# Differentiating Instruction in Geometry and Pre-Calculus

Bryan Coe, Math Teacher, Rush-Henrietta High School. bcoe@rhnet.org

**Background**

I have been teaching mathematics in Rush Henrietta for the past 13 years. Over the years, I have taught many different levels of mathematics, ranging from Algebra at the ninth grade level to Pre Calculus in the high school. I have always been interested in learning new approaches to use in my classroom to improve my instruction and have a better impact on my student’s success.

When I first started teaching, I was primarily a teacher who used direct instruction. I was big on having a routine. My class typically started with a warm-up. While the students worked on their warm-up, I would check student’s homework. I was not looking at accuracy but just checking for completion. After going over the warm-up, I would ask if there were any questions from the homework, and then go over any questions the students had. I would then work through the examples from my notes on the overhead and then have the students practice similar problems. Occasionally if it fit in my schedule, we would do group work. But this consisted of students practicing problems they had already learned.

This method of instruction worked for me and I got typical results in my classroom. The students who did their work outside of the classroom or came in for help were successful on tests and quizzes. Some years results were better than others depending on the caliber of student in my classroom. As I matured as an educator, I started thinking more about student learning and asking about some important questions. Is the method of direct instruction the best method for student learning? How do I know if my students understand the lesson? What do I do about the students who do not write down notes or fall asleep during class?

One of my beliefs about learning mathematics is that students really learn math by doing. They need to have time to practice the skills, discover the big ideas through inquiry, and see multiple methods for solving specific problems. Only then will they truly understand it. The challenge is how to do this with the time crunch we all fall under. We have so many performance indicators that the students need to learn before the regents exam in June. At the time when I was having this internal debate, I was teaching Algebra 2 and Trigonometry and Pre-Calculus. Algebra 2 and Trigonometry is an intense class with many topics to cover and ends with a Regents exam. I did not feel comfortable changing my teaching approach. I stuck to primarily direct instruction, but I made sure I offered plenty of opportunity for students to practice the material. Pre Calculus on the other hand was not a Regents course and it offered me much more flexibility. I shifted my teaching style from direct instruction to a student centered approach. I created student workbooks for each unit. I did a little direct instruction but the students primarily worked in groups and I was able to give mini lessons when the students needed them. This worked great for me. I had a much better understanding of my students needs and their understanding. Students scores improved and I was inspired and re-energized as an educator.

I continued this approach in Pre-Calculus for the next few years, really perfecting my work books. I believed I was differentiating instruction for my Pre Calculus students. I had an eye opener when I went and observed the classroom of my school’s differentiation coach, John Prouty. John taught Chemistry part-time and worked with teachers on differentiation. His classroom was impressive. The students were working on a variety of tasks, and John was monitoring student’s progress. The most impressive thing was that all of the students were engaged. I decided I wanted to work with John and try to differentiate in my classroom. I worked with another teacher and we planned a differentiated lesson in Algebra 2 and Trigonometry. We created a tiered lesson for a topic that students usually struggle with. It was very time consuming, but our results were phenomenal. Students went above and beyond what I expected and their test scores were great. For the rest of the year, I differentiated a few more lessons, created mostly tiered lessons based on student’s readiness. When I joined the MCC Fellows Program and we started discussing possible action research plans, I knew I wanted to do something with differentiated instruction. I have had some success with it but I would like to try to do it more often and also try some different approaches with differentiating. My inquiry question is what impact will differentiating instruction have on my Geometry and Pre-Calculus students? I wanted to look at the social, emotional, and academic impact it had on them. I also wanted to look and see if my students demonstrated growth in their understanding of the concepts that I differentiated and if their test scores improved.

**Inquiry Design**

I teach a total of 98 students dispersed among my 2 sections of Geometry and 2 sections of Pre- Calculus. My Pre-Calculus class is at Rush Henrietta High School and consists of juniors and seniors. My Geometry class is taught at the Rush Henrietta’s Ninth Grade academy, and consists of 9th grade accelerated students. My plan was to differentiate at least one lesson in each of my units in Geometry and Pre-Calculus for the first semester. Differentiation has been linked to increased student engagement and better content understanding. **My goal was to try a variety of approaches for differentiation and see which method was the most effective, which method I preferred as an educator and to survey the student to see what method they preferred as well.** I incorporated some 21st century learning tools and the use of technology within these units as well. I followed this plan in both my Pre-Calculus and Geometry courses.

**Pre-Calculus**

For my action research plan, I used my Pre-Calculus classes to analyze what impact the differentiation had on students test scores and overall achievement. I have taught the class for the past five years and I figured I could compare test score between this year’s students and last year’s classes. I also looked at what happened when I differentiated in one class and not the next.

For my first unit in Pre-Calculus, I decided to give my students a version of the unit test as a pre- assessment. I did this on the very first day of school. It was funny to see the student’s reaction to finding out that they were going to have to take a test on day one. I decided to do this because the first unit in Pre-Calculus is for the most part review from previous math classes. My plan was to use the student’s results to differentiate the topics of study based on need. I figured the pre-assessment would show which concepts each student needed to work on and I would be able to group students based on similar needs. I also offered my students the incentive that if they scored above a 90% that they could use that as their actual test score. The test scores were much lower than I expected, but I was able to see some glaring needs for the entire class. The biggest issue of concern was the concept of inequality, especially when viewing a graph. **The pre-assessment results forced me to change my plan. I needed to completely change my note packet that I was planning on using**. I changed the primary focus of the packet to solving inequalities, specifically when looking at a graph. I did not see the need to differentiate for individual students because the whole class needed more work with inequalities. After the test, I compared this year’s test results with my classes from last year. This year’s class averages for the test were 86.4% and 81.3% and last year’s averages were 81.9% and 82.9%.

For my second unit in Pre-Calculus, I decided to only differentiate in the class that had the lower average on the first unit test. I wanted to compare my differentiated plan with what I have done in the past in Pre-Calculus. For the past three years, I have been using a student centered approach where the students worked in groups on a chapter workbook. I would give mini lessons to each group. I wanted to see if the differentiation activity had enough impact in my one class to narrow the gap between their test scores. For this unit, I gave a pre-assessment before we started the unit and then I gave an **exit ticket** at the end of each lesson. **An “exit ticket” is a tool used at the end of a lesson to obtain feedback from students on their understanding of the lesson content.** Exit tickets provide student-generated data which teachers can use to differentiate instruction. Exit tickets provide immediate feedback to teachers and help clarify if students are uncertain or confused, if they need more practice, or if they have mastered key concept(s) from the lesson.

I kept track of the students individual needs based on the exit tickets. Midway through the unit, I had a review day planned for both classes. For the class that I differentiated in, I assigned individual problems to each student based on needs. The other class was expected to do the entire assignment. I was able to work with each kid on their needs during the lesson. The lesson went well and I felt both classes were ready for the unit test. The class that received the differentiation test average was a 76% and the non-differentiated class average was an 82%. Both classes test average decreased from the first unit. This is common because the material taught in the second unit is much more challenging than the first unit. The differentiated lesson did not narrow the test average gap between my two classes; my one class average is still 6% higher. I wonder if I would see a similar difference if I compared my class average from their Algebra 2 and Trigonometry Regents scores.

