I propose here ideas teachers need to know and pay attention to when teaching mathematical proofs and how to prove.
A. What is a (mathematical) proof?
I define proof as a relational network of claims (propositions and conclusions), substantiation (established knowledge that makes the claim legitimate) and appropriate connectives so sequenced to justify why the conclusion is a logical consequence of the premises. Continue reading “Tips for Teaching Proofs and Proving”
The following problem (or proving activity, if you like to call it that) is a typical textbook geometry problem. It is tough and guaranteed to scare the wits out of any Year 9 student.
When I used the given condition to construct the figure using GeoGebra, the only thing I can move is A or B and what it does is simply to reduce or enlarge the circle. Pretty boring. So I thought of making C dynamic. The way to do this is to construct point C along the circle and then construct a perpendicular line to AB. With C moving along the circle, the once static and close task is now a dynamic, exploratory and task.
Your students will observe that for triangle ABC to be an equilateral triangle, CD must be the perpendicular bisector of AB. You can now ask them the problem: Given that CD is the perpendicular bisector of radius AB, prove that ABC is an equilateral triangle, which is what the textbook is asking them to do.
In presenting the problem the way I’ve shown above, you did not only make the problem more interesting (hopefully) and accessible to the majority of the learners (I’m sure most of them can answer the questions), you have also given learners the chance to explore the problem first and be familiar with the situation.
Note that you will be doing a disservice to your students’ geometry life if you will stop at #5 and not give them the opportunity to prove. Proving is what makes mathematics different from other disciplines. It would be a shame if they will go through life only complaining about x and not of proving as well. I’m joking but you know what I mean. You may want to check some of my favorite post about teaching geometry through problem solving: Unpacking mathematics – a geometry example and Problem Solving Involving Quadrilaterals.
Engineers, mathematicians, and mathematics teachers all deal with mathematics but it is only the math teacher who talks about math to non-mathspeakers and initiate them to ‘mathspeak’. To do this, the math teachers should be able to ‘unpack’ for the students the mathematics that mathematicians for years have been so busy ‘packing’ (generalising and abstracting) so that these learners will learn to do the basics of packing by themselves. This is in fact the real job description of a mathematics teacher. I won’t comment about the remuneration as this is not this blog is about. I thought it would be best for me to continue sharing about the ways we can unpack some of the important ideas in mathematics as this is the mission of this blog. Just in case you haven’t read the blog description, this blog is not about making mathematics easy because math is not so stop telling your students that it is because that makes you a big liar. What we should try to do as math teacher is to make math make sense because it does. This means that your lesson should be organised and orchestrated in a way that shows math does makes sense by making your lesson coherent and the concepts connected.
Today I was observing a group of teachers working on a math problem and then examining sample students solutions. The problem is shown below:
The teachers were in agreement that there is no way that their own students will be able to make the proof even if they know how to prove congruent triangles and know the properties of a parallelogram. They will not think of making the connection between the concepts involved. I thought their concerns are legitimate but I thought the problem is so beautiful (even if the way it is presented is enough to scare the wits out of the learners) that it would be a shame not to give the learners the chance to solve this problem. So what’s my solution to this dilemma? Don’t give that problem right away. You need to unpack it for the learners. How? To prove that AFCE is a parallelogram, learners need to know at least one condition for what makes it a parallelogram. To be able to do that they need to know how to prove triangle congruence hence they need to be revised on it. To be able to see the necessity of triangle congruence in proving the above problem, learners need to see the triangles as part of the parallelogram. So how should the lesson proceed?
Below is an applet I developed that teachers can use to initiate their learners in the business of making proofs where they apply their knowledge of proving triangles and properties of quadrilaterals, specifically to solving problems similar to the above problem. Explore the applet below. Note the order of the task. You start with Task 1 where the point in the slider is positioned at the left endpoint. Task 2 should have the point positioned at the right end point. You can have several questions in this task. Task 3 should have the point between the endpoints of the slider. Of course you can also present this using static figures but the power of using a dynamic one like the geogebra applet below not only will make it interesting but the learners sees how the tasks are related.
Task 1
Task 2 – Which pairs of triangles are congruent? Prove your claims
Task 3 – What can you say about the shape of AFCE? Prove your claim.
[iframe https://math4teaching.com/wp-content/uploads/2013/05/Parallelogram_Problem.html 550 500]
Here’s the link to the applet Parallelogram Problem
Note that Task 3 has about 4 different solutions corresponding to the properties of a parallelogram. I will show it in my future post.
More of this type: Convert a Boring Geometry Problem to Exploratory Version
A mathematical model is an abstract model that uses mathematical language to describe and understand a situation. Here’s a nice video that presents a model for the Lunch Date Problem. The video below shows a step-by-step tutorial using SketchPad to build a geometric model of the problem. What is nice about this video is that it still leaves the solving to the students. The resulting geometric model leaves enough information for students to figure out a solution.
You and a friend arrange to meet between 12:00 and 1:00 in the afternoon. After a week neither of you remembers the exact meeting time. As a result, it is possible for you arrive at random between 12:00 and 1:00 and waits exactly 15 minutes for your friend to arrive. After 15 minutes, each of you leaves if the other person has not arrived. What is the probability that the two of you will meet?
Mathematical modelling involves the use of mathematics to understand a real world problem. Click What is mathematical modelling? to learn more about it.
