Analyzing Student Learning

Student Work (ignore the number line on the bottom, that is part of the second question not analyzed with this learning objective and data)

The rubric provides a guideline for categorizing student mastery. If students scored a 4, they were able to correctly order all 5 fractions from least to greatest and show correct reasoning through a strategy that they were comfortable using. If students scored a 3 it could mean two things. Either students misplaced one fraction (but knew how to correctly order all the other fractions from least to greatest) or students correctly ordered 3 fractions but mixed up the other 2 fractions. If students scored a 2 it meant that they could only correctly order two of the fractions but mixed up the other three fractions and could not put them in the correct order in relation to the other three fractions. If students scored a 1 it meant that they could possibly identify the largest or smallest fraction but could not order the rest of the fractions or they could not correctly order any of the fractions.

The data shows that over half the class is correctly able to show a strategy to order all 5 of the fractions from least to greatest. Looking at Andy’s work which is very similar to the other students who scored a 4 on the rubric, they understood three big ideas that helped them place the fractions in correct order. First, looking at Andy’s work, he knows that whenever you have 1 out of a certain number  (such as ¼), it will be less than only having one left (such as ⅞). Therefore, he knew that ¼ and ⅕ had to come before ⅞ and 8/9. Second, he understood that fourths were larger than than fifths so when you eat 1 out of 5 pieces, you are eating a smaller amount. That thinking is shown through his visual representation when he adds dotted lines to the fifths representation to turn it into fourths and writes “smaller” to show that a fifth is smaller than a fourth. Third, he was able to successfully convert the improper fraction into a mixed number and then understand that half is bigger than the fifth and fourth but less than the ⅞ and 8/9. Both the data and student work show that over half of the class is able to do these three items to show their thinking.

When looking at the data, it looks like at first glance 18% of the class is having trouble with ordering one or two fractions. However, when looking closer at student work, there is a consistent misconception. Students did not understand how to convert an improper fraction into a mixed number or didn’t understand what the improper fraction meant. All of the students made the same error and had the same misconception that 39/6 was the largest because of the 39 on the top. This can be seen in Maggie’s work. To clarify this misunderstanding, I will have these students draw out what 39 sixths looks like with pre-drawn bars cut into six pieces to save time. This will translate into understanding that this fraction really represents 6 and 3/6 and is not the largest number when compared to the other drawings.

Last, when looking at the data that shows rubric scores at a 1, it looks like over 20% of students cannot order fractions at all. However, when digging deeper nearly all the students had a conceptual understanding of whether ¼ vs. ⅕ was larger and ⅞ vs. 8/9 using the same reasoning as above. However, when it came to comparing ¼ and ⅕ to ⅞ and 8/9 students did not know which one should go first or last. They did not have an understanding that ⅞ and 8/9 is closer to one whole so it is larger than ¼ and ⅕ which is just one piece out of a whole. They also did not have an understanding what 39/6 represented. Most of them placed this fraction in random spots but nearly all of them consistently put ¼ after ⅕ and 8/9 after ⅞. To address this common error I will have students draw out each of the fractions to visually see the difference between taking only one piece out of a whole vs. only having one piece left. You can see in Irvin’s work that he was not making connections across the fractions but just for the fractions that look similar (¼ and ⅕   ⅞ and 8/9). Another strategy is to use paper strips to have students engage with the fractions in a hands on way to see what each of the fractions look like. I would have these students draw out 39 sixths as well, similar to the group of students scoring a 3 from the rubric. After the visual representations, for the group scoring a 3 on the rubric, I will hand out two student copies of correctly placing a different improper fraction in order and one that incorrectly places it. I will have students work in pairs to justify which paper was correct and which paper was incorrect. I will do the same for the group scoring 1’s on the rubric but orient it more towards misplacing the 1 out of a number and having only one piece left out of a whole.

The last noticing from the data was that 2 of my ESL students who scored a 1 on the rubric scored a 1 due to placing the fractions from greatest to least (in addition to not knowing what to do with the 39/6). These students might need additional support with language of “greatest” and “least” to understand how to order in that direction.