For my third unit in Pre-Calculus, I wanted to differentiate in both classes. I felt that my differentiated lessons were having a positive impact on my students learning. I know that the test results may have not shown it but from my conversations with the students and the results on their daily exit tickets, I felt it was worthwhile for both classes. But I wanted to have the students have some say in how the material was presented to them. For all of my previous experience with differentiation I had made decisions based on student’s readiness.** I wanted to see if giving the students some choice would improve achievement.** I gave my students a survey prior to the unit. The students had 4 options for instruction during the unit. Option 1 had the students working in groups on the problems in the workbook. The workbook had examples already done out for the students. Option 2 had the students starting in a small group with me. They would receive a mini lesson and then work on the packet with their group. Option 3 took the reverse classroom approach. I created some online videos for the students with my Pencast software (. The students would have to watch the videos prior to class and then work on the workbooks with their group. Option 4 allowed the students to work in groups on the workbook but if they needed it, computers would be provided for the students to watch the online videos. I expected the results of the survey to be pretty diverse, but the majority of the students wanted option 1 or 2. A handful of students wanted option 4, but none of the students wanted option 3. I wonder why the students were reluctant to try the reverse classroom approach. **What can I do to get them to step out of their comfort zone?** And if it did, what impact would it have on their learning? The unit went pretty well. My test results were inconsistent between both classes. My class that had scored higher on the first two units did much better than my other class. Especially the students who went with option 1 and 2. As I looked at individual students grades most of the students who struggled were the ones who choose option 4. Were my expectations not clear for these groups? Did I not spend enough time in class checking in with them? My other classes test results are a little skewed. The class had to take the test after a week off for Thanksgiving. Even though my test scores were not great during this chapter of my research, having the student choose their method of instruction has merit. The majority of my students were engaged during class and it allowed them the opportunity to learn based on their specific learning style. I plan on using this method of differentiation in the future in Pre-Calculus, but I am going to make sure I use exit tickets so I can assess my students understanding after each lesson.

**Geometry**

This is my first year teaching Geometry, so I do not have any prior classes to compare my results with. However I still wanted to differentiate in each unit and see what impact it would have on student achievement. The first three weeks of Geometry we review some of the major concepts from Algebra. We focused on factoring, quadratic equations, algebraic fractions, and radicals. For each concept I had the students take a pre assessment and I used the results from the pre assessment to group my students. By using this method to group the students it was really easy to address any needs the students had. Most of my students had a really strong background on these concepts and it was easy to extend and challenge them with more challenging problems. During this review, I decided that I needed to extend the algebra review into pre-teaching some of the concepts from Algebra 2 and Trigonometry. My most rewarding differentiated lesson came with Radicals. I created a 3 tiered lesson based on the students pre assessments. The tier 1 examples were radical problems without variables. The tiered 2 examples were radical problems with variables. The tiered 3 was a complete extension from what the students had previously learned. It was an A2T lesson on simplifying radicals involving cubes. My goal was for every student to get through at least tier 2; about half of my students were able to get introduced to tier 3 problems. I plan to incorporate some pre-teaching of Algebra 2 and Trigonometry whenever possible in Geometry. My students this year are pretty strong mathematically and I want to challenge them to go beyond the Geometry curriculum. Many teachers at Rush Henrietta consider Algebra 2 and Trigonometry the most challenging course for our students, so the more practice they get with the material the better off they will be in the long run.

My plan for the second unit of Geometry, Transformations, was to do a reverse classroom model for the entire unit. Unfortunately my Pencast software (http://www.livescribe.com/en-us/pencasts/) did not come in time, and I did not find any online video on you-tube or Khan Academy. I gave my students a pre-assessment for the unit and most of them had strong foundations of the basic transformations. I was able to create a student-centered environment, with some tiered lessons for the students. I was able to give individual mini-lesson throughout the unit. The test scores for this unit were excellent. The majority of my students were above 85% on the unit test.

For the third unit, I was finally able to do a reverse classroom lesson. I did it with the concept of Midpoint. Using my Pencast software, I created an online video for the student to watch for homework. The students then came to class and worked on a tiered lesson on working with the midpoint formula. The majority of my students watched the video for homework and did an excellent job on the tiered problems. For the students who did not watch the video for homework, I had a class set of laptops for the students to use in class. They watched the video and then worked on the tiered lesson. I surveyed the students after the lesson and the majority of the students appreciated the video. They liked being able to watch the video at their own pace and then rewind the video when they needed it. In addition to the reverse classroom lesson, I also did several tiered lessons during the unit. Again my test results were excellent.

