Ashlie I. Christian This paper was completed and submitted in partial fulfillment of the Master Teacher Program, a 2-year faculty professional development program conducted by the Center for Faculty Excellence, United States Military Academy, West Point, NY, 2015. Take boards. Those words have been used in classes at the United States Military Academy on a near daily basis for over 200 years, and are often still used. It has frequently been said that perhaps no one method has so influenced the quality of the instruction of the cadets as the blackboard recitations. Major Thayer, Father of the Academy, insisted on this form, although the use of chalkboards for students to practice problems is credited to Mr. George Baron who taught the first mathematics lesson at West Point, which was the first lesson in any subject at West Point, on 21 September 1801, six months before the Academy was legally founded. He used a standing slate to teach algebra to a few cadets, this being the first recorded use of the blackboard in the United States. (Joe Albree, 2000) Math classes at West Point still run following those guidelines that have been used for well over two hundred years. For the purpose of instruction and recitation in the mathematical department each class is divided into convenient sections. Originally these sections numbered from ten to fifteen cadets; now there can be more, but no more than 18 cadets in any section. Taking boards meant every member of each section is, if possible, required to daily explain at the blackboard, or wall slate, one or more key concepts of the lesson given out on the previous day. It is then thoroughly examined by practice questions on a portion or all of it. Points not well understood are carefully explained by the instructor or professor. For the purpose of testing this understanding various practical examples illustrating the principles of the course are required to be worked by the pupil at the blackboard or slate (United States Military Academy, 1902). The Project Three sections of MA103, Math Modeling and Introduction to Calculus, the first semester core math course at West Point, were used in this project. Each class, upon entry to West Point, is assessed on their fundamental math skills with the Fundamental Concepts Exam (FCE). Cadets enter the U.S. Military Academy with different math backgrounds. To successfully begin work in their math/science/engineering (MSE) courses at USMA, incoming cadets must arrive with the knowledge of certain mathematical skills and concepts. The FCE is designed to assess those skills. All cadets are required to pass the test and demonstrate mastery of its topics before moving on to higher math courses (United States Military Academy West Point, 2015). The cadets are required to take the FCE again in the middle of their first semester. If they do not pass it, they are given another chance toward the end of the semester. I chose to use this exam in order to assess whether practicing problems at the chalkboard, or at their desk, was a more effective method of learning and retaining those fundamental math concepts. 1 P a g e
In my study I looked at my three sections of MA103, and did at least one fundamental concept problem a day for review. My first section (n=17) worked exclusively at the chalkboard for these problems, my second (n=16) only practiced problems at their desk, and my last section (n=18) alternated between the desk and chalkboard each day. From an instructor standpoint, I enjoy having students work at the boards. First and foremost they are awake; they are also engaged, and actively trying to answer the problem. After the majority of students have completed the problem, I task one of the students with briefing the problem and describing to the class how they solved it. This teaching technique I believe allows for multiple learning styles to be addressed: visual learners and verbal learners get the explanation portion; while active learners get to work through the problems and can work with their classmates to try to figure things out, and reflective learners can work through the problems on their own. Also it allows me to see where cadets are making mistakes and determine if there are any patterns of error class-wide or if it is just one or two students struggling. The downside is that students sometimes feel intimidated by putting their work up on the chalkboard and being asked to brief it. If a student is completely lost they tend to stand and stare at everyone else s boards sometimes copying their classmate s work, or just leaving it completely blank. I remember the feeling of dread I would sometimes have myself as a cadet when I heard the instructor say, Take Boards, never knowing if I would be the one called out to brief the problem that day. Some departments and instructors adapt a very formal, somewhat scripted, briefing style for briefing the problems to the class. Since it is first semester plebes and the goal of my study was to see which method produced the best learning, I wanted to try to mitigate that fear factor of briefing the boards. I would take a more relaxed approach and often let students volunteer to brief, or help those that were struggling with the problem first, and then have them explain it to the class. I wanted to try to alleviate any undue stress that presenting in front of the class might have on their learning. Another downside is then the students do not have the problem written in their notes. I tried to give them time to copy it down, but I do not think they always did it. For the class that only worked at their desk I could not see their mistakes or if they were struggling as easily. It was harder to keep students on task, or tell if they were on task, but the plus side was the problem was already in their notes for the day. Another downside from an instructor perspective was that I could not tell if the students had completed the problem as easily, and there was not as much cooperation and teaching among the students to help each other. If a student was stuck, they would often just sit there and pretend to work, not asking for help. Conversely, if a student finished early, they would also just sit there and not offer help to their classmates as readily as when they worked at the boards. When it looked as though most of the students were done I would write the solution up on the board and ask one of the students to explain it, or if there was not enough time, I would walk through the solution. For my last section, I alternated between the two methods each day, allowing them to work at the boards and at their desk on a regular basis. 2 P a g e
Number of Students Take Boards: Does Working at Chalkboards in Math Class Actually The results The initial assessment scores for the 51 students in my three sections followed a fairly normal distribution (Figure 1), with an average score of 64.76, median of 65.7, and a standard deviation of 13.35. 7 Initial Assessment Scores Gateway Exam Scores 6 5 4 3 2 1 0 22.9 37.1 48.6 51.4 54.3 57.1 60 62.9 65.7 68.6 71.4 74.3 77.1 80 82.9 88.6 91.4 94.3 Gateway Exam Score Figure 1. Distribution of Initial Assessment Scores by number of students scoring each percentage. All three sections had an overall increase in the average score after the FCE given in the middle of the semester; the placebo sections also saw an increase in their average. Overall the largest gain was seen in the section that alternated between board work and deskwork. The average score increase was 16.2167%, bringing the class average on the gateway from a 65.56% up to almost 82%. Working at the chalkboard only saw an increase in the average by just over 10%, while merely working problems at the desks saw an increase in only 8% (Table 1). 3 P a g e
Table 1. Summary of Results showing the Gateway Exam average, FCE average, average gain, and standard deviation for each of my three sections. Gateway Average FCE Average Average % Gain Standard Deviation Chalkboard Only 61.8471 72 10.1529 10.4220 Deskwork Only 66.9688 75 8.0313 9.7102 Alternating 65.5611 81.7778 16.2167 13.7087 The results lead me to believe that having students merely practice problems at their desk has the lowest impact on retention and improving fundamental math concepts. The section that conducted deskwork had the highest average on the initial concepts exam, but then ended up seeing the lowest gain compared to the other two sections. This could be due to lack of effort, or conscientiousness by the students, but alternating between the boards and the desks doubled the average gain. This leads me to conclude that the use of chalkboards definitely does have its place. In the score comparisons you can see that the largest impact was on those students who initially scored between a 48% and 65% on their initial assessment (see Figures 2-4). With the chalkboard use (and alternating in the chalkboard use) these students all saw a significant increase in their scores. Those students that already had stronger fundamentals did not benefit, but they were