Technology is integral to modern education, supporting exploration, computation, assessment, communication, and motivation. National curriculum and teacher preparation standards, such as the Common Core State Standards for Mathematics (NGA & CCSSO, 2010) and Standards for Preparing Teachers of Mathematics (AMTE, 2017), emphasize the strategic use of technology in teaching mathematics across grade levels. At the college level, the Committee on the Undergraduate Programs in Mathematics recommends increasing the sophistication of technological tools used by mathematics major students (Zorn, 2015). The Mathematical Association of America Instructional Practices Guide views technology as a constant theme across instructional practices, promoting student engagement and learning (Abell et al., 2018).
As GeT instructors, our collaborative interest in technology emerged from our exploration of technologies such as Dynamic Geometry Environments (DGEs) and Digital Proof Tools (DPTs) in our teaching of geometry. We believe providing GeT students with initial exposure to technology tools can help them develop a positive attitude and appreciation for these tools and thus inspire them to incorporate such tools in their future classrooms. Our initial work on technology was the development of the narrative of SLO 6 – Technologies, calling for students to effectively use technology in GeT courses to explore, conjecture, and strengthen their understanding of geometric concepts and relationships. Working with DGE tools like GeoGebra, we observed how students naturally discovered geometric relationships through hands-on digital manipulation, while DPTs, like FullProof, could support both proof construction and proof evaluation with instant feedback (Baccaglini-Frank, 2011; Buchbinder et al., 2023; Bülbül & Güler, 2022). Combining research-informed practices and our own teaching experiences, we contributed a chapter on the topic of the importance and application of technology in GeT courses to the upcoming book entitled GeT Courses: Resources and Objectives for the Geometry Courses for Teachers. In the chapter, we further elaborated on SLO 6 and shared how DGE and DPT-incorporated activities could cultivate students’ geometric habits of minds (Driscoll, 2007) in various geometry contexts, such as proof writing, constructions, transformations, Euclidian and Non-Euclidian geometries.
Looking forward, we see tremendous potential in emerging technologies (e.g., generative AI and augmented reality). GeT instructors should stay updated on the latest technology trends and developments, be aware of potential challenges and solutions, and reflect on the use of technology in their teaching, in order to create supportive and engaging learning environments to foster students’ mathematical thinking.
References
Abell, M., Braddy, L., Ensley, D., Ludwig, L., & Soto-Johnson, H. (Eds.). (2017). MAA instructional practices guide. Mathematical Association of America. https://maa.org/resource/instructional-practices-guide/
Association of Mathematics Teacher Educators [AMTE]. (2017). Standards for preparing teachers of mathematics. https://amte.net/standards
Baccaglini-Frank, A. (2011). Abduction in generating conjectures in dynamic geometry through maintaining dragging. In Proceedings the 7th Conference on European Research in Mathematics Education (pp. 110-119).
Buchbinder, O., Vestal, S., & An, T. (2023). Lessons learned using FullProof, a digital proof platform, in a geometry for teachers course. Proceedings of the 25th meeting of the MAA special interest group on research in undergraduate mathematics education. Omaha: RUME.
Bülbül, B. Ö., & Güler, M. (2022). Examining the effect of dynamic geometry software on supporting geometric habits of mind: A qualitative inquiry. E-Learning and Digital Media, 20427530221107776.
Driscoll, M. J., DiMatteo, R. W., Nikula, J., & Egan, M. (2007). Fostering geometric thinking: A guide for teachers, grades 5-10. Heinemann.
National Governors Association Center for Best Practices [NGA] & Council of Chief State School Officers [CCSSO]. (2010). Common core state standards for mathematics. http://www.corestandards.org/Math/Zorn, P. (Ed.). (2015). 2015 CUPM curriculum guide to majors in the mathematical sciences. Mathematical Association of America.
When I found out that the RUME conference was going to be in Omaha, I was probably the only member of the GeT: A Pencil group who was excited. For me, it was a 4-hour, 235-mile trip down Interstate 29. The Embassy Suites was the perfect venue, located within walking distance of several restaurants, with a friendly staff, and a great evening happy hour! Many of us from the GeT group spent a lot of time visiting with each other during these happy hours.
The GeT Together on Wednesday started around 9 a.m. with a welcome from Pat. Then Laura and Amanda updated us on the book, and we got to see a list of accepted chapters. It was valuable to all of us that are writing chapters to see what others are writing. Then Dorin talked about his proposed chapter on the archetype work that we did in 2018-2019. I do believe that work was instrumental in getting us to the GeT Student Learning Objectives (SLOs) so I am thankful that he has gathered a team to write that chapter. Michaela from the ESLO group shared what they have been doing and some of their thoughts after discussing the first three SLOs. Engaging with Michaela, Mara, and Younggon from the ESLO group was nice. I appreciated hearing their perspective on the SLOs, especially Mara’s sharing on how she has changed her GeT course and feels that the course has improved with the addition of the SLOs.
The opportunity to split into different groups and talk about the proposed chapters in the afternoon was very useful. When you are meeting with your co-authors, everyone often has a similar perspective so getting ideas from an outside person was helpful. I think that some of the author teams made progress on their chapters. Thanks to Inese and Carolyn for all they did to organize and keep us on task!
During the Thursday morning RUME Teaching Geometry for Secondary Teachers working group, we were joined by a couple of new people interested in the GeT course. Nat started us off by showing the SLO website. We are still hoping that more people register for the Forum and share their thoughts on the SLOs there. If people join the forum and leave comments there, anyone registered for the forum can read their comments and respond.
Then Mike presented the results of a survey done by the GRIP lab, where they listed the ten SLOs, along with eleven distractor SLOs, and people had to rank them 1 through 7. The survey results are in the image at right. These survey results were very satisfying for me (and likely others in the SLO authoring team). Nine SLOs were ranked in the top ten slots. Notice that SLOs 1, 4, 2, 5, and 7 were ranked #1 at least ten times. In addition, all of the SLOs had very few #7 rankings, with SLO 2 having the most with three rankings at #7. While SLO 8 is lower in the rankings, it was only ranked #7 twice. SLO 13 was “Understand the ideas underlying advanced geometric topics in Euclidean and absolute geometry.” I feel like these survey results solidify that we were on the right track in selecting and writing the SLOs.
Thursday afternoon I was able to see some of my GeT colleagues present about the Adinkra lesson that they have been using to teach transformations in their courses. It is always great to support other colleagues in their teaching and research. In addition, Tuyin, Orly, and I presented our FullProof project, including data from both fall semesters that we have used it. While we had done a similar presentation at AMTE in New Orleans, this one had better attendance and audience interaction. The best part was seeing our GeT colleagues in the audience supporting us and asking questions.
As a first-time attendee of RUME, I really enjoyed it. I teach the proof course at SDState so I was able to find a lot of great sessions on teaching and learning proof. Another thing that I liked about it was that it is a smaller conference, so everyone knows each other. Dinner with the GeT: A Pencil community on Thursday evening was a lot of fun. Our table was lucky enough to have Joe Cole, the magician comedian, come and show us one of his tricks. The best part of my trip to Omaha was hanging out with the GeT: A Pencil community—what an awesome group of people. This community has been wonderful for me professionally, and I have also found amazing and supportive friends!
Proof-writing is a core disciplinary practice of mathematicians and a crucial skill of all mathematics majors and future mathematics teachers. Developing students’ facility and comfort with proofs is an important objective of undergraduate Geometry for Teachers (GeT) courses (An et al., in press; Grover & Connor, 2000). GeT courses cultivate robust and flexible knowledge and skills such as writing, analyzing, and critiquing proofs that allow prospective secondary teachers (PSTs) to support their students’ learning of geometry (AMTE, 2017; González & Herbst, 2006).
Technology plays an important role in supporting PSTs’ engagement with proof. Dynamic Geometry Environments like GeoGebra, Geometer’s Sketchpad, and others provide opportunities to explore geometrical properties and make and test conjectures (Jones, 2000; Mariotti & Baccaglini-Frank, 2018). These tools have been beneficial for fostering PSTs’ attitudes towards proof and comfort with proof and teaching it to secondary students (Abdelfatah, 2011; Kardelen & Menekse, 2017). However, they offer little support for writing a deductive proof of a conjecture or a theorem.
