As we finish our second wave of state testing, I am reminded how critically important it is that teachers know and teacher their standards for each math concept. Whether your standards are the CCSS or the Texas TEKS, they are vertically aligned to ensure that concepts are introduced in a way that allows students to build upon a solid foundation. Let’s look, for example, at fractions. I am paraphrasing the Texas TEKS, but you’ll find that the CCSS are very similar.
At the most basic level, students learn to partition (divide) shapes into two or four equal parts and describe the parts using words. They also learn to identify examples and non-examples of halves and fourths.
Students continue to partition objects, but now they add eights to their repertoire. In addition, they understand that the more parts a whole is divided into, the smaller the parts. That is a huge concept! Up until this point, 8 has always been bigger than 4. But now, we tell kiddos that 1/8 is smaller than 1/4. Students needs lots of concrete experiences to truly understand that concept. Finally, students understand how many fractional parts it takes to make one whole and count fractional parts beyond one whole. In other words, it takes 4 fourths to make a whole, so if I have 6 fourths, that’s more than a whole. In 2nd Grade, students do not use formal fraction notation at all. Everything is done using words only–one fourth, two eighths, etc.
In 3rd grade students are introduced to formal fraction notation–the numerator and denominator. They learn about unit fractions–the idea that 1/b is one part of a whole divided into b parts. So 1/4 is one part of a whole divided into 4 parts. Their understanding of fractions goes beyond partitioning a whole and extends to identifying fractions on a number line and with sets of objects. Equivalent fractions are introduced with concrete materials, pictures, and number lines. Students compare fractions with either the same numerator or denominator, further reinforcing that the larger the denominator, the smaller the pieces.
Students learn to compose and decompose fractions with objects and pictures. Essentially, they are learning the concept of adding and subtracting fractions with like denominators. This concrete foundation helps them to avoid the common misconception that 1/4 + 1/4 = 2/8. If I put together 1/4 and 1/4, now I have 2/4. Equivalency and comparison are now accomplished using a variety of methods, which should be built on fraction sense. Understanding, for example, that 7/8 is very close to a whole, while 2/6 is much closer to 0. Or that 4/6 is greater than 1/2, while 3/8 is less than 1/2. The relationship between fractions and decimals to the tenths and hundredths is introduced.
At this point students are asked to add and subtract fractions with unlike denominators–a skill totally dependent on an understanding of equivalency. They are multiplying a whole number and a fraction (1/4 x 3 or 3 x 1/4). There is no way this will make sense to them without an understanding of unit fractions and addition of fractions with like denominators (3 x 1/4 is the same thing as 1/4 + 1/4 + 1/4). Students divide a unit fraction by a whole number (1/4 ÷ 2) or a whole number by a unit fraction (2 ÷ 1/4). The standards stress that this should be taught with objects and pictorial models.
Each and every step in this process is crucial. Personally, I like seeing the vertical progression of each math concept from Kindergarten up to 5th Grade. As a classroom teacher, of course you must be most familiar with your grade level standards. Keep in mind that often includes not only what you will teach, but how you will teach it (concrete, pictorial, abstract). But I would suggest you take it a step further and familiarize yourself with the standards of the grade level before and the grade level after. Knowing the standards the students should come to you with will help you do a quick preassessment prior to teaching and it will also help you pinpoint and fill in the gaps students might have. Understanding what is to come will give you a better understanding of the standards you are supporting.
The standards might not always be easy to understand, but unpacking them must be a part of planning for instruction. Make sure your step up that staircase is solid!