe to an even higher plan
   1. Getting kids to think about Second Order Consequences
   2. You can't do that (or introducing negative numbers)
   3. "History" lessons
      1. Rene Descartes and "Mom, there's a fly in my room."
      2. Why on earth did it take so long to discover Zero and why is it so important anyway. Can you figure it out? If not ask your kids, its okay if they don't have an answer, some of the best questions are ones that it takes a while to figure out.

Thanks to:

• Robert B. Davis - Bob Davis was my professor at Rutgers in the Masters in Mathematics Education Program. This is based on a warm up he taught teachers for the Madison Project. Bob was a true master teacher and a heck of a nice guy. He also ate the second worst banana I ever saw.
• You, for taking the time to learn how to help your child learn by doing.