I Remember That! Recognizing and Using Prior Knowledge to Solve Problems
LHSS Math Department: Annie Adams, Scott Denoyer, Trey Fisk, Robert Ladage, Dawn Martin
Students learn new mathematical concepts within the context of their own prior knowledge. This knowledge includes ideas and concepts acquired in a previous unit or lesson as well as concepts acquired in previous grade-levels, courses, or daily life. Helping students recognize and adapt this prior knowledge is a central ingredient for solving problems and learning new concepts. It’s also a central challenge for math teachers in any grade-level or curriculum, elementary through higher education.
Our Lutheran South math team launched a new project to tackle this challenge beginning with a Finite Math and Pre Calculus lesson on solving systems of linear equations using matrices. The lesson was structured around a pre-post problem design involving a system of three equations and three unknowns. To solve the problem, students needed to recognize and adapt several discreet elements of prior knowledge from previous math lessons:
Identifying key elements to draw from word problems
Writing linear equations from a word problem
Taking a system of equations and solve it using matrices as a tool
Interpreting results to answer the question
Students first attempted the problem at the beginning of the lesson prior to any teaching, review, or explanation. Our team anticipated that many students would struggle to complete a solution and would not recognize the prior knowledge they already had to tackle the problem. By using the same problem for a second attempt at the end of the lesson, both the teachers and the students had an opportunity to observe the difference in their problem-solving experience after some teaching/coaching about use of prior knowledge.
Learning Opportunities
Between the pre and post problem attempts, we constructed two pivotal learning opportunities for students.
The first was a “making connections” activity where students worked in pairs to analyze and identify relationships between a set of pre-selected topics (tools, vocabulary, types of problems, previous learned mathematics). We anticipated this would heighten students’ consciousness of the important connections between various topics and ideas that reside in their prior knowledge. We also hoped it would help students gain some practice in looking for important relationships that might not be readily evident, including the relationship between matrices and systems of equations (the central topic of this lesson).
The second activity was a coaching and modeling segment on a different system of equations problem, which was slightly less complex (two equations and two unknowns). The goal of this activity was to help students break down the stages of problem solving while also using guiding questions to introduce a problem-solving routine students could use independently for self-questioning about prior knowledge. Our rationale for using a less complex problem during this modeling segment was to preserve some productive struggle at the end of the lesson when students again attempted the more challenging problem with three equations and three unknowns.
Pre-Post Problem Solving Results
Assessment results from the pre-post problem attempts are listed below, based on work samples from twelve representative case study students.
The most significant change was the increase from 4 to 9 students who were able to “write linear equations from a word problem” on the 2nd attempt. Administrators and team members who observed the lesson noted that a good number of students remembered to apply guess & check as a methodology during both problem attempts, but only two students used matrices.
Pre-Post Student Reflection
In addition to collecting evidence from observations and students’ pre-post problem-solving work, we also asked students to rate their own comparative perception of the problem’s difficulty at both points during the lesson and to complete a pre-post self-evaluation on their own level of mastery with using of prior knowledge. We used this data, combined with the other evidence we collected, to better understand students’ level of ability and confidence with applying prior knowledge to solve problems.
The self-reflective difficulty ratings showed:
10/12 case study students rated “determining whether number of variables matches number of equations” as Easy on the 2nd attempt. 6 students changed their ratings from “hard” to “medium” or “medium” to “easy.”
5 students changed their ratings from “hard” to “medium” or “medium” to “easy” on “interpreting the solution.”
Overall, the case students who describe themselves as “low ability level” from Exit Slip #1, showed relative growth in confidence (difficulty ratings) on the 2nd attempt, although actual work showed little or modest improvement with solving the system of equations.
Most students rated “identifying what was being asked” as “easy” on the 1st attempt and all students rated this step as “easy” on the 2nd attempt.
Finally, the teachers asked students to do a pre-post self-evaluation of their ability-level with problem solving and using prior knowledge (see Exit Slip results below). No students rated themselves as “low” and the number of students rating their ability as “high” changed from three to seven.
Teacher Reflections
Based on these results and additional evidence recorded by observers, our team identified three major findings from the research lesson.
1) Teaching and reinforcing routines for applying prior knowledge
Introducing a list of problem-solving stages and modeling this process helped give students a plan for approaching the problem and a routine for applying problem-solving steps they had learned in the past. Reviewing and practicing these steps also seemed to give students more confidence with solving and interpreting problems, even though many of them still struggled to accurately solve the problem and interpret a solution. The confidence shows they may feel more comfortable with recalling and applying the problem-solving framework even though they missed some of the details in their solutions. This was an encouraging sign of student tenacity and persistence which seemed to increase when they realized they had the prior knowledge and skills to tackle the problem.
Going forward, our group plans to reinforce the use of this routine and provide more opportunities for students to practice “making connections.” We need to help students not only develop proficiency with individual mathematical tasks, but also help them recruit those skills to make connections with new problems. We are working on ideas such as a classroom “word wall” where students could readily connect daily work to the larger scheme of topics and concepts learned throughout their math careers.
2) Matching problem-difficulty with methods
Many students chose to use guess and check on both problem attempts because they found this to be sufficient for solving the problem. They were using prior knowledge, but simply chose the method they found most practical for solving the assigned task. If we want to encourage students to use prior knowledge with more difficult methods (e.g. matrices), we should design sufficiently rigorous problems that don’t have “nice numbers” (e.g., using an unfamiliar money system with odd denominations of bills such as $3.21, $7.63, $19.21). It doesn’t make sense to ask students to use advanced tools like matrices on simpler problems where guess and check might be sufficient.
3) Understanding and assessing prior knowledge
Finally, as we reflected on results from this lesson, it was difficult to discern whether students were merely repeating a set of steps they just saw on the board, or whether they were actually making a connection to key concepts and skills from previous learning. It will be helpful in the future to create more time and space between practice and assessment, and to find ways of listening to or recording students as they verbalize thinking. It will also be critical to develop a more precise understanding of how to define and assess the application of prior knowledge in mathematics.
We hope to build on this experience with other courses and help students continue developing both effective strategies for using prior knowledge and increased confidence for tackling problems in unfamiliar contexts.