Applying past knowledge
Accessing prior knowledge; transferring knowledge beyond the situation in which it was learned.
Introduction
Applying Past Knowledge to New Situations is a critical part of the learning process in any math classroom. This Habit of Mind is embedded in my instruction more so than any other. The curriculum is linear, and each topic builds on the previous. In a successful math classroom, students are building connections between units and developing a logical progression of mathematical thinking. Introduction of new content centers around the idea that students are already equipped to handle the problems, they just need to be shown how. Convincing them of this has been my goal. My students need an extra push to apply what they learn to new situations, and this Habit of Mind serves that purpose. Students should be intentional about how they approach new challenges and call upon past knowledge to solve them.
Table of Contents
Explicit Teaching
Being Intentional
The breadth of content covered in Algebra II can be overwhelming for students, so explicitly teaching them how to apply past knowledge can help minimize the frustration with learning something new and challenging. This introductory lesson focused on the mindset students bring into the classroom and how it impacts their ability to learn the content. Learning is not a passive activity, but my students were never given the tools to actively engage in their own learning. This lesson begins to rectify that.
Before applying this Habit to a math context, I wanted students to consider how their life outside of school could benefit from Applying Past Knowledge to New Situations. The personal questions they answered made discussion natural. To close out, I pointed out to students that they could always look for ways to apply what they know. As we transitioned into the content lesson for the day, I frequently referred back to discussion we had at the beginning of the day. There are many math skills that students take for granted, and I took every opportunity to point out that students already know a great deal.
My classroom is discussion-focused, but I wanted students to spend time reflecting individually on their own experiences before we started the discussion in this lesson. This slide was shown at the front of the room as students recorded their answers on a google form.
This activity was effective at getting students to reflect on their personal growth as individuals. This student identifies that they have a clearer understanding of themselves in relation to others because of the experience they had.
These responses are good examples of what students were learning from the mistakes they wrote about. It shows that they are already learning from past experiences, but they may not be aware of it without attention being drawn to their growth.
Both of these students are identifying important aspects of their high school experience that they can take into other aspects of their lives. While their takeaways are almost complete opposites, both demonstrate that students are thinking forward about how they will apply these experiences in new ones.
Teacher Reflection
This lesson helped students recognize times when they have learned. Seeing how an experience impacted them can change the way they feel about it. The questions reframed things that students know in a way that they can use elsewhere. Their negative experiences take on a new meaning, and they already use them to inform how they act going forward.
This lesson served as the important first step for shifting students’ mindsets when it comes to being intentional about learning. I think students engaged with this lesson, but I do not feel like the mindset registered as deeply as I would have liked. After this lesson, I planned additional reinforcement lessons and implemented ways to more regularly enforce this Habit.
Reinforcement mini-unit
Content-based Reinforcement
To reinforce this Habit in content with my students, I taught sequences to build on what they already know about linear and exponential growth. This is something many people intuitively understand, but usually not in a formal mathematical way. This mini-unit began with this idea and used the Habit of Applying Past Knowledge to challenge students to master this content in a new way. Students were refreshed on their experience with linear and exponential functions so that they could explicitly identify what “new” things that have already learned before. They were anticipating that the same rules from linear functions would apply to arithmetic sequences. Rather than tell students about the relationship between arithmetic/geometric sequences and linear/exponential functions, they discovered them by being intentional about the way they learned new content.
I led with an exploration for a simple arithmetic sequence that was framed as a linear function, a problem that students are used to seeing. This is where I made the explicit recall to past knowledge students already possess. As we learned terminology for sequences, students were actively identifying connections to linear functions. In the notes template I gave students, there was a section for students to draw an explicit connection to the linear function. In the end, they had a linear function and an arithmetic sequence side by side, with the proper terminology for each.
This student was able to correctly translate their knowledge of linear functions to the new language they learned about sequences. They used what they learned previously to solve new challenges.
Student Data
As part of the diagnostic and the beginning of the mini-unit, students completed a questionnaire about their use of Applying Past Knowledge to New Situations. For the most part, the data indicated that students were not actively using this Habit. When comparing this to the summative data from when they took the survey again, it was clear that students made progress on this Habit of Mind.
