To conduct this research, I gathered data from two class discussion activities, homework, quizzes, and self-reflection audio journals across three weeks, from November 16, 2020 to December 7, 2020. During this time students explored a few geometry topics: Inductive Reasoning and Conjectures, Compound Statements, and Conditional Statements. I want to note that the students I conducted this research with are in a 1st hour class that I had recently started teaching in early November.
Discussion Activities
Two class discussions were held for students to engage in debates over the validity of math statements. In the first discussion called Math Debates #1, students viewed various representations of math statements on a PowerPoint. Brodie (2010) shares that proposing too many challenging tasks in a limited time may easily demotivate and disinterest many students. I determined that it would be most beneficial to include pictures and audio statements initially and eventually increase the level of difficulty through providing written math statements last since these would take more thought to determine their falsity. Some statements were given in pairs of before and after photos, some statements were given in audio form, and other statements were given as written math sentences. I want to add that the pictures and audio statements were intentionally chosen to be related to real-life scenarios rather than explicit math sentences. An example of a real-world scenario statement used in one of the audios reads, If you own a driver's license then you are at least eighteen years old. An example of an explicit math sentence that was given is, If a figure is a right triangle, then it has one acute angle. The reason that majority of the statements that given were real-world scenarios is because I anticipated that students would take a longer time to discern the falsity of the explicit math statements. These types of statements typically require more analysis and breakdown of given information to conclude than determining the falsity of pairs of pictures or real scenarios that students might have experienced and be more knowledgeable about. Also, there were true and false statements among the three representations presented. I did not want to spend too much time on every problem since there was a lesson planned for the day and I anticipated that false statements would take longer to discern than true statements. In class discussion students had minimal difficulty discerning the falsity of the picture and audio statements which was something I had predicted. When it came time to discussing the more explicit math statements it took some probing students for them to reach a solid conclusion. For the statement, If a figure is a right triangle, then it has one acute angle, students took a second to recall what they know about triangles. One student recalled that right triangles, in particular, have one angle that is 90 degrees and that the angles of all triangles add up to 180 degrees. From there I was able to present a case for students to think about, "What if none of the angles of a right triangle is acute? Is this possible?" Another student responded that two of the angles would have to be acute if the right triangle has one 90 degree angle and since the angles have to add up to 180 degrees, the two smaller angles would need to add together to make 90 degrees.
Math Debates #2 was created with two pictures that included math statements and three explicit math sentences. This time around I anticipated that the activity would take a longer time to complete since most of the statements given were explicit math sentences. Additionally, I want to add that the statements that were chosen in the first activity were a combination of false and true statements. The statements that were given in this second activity were all false and I made this known to students upfront. The reason for this was that I wanted students to spend less time worrying about falsity and more so on the reasons behind the falsity of the statements. The hope in doing this was that students would gain more practice in generating specific cases, something I realized students struggled with in homework #1. One statement I would like to expound on is, If a times b is less than zero, then a is less than zero. One student, Student A, shared that they believed this statement must be true because when a is a number less than zero and multiplied to b, another number, a number less than zero would result. Although this statement was valid, I asked the student to consider what the outcome would be if b is less than zero. Another student, Student B, followed up on this case and claimed that if both numbers were zero then if you multiplied them you would get zero, which would not work because zero cannot be less than itself. I acknowledged the truth in this student's analysis and I probed further, "So does that mean that b can't be less than zero?". Student A revealed, that b can be less than zero if a is not less than or equal to zero. In general I believe the reason that these students struggle with counterexamples is because they do not consider all types of possibilities, given the initial conditions of false statements. This is something, however that can be improved with practice. This activity took significantly longer than the first activity for reasons previously mentioned.
Homework Assignments
After looking through student responses in the Inductive Reasoning homework #1, I noticed that majority of the students lacked use of formal math language. For the questions where students had to discover the pattern of a sequence and then write a conjecture they had to use a sentence starter modeled in the notes, To generate the next term, and then complete the sentence. However many failed to provide this in their explanation. For example the answer to one of the questions should have read, To generate the next term add five to the previous term. Student R's response read, the number's being added up by five. This reveals majority of my students might not have practiced formal math language prior to my class. My focus was identifying their understanding of class concepts and terms within their written work, so informal language would not have been the sole factor in me grading their responses. The average score for this homework was 17/32.
For Compound Statements homework #2, given in the lesson after quiz 1, students were given a diagram made up of interconnected rays. There were four statements given with the diagram for students to write the indicated compound statements and determine the truth value for. These statements were: Points C, E, and B are collinear, AEC ≅ DEB, EF is the angle bisector of AED, and BEC is an acute angle. The average score for this homework was 20.5/26, which is a significantly higher average than the homework #1. This homework may have been a little easier than homework #1 for students since they did not have to provide counterexamples for their false statements and actually do math calculations to obtain their answers. Also, the truth tables on page 2 of the homework were straightforward and modeled the truth tables from the compound statements notes closely.
In, Conditional Statements homework #3, students had to determine the hypothesis and conclusion of some given statements, write sentences as conditional statements, provide the other forms of conditional statements for given sentences, and write the proper form of conditional statements given the symbolic notation. The average score on this assignment was 19.4/33. After viewing student work, I determined that the students needed more time to practice using each conditional statement form and recognizing their respective symbolic notations, so I decided to give their quiz the following week on Monday, December 7th instead of the previous Friday.
Assessments
The assessment, given on 11/23/2020, was designed to ascertain which math concepts from Inductive reasoning, conjectures, and Compound statements were my geometry students able recall and apply from the week prior. The tool I used to create my assessment is google forms. The assessment is made up of five total questions. All of the questions are short answer questions and majority are two-part questions. This allowed students to share more of their thoughts and for me to see how they think within the context of each question. In the assessment I provided images for some of the questions that I felt would take students more thought and application to come to a valid conclusion. The class average score was 8.6/12 which I would consider to be an adequate grade based on how well they did on homework #2.
The assessment, given on 12/07/2020, was designed to ascertain if students could recall the different forms of conditional statements: converse, inverse, contrapositive, and biconditional, and determine the falsity of their statements. I created this quiz on a word document, because it was simpler and took less time to create than it would have on google forms. On the quiz students were given one conditional statement that they had to use to determine the other forms and find their truth value. The average on the quiz was a 6.9/9. This average is slightly higher than the average of quiz 1. I believe that this may have been because students had more practice since the first quiz with determining truth values for statements. So, majority of students had less difficulty this time with determining truth value of statements than the previous quiz. However, I did notice that students struggled with deciphering the proper symbolic notation such as p implies q or q implies p when determining the indicated conditional forms. Some students also gave the inverse statement when asked to given the converse and vice versa. This quiz was evidence that students had improved in determining truth value, but also reveals that students needed more practice working with symbolic notation and memorizing multiple math terms at once.
Teacher Audio Journals
Teacher audio #1 is a reflection detailing class discussion in Math Debate #1. Teacher audio #2 is a reflection on class discussion from Math Debate #2. Although some details of the class discussions have been mentioned earlier these audios provide more details of how the discussion was organized and functioned. Unfortunately, due to time constraints, I was not able to implement Math Debates #3 like I had initially planned.
Conclusion
Once I implemented Math Debates #1 and looked over the first two homework assignments I was able to anticipate the type of statements students would have most trouble with in Math Debates #2. These were the problems that involved computation and included a myriad of conditions for students to think about when determining their falsity; such as the statement mentioned in Math Debates #2 above. Majority of the feedback given to students was verbal through the discussion activities and class lessons. I noticed that during Math Debates #1 I probed minimally and responded almost automatically between every students' input into the conversation. I realized that more student thoughts would come out of having students reply to one another. So, for Math Debates #2, I probed a little more using, Why? to get students to justify their claims. I also initiated conversations between students by asking them to add onto what a fellow classmate had to say on a given statement. This research highlighted some types of students that engage in math arguments. Some of my students struggled the most with recalling math concepts and applying them in given statements. Other students had difficulty looking at specific examples to make general claims. I claim that engaging in math discussions helped model for students how to explore details in a statement and apply their prior knowledge to develop solid counter examples. I plan to continue engaging in math debates with my students to increase their math confidence and competence across future math ideas and concepts in new units. After seeing their comfortability increase significantly from Math Debates #1 to Math Debates #2 I claim that continuing these discussions will, result in more thoughtful student-to-student conversations and eventually predominantly student led Math Debates.
References
Brodie, K. (2010). Teaching mathematical reasoning in secondary school classrooms. New York: Springer