My next unit in Geometry was Logic and I decided it was not worth differentiating. I tried to come up with some areas to differentiate but I had no luck. The concepts that we teach in the logic chapter can really only be learned through memorization, so I taught the unit with direct instruction. My results for the unit exam were again excellent.

For the next unit in Geometry, I gave the students the same survey that I gave my Pre-Calculus students. I wanted to differentiate the lessons based on student preferences. Most of the students wanted option 1, and a few students wanted option 2 and 4. Again none of the students wanted option 3, the reversed classroom. My students appear to be reluctant to choose a method that is different than their normal routine. As the unit progressed, none of the students used the computers to watch the online videos; they stuck to working in their groups. My test results were again good. I plan to try to differentiate more in Geometry as we move further in the curriculum.

Class averages on unit tests for two Geometry classes involved in this action research are as follows:

Unit 1: 90.4%/90.9%

Unit 2: 92.2%/85.1%

Unit 3: 88.8%/87.8%

**Reflection**

My goal for my action research on Differentiated Instruction was to show that it had an impact on student achievement and that I would be able to see improvement in test scores. After doing my research I have come to the conclusion that it is very hard to show this quantitatively. I did not differentiate every lesson in the units, so to jump to the conclusion that the single differentiated lesson changed test results may be unrealistic. But I did try several different strategies to analyze the impact it had on student’s achievement. I compared my first unit test scores from this year’s class with last year’s classes. This year’s class averages were 86 and 81 compared to 81 and 82. My test scores did improve from last year, I would like to think it was due to the changes that I made, but it is a completely different group of students. I also, for my second unit, differentiated in one class and not the other. In the class that did not receive the differentiation, test scores were higher. But again I am comparing two different classes and they have different students with different levels of mathematical ability. I also wanted to see if giving the students choices in their differentiation if it would improve test scores. The class averages for the 3rd unit test were actually lower than the first two units, but the content in the 3rd unit is much more challenging than the first two units. Probably the one area where I can quantitatively show growth in student achievement based on differentiation is when I looked at 4 of my current students who I also taught last year in Algebra 2 and Trigonometry. Last year these students primary source of instruction was direct instruction. This year in Pre-Calculus the instruction is student centered and they have received several differentiated lessons. Student 1 overall average last year was 89%; his first quarter average in Pre-Calculus was 94%. Student 2 overall average last year was 82%; his first quarter average in Pre-Calculus was 87%. Student 3 overall average last year was 82%; his first quarter average in Pre-Calculus was 84%. Student 4 overall average last year was 76%; his first quarter average in Pre-Calculus was 88%. All of these students’ averages have improved from last year. I know it is a different course, but my teaching practices are completely different.

I also wanted to hear from my students on the impact of the differentiated lessons. I surveyed both my Geometry and Pre-Calculus students several times during the semester. The results were overwhelmingly in favor of a student centered classroom versus a teacher centered classroom. The majority of the students appreciated when I differentiated based on needs from pre assessments. They liked not having to do problems that they already understood, they felt when teachers do this it is overkill and they lost interest in the class. The students also appreciated when I allowed them to choose which method of instruction they followed. They felt I was allowing them to take ownership on their learning. They also appreciated the videos I created with the Pencast software, but were reluctant to watch them for homework and have a reversed classroom. They found them useful as a resource to access when needed.