extremely helpful in assisting their classmates while working at the boards. Upon conducting further analysis the only real conclusion that can be drawn with statistical evidence is that the combination of working at the desk and the chalkboards is more effective than any of the other methods (see Appendix A). The confidence interval showing a statistical significance between working at the boards and alternating (Table 3, and between the placebo group with no FCE preparation and the section that alternated (Table 4), is about 88%. Given that the sample size is so small, that might prove to have statistical value, showing that there is a significant difference in the average gains of the two groups. The difference between the gains of the students who worked only at their desk, and those that alternated between the desks and at the board is much more significant (Table 3) with a confidence interval of just over 94%. Student Feedback In the course end feedback, I asked the students two questions regarding their preferences and insights on working at the chalkboard versus at their desks. 1) In regards to preparing for the FCE at the beginning of the semester did you find working problems at the board, or problems at your desk, more helpful? 4 P a g e
2) Overall in MA103 (and math class in general), what is your opinion on working problems at the chalkboard versus at your desk? Did you feel one method was more beneficial to your understanding of the concepts? The feedback that I received was interesting. For the first question 49% said that working at their desk was more helpful, 34% said working at the board, and the remaining 17% took the neutral route and said both methods were helpful. Surprisingly, for the second question 60% of the students said that learning at the board was more beneficial, with 37% preferring to work at their desk. The consensus among those that preferred working at their desks was then they had the material in their notes already, while those that said they learned more from working at the boards had more insightful responses. I preferred working at my desk, so I can work problems right into my notes. But the occasional board problem was fun and engaging. I think working at my desk is better because I then get to use the problems as notes. Also many of the board problems require technology, so I have to be at my desk at some point. Working problems at my desk was more beneficial to me because I had the problem as a reference for studying. I felt that working the problems on the board was more beneficial. The teacher was able to see mistakes in our work or guide us in the right direction if we didn't know what to do next. Fellow cadets could also help each other as well. I like board problems. We were able to take a step back and really analyze things. Once we saw the big problem, we were able to go in and do the little/sometimes long calculations. We also got to see what others were doing, and help each other. The only thing is we can't take the boards off the wall, and put them in our notes. Chalkboard over desks because it enables the instructor to see all your work and help multiple students out at one time. I'm a visual learner, so standing up at the boards and writing my thoughts big and clear was extremely helpful. Overall working at the boards was more helpful. I also paid attention more because I was standing. 5 P a g e
Conclusion Overall I believe that there are several benefits to working at the chalkboards, but there are also benefits to allowing students to work at their desks. Sometimes working at the desk might even be a necessity, based on the use of different technological platforms such as Mathematica or Excel. The primary data showed that alternating between the two methods had the highest measure of success; almost double that of any one method alone. I was pleased with this outcome because from my observations I felt combining the two methods was most beneficial to the cadets and their learning. It also seemed to please both sets of students, those that like working at the board and those that want to work at their desk, while still getting everyone up at the boards in order for me to evaluate them more easily. 6 P a g e
Works Cited Joe Albree, D. C. (2000). A Station Favorable to the Pursuits of Science: Primary Materials in the History of Mathematics at the United States Military Academy. American Mathematical Society, 11. United States Military Academy. (1902). The Centennial of the United States Military Academy at West Point 1802-1902. West Point, NY. United States Military Academy West Point. (2015, February 6). Department of Mathematical Sciences. Retrieved April 2, 2015, from United States Military Academy: http://www.westpoint.edu/math/sitepages/prospective%20students.aspx 7 P a g e
