2. Instruction – The teacher uses research-based instructional practices to meet the needs of all students.
2.3 Reflecting on Teaching
Teacher makes an accurate assessment of a lesson’s effectiveness and the extent to which it achieved its instructional outcomes and can cite general references to support the judgment.
Math lessons, whether they are provided by a curriculum or of the teacher’s original design, may not cover all of the essential understanding required by a unit assessment. In other words, each lesson’s effectiveness will vary based on the population of students and their current background knowledge. While I’ve been teaching elementary mathematics, I’ve noticed opportunities where I can expand on students’ knowledge. For example, when recently grading math assignments, it was clear that several students were exhibiting difficulty with writing about mathematics. While most students showed that they were developing computational fluency they typically responded to a story problem with a picture or an equation rather than writing explanations. Also, most students were not using academic vocabulary in their writing. In other words, assignments would specifically call for explanation and students would write very simplistic responses.
When evaluating the math curriculum, there weren’t many strategies in place to teach these skills, at least at this point in the school year. I decided to utilize this evidence to design a lesson on problem solving. This fell in line with the curriculum and provided an opportunity for students to engage in some deeper level thinking about problem solving. After I introduced this lesson, the learning target, I can utilize models to represent equations with mixed numbers, was discussed as a class, providing a deeper understanding of why models are helpful to use in mathematics while also reviewing academic vocabulary. In a previous lesson, students were introduced to the vocabulary Proper Fraction, Improper Fraction, and Mixed Number. During the introduction to this lesson, students were reintroduced to these terms but with visual models associated with them (Figures 1 and 2).
After some direct instruction presenting a background on problem solving (see Figure 3) and assessment of student understanding of the learning target, I then passed out fraction strips; students had used fraction strips on previous assignments.
Students were also given a packet containing four story problems, presenting each problem on a graphic organizer along with steps for students to explain their thinking. Using the document camera, the first problem was reviewed and solved with students, evaluating their understanding of the assignment and presenting them with one strategy to solve the first problem. After circulating, I realized that students were also struggling with the second problem. I reviewed this problem with them as well, noting that it required some higher-level thinking (students couldn’t solve this as easily because they had to convert mixed numbers to improper fractions, something that was a newer concept to them at the time). Students continued to work in pairs to solve the remaining two problems. As I circulated, I noticed that several students were really enjoying the lesson. I also noticed that some were struggling with drawing out the fraction strips. I continued to provide help to individual students when necessary.
After the lesson and while reviewing student work samples, I noticed that most students were able to solve problems utilizing computational procedures, yet some still struggled with the deeper thinking processes of using fraction strips and drawing models. Also, many students struggled with explaining in written form how they would use fraction strips to solve a problem as represented in Figures 4 and 5.
Being the first time they had been introduced to an assignment like this, I decided I will reintroduce a similar assignment again. As described by Marzano (2007), repeated exposure and revisions to content will help them retain and build on prior knowledge. Rather than provide this exact format, I plan to make changes to the assignment. First, students will be presented with one or possibly two problems. Four problems presented too much work and less of the deeper level thinking I was hoping for. Students should also be able to use a visual model that works for them, communicating that there is more than one way to solve a problem. This will also provide opportunities for students to present different ideas to their classmates. I may introduce one problem prior to the start of a math lesson, to get students in that frame of mind. Lastly, I will utilize even more opportunities to gather evidence of student voice, reflecting on the problem solving activity with them, after it is complete and how they accomplished the goals of the learning target.
In conclusion, due to the evidence provided by student work samples, it is clear that this class needs more instruction on how to explain their thinking. Similar to their writing assignments, students will be asked to describe their thinking using academic vocabulary words. During assignments containing story problems, I will edit the instructions with sentence frames and prompts to assist students in acknowledging the expectations provided by the verb “explain”. The more they practice explaining mathematic strategies, the more likely they’ll be to write this thinking on paper. I will also introduce a representation of this activity again, providing more practice after students gain more understanding and instruction in regard to their writing about mathematics.
Marzano, R. J. (2007). The art and science of teaching. Alexandria: Association for Supervision and Curriculum Development.