To truly assess what impact the differentiated instruction had on my students, I really have to use my own personal reflections. As I reflect back on my lessons from the first semester and the different approaches that I tried, I truly feel that the differentiation had a positive impact on my students. My students showed a deeper understanding of the topics that were differentiated. The evidence came from conversations I had with the students during the lesson and also from the exit tickets at the end of the lesson. One of the things that I will take away from this experience is the importance of pre assessments and exit tickets. The pre assessment really allows me to see what prior knowledge the students have and should truly guide lesson plans. It can really help me decide when to differentiate and when not to. The exit tickets allow me to get immediate feedback on student understanding after the lesson. It shows me which students mastered the concept and which ones need more clarification. I can use this data to differentiate the next lesson.

This action research experience with differentiation really had a huge impact on me as an educator. I learned a lot about myself and my beliefs in education. I believe that students get the most impact out of a class when they are actively involved in their own learning. They truly learn mathematics by doing and experiencing the problems. By differentiating, I shifted my focus to presenting the material based on the individual student needs and less on finding the most effective way to present the content to the entire class. One of the things that I learned is that differentiation is a time consuming process. It takes a lot longer to plan these types of lessons, but the impact it has on students is totally worth it. My students were much more engaged during the lessons. I was able to really focus on each individual student and what they needed. I was much more informed on how long I needed to focus on a specific topic, based on the exit tickets and conversations I had with each individual student.

I have learned that there are a variety of ways to differentiate and I don’t always have to differentiate based on readiness. Differentiating lesson based on student’s preferences for learning can be just as effective. Student’s motivation increased when they were given some choices. It also allowed me to better understand what type of learners I had in my classroom.

I also learned that tiered lessons are a great way for me to enhance the learning experience for the gifted students in class. These students typically are not challenged and are bored with the typical lessons. By using a tiered lesson, not only do they get to practice the required outcome, but I can go much deeper. The gifted student is not the only student who benefits from a tiered lesson, other students can go farther in their math experience than they normally would as well. My students took the tiered lesson as a personal challenge and they aimed to get through all of the tiers.

I plan to continue to differentiate in both my Geometry and Pre-Calculus classrooms. I will not do it every day because that is not a feasible approach, but I will do it whenever possible. In my Pre- Calculus classes, I will continue to allow students the choice on their differentiation. But I am going to use more exit tickets and set specific classroom outcomes for each group. I also hope to challenge my students with tiered lessons. **Some things that I am left thinking about are what units in Geometry and Pre-Calculus are best suited for differentiation? When should I differentiate based on readiness? What about based on student’s learning styles? Which option has the most impact on my student’s learning and understanding?**

I want to do more with the reverse classroom. I have the technology resource to use with the Pencast software, now I need to get the students on board. **What can I do to get my students to buy into trying the reverse classroom? Is the only way to get them to do it is to assign it to all of them?** I think I will try it with my Geometry classes first. These students seemed to buy into the flip classroom lesson more the first time I tried it and I also have more time with them.

**Conclusion**

In conclusion, I feel that differentiation has a positive impact on both students and me as an educator. It allows me to have a better understanding of my students’ needs and learning styles. It allows me to focus my instruction to each individual student. Pre-assessments and exit tickets are vital tools to use during differentiation. They give me concrete evidence that shows students growth in their understandings. It also allows me to focus on my students’ needs when planning lessons.

**Resources**

Adams, C. *“Tiered Lessons: One Way to Differentiate Mathematics Instruction.”* Gifted Child Today 27.2 (2004): 50-65

Anderson, Kelly M. *“Differentiating Instruction to Include All Students.”* Preventing School Failure 51.3 (2007): 49-54.

Burgin, S. Helping PLC Members Share Their Work With Others, pages 149-155: *“A Demo-a-Day in High School Chemistry.” *

Tomlinson, Carol Ann. *“Mapping a Route Toward Differentiated Instruction.”* Personalized Learning 57.1 (1999): 12-16

Pine, G. Teacher Action Research: Building Knowledge Democracies (2009, Sage Publications), pages 254-263: *“Conducting Teacher Action Research.”*

**Sample Exit Tickets:**

Pre-calculus chapter 2 exit tickets

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