Math 120R is a Precalculus/Trigonometry course that prepares first year students majoring in STEM for Calculus. Out of this class of 31 students, the breakdown for majors is as follows, Biology/Physiology (18), Engineering (5), Computer Science (3), Economics (2), Astronomy (1), Geosciences (1), No Major (1). With the exception of 1 student, Calculus is a core course these students must succeed in for their academic and career goals, which is why this course is very important for them. 

Through initial questionnaires given as homework, students in Math 120R come from a diverse math background. Some students have not taken any math courses since their sophmore year of high school due to the Covid-19 pandemic, while other students have been introduced to Calculus concepts such as derivatives but did not achieve satisfactory results on a Calculus placement exam due to unfamiliarity with Trigonometry. A negative consequence of this diverse background is that students will learn at different paces. 

An initial question while working on designing this TAR project is how can I reach these students with a diverse mathematical background caused by the pandemic? This led me to The Wisconsin Department of Education website that listed the Scaffolds and Supports approach as a possible way to "address unfinished learning" caused by the pandemic and "accelerate learning". Further reading into education literature led me to the findings of Aaron and Herbst (2019) which states that "participants (teachers) valued discussion as an opportunity for students to discuss and expand their understanding of mathematical objects". The question that followed was what type of lesson plan can I come up with to utilize the ideas of Scaffolds and Supports while also getting students to have discussions? This led me to the findings of Marriotti (2006), who found that a "persistent separation of the activities of conjecturing and proving leaves students with an incomplete view of the discipline of mathematics and robs students of the complex and rewarding experience of mathematical discovery". 

With this in mind, I came up with a lesson plan where students, working in groups, will conjecture their own ideas about mathematical principals, followed by a classwide discussion on each groups findings.

The desired learning outcomes for this TAR project are 1) students can utilize the key properties of a logarithmic function in computations, and 2) students express a positive attitude to math learning when they can participate in the education process. 

From the findings of Aaron and Herbst (2019), I wanted to design a lesson plan where small group discussion and class discussion were both integral. This also leverages the diverse math background these students have, where the group discussions act as a soft Scaffolds and Supports approach. The classroom for this class is a collaborative environment, in that four students sit at a larger table with a white board at each table, so the classroom setting made it effortless for students to work in small groups and have discussions.

With this in mind, I designed a guided worksheet revolving around logarithmic functions for each group to work together on.  The worksheet includes many numerical examples, from which students are asked to try to determine what properties and laws of logarithmic functions they might derive. The idea here is to simulate how scientific discoveries are made, through observations of nature and formulating conjectures. I believe this approach catered to these students will be beneficial to their academic career as more than two-thirds of them are majoring in physical sciences. 

For assessments to the desired learning outcomes, I gave students a survey of free response questions on their familiarity with the topic of logarithmic functions, and a quantitative assessment of their familiarity with the topic solving numerical problems. Then after the activity, students answered another survey of free response questions on their familiarty with the topic and their feelings on the activity, combined with another quantitative assessment. In addition, 1 problem on their midterm exam tested specifically their familiarity with the materials on the guided worksheet, which served as an additional quantitative assessment. 

For analysis of the data, I went through the free response surveys and focused mainly on student feelings on the expectations pertaining to logarithmic functions for this course. I grouped students based on their responses after the guided worksheet activity, and further grouped them based on their pre-guided worksheet activity responses. In addition, I analyzed the average midterm score (out of 10 points total) for each of these subgroups, and the percentage increase on the quantitative assessment (worksheet) they were given before and after the guided worksheet activity (out of 12 points total). 

There were 12 students who felt "Confident/Great" on the stated expectation, with an average midterm score of 7.25 and a 52.7% increase on the quantitative assessment after the activity.

1. 8 out of the 12 students felt they needed more practice on the topic prior to activity

   • Average midterm score of 7.25

   • 62% increase on the quantitave assessment after the activity

2. 4 out of 12 students felt confident on the topic prior to the activity

   • Average midterm score of 7.25

   • 33% increase on the quantitative assessment after the activity

There were 16 students who felt "Okay/Comfortable" on the stated expectation, with an average midterm score of 8.87 and a 37.5% increase on the quantitative assessment after the activity. 

1. 9 out of 16 students felt they needed more practice on the topic prior to activity

   • Average midterm score of 8.11

   • 37% increase on the quantitative assessment after the activity

2. 4 out of 16 students felt okay/comfortable on the topic prior to the activity

   • Average midterm score of 10

   • 48% increase on the quantitative assessment after the activity

3. 3 out of 16 students felt confident on the topic prior to the activity

   • Average midterm score of 9.6

   • 25% increase on the quantitative assessment after the activity

There were 3 students who felt they "needed more practice" after the activity, with an average midterm score of 9.33 and a 25% increase on the quantitative assessment after the activity.

With an average midterm score of 8.29 and a median midterm score of 9, I believe students effectively learned the Laws of Logarithms through this activity. In regards to whether students had a positive experience :

1. 16 students found the activity to be "helpful/useful", with an average midterm score of 8.06 and a 36.97% increase on the quantitative assessment after the activity.

2. 11 students preferred the activity with the guided worksheet over taking their own notes, with average midterm score of 8.54 and a 55.3% increase on the quantitative assessment after the activity.

3. 4 students preferred taking their own notes and doing more computation examples than the activity, with an average midterm score of 8.5 and a 27% increase on the quantitative assessment after the activity.

Based on these free responses, I believe students in general had a more positive experience in learning math when they were able to discover math principals on their own through discussion with their peers. 

I was suprised by the average midterm score based on their survey responses. I had expected those feeling confident to get the higher averages compared to those feeling less confident, but instead the reverse happened. A colleague of mine suggested that this could be due to students having a good intuition for the amount of effort they need to put in to get their desired grade. Some students might be okay with getting a letter grade of C, and they felt confident enough to get that grade. While on the other hand, students who are aiming to understand everything, might always feel there is something else they don't fully understand yet. 

Creating this particular guided worksheet activity made me realize that not all topics are suitable for this type of teaching approach. Some topics are better learned when it comes from a traditional lecture, while some topics are better learned when it comes from group discussion and discovery. In addition, this experience has influenced my teaching philosophy. Knowing your audience is extremely important when it comes to lesson design, as it can lead to more effective communication with students. In particular, I find that it helps students find math exciting to learn, thereby changing their perception of a subject that traditionally has not been well received by students. 

• Mariotti, M. A. (2006). Proof and proving in mathematics education. In Handbook of research on the psychology of mathematics education (pp. 173-204). Brill.

• Aaron, W. R., & Herbst, P. G. (2019). The teacher’s perspective on the separation between conjecturing and proving in high school geometry classrooms. Journal of Mathematics Teacher Education, 22, 231-256.