With the advancements in artificial intelligence and machine learning, new tools have been created that support automated and interactive proof writing and verification (e.g., Lodder et al., 2021; Matsuda & VanLehn, 2005; Wang & Su, 2017). FullProof is a software developed to advance students’ proof-writing skills in Euclidean Geometry. Using FullProof, students write a step-by-step two-column proof, using an interactive diagram, an equation editor, and a library of theorems/postulates and definitions. The software checks the proof and provides feedback and/or hints to improve their work.
Our project aimed to explore the potential benefits of FullProof for GeT students; our research question was: How and to what extent does engaging GeT students with the FullProof software affect their competence with, and beliefs about geometry proofs?
Methods
Research Setting
In Fall 2021, we conducted an exploratory study by integrating FullProof into our GeT courses, which all have a large portion of the curriculum devoted to Euclidean Geometry from a synthetic perspective. Two GeT courses had a mixed student body of PSTs and STEM majors; one GeT course only had PSTs. Thirty students agreed to participate in the study across the three GeT courses.
FullProof served as a support system for writing proofs in class and homework. Each instructor used about 20 proof problems from FullProof ‘s collection, on a variety of topics, at three levels of difficulty: high, medium, and low. Figure 1 shows a screen capture of a sample problem (medium difficulty) about triangle midsegments.
Figure 1. Sample proof problem
Each element in the figure can be highlighted or marked when pointed to, and auxiliary lines can be added as needed (Fig. 2). The right side of Figure 1 shows the givens and the statement to be proven. The students write the proof by typing statements in the numbered lines and justifying them. The justifications can be searched by keywords or by browsing the FullProof justifications library.
At any point in the process, the students may ask for a hint by clicking the hint button. The system will produce hints in order of increased specificity from a vague “try using triangle congruence,” to suggesting a certain theorem to use, to proposing a particular step–like “try proving ∆BFO ≅ ∆EGO.”
Once finished, the students click the “check” button, and in a few seconds, the algorithm checks the proof and provides feedback. Correct proof lines get a green check mark. Incorrect or partially correct lines are marked down with an explanation of the mistake. Figure 2 shows a solution with two mistakes – a missing proof step and a missing justification for the last step. Clicking on the notifications will show a student what the missing step was (here ∆BFO ≅∆EGO by angle-side-angle theorem) and how many points were deducted for each mistake.
Figure 2: Sample solution in FullProof and the feedback provided by the software.
Clicking the “try again” button allows a student to improve their work, resubmit, and receive more feedback. The number of hints and submission attempts is unlimited (except for the test mode), allowing the students to achieve a perfect score eventually. The information about the number of hints and submission attempts is stored in the system and is available on the instructor’s interface (Fig. 3).
Figure 3: Excerpt of instructor interface in FullProof
Data Collection and Analysis
To respond to our research question, we administered pre- and post-surveys via Qualtrics, which took about 20 minutes to complete. Most questions were Likert-type on mathematical identity and comfort level with writing proofs. The items were adapted from the literature (e.g., Kaspersen & Ytterhaug, 2020; Stylianou et al., 2015). Here, we focus on responses to the bolded open-ended questions, developed by the researchers for the post-survey, about student perceptions of FullProof (Fig. 4). We analyzed the data qualitatively, using open coding and thematic analysis (Patton, 2002) to reveal recurring themes in students’ responses, specifically the types of positive and negative appraisals about using FullProof. To capture all themes, comments that included multiple ideas were coded multiple times.
Figure 4. Open-ended Items from the FullProof Survey.
Student Appraisals of the FullProof Platform
The analysis of the positive and negative appraisals in students’ written comments on the post-survey revealed three main themes that emerged from the 61 positive appraisals of FullProof identified across all seven open-ended questions. The largest category (41%) described the affordances of FullProof for supporting students’ writing and understanding of proofs, including the searchability of reasons for proof steps, clear structure that supports communication and comprehension, interactive feedback, and the ability to pursue multiple solution paths. One student wrote, “FullProof has helped me a lot when writing proofs. I like how I can search for reasons if I am not completely sure about a reason/theorem.” Another student wrote, “FullProof made writing proofs easier not only to write, but also to understand.”
