The Data Results:First, for the data geeks, the data set is 400-450 students per grade level and over 50% were scored for each level.
As hoped, the proficiency of students increased on the correct answer by grade level with the exception of 8th grade. However, the statistical significance may be irrelevant. Regardless, we are at 90% correct. That is pretty good.
The more interesting data is the quality of the explanations. Here is how the scoring worked.
0 - No explanation
1 - Significant logical errors in the explanation
2 - Logic was acceptable but the explanation either had errors, was only calculation
based or was guess and check
3 - Explanation was the process used. It either explained how they achieved the
solution (algorithm) or implied an equality of the two sides.
4 - Student specifically stated that the left side of the equation had to equal the right.
It was not just implied.
The understanding of what the equal sign means varied greatly not only by grade level but by school. The chart below is bothersome for a few reasons. The first is the red square which represents almost 1 in 10 students didn't even bother to put an explanation down. How would those students perform on the SMARTER Balanced Assessment? The remaining parts that bother me have more to deal with the quality of the explanation than anything. Students, at minimum, should have been able to explain to a level of 3. The amount of students that only put down calculations increased as they got older (Schools A and B). Also, the clarity of how each building emphasizes how to explain something is supported in this question.
Students who were incorrect most commonly answered 18 or 23. However, other incorrect answers were 0, 1, 6, 7, 8, 9, 10, 11, 12, 15, 16, 17, 19, 20, 22, 24, 25, 26, 28 and 78. A challenge would be to figure out how they worked out all those answers.
Where do we go from here?
The number of students who answered 18 and 23 is alarming and indicates a teaching error. In other words - WE OWN THIS PROBLEM. Students did not show a clear understanding of what the equal sign means. One can only assume that when we teach the math operations we run a string and continue to add numbers to the right. Mathematically - this is incorrect.
12 + 6 = 18 is a correct statement
However, if we add more: 12 + 6 = 18 + 5 = 23
This is no longer correct because the values are not equal.
12 + 6 is not equal to 18 + 5
To write this correctly, we would make a new row.
12 + 6 = 18
18 + 5 = 23
The other portion that we own is how our students explain things. Explanations in math should not just be the math process in words. See the exemplars for more clarity.
So, I pose to you...how are you teaching your math?