Diagnostic Data
Summative Data
A trend I noticed in both the diagnostic and summative data is that students better know how to use past knowledge for planning new learning plans. This was something It actively led students to doing in the mini-unit. On the other hand, there was little to no improvement in the third row. Going forward, I know to be more intentional about how I guide students to think about how their past knowledge affects new learning.
Teacher Reflection
The purpose of this mini-unit was for students to realize how mathematical concepts build on one another. There is an explicit and straight-forward connection between linear functions and arithmetic sequences, and I taught students how to recognize that relationship. Similar relationships exist throughout mathematics, and students are better equipped to build upon prior knowledge.
Above is the aggregate data from the diagnostic and summative survey. The total number of students who selected each response is given with the percent in parentheses.
The majority of students indicated on the diagnostic that they were not actively implementing this Habit of Mind. The content taught between the surveys was focused on teaching students to develop learning plans for approaching new content. The only content taught came when I introduced students to the concept. My role was facilitator: I assisted in connecting their prior knowledge to a new topic. Despite not explicitly teaching arithmetic and geometric sequences, the class average for the assessment increased from 54% to 80%. This demonstrates that students are taking ownership of their learning and are making academic progress through the Habits of Mind. At the end of the unit, nearly 80% of my students indicated that they were understanding their previous learning in new ways beyond the situation in which it was learned.
Regular reinforcement
Engineering Design Process
In terms of applying this mindset, I don’t believe there is a better, more explicit method than the Engineering Design Process (EDP). The EDP highlights the iterative nature of solving large problems. In order to create the best possible solution, it is imperative to learn from previous attempts and past knowledge. The cycle of improvement relies on being very intentional about why certain decisions are made in the search for a better solution. This poster hangs in my room for me to reference as students are tackling difficult problems and large projects in my Challenge-based Learning units.
These challenges are perfect for students to call about their stores of knowledge and think creatively to deliver the best product. Students use the Habit of Applying Past Knowledge to continually refine their product by going through the same process of evaluation multiple times. "Gathering Information" fundamentally changes meaning after students have gone through the rest of the process. The focus shifts to abstracting meaning from their initial results and understanding why they got the results they did.
The emphasis on communication in the EDP benefits the teaching of Applying Past Knowledge to New Situations. Understanding how to communicate complex data comes from developing this Habit. Clearly identifying past knowledge creates a through-line for the audience to follow the student's logic.
Student Work
In the project students completed for sequences, the benefits of Applying Past Knowledge can be seen in the improvement in the solution as they built upon their initial ideas. In this project, students were instructed to find any solution that worked first, then look for ways to improve that answer. As a result, their first drafts did not achieve high scores. However, they were more quickly able to recognize areas to use what they learned by going through the process. Their scores improved rather dramatically at first, then it became a challenge to find ways to make smaller and smaller improvements as they approach the maximum score of 49.10%. Both of these students made significant progress.
Scores move from right to left as students implemented different solutions.
This student had a higher initial score of 35.66%. It took them less iterations to get to their highest score of 48.05%. In the end, their best score was lower than Student 2's.
Applying Past Knowledge is naturally embedded in the work students do in my class. The Engineering Design Process allows me to make this Habit more explicit to my students. As they internalize the principles of the EDP, they are developing a stronger Apply Past Knowledge mindset in addition to a plethora of other useful problem solving skills.
In these students’ write-ups about the project they completed, they demonstrate that they have internalized this Habit. Due to the nature of the project, each attempt was essentially a new situation. Every change they made had a significant impact on the options available, so they had to be intentional about the choices they made. In analyzing their process for solving the problem in the project, they are walking through how they learned from and applied their knowledge from previous attempts.
Student Data
Students completed a survey at the end of the engineering project to gather data for The University of Cincinnati. The survey was focused on measuring the degree to which students engaged with principles of engineering, including the Engineering Design Process that I incorporated into this Habit of Mind. There are multiple questions that explicitly relate to Applying Past Knowledge to New Situations that were of particular interest to me. A portion from multiple students' surveys is included below.
