Qualitative
Academic gains are the ultimate door opener – they are the foundation of a truly transformational teacher. Students make dramatic levels of academic growth (that is measurable and rigorous). Families know the level of rigor necessary for college and career readiness in the 21st century.
Introduction
In this section, I compare writing samples from students to demonstrate their academic growth in understanding and explaining mathematical concepts. Literacy is a critical and often overlooked aspect of mathematics education. My instruction focuses on using multiple strategies to reinforce literacy skills in a math context. Students do more than answer questions; they analyze complex problems to create heuristics for learning new and challenging topics. In these assignments, students must understand and provide context for mathematical thinking. Simply showing work for a solution does not convey genuine understanding of the topic.
These assignments are excellent for remediating concepts with which students are struggling, as well as for pushing students to summarize information efficiently and thoroughly. They provide a clear illustration of the depth to which students understand a topic. Students learn how to clearly define a process, pulling in skills from previous math classes and connecting them to new contexts. This the transferable skill that on-demand writing assignments build. The growth students demonstrate here speaks to their broader academic growth and transition readiness for employment or enrollment beyond high school.
In the samples below, I compare students’ On-demand Writing answers from September to their answers in March. The rubric used to measure student growth has been used and referenced consistently throughout the year. As the year went on, the focus shifted from solving equations to broader mathematical reasoning, but the emphasis is still on explaining the process. The directions remained the same, but I tailored my support and intervention based on students' individual needs.
Table of Contents
On-demand writing
Rubric
On-demand Writing assignments include two portions. First, in the left column, students show their work solving the problem at the top of the paper. The typical expectations for showing work in a math class apply. Next, students are responsible for explaining the rationale for the steps they took. They use the right side of the paper to express mathematical reasoning in complete sentences. I push students to include proper mathematical vocabulary and proper grammar in order to close the significant literacy gaps. Students are supported in this endeavor through the inclusion of exemplars and with lists of terms helpful to include.
Overall, the rubric has sixteen points across four categories. Three of the four categories in this rubric deal with students’ explanation of ideas. Mathematical Correctness tends to what is typically expected in a math classroom, though it makes up only 25% of the grade. Pushing students to us appropriate mathematical Vocabulary formalizes concepts that students are somewhat familiar with. Organization and Fluency measure the students ability to explain the problem at the conceptual level, while Grammar measures how well their writing conveys that understanding.
Along with the rubric, I provide students with 2-4 terms they should use in their responses. This is a hint for students who are struggling with the assignment, as well as a challenge for students who understand the content in informal ways. This allows me to develop students' literacy skills while pushing them to understand math content at a deeper level.
Context
At the beginning of the year, my students struggled to solve basic equations. They stumbled through simple mathematical operations and lacked the foundational skills necessary to master Algebra II level content. Rather than simply drill-and-kill math problems, I wanted to take another approach to remediating and building upon these skills. An important variable in this approach is the fact that my students' average reading level is 6th grade, and this deficit shows up in their ability to express their thinking in writing. A supplementary focus of these assignments is improving students' clarity of thought by giving them the tools they need to grapple with math problems at all levels. Students have not demonstrated mastery of a concept if they cannot explain it, and they have needed a great deal of support in learning how to do so. Students take these assessments monthly, allowing me to provide the explicit and regular feedback necessary for students to understand how to explain mathematical thinking in writing.
These On-demand Writing assignments serve as check-ins for student progress. My instruction is driven by productive student-centered discussions, and these assignments provide regular insight into how students are internalizing the skills they are learning. Speaking about math problems is much easier than writing about them, and I am pushing my students to translate their informal understandings of math into formal written responses. The skills they learn explaining simple mathematical concepts like solving equations extrapolate to learning the challenging content they will engage with later in the year.
Three student's responses are included below. While they each came to Algebra II with varying mastery of prerequisite knowledge, all three students below were challenged by these problems. Each shows significant growth in their abilities to express mathematical thinking. For context, Student #1 initially scored 15 on the ACT and is concurrently enrolled in Algebra I to make up a failed credit. Student #2 scored 17 on the ACT in September, and Student #3 scored 21. The goals these students set for themselves were based on reaching the next “gate score” of 18, 22, and 27 respectively. Every student met their goal by the December ACT and moved to at least Proficient on the March On-demand Writing task.
Due to the COVID-19 pandemic, students' responses in March were collected via a Google Form. I asked students to submit a photo of their work along with their typed submission.
Student #1
September
Student #1 understood inverse operations, but did not explain why he made the decisions he did. He reached an incorrect conclusion by not isolating the variable entirely. It is clear that this student has an idea of what is going on, but lacks some of the fundamental knowledge necessary to solve the problem correctly. I expect that this student will benefit from learning and practicing basic mathematical vocabulary so that he can better organize his thinking in these types of problems.
March
Student #1 benefited from practicing mathematical vocabulary related to the content in the problem. Using the relevant terminology is where he improved the most. His answer does not thoroughly explain the steps that he took, but it is clear that he understands what he did. In another context, this would be an excellent answer. However, the purpose of these assignments is to explain mathematical thinking in writing
Student #2
September
Student #2 clearly understood the process for solving the problem despite making a common error when taking the square root (x = -3 as well). She used some of the vocabulary I suggested in class, but otherwise offered very little in terms of a reasoning for what she did. She merely summarized the steps she showed on the right, and would likely struggle to explain this concept to someone who didn’t already understand it.
March
Student #2 demonstrated an incredible amount of growth on the March assignment. She highlights significant steps in her answer, making it clear how she organizes her thinking. She does a much better job explicating her solution and develops an answer that would be useful for teaching the concept to another student.
Student #3
September
Student #3 makes the same error mathematically as Student #2, but he offers more of an explanation for his decisions. There were several places where the grammar is lacking, but this answer does the best job of explaining a step taken. This student would benefit most from learning to separate their thoughts into distinct steps, so that they have less run-on sentences interrupting the fluency of their writing.
March
Student #3’s response in March shows better organization of their thoughts..His use of mathematical vocabulary improved and it is more clear where his words match the work shown.He used the feedback I provided to better explain why he took each step that he did. He still has an issue with comma use, but they don’t detract too much from the quality of the answer.
Conclusion
Literacy and writing skills are cross-curricular. My students are stronger writers because of the work they have done in my classroom. My teaching oriented toward improving students' mathematical thinking and expression. Students at every level of mathematical readiness demonstrated growth in their abilities to explain and write about mathematical concepts. Emphasizing this growth in my classroom shaped the delivery of content and how students were supported. They learned skills for problem solving beyond what it is typically expected in an Algebra II class. The areas they struggled in with On-demand Writing assignments correlated with their struggles on the ACT. The growth my students saw in literacy and numeracy skills was borne out in their ACT scores. Each student included above made significant progress during the first semester of the year. Student #1 improved from 15 to 19, Student #2 improved from 17 to 22, and Student #3 improved from 21 to 27. This dramatic level of growth captures only one semester of the year, and my students are prepared to extend this growth through the end of the year.