In order to assure that each child is learning, we must must determine what a child knows both before and after the lesson. Nationally, we try to assess learning by standardized test scores. The large scale of most school systems requires assessments that can be recorded in databases to be tracked and analyzed. Parents often feel that their child already knows something because they were exposed to material and children often claim that they ‘already know that’ because they’ve seen something similar before. None of these methods can provide a comprehensive answer to the question: "Did the individual child gain or acquire knowledge of or skill in something?"
Just because a child has been exposed to material, does not mean that the child learned it. Scoring well on a multiple choice standardized test does not guarantee mastery. Children can parrot back information or memorized steps without full comprehension. What many parents fail to understand is that there are degrees to understanding each concept. In my classroom I strive to make sure that each child walks away with a greater degree of understanding than when they entered my classroom. This often requires me to adjust my learning objective as the lesson progresses.
I strive to make sure that each child walks away with a greater degree of understanding than when they entered my classroom.
For example, this past weekend my third grade classes began a unit on fractions. My first goal was to make the connection between division and fractions. I also wanted my students to understand what a fraction is, how the denominator relates to the parts of a whole, that fractions can be added or ‘counted’ like any other object and to relate whole numbers to equivalent fractions
I started with a review of division:
Miss Danielle bakes 48 cookies for her class. If there are 9 students in her class, how many cookies can she give to each student? How many cookies will she have left over?
Then, I transitioned to an almost identical problem to introduce fractions:
Our class is going to have a pizza party. If I order two pizzas for the class, how much pizza should I give each student?
We discussed the difference between these two problems until I was convinced that every student understood the difference between division problems with remainders and ones with fractions. While we CAN turn every remainder into a fraction, sometime it simply does not make sense. Each child provided an example of something that they could divide into fractions and something that could not be divided into fractional parts. Following my belief that laughter helps children remember, I celebrated their often silly examples such as:
“We couldn't each have 1 1/4 dogs. Someone would have an extra puppy to take home.”
It wasn’t until the children started to play with the manipulatives that I realized that they were going to learn much more than I had originally anticipated.
First, we played with magnetic fraction bars. My original intent for this project was to allow the students to see the connection between the denominator and the number of pieces required to make a whole. I also wanted them to see that larger denominators meant smaller pieces. But then the students started moving the pieces around and reported on their own discoveries. Two quarters were the same size as one half. One half was the same size as three sixths. You could replace 1/5 with 2/10 but if you tried with 1/12ths it was always too short or too long.
They were discovering equivalent fractions on their own, not through formal study or by being taught a formal lesson but through experience. The fractions weren’t just numbers on a worksheet that needed to be completed but an abstract concept that was now real and comprehensible. The greatest learning moments are when children experience the joy and excitement of making their own discoveries. It's moments like these, when their faces light up with each new discovery, that I live for.
My students presented me with another surprise when we played with fraction circles. Not only were they able to use them to perform division with fractional remainders, but they again were drawn back to equivalent fractions. By the end of the period each child had created an addition problem that required at least one conversion to common denominators. One boy even predicted, and then proved, that he needed a 1/20th to complete the circle that he started with ½ + ¼ + 1/5.
“The use of equivalent fractions as a strategy to add and subtract fractions” is a Common Core Standard for 5th grade. Yet all of my 3rd Graders were happily discovering this strategy on their own through guided play with manipulatives.