I just came back from a training in Austin with the Charles A. Dana Center, where a group of math education leaders came together to prepare to facilitate training of elementary teachers around the world. We spent four days learning, doing math, exploring the Standards for Mathematical Practice and focusing on Common Core Math Standards of place value and fraction conceptual understanding at the elementary level (Grades PreK – 5th).
One of the really powerful “aha’s” to come out of this is the progression and building of fractions that starts at PreK. We explored specific standards related to the idea of unit fractions and looked at the building of this using a Vertical Alignment from PreK through 6th grade. What I got out of the whole process is how focused grade-level standards are (so for example, in first grade, students work with fractions of halves and fourths using area models – really developing visual models and understandings 1.GA.3). Each grade level builds on this – and there is no ‘algorithms’ until 5th/6th grade. Unit fraction and thinking of fractions as numbers on a number line, just like whole numbers, is a huge emphasis as students progress through elementary grades, so that students think fractions are just numbers – not ‘things’ we have to operate on to get numbers. There’s an emphasis on the geometric/visual as well, through area models and number lines.
One thing that is emphasized is using benchmark fractions and number lines to really help students think about fractions as numbers and what the whole is and what the fractional numbers look like compared to this whole or benchmarks like 1/2. If students get a strong conceptual understanding of this in the early grades, then this idea of benchmarks helps them make sense of fractions with larger numerators and denominators in the upper elementary grades (and beyond) and the eventual use of algorithms make sense as a way to simplify the process of working with fractions (i.e. multiplying/dividing).
As an example, here’s a number line with a benchmark fraction:
So – asking students to think about fractions – so for example 6/8 – and identify whether that might be closer to one or closer to 1/2 and EXPLAIN their reasoning requires students to think about the whole, how many pieces the whole has been cut into (so the unit fraction) and how many of those units we have and how does that compare to the benchmarks of 1 and 1/2.
The visualization and explanation are components of learning fractions that I think are often rushed over or skipped, especially as we get into upper grades where we rush to give them the algorithms. If we spent more time with these ideas, and added on more complex fractions, as it does in the CCSS, when students got to the point where they were working more abstractly with fractions using algorithms they wouldn’t struggle as much as they do now. If students have had a lot of practice using visuals and thinking about fractions with the number line, then in the upper grades you can continue this idea of benchmarks using something like the Rand# generator on a calculator – such as the FX -55plus, which allows for random fraction generation (with larger numerators and denominators). Asking the simple question is it closer to 1/2 or 1 and EXPLAIN your reasoning provides an understanding of a students conceptual understanding of fractions.
Sample using a calculator to generate random fractions (can still use a number line to support students explanations):
Are these fractions closer to 1/2 or closer to 1? EXPLAIN your thinking.
As you can see – the explanation really reveals what a student understands about the whole, the size of the pieces, the benchmarks. Obviously explanations could include drawings and manipulatives as well, but you get the idea. Explore the progression of learning vertically and really focus on visuals, explanations, and building understanding.