In exploration math, we've begun to work with fractions, decimals and percents. For many students (and adults) fractions are an enigma. Indeed, they are sometimes taught as an inherently counter-intuitive subject. I know that "the bigger the bottom number, the smaller the fraction" was taught in my elementary school as a way to remember the impact of the denominator. Unfortunately, it often just solidifies children's confusion.

It can be easy to think of math and number systems as a set body of knowledge but they're not. As humans, we are continuously expanding the ways we can express our world in numbers. The development of fractions is very recent mathematically. While many ancient cultures had unit fractions such as 1/2 or 1/4,fractions were primarily conceived of as a way to represent equal sharing. Common fractions (such as 2/3 or 4/5) are much more recent - around 500 BCE. Decimals are an even more recent mathematical development. The Hindu-Arabic base 10 system was not adopted widely in Europe until 1300 CE (1300!) and the use of a decimal point to show fractional parts came *after* that.

Why do I share all of this? Because it illustrates that it's taken humans a long long time to figure out fractions. It's no wonder that children need a lot of different experiences to come to understand them and use them fully.

I've been re-reading Young Mathematicians at Work - Constructing Fractions, Decimals and Percents that dissects the challenges many students (and adults) experience with fractions. Often, we move to the manipulation of pure numbers far too quickly. Procedures and algorithms are taught independent of the context that makes them useful. (I've actually read about a teacher whose mantra about fraction division was, "Don't ask why, flip and multiply!" - here's a blog about a fraction lesson I did in response.) By proposing real world problems that necessitate the use of fractions to come to satisfactory answers, we give the concept of a fractions and decimals meaning and utility. This leads to a deeper understanding and an ability to generalize understanding across a range of situations.

The authors, Catherine Twomey Fostnot and Maarten Dolk, propose that a child's understanding of fractions works like a landscape in which the understanding of one skill exposes new skills on the horizon which, when arrived at, open up further skills. It is not a linear journey and different children reach epiphanies from different directions. It's a good description of the work that is going on in all three classrooms as we explore fractions. These are the concepts that are on the landscape. Your child is moving among them and solidifying his or her understanding of them through carefully constructed experiences:

- Part/whole relations - the whole matters when one is comparing fractions. This concept is revisited often as students extend their use of fractions. Finding parts of wholes, extending ratios, and eventually working adding, subtracting, multiplying and dividing fractions.
- Equivalency vs. congruency - fractional pieces do not have to be identical, only equivalent. 1/2 + 1/4 = 3/4
- Extending concepts of equivalency to decimals and percentages - students see the relationship among these different ways of recording fractional parts and can apply them fluidly
- Connecting multiplication and division to fractions - children understand fractions as "fair sharing" and see the relationship between fractions and division (splitting apart) and eventually multiplication 5/8 = 5 * 1/8.
- Relations on relations - when multiplying and dividing fractions there are two wholes to keep track of. For example, when I divide 2 1/4 cups into half cup scoop measures, I end up with 4 and 1/2 half cup scoops. My whole for the dividend is cups. But my solution's "whole" is the half cup measure.

These are the big concepts on the horizon. They enable students to identify fractions, order and compare fractions and decimals, find equivalent fractions, find equivalent decimals (and percentages), combine fractions with like denominators, combine fractions with unlike denominators, and eventually multiply then divide with fractions. The conceptual understanding enables the skill work which becomes more efficient through experience.

Whew! If you've made it through to this point in the blog, you are very dedicated to your child's mathematical development and might be wondering what you can do to support understanding of fractions and percents at home. Here are some ideas:

- Cook! Read recipes. Then pretend you can't find some of your measuring cups.
- Connect computer downloading bars to fractions and percents.
- Play games of chance and talk about probability.
- Compare things.
- Encourage "rate" thinking - i.e. "We've gone 40 miles since we got gas and we've got half a tank left, how much farther can we go before we get gas?"
- Talk about time as fractions - quarter of an hour, 3/4 past.
- Count down using decimals or fractions instead of just whole numbers.
- Play fraction dice games - in the simplest you try to make the largest (or smallest) fraction by rolling a die and deciding if you want to use the roll as the numerator, denominator or one of your two throw away rolls. Works with decimals, too.
- Put your child in charge of distributing anything fairly - goldfish crackers, screen time, hot dish.

"Mathematizing" is a word in education which makes me cringe a little but the idea behind it is very, very important. It means finding ways to explore situations mathematically - seeing relationships and using numbers to explain the world around you. Students who do it regularly feel like math makes sense and, importantly, that working with numbers is a creative endeavor. Students who don't mathematize see math as a set of arbitrary rules one follows to reach an answer that is correct. Math topics are not related and don't make sense.

The more you and your child use fractions, decimals and percentages to describe the world around you, the better the chance your child will begin to see numerical relationships and begin to mathematize her world.

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