The forced-choice portion of the survey is included above. The questions that specifically relate to Applying Past Knowledge to New Situations are highlighted in the red box.
Students with varying levels of success on the project are included to demonstrate the breadth of students my teaching this Habit of Mind has reached. Students at all levels are implementing the Engineering Design Process and are recognizing their own personal growth. Several students were frustrated with the iterative nature of the project while they were completing it, but the conversations we had as a class after it was finished made them realize the purpose. While students were reflecting on the process, they were able to identify the benefits of the approach they took. The information they gathered informed new attempts, and they saw how their intuition played into their success on a challenging project.
Aggregate data for the questions of interest are included below.
These are the responses to Question 19 of the survey students took at the end of the unit. This question asks students to reflect on their application of Applying Past Knowledge to New Situations.
Overall, the students surveyed indicated that they are thinking about their prior knowledge as they tackle new experiences. In the Engineering Design Project, students drew from a variety of experiences to shape the attempts they made for solving the challenge. Applying Past Knowledge to New Situations is an explicit part of that process, and students demonstrated awareness of their own development of that skill in these survey responses.
Student Reflections
Written Reflections
In March, students reflected on various aspects of their personal growth. I have explicitly taught this Habit of Mind throughout the year, and students were asked to write about the extent to which they use Applying Past Knowledge to New Situations in their lives. As we have entered non-traditional instruction, students are connecting skills they learned earlier in the year to tackle new content independently.
Student #2 recognizes the hidden curriculum behind mathematics instruction that relates directly to this Habit of Mind. They see how their prior knowledge prepares them to learn new things. Student #3 builds on this, writing that this Habit of Mind has helped them solve questions more efficiently. These students' awareness that they are using past knowledge is a critical step in internalizing this Habit of Mind. New learning going forward will be informed by this understanding, and students will master new content by seeing how it utilizes what they already know.
Embedded Reflection
Students are regularly reflecting on their growth in Applying Past Knowledge to New Situations within content lessons. As students are engaging with new content outside the classroom environment, it is incumbent on them to play multiple roles in their education. They are largely teaching themselves, and it is vital that they are actively reflecting on their learning as they tackle challenging content. To accomplish this, students write about their understandings of the topics they engage with.
In this online lesson, students practiced problems around multiplying and dividing rational expressions. Operations with rational expressions can be one of the most difficult topics for students to grasp in Algebra II. It is a higher level concept that requires a strong foundation of the underlying skills, and my students are demonstrating awareness of the connections between old and new content.
First, I explicitly ask students how they saw these problems connecting to work we have already done. More importantly, students are recognizing where the challenges are with a topic and proactively working around them.
This student points out where rational expressions connect to what they already know about fractions and properties of exponents. They can see how the underlying skill is important to understanding rational expressions.
This student sees how their past knowledge relates to the skills needed to complete this assignment.
Second, I ask students to consider what helped them solve the problems. Since they were engaging in this practice with minimal direct instruction, it is important for me to see where students are having success. For most of my students, my pushing them to apply past knowledge is referenced as the key to their success.
I reminded students that they learned Keep, Change, Flip from their middle school teachers in the Classroom assignment. That helped this student with the trickier part of the problems - dividing.
This student has internalized Applying Past Knowledge to New Situations, and is using it to find ways to learn new content.
Reflecting on this Habit of Mind has made my students stronger learners. They are seeing connections between their study habits and success on assignments. They are not simply talking about this trait but are using it to grow personally and as a student.
Conclusion
Learning in math is not discrete, but it can certainly feel that way to students. Planning units of study with this Habit in mind shifts the approach I take to introducing new content. Each unit builds on the foundation from the previous to take students to new levels of mathematical thinking. Incorporating this Habit of Mind in the classroom brings students into the curriculum designing process just enough that are equipped to explore their own thinking in new ways. My students have begun to learn new content knowing they will have to apply it in higher level math courses, and it has changed the way they have engaged with the material. Understanding that knowledge transfers beyond the situation in which it was learned makes every experience more impactful.