Do you remember fractions from your elementary school days? I remember them only as memorized algorithms- a set of rules to do this when that happens- but not knowing why those were the steps. It wasn’t until my Montessori training that I gained a real understanding of what a fraction is and how to perform arithmetic with them.
What is a Fraction?
To give you a refresher: A fraction is made when a whole is divided into equal pieces. Each of those pieces is a fraction. The word “fraction” comes to us from the Latin fractio which means “to break,” which is the same root for “fracture.” The denominator (bottom number) denotes how many pieces the whole is divided into and the numerator (top number) marks how many parts you have.
In public schools that use common core standards, fraction work begins in third grade (9-year-olds). A recommended introductory lesson goes like this:
The teacher collects from the classroom four discrete equivalent objects, such as chairs, and explains to the students “This group of four chairs is my whole. I have a whole made of four pieces.” He would then select one of the chairs and say, “This is one of the four chairs, or one fourth of the whole.”
This demonstration is mathematically and conceptually sound- however, with this as the child’s first concept of a fraction, it has a very limited applicability. For example, with this as the child’s foundation how are they to compare the magnitude of a fraction?
Which is larger 1/4 or 1/8?
Most 9-year-olds will say that 1/8 is bigger because the group of eight chairs is bigger, but a very clever child might say 1/8 and 1/4 are the same because the chairs are the same size.
With this as their conceptual basis, the problem solving process is hopeless. Understandably, the instruction moves quickly to rote memorization of rules, such as “The fraction with the larger denominator is smaller.” This happened to me in my school days.
The Number Line
The next tool concretizing fractions that Common Core recommends is the number line. The idea is that fractions are numbers that are smaller than one, but larger than zero- they are the numbers between 0 and 1.
On graph paper, the student would draw the number line – effectively replacing chairs for the more manageable graph squares. A number line to show 1/2 would have one dash right in the middle to make two sections. A number line that represents the tenths would have nine dashes across, to make ten sections.
Now, let us have the student solve a basic equivalency problem: Are 1/2 and 5/10 equivalent?
The number line does not really help the student solve the problem, again instruction would necessarily move to working out algorithms on paper as soon as possible.
This is how I learned fractions when I went to public school. It wasn’t until my late twenties, when I took my Montessori Training, that I actually understood fractions.
Teaching Fractions the Montessori Way
Now let’s look at the Montessori approach to fractions. The initial fraction material consists of ten metal congruent circles in matching green frames. The first red circle is one whole and the remaining are divided into fractions from 1/2 to 1/10.
Let’s revisit the first problem: Which is the larger fraction 1/4 or 1/8?
With the materials, the student would select one piece from the “fourths family” and compare it to one piece from the “eighths family.”
The answer is self-evident. A Kindergartener can solve this problem and they do. In the Montessori classroom we begin fraction work at 6 years old and they continue this work through at least fourth grade. The only prerequisites are a familiarity with whole numbers and the ability to count to ten.
If we look at the second problem: Are 1/2 and 5/10 equivalent?
The student brings down halves fraction circle and the tenths fraction circle. They select one piece from the “halves family” and five pieces from the “tenths family.” They are equivalent and the student can check their own work. The child can place the 5/10 into the halves frame and place the 1/2 into the tenths frame. Concretely the student can prove to themselves that the answer is correct.
Does this really work?
I presented the Equivalent Fractions lesson a few days ago to a group of first graders. For them, this lesson was pure FUN. They cheered, waving their arms about, when we found equivalent fractions and literally whined when the lesson was over, “Just one more fraction Ms. Emily!” Tiny 6 year olds bounded out of their seats when the lesson was over to go on and discover for themselves which fractions were equivalent to 1/3.
Throughout all this joyful work, the child will write all of the problems and answers into their work journal. These are reviewed for accuracy by the teacher and lessons are re-presented if a key concept is missing. And well before third-grade, when public school students are just getting started, students in my class rush up to me with jubilance, “I don’t need the materials anymore!” They have discovered the algorithms, that other children simply rote memorize, all on their own.