Response to "Why you should learn geometry"

In Walter Whiteley’s article titled “Why you should learn geometry” he refers to an article published by David Eggenschwiler titled “Why you should learn algebra”.  Eggenschwiler’s article argues that high school algebra “teaches the mind how to think”.  He states that “what you learn to think about it not as important as how you learn to think” and that learning algebra lets you think more abstractly and creatively, which then can be applied to other areas such as music and critical writing.  While I would say that Whiteley agrees with Eggenschwiler’s ideas, he also wants to broaden the ideas to include geometry.  Whiteley makes a strong point that if people continue to view algebra as the central mathematical knowledge and geometry and associated visual reasoning as marginal, then we will exclude people with “enormous potential to contribute to science, engineering and mathematics”.

While I agree with the ideas behind both Eggenschwiler and Whiteley, I believe that the learning of mathematics should not be limited to just algebra and/or geometry but should also include things such as the arts and history.  To relate to Whiteley’s ideas – if we do not include many different entry points into the learning of mathematics (algebraically, visually, musically, creatively, historically, etc.), we will end up excluding people who can bring new ideas and insights to the fields of mathematics and mathematics education.

To answer Whiteley’s question about if I feel as if I know enough geometry to bring it in when it is relevant in the math I am teaching, my answer would be yes and no.  Do I always know the correct way to explain things in terms of proofs or terminology? No, I do not.  But do I know how to explain things using shapes or geometrical ideas?  Yes, I would say I do.  An example of when I use geometry or visual representations the most is if I am solving word problems.

My input on the recent trends to reduce or remove geometry from many math programs is that I think that geometry is something that is very important to use in the classroom for the same reasons Whiteley stated in his article.  I think geometry allows students who are visual learners a way to access the concepts they are learning.  However, I think requiring students to know so many different postulates and theorems might be a bit much.  It is like with anything in education – if teachers are required to do x amount of things with geometry, the purpose behind incorporating geometry is often lost.  This is because teachers are told exactly what to do even if it is not the most natural or best approach.  It is similar to an article I read on the use of manipulatives.  The ideas behind using manipulatives are great, but often times teachers feel as if they have to use manipulatives when it may not be the best approach for them in particular, and therefore the intended benefits of using manipulatives are lost.

Response to "Learning from learners: robust counterarguments in fifth graders’ talk about reasoning and proving”

In Vicki Zack’s article titled, “Learning from learners: robust counterarguments in fifth graders’ talk about reasoning and proving” a group of fifth graders are given a problem that has been incorrectly solved by someone their age.  The students were asked to formulate counterarguments to the incorrect solution first by themselves by writing in a math log, then with a partner, then eventually with 12 other classmates (there were 26 students in the class, so the teacher split the class in half and within each half of 13 students, each student had a partner to share ideas with before their shared with all 13 students in their half).  The problem they were given was:

Task 1: Find all the squares [in a four by four grid given as a figure]. Can you prove you have found them all?

Task 2: What if ... this was a 5 by 5 square? How many squares would you have? Extensions were subsequently posed.

Task 3: What if this was a 10 by 10 square? What if this was a 60 by 60 square? How many squares would there be?

The incorrect solution was:

Imagine that two of your classmates, Ted and Ross, came up with the following solution for the 60by60: The answer for the 10by10 grid is 385 squares. So take the answer for the 10by 10 square (385) and 10 x 6 = 60 so multiply 385 x 6 = 2310 and you have the answer for the 60by60. What would you say?

There were about six different counterarguments the students came up with, which I will not try to describe here.  What I will do is list a couple of ideas that I thought were really interesting in reading this article.

The first idea was:

A look at the counterargument helps me as a teacher to come to a better understanding both of the mathematics, and of the children's understanding of the mathematics.

This is something that I think we often times can forget as educators.  Obviously we try to meet students where they are at with their mathematical knowledge, and help them grow from there, but it is so easy for us to impose our way of thinking onto our students.  I really like the idea of using counterarguments as a way of discovering how students understand different mathematical concepts.

The second idea I thought was interesting was:

When Adele keeps saying that the Ted-Ross strategy sounds good, stating: "I think it's right but I don't know how it works," Maggie (who has just finished presenting CA #1 to her group of five) insists: "Then you have no reason to think it's right."

It is amazing how often we just teach concepts to students and tell them that it is correct and the one way to do it, but we don’t let them necessarily figure that out for themselves.  We tend to make “Adele’s” who just say things like “I don’t know why this works, but my teacher said it works this way.  So this is what I am going to do” instead of making “Maggie’s” who ask questions like “Why does it work that way?  How can we be sure?”

Response to "On Culture, Geometrical Thinking and Mathematics Education"

Paulus Gerdes’ article titled “On Culture, Geometrical Thinking and Mathematics Education” