Qualitative growth in my classroom is highlighted by our classroom "Write about Math" lessons. Students, once a week, focused on looking at math strategies, proposed solutions, and explaining why those solutions were correct or not. Being able to look at mathematical strategies and highlight why someone is wrong in their thinking, fixing it along the way, is an extremely rigorous and difficult task. At the beginning of each lesson, I give students a problem and someone's explanation of how they completed the problem. Scholars then have time to talk through the process with a partner and then get individual think time to agree or disagree with the strategy used to solve the problem. Once they have made that decision, they then write their explanation of how they agree or disagree with the strategy, correcting it if it is wrong.
"Write about Math" Beginning of Year
Below are directions that I gave students on our very first "Write about Math" lesson of the year. Scholars were given these instructions on the board, along with at their seats. They then talked in partners about the work for ten minutes and came up with their first decision of agree or disagree. They then were told to explain why they agreed or disagreed with the strategy, and to show their thinking of how they solved it. Outside of those directions, scholars were not told how much to write, how much to explain, or how they were to show their thinking.
This is the prompt that was given to scholars on the first day of "Write about Math." You can see that the problem is solved in a certain way, and asks scholars what the student did incorrectly. They had to explain the error, correct the error, and solve.
The following images are samples of two students who came into my room at the beginning of the year and solved the problem by simply reading the prompt. This prompt includes 4th grade standards of adding fractions together, but goes on to 5th grade standards when they are asked to complete an error analysis of the problem. These scholars, per the samples, both misunderstood the questions to be answered, along with the actual mathematics of the work.
Most of the writing in math looked like this to start the year. Within my intervention groups for math, we began each day with a constructed response question like the one represented below. Scholars would "Make it RAINS," meaning Read the Question, Annotate Important info, Number the questions it asks you, and Split the scratch paper into that many parts. After scholars answered, I would take a few responses and share them on the screen anonymously. This gave us the opportunity to talk through "effective" answers to the question and "unsatisfactory" means of answering. When scholars saw the two types, they became gradually more concise and clear in their work and explanations. As the year went on, students were more and more effective in their responses to constructed response questions.
This scholar simply took every fraction and added them together. He added uncommon denominators and added all fractions up. He didn't understand the three tasks he should have answered.
This scholar multiplied the first two fractions he saw. He clearly didn't read the directions to the problem, and didn't perform the correct mathematics needed to solve the problem.
The following are the two rubrics that I used to score each of the tasks above. They both receive a "0" for a score. Both responses are "incorrect or irrelevant." Neither sample shows the correct work, correct explanation, or correction of the mistake made.
"Write about Math" End of Year
For our last "Write about Math" lesson of the year, I gave my scholars the exact same problem they were given at the start of the year. Notice that these samples include ALL elements of the task. The same scholars name the error, correct the mistake, and solve the problem correctly. This is a very high level of math understanding. For students to name and correct a mistake, as well as solve the problem means that they are reaching beyond procedural mathematics and into the conceptual. It is also incredible that this work is from the same students who had no idea what they could do to solve at the start of the year.
Below are the rubrics associated with the two samples shown above. Notice that both receive a score of "3." Both scholars gave the explanation of the error. They then explained that they had to "Change 3/8 to 6/16" before they could subtract. Finally, they both found the correct answer of 3/16.
Conclusion
When reviewing the starting and ending points of these three scholars, it is clear that they made dramatic academic growth. Simply looking at the writing can tell a casual observer that they comprehend much more than they did previous to the year. With a more sophisticated observer, scholars are seen providing in-depth analysis of a mathematical strategy and providing their opinions on how they would correct the math. This level of conceptual understanding of math highlights the level of rigor that they learned at this year, and further challenges me to continue providing writing opportunities in my math class for students.