Appendix A (Scores and Data) Table 2. t-test analysis comparing the grade percentage gains of working at the chalkboard versus alternating between the boards and desk. Board vs. Combo Gains t-test: Two-Sample Assuming Equal Variances Variable 1 Variable 2 Mean 9.622222222 16.21666667 Variance 107.2983007 187.9273529 Observations 18 18 Pooled Variance 147.6128268 Hypothesized Mean Difference 0 df 34 t Stat - 1.628311237 P(T<=t) one-tail 0.056345757 t Critical one-tail 1.690924198 P(T<=t) two-tail 0.112691514 t Critical two-tail 2.032244498 Table 3. t-test analysis comparing the grade percentage gains of working at the desk versus alternating between the boards and desk. Desk vs. Combo Gains t-test: Two-Sample Assuming Unequal Variances Variable 1 Variable 2 Mean 8.03125 16.21666667 Variance 94.287625 187.9273529 Observations 16 18 Hypothesized Mean Difference 0 df 31 t Stat - 2.025362157 P(T<=t) one-tail 0.025756818 t Critical one-tail 1.695518742 P(T<=t) two-tail 0.051513637 t Critical two-tail 2.039513438 8 P a g e
Table 4. t-test analysis comparing the grade percentage gains of not preparing for the FCE versus alternating between the boards and desk. Combo vs. Placebo Gains t-test: Two-Sample Assuming Unequal Variances Variable 1 Variable 2 Mean 16.21666667 10.41046512 Variance 187.9273529 154.2367127 Observations 18 86 Hypothesized Mean Difference 0 df 23 t Stat 1.660008754 P(T<=t) one-tail 0.055243622 t Critical one-tail 1.713871517 P(T<=t) two-tail 0.110487244 t Critical two-tail 2.068657599 9 P a g e
100 90 80 70 60 50 40 30 20 10 0 Score Comparison with Chalkboard Use 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Gateway Exam Scores FCE Scores Figure 2. Chart showing initial Gateway Exam scores and FCE scores for cadets in the section doing only chalkboard work. 100 90 80 70 60 50 40 30 20 10 0 Score Comparison with Deskwork 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Gateway Exam Scores FCE Scores Figure 3. Chart showing initial Gateway Exam scores and FCE scores for each cadet in the section doing only desk work. 10 P a g e
120 Score Comparison: Alternating Boardwork and Deskwork 100 80 60 40 20 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Gateway Exam Scores FCE Scores Figure 2. Chart showing initial Gateway Exam scores and FCE scores for each cadet in the section alternating between desk and board work. 11 P a g e
Table 5. The breakdown of scores by individual in the chalkboard only section. Final Course Grades are also included for comparison. Section Gateway Exam Scores FCE Scores Gain/Loss Final Course Grade Chalkboard Use 37.1 44 6.9 63.45 48.6 52 3.4 86.78 48.6 60 11.4 76.5 48.6 64 15.4 79.7 54.3 84 29.7 86 57.1 68 10.9 80.69 60 72 12 86.8 60 76 16 82.68 60 80 20 83.75 60 84 24 RESIGNED 65.7 80 14.3 83.8 65.7 84 18.3 86.22 71.4 64-7.4 80.05 74.3 68-6.3 86.03 74.3 72-2.3 87.9 77.1 84 6.9 90.23 88.6 88-0.6 90.03 Average 61.8471 72.0 10.1529 83.1631 Standard Deviation 12.7757 12.4097 10.4220 6.4974 12 P a g e
Table 6. The breakdown of scores by individual in the desk work only section. Final Course Grades are also included for comparison. Section Gateway Exam Scores FCE Scores Gain/Loss Final Course Grade Deskwork 51.4 48-3.4 49.47 51.4 76 24.6 84.88 51.4 60 8.6 50.92 54.3 72 17.7 73.48 57.1 60 2.9 79.85 62.9 76 13.1 90.68 65.7 68 2.3 86.28 65.7 72 6.3 72.58 68.6 60-8.6 74.9 68.6 84 15.4 77.85 68.6 92 23.4 83.1 74.3 80 5.7 83.1 74.3 88 13.7 81.08 80 84 4 85.08 82.9 92 9.1 90.18 94.3 88-6.3 80.9 Average 66.9688 75.0 8.0313 77.7706 Standard Deviation 12.3583 13.1047 9.7102 11.9844 13 P a g e
Table 7. The breakdown of scores by individual in the section that alternated between board and desk work. Final Course Grades are also included for comparison. Section Gateway Exam Scores FCE Scores Gain/Loss Final Course Grade Alternating Board/Desk 22.9 64 41.1 62.65 51.4 80 28.6 86.05 54.3 76 21.7 65.03 57.1 52-5.1 70.28 60 76 16 78.18 60 84 24 78.79 62.9 72 9.1 77.78 62.9 92 29.1 90.68 62.9 96 33.1 82.35 68.6 84 15.4 92.65 68.6 92 23.4 90.05 71.4 96 24.6 86.63 71.4 96 24.6 91.48 74.3 72-2.3 83.35 77.1 76-1.1 88.79 80 88 8 80.1 82.9 84 1.1 87.85 91.4 92 0.6 89.93 Average 65.5611 81.7778 16.2167 82.3678 Standard Deviation 14.9015 12.0565 13.7087 8.9773 14 P a g e