Introduction to Fractions

Last week and this week, both Astroboy (4) and Thumper (7) got their introduction to fractions.

In training, I was told that fractions come at the end of the series and that some kids don’t even get to it.  But reading my album I see that fractions can be introduced as young as 3 since it’s just a Sensorial experience.  For Thumper, we verbally introduced the idea pretty young with half, and half of half, etc.  I’m pretty sure she learned them in school too.

The Fractions curriculum is basically 6 parts: naming, equivalence, and the 4 operations. It appears that fractions are taught like other Montessori math, first manipulating concrete materials and then moving toward abstraction. The main materials are the fraction metal insets, and fraction insets in other shapes, plus lots of cut out fraction pieces, and reference charts. Fraction is something that is introduced in first grade.

I love seeing the difference in development and how that affects the way the children learn. For Thumper, I did the classic first presentation of all 10 fraction metal insets, naming them, and showing her how to name them in both Chinese and English.  In Chinese, you read the denominator first then numerator, 三分之一, for example for 1/3.  The nice thing about it is that you don’t have different names for the numbers like in English: halves, thirds, fourths, etc.  In Chinese the denominator is called 分母 and the numerator 分母.  分 means to divide, 母 is mother and 子 is child.  So super easy to remember.

As an older child, Thumper got the concept of fractions easily. We even started on equivalence. So I didn’t need to spend much time on the IDEA of fractions. But I could tell that she has trouble with the difference between the way we name fractions in English and Chinese. She keeps wanting to flip them when she says it in Chinese. I remember I did the same thing after years of naming fractions in English.  It’s making me wonder if I need to teach fractions in Chinese at all.

Astroboy really wanted the same presentation, so I did it for him.  But after the presentation, he couldn’t really name the fractions. So the next day I pulled out my fraction skittles instead.  These look like little people, color coded inside; compared with the insets which are just all red. It’s meant to be handled and “played” with.

We first did the sensorial presentation.  He was much more interested in handling the skittles material. However the presentation is kind of short and after comparing the bottoms of these skittles, rolling them around, feeling them, we seemed kind of done. I could tell he was kind of underwhelmed and probably wouldn’t pull them out again. Watching him handling the skittles hit home to me that he needs to really “play” and handle the materials for a bit in order to get the concept. So I made up some games on the spot.  We walked the skittles around the rug because they were going to “school” and I asked him to hand me specific ones. I want to note that this is different from the 3 period lesson I typically am shown, where you’re asking them to put this skittle under the rug, below another skittle, over your head, etc. Because there’s now a narrative, a game. And for a 4 year old, he likes games.

I think pretend play initiated by teacher is frowned upon in Montessori. It’s not discouraged in children, just that adults don’t need to initiate it and impress upon the children an adult fantasy, they’ll do it naturally. But I have never seen how this is done in action. How do you make something interesting so children want to play with it on their own? (PS, I’m reminded that I heard Mario Montessori used to play/manipulate with a material for two weeks at a time in order to find that magicalness of each material, as the teacher needs to show a joy and wonder with the materials to incite the children’s interest. I don’t think I’m there yet.)

Fraction Skittles with labels

Eventually I showed him the labels for the fractions and we labeled them.  Then we played the mystery bag game, which I’m discovering is what you do when the child gets bored with any presentation.  They just LOVE pulling things out of a bag, or out of your hands and opening them.  So we pulled up some fractions and tried to find the equivalent skittle.  With this practice, the names rolled off his tongue much easily.  He then wanted to line them up like a train to go somewhere.  I just left him along to play.

Lining up Fraction Skittles

Today, he brought out the fraction skittles again and on his own (he usually wants me to play with him), lined them up so they can go to school on the mat.  Once again I let him be because I figured any sensorial play is good. After awhile, we played the mystery bag game again, where he pulled out fraction names from my hands and found the equivalent skittle to line them up on the tray “to go to school”.  We also played a variation where I lined up different fraction labels in a row randomly, and he had to find the equivalent fraction skittle. He loved both games and though couldn’t remember the names at the beginning (they forget so easily!), was able to name them at the end.

Followups
Pouring through albums and online blogs, I tried to figure out what are the follow up materials and extensions for really internalizing fraction names and step 2 of learning equivalence (1=1/1=2/2, etc).

I’m going to need both more concrete and abstract materials. Whatdidwedoallday has a fraction cabinet based on the 45 presentations in the Montessori Research and Development album. I cross referenced with mine and saw that we have the same type of work but it’s not detailed and divided into 45 presentations; they’re listed as extensions with no clear guidelines of materials. So I may not do a cabinet. It seems more useful for younger children as they don’t have to write much, just manipulate the strips of paper.  I will go back and make them as I need to, otherwise start with worksheets for Thumper.  (I note that the problem with worksheets is that children think they’re done after a page, whereas strips of paper makes it more “open ended”.  But I’m pressed for time right now.)

For materials it appears I definitely need to make:

• Fraction charts
• Fraction lables (1, 1/2, 1/3, or individual numbers plus a black line made out of construction paper)
• Fraction manipulatives (plastic/paper version of metal insets)
• Equivalence worksheet
• Operations worksheet/problem sets (way down the line)
• Nomencalture cards (for younger kids but I like the booklets better)
• Square and triangular insets or paper material for equivalence.

I’m thinking of these extensions for learning fraction names and equivalence for both kids, in the following order:

1. Making fraction booklet.  For Astroboy, he can trace the fraction inset, cut them, and paste them into a booklet.  For Thumper, she can trace them (may be too easy for her but she loves arty activities), pull out a fraction label, and color them in, label them, also making a booklet.
2. Fraction match-up.  Match different types of things that look the same and match.  For example, matching the skittles to constructive triangles, to the metal insets, to the labels.
3. Worksheets.  I found some on math-aids.com where Thumper can either label fractions she sees or color in fractions.  It’s actually similar to the booklet but for me it’s faster.  I’m going to see which one suits her current age.
4. An equivalence research worksheet.  I’ve seen various versions of this online.  Some are strips of paper, some are worksheets partially filled out (1/2 = x/4, 4/x, where x is blank).  I’m going to start with a “worksheet” similar to the one you see at Nienhuis, where I will list the starting fraction and she has to find all the equivalent fractions.

Worksheets are bad for Thumper sometimes because they make it appear like there is A LOT of work.  So we’ll see how this goes.

Super excited!  There are so many materials to make right now and I’m behind.  So it’s always good when I have a clear idea of what I need to do and it doesn’t “seem” that they will take a lot of time.