The second theme (26%) described the advantageous technical features of FullProof, such as hints which help them move forward if stuck, interactive feedback pointing to errors, visual clarity, and color-coded elements of a diagram. The third theme (25%) described the affordances of FullProof as a pedagogical tool for teaching others. In this theme, the participants highlighted the elements of the software that would support them as teachers in the geometry classroom. Students wrote that FullProof presents “a good instructional strategy to implement in the classroom,” and the platform “makes it easier for the students to see what their errors were.”
The 23 negative appraisals constituted 29% of the codes and were distributed rather uniformly across five themes. The main critique (7 out of 23 appraisals, 30%) concerned the discrepancy between the wording of the theorems in the FullProof platform and in class. For example, FullProof does not have the angle-angle-side triangle congruence theorem, so students need to complete an extra step and use the angle-side-angle congruence theorem. Another category of critiques (22%) described the variation in standards of rigor employed by FullProof vs. classroom instructors. The students wrote that the software allowed them to skip steps and earn full points for their solution, while they were aware that their instructor would have likely deducted points. One student wrote: “It allows some things to slide, that should be wrong.”
Additional patterns emerged from the analysis of specific survey questions. When asked, “Has FullProof changed the way you write proofs,” the responses split almost evenly between yes and no. But in the written comments clarifying the forced choice, 83% of comments were positive regarding FullProof. A similar pattern was observed in the question: “How has FullProof changed your understanding of reasoning and proof?” Sixty-one percent of students wrote that FullProof positively influenced their understanding, but interestingly, 91% of their comments contained some specific description of the positive effect of FullProof. When asked, “Would you use FullProof in your future classrooms,” 14 out of 17 students who answered this question responded positively citing the advantages of theplatform. When asked to describe some of the successes and challenges students had with FullProof, 65% of participants’ comments described challenges, as opposed to 35% that described successes. However, the overall impressions of FullProof were very positive with 72% of comments containing positive reviews of the tool. These results show positive appraisals of the FullProof platform outweigh the critiques of the tool.
Discussion
Our quasi-experimental research design has several limitations. Due to the lack of randomization and a single experimental group, potential confounding variables were not controlled (Johnson & Christensen, 2012). Each course was taught by a different instructor, with no common curriculum or textbook across the institutions. There was also some variation in how instructors used FullProof in their courses, with one institution teaching in a hybrid mode due to the Covid-19 pandemic. Because of these variations, our study design emulates natural conditions of how instructors might use any technological tool within the unique constraints of their institutional environments. Against this backdrop, we find it encouraging that the qualitative analysis showed more positive appraisals (69) than negative appraisals (27) about using the platform which indicates the overall positive impact of using FullProof in the GeT course. Moreover, since our analyses did not distinguish between PSTs and other STEM majors, the observed potential advantages of FullProof may apply to all GeT students.
As instructors, we note several advantages to using FullProof in our GeT courses. The availability of multiple proof problems at various difficulty levels and the feedback provided by the FullProof platform allowed assigning a greater number of challenging problems, compared to the past. Since FullProof offers hints and multiple submissions, students were able to get assistance from the software itself. However, this also presented a challenge; some students abused this functionality to get hints on every proof step. In the future, we plan to limit the number of hints and submission attempts. Another limitation of FullProof was the observed lack of rigor in its auto-grading system. We observed cases that FullProof issued students three stars (excellent) on their proof when certain necessary proof steps were missing or the reasons were not matching the statements. This limitation makes FullProof more suitable as a learning tool instead of a formal assessment tool. However, the discrepancies in the grading standards between the software and the instructor can be leveraged to engage students in analyzing and critiquing proofs, which is an important learning objective of GeT courses.
Based on what we learned from the pilot study, we modified the research design and are currently collecting the second round of data in our GeT courses. Some changes include shortening the surveys and assigning a few common FullProof problems across courses. While we feel FullProof is overall an efficient tool in facilitating the learning of proof, we want to further study how and to what extent it improves GeT students’ proof learning experience.
References
Abdelfatah, H. (2011). A story-based dynamic geometry approach to improve attitudes toward geometry and geometric proof. ZDM-Mathematics Education 43, 441–450. doi.org/10.1007/s11858-011-0341-6
Association of Mathematics Teacher Educators. (2017). Standards for Preparing Teachers of Mathematics. https://amte.net/standards
An, T., Boyce, S., Brown, A., Buchbinder, O., Cohen, S., Dumitrascu, D., Escuadro, H., Herbst, P., Ion, M., Krupa, E., Miller, N., Pyzdrowski, L., J., Sears, R., St. Goar, J., Szydlik, S., Vestal, S. (in press). (Toward) Essential student learning objectives for teaching geometry to secondary pre-service teachers. AMTE Professional Book Series, Volume 5: Reflection on Past, Present and Future: Paving the Way for the Future of Mathematics Teacher Education.
González, G. and Herbst, P. (2006). Competing arguments for the geometry course: Why were American high school students supposed to study geometry in the twentieth century? International Journal for the History of Mathematics Education, 1(1), 7-33.
Grover, B., & Connor, J. (2000). Characteristics of the College Geometry Course. Journal of Mathematics Teacher Education, 3, 47–67.
Hanna, G., Reid, D. A., & De Villiers, M. (Eds.). (2019). Proof Technology in Mathematics Research and Teaching. Springer International Publishing.
Johnson, B., & Christensen, L. (2012). Educational research: Quantitative, qualitative, and mixed approaches (4th ed.). SAGE Publications, Inc.
Jones, K. (2000). Providing a foundation for deductive reasoning: Students’ interpretations when using dynamic geometry software and their evolving mathematical explanations. Educational Studies in Mathematics, 44(1), 55–85.
Kardelen, G., & Menekse, S. T. B. (2017). Views of pre-service teachers following teaching experience on use of dynamic geometry software. Educational Research and Reviews, 12(24), 1208-1219.
Kaspersen, E., & Ytterhaug, B. O. (2020). Measuring mathematical identity in lower secondary school. International Journal of Educational Research, 103, 101620. https://doi.org/10.1016/j.ijer.2020.101620
Lodder, J., Heeren, B., Jeuring, J., & Neijenhuis, W. (2021). Generation and Use of Hints and Feedback in a Hilbert-Style Axiomatic Proof Tutor. International Journal of Artificial Intelligence in Education, 31(1), 99-133.
Mariotti, M. A., & Baccaglini-Frank, A. (2018). Developing the Mathematical Eye Through Problem-Solving in a Dynamic Geometry Environment. In Broadening the Scope of Research on Mathematical Problem Solving (pp. 153-176). Springer.
Matsuda, N., & VanLehn, K. (2005). Advanced Geometry Tutor: An Intelligent Tutoring System for Proof-Writing with Construction. Proceedings of the Japan National Conference on Information and Systems in Education, (Vol. 125, pp. 443-450).
Patton, M. Q. (2002). Qualitative research & evaluation methods. Sage.
Stylianou, D. A., Blanton, M. L., & Rotou, O. (2015). Undergraduate students’ understanding of proof: Relationships between proof conceptions, beliefs, and classroom experiences with learning proof. International Journal of Research in Undergraduate Mathematics Education, 1(1), 91–134.
Wang, K., & Su, Z. (2017). Interactive, intelligent tutoring for auxiliary constructions in geometry proofs. arXiv:1711.07154v1
Four questions with Sharon Vestal, Associate Professor at South Dakota State University
What is special about your GeT course? In 2-3 sentences, describe your GeT course.
The GeT course at South Dakota State University is unique in a couple of ways—it is actually called Geometry for Teachers, and it only includes students planning to teach secondary mathematics. My course includes both content and pedagogy, with a large emphasis on best practices for teaching geometry.
Who are your students?
My students are typically sophomore or junior majors in our department, seeking a Mathematics with Teaching Specialization degree. Frequently students are taking the GeT course at the same time as they take our Logic, Sets, and Proof course. Since I teach both courses in the fall semester, many students have me as their instructor for 6 credits of mathematics.
What are you most interested in learning/achieving through participating with the GeT: A Pencil community?
I was among the group to attend the conference in Michigan in June 2018 so I have had the privilege of being involved in this group since it began. One of the best parts of this group has been building relationships with colleagues teaching GeT courses at institutions across the country. These relationships have led to other opportunities and projects that have helped in my professional development.
The GeT: A Pencil community has helped me grow as a teacher and scholarand has provided me with a group of people who have similar beliefs and struggles. I find myself looking forward to our meetings as we have meaningful conversations, and we all listen with an open mind. These thoughtful meetings happen because of mutual respect.
What is your favorite book you have read in the last few years?
I have read a lot of books on teaching in the past few years, but one of my favorites is Instant Relevance by Denis Sheeran. I coordinate an NSF Robert Noyce Scholarship Program so each summer we have a conference and bring in an outside speaker, usually an author of some book on teaching. We invited Denis to come during the 2017 summer conference, and I have continued to stay in touch and communicate with him. Instant Relevance is about using current events or trends to teach content in your classroom. For example, one of our sessions was on how we might be able to use fidget spinners in the classroom because they were popular at the time.
When I joined the Geometry for Teachers (GeT) group in the summer of 2018, one thing that was always clear to me was that the GeT course at various institutions is very different. During the 2019 – 2020 academic year the Teaching GeT group, led by Dr. Nat Miller, made a huge effort to create a common list of Student Learning Objectives (SLO) for the GeT course. The list needed to be concrete enough to help a faculty member teaching the course for the first time, yet flexible enough that it works for different institutions.
Since I feel that proof should be the focus of the GeT course, I will elaborate on the first Proof SLO, “Derive and explain geometric arguments and proofs in written and oral form.” In the study of mathematics, true understanding is attained when one can read and write a proof of a theorem, explain it to another person, and apply the theorem. Geometry is an area of mathematics for which this skill may be more easily attained because figures are frequently used to illustrate theorems. While one should never rely on a picture when completing a geometry proof, having an image is often helpful when writing a geometry proof.
Why is it important that future teachers have a solid background in proof? While not every state has adopted or uses the Common Core State Standards for Mathematics (CCSSM), many have written new mathematics standards that are very similar to these standards. In the CCSSM, the first mention of proof is in the 8th grade geometry standards. These standards focus on using rotations, translations, and reflections to demonstrate that two figures are congruent to one another, and students are expected to be able to explain a proof of the Pythagorean Theorem and its converse. In the CCSSM high school geometry standards, students are expected to “understand congruence in terms of rigid motion” and “prove geometric theorems,” (National Governors Association, 2010).
Can we expect secondary mathematics teachers to teach proof without undergoing it for themselves? Mathematics teachers need to experience geometry proofs from the student perspective so they can empathize when their own students struggle. They need to be able to understand different types of proofs, such as synthetic (from axioms), analytic (using coordinates), and proofs using transformations or symmetries. They should be able to communicate proofs in different ways (two-column, paragraph, or a sequence of transformations). It is also essential that they can choose the most accessible type of proof for the situation.
Figure 1: Euclid’s Elements of geometry by University of Glasgow Library
In an ideal world, students would enter the GeT course with a strong background in proof, but that is generally not the case. Some of the variation in their proof background depends on the level of the GeT course (200, 300, or 400), but much of it depends on their own high school geometry experience. Based on what the GeT students tell me, high school geometry likely has the most variability of all high school mathematics courses throughout our country. Some students never had to do proofs in their course, while others did a lot of proofs in various forms. This is one of the reasons that the GeT SLOs are so important—we cannot help improve the teaching of high school geometry without looking at our own teaching of geometry. Great teachers are reflective practitioners—what better way to produce strong mathematics teachers than to model this practice for our preservice teachers.
Writing a synthetic proof requires the ability to put together a logical argument in a systematic manner. This skill leads to growth in critical thinking and reasoning. Daily we encounter situations where critical thinking is needed. Geometry is the best course to use proof to help students build problem solving skills, which is why I believe that proof should be the heart of the GeT course.
References
National Governors Association Center for Best Practices, Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC.
Geometry for Teachers (GeT) is a course at South Dakota State University (SDSU) that is entirely made up of mathematics majors who are planning to become certified to teach middle and high school mathematics. It is typically the first mathematics course that our preservice teachers take that includes pedagogy. Since I want the students to be well prepared to teach high school geometry, we focus on Euclidean geometry throughout the course.
This fall when I walked into class on the first day, I heard one student say to another, “we just need to memorize it.” I informed the student that the word “memorize” was not to be used in my classroom—our goal should always be to understand mathematical concepts and to teach our students in a way that develops their understanding. In this article, I will outline some discovery activities that I used with my GeT students to help them understand formulas rather than relying on memorization of formulas. My goal is that they will use these experiences when they are teachers to help their own students learn with understanding rather than just memorizing.
Sum of Angles in a Triangle
Throughout the semester my students used the theorem for the sum of angles in a triangle, but they didn’t really understand how we knew it. So in class, I gave each of my students a half-sheet of paper and asked them to use a straightedge to draw a triangle and a quadrilateral and to cut out their figures. Next, I asked them to cut or tear the corners as shown in Figures 1 and 2. Once they had torn off the angles, I asked them to put the vertices of the triangle together so that they are touching. With the triangle, the students immediately saw that together these angles formed a straight angle, which demonstrates that the sum of the measures of the angles in a triangle is 180°. I had them repeat this same process with the quadrilateral in order to recognize that the sum of the measures of the angles in a quadrilateral is 360°. While this was a quick activity to do with the students and required few materials (half sheet of paper, straightedge, and scissors), it gave them a physical representation of facts that they had been using throughout the semester.
Figure 1
Figure 2
Area Formulas
Another discovery activity that I have used involves finding the areas of rectangles, triangles, parallelograms, and trapezoids using only the fact that the area of a rectangle is base*height. I gave students a piece of cardstock with Figures 3, 4, 5, and 6 printed on it, and asked them to cut out each of the figures. Then I had them take the rectangle (shown in Figure 3) and cut along the diagonal so that they could “see” that the area of a triangle is ½ base*height.
Figure 3
Figure 4
Figure 5
Figure 6
Next, they cut the parallelogram (shown in Figure 4) along the dotted line. Then they translated this right triangle to the left so that the hypotenuse lined up with the side of the parallelogram, creating a rectangle. Looking at this rectangle, they saw why the area formula for a parallelogram is base*height.
To understand the formula for the area of a trapezoid, we looked at it in two ways. Using the trapezoid shown in Figure 5, students cut along the dotted line and reflected the triangle over the bottom base of the trapezoid. Next, they translated the triangle left and vertically so the hypotenuse of the triangle lined up with the non-parallel side of the trapezoid, forming another rectangle. The rectangle clearly had height h, but the length of the base of the rectangle wasn’t obvious. They observed that the length of the base was a number between a and b, and then eventually came up with the base length of , the average of the bases. Again using the formula for the area of a rectangle, they
concluded that the formula for the area of a trapezoid is
Now using Figure 6 and cutting along the dotted lines, they created another rectangle by rotating these small right triangles 180° about the midpoints of the non-parallel sides of the trapezoid. Once again, they created a rectangle and “saw” the formula for the area of a trapezoid.
Distance Formula
When we started the discussion of the distance formula, I asked my students how the formula was explained to them. Some of my GeT students said that their teacher wrote the formula on the board and told them to memorize it: . Again, this idea of having students memorize formulas without understanding them is not what we want our future teachers doing. So, I plotted the points and on a coordinate plane, drew the right triangle (shown in Figure 7), and marked the hypotenuse of the right triangle, d.
Figure 7
Then the students found the lengths of the legs of the right triangle and used the Pythagorean Theorem, giving them:
Now rather than memorizing the distance formula, my GeT students understood its origin.
Using these discovery activities in my GeT course to cover basic concepts in geometry gave my students active learning strategies to use in their own classroom. For some of these students, it was the first time that they experienced active learning in a mathematics course. In addition, completing these exercises illustrated the importance of teaching mathematics for understanding rather than telling students to memorize formulas. Mathematics education research indicates that memorizers are the lowest achievers in mathematics (Boaler, 2015).
Throughout our mathematics courses for preservice teachers, we need to model best practices. These discovery activities facilitate meaningful mathematical discourse, connect mathematical representations, and build procedural fluency from conceptual understanding, which are some of the mathematics teaching practices found in NCTM’s Principles to Actions (2014). As we prepare future mathematics teachers, we need them to understand the importance of what they do every day and the impact that they have on their students’ learning. Many of my GeT students dislike geometry at the beginning of my course, frequently because they had had a bad experience in their high school geometry course.
By the end of the course, most feel prepared to teach geometry and some even enjoy geometry.
Sharon Vestal is an Associate Professor at South Dakota State University
