Assignment, due Monday, 4/4/22 (300 words minimum): We next return to a topic that we introduced earlier in the semester, Geometry. This is a subject that tends to provoke strong feelings in many people (teachers and students!). Our current reading from Wu directly addresses one of the most important discussions around Geometry – in TSM, it is unique in being the one area of mathematics in which axioms and formal proofs are the focus. Read the three-page except from Wu, and respond to the questions below (300 words minimum).
- What does Wu think of the current state of affairs in TSM with regards to teaching Geometry? What are some of the solutions he proposes? Explain.
- On page 334, paragraph 2, Wu describes what he considers the most important skill that students must acquire. What is it? When you think about your own teaching, what is one thing you can do to help students acquire this skill?
- Extra Credit. Respond to one of your fellow students’ comments. Do you agree? Disagree? Did their comments make you think or provoke additional questions? Reminder: Be respectful, be kind.
The current TSM has a curriculum for Geoxmetry full of proofs. It begins with axioms and makes students prove every theorem using axioms. Wu is concerned that students don’t have enough opportunities to achieve mathematical reasoning until they learn Geometry in high school. Students’ lack of practicing how to prove can discourage them from learning Geometry. They may get overwhelmed with the new field of mathematics and a new method using axioms. To solve this, Wu suggests inserting proofs for major theorems in every school mathematics course. Such exposure to proofs allows students to get familiar with proving mathematical statements with axioms when they learn and prove theorems in geometry.
Wu points out another issue with TSM is Euclid’s model of “proving every statement from axioms”. It seems too unrealistic for high school students to prove every statement from axioms because it is time-consuming. Also, deductions from axioms are problematic since they make it dull and difficult to learn. Wu recommends discussing axioms at the end of geometry rather than at the beginning. Teachers can start with brief axioms and familiarize students with the concepts. Proving axioms can be positioned as the last portion of geometry. This encourages students to approach to proofs with better understanding.
Wu narrates that acquiring how to move from a hypothesis to a conclusion using logical reasoning is the most essential skill in school mathematics education.
As a pre-service teacher, students can develop logical reasoning during the open discussion which can be implemented almost every lesson after presenting the mathematical concepts. It allows more students to discuss the question by building a hypothesis to a conclusion with logical reasoning. In this method, students can participate and propose their hypotheses with mathematical reasoning to support their arguments when I ask a question. Few more students can suggest their ideas with evidence.
Geometry in TSM is littered with proofs everywhere which start with axioms and build from there. The problem Wu and I have with this model is before geometry students are not made to write proofs and after geometry proofs are not touched on for a while in some cases or even not at all depending on what students go onto do. Something that I would like to highlight is we as math ed students have to take an intro to proofs class and then the concepts are applied to our senior courses. High school students are just thrown to the wolves and told to figure it out which is not fair by any stretch of the imagination. Of course students get overwhelmed by this new way of doing things and can end up disengaging.
Wu advises to make the transition less of a shock showing proofs in other courses leading up to geometry so it doesn’t hit students like a ton of bricks. Another thing Wu mentions is a prove everything approach for isn’t a good way to start out I would ask what notions students have already about geometry. I would focus on applications such as architecture or fashion from a more real world context I would incorporate the axioms. The main goal would be to have students develop a hypothesis and then learn to use what they are given to problem solve in this case axiom and theorems.
Hi, Irina. I really like what you brought up about getting students to ask real-world questions, and then you’re connecting the questions they ask with geometry. I think this will help them understand better and make them more interested and engaged. I think your method can better promote students’ learning thinking so that they can think and discover more actively.
TSM’s present situation in geometry is to teach students to prove everything with a few axioms. The proof method in TSM is very difficult for students to understand. TSM doesn’t teach students how to prove it correct. Wu’s solution is that we should focus on educating students about the proof of mathematics. Everything we can use can be taught in a separate chapter to teach students how to prove it, and students can learn how to prove it with an acceptable understanding method. I think Wu’s idea is very correct! In the mathematics education I received in the past, I found that I didn’t think carefully about the origin of every formula and theorem, which led me to forget it after I didn’t study it for a period of time. But when I thought and reasoned every formula and theory I used through my own brain, I could better understand and remember it, and I would be handier when I needed to apply it to them in the future, and I could also connect different rules to solve more difficult problems.
Wu believes that the most important skill that students must master is how to use logical reasoning to draw deductions from assumptions. When students encounter two statements, they can find the connection and relationship between the two statements and apply them together.
Generally speaking, I think Wu’s ideas are worth learning from every teacher. If I become a teacher in the future, I will definitely teach my students the ability to think first. I think learning to prove and think is the process of really learning things. You should teach students the ability to fish, instead of giving them the fish directly.
“The best way to learn something is to discover it for yourself”, which means that the most productive learning is when students explore and discover for themselves.
“The best way to learn something is to discover it for yourself”, I agree with that. however, there is a certain thing according to the ‘zone of proximal development’ students cant do by themselves.
Wu believes the current state of affairs in TSM is filled with proofs using axioms when it comes to Geometry. The current main objective of TSM is to introduce axioms at the beginning of the course and make students prove every theorem no matter what. However, since students are new to the environment of proofs it will be very difficult for them to progress. They are first introduced to proofs in geometry and have no second hand knowledge, thus the goal of teaching students with reason fails. Wu suggests proofs deserve to be an integral part of the school mathematics curriculum. Proofs can be taught in a way that students will understand. If proofs can be incorporated in every math course, it will decrease pressure on students and help them get a better view on proofs using axioms.
Wu considers the most important skill that students acquire is whether one learns to move from A to B by the use of reasoning. Which means students must be able to successfully formulate a hypothesis based on their surroundings. Then students will be able to come up with a conclusion based on what makes sense. One thing I can do to help students acquire this skill is to encourage them to make discoveries on their own. In other words, just serve them as a guide, and help them progress when they are having difficulties. Based on students’ discoveries they will get a realistic grasp on what was learned, and that will help them formulate conclusions.
I like the way you help students after they try it on their own instead of showing them how to do it. I believe that this approach has more benefits rather than giving them an answer. During the peer leading and tutoring, I implemented this in every session and I found out that it was pretty helpful for students to understand better. Sometimes little struggles needed to learn…
In Higher levels of mathematics courses, students require to learn different types of proves where axioms involve. Based on the curriculum students learn about proof in their geometry classes with axioms and the education system believes it’s the best way to learn proof. However, Mr. Wu believes that students don’t have enough opportunities to practice problems involving mathematical reasoning before they take their geometry class. consequently, students often can get overwhelmed by the proof in their geometry or other higher-level classes. wu suggested that every shcool mathematics course should include some proof with the use of axioms. therefore most of the students will feel much more comfortable and confident when they are proving theorems in their geometry classes and beyond. Also from my own experience doing proof in my geometry classes for the first time wasn’t that simple due to the fact I didn’t have something similar in my previous classes . however, that experience from my geometry class helped me to understand problems in my proof and logic class.
Mr. Wu said, ” In terms of school mathematics education, the most important skill that students must acquire is how to move from a hypothesis to a conclusion by the use of logical reasoning.” I believe the only way students can get better with that is by practicing more and more unless they can find the pattern between similar hypotheses. They should start with a hypothesis that is easy to understand and then they will progress to a more complicated hypothesis. moreover, it is important for students to know a different way to prove the same hypothesis and make a valid conclusion. Groupwork can be helpful for students as well where they can effectively discuss the hypothesis with their classmates which can help them to have better comprehension.