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
- What do the markings in the diagram tell you about the figure ABCD? What kind of shape is ABCD? Tell us how you know.
- Do you think the two triangles formed by the diagonals are congruent? Can you prove your claim?
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
Like the post. I think the idea of unpacking is helpful here. Wondering if the students need some support to conjecture AFCE is a parallelogram first. Either show length of the sides (for an opp sides are congruent approach – goes with the proof method) or the slope of the sides (if you want to emphasize the parallel). Sometimes I think about a process of formalizing the logic: notice, justify, prove. What is true about the diagram? Why is it true? Is it always true?
It might be helpful to have a link to the sketch on GeoGeraTube, or a download link for the file. Sometimes my browser just won’t do the java for embedded GGB. On the Tube there would be the option for mobile-ready version, too.
I was thinking of putting that scaffold, too as to the shape of the AFCE but the applet shows that E and F were both coming from the midpoint of the diagonals. Also DE and FB are congruent. Maybe it’s enough hint for them to make the conjecture. If the static figure will be used then yes maybe students need additional support.
I have a problem uploading in GeoGebra tube. I will just put a direct link to the applet so readers can open it in their browsers.
The idea of ‘unpacking’ is from one of my readings on math teachers knowledge. I could not anymore recall the title but I’m sure I got the ‘unpacking’ term from one of Deborah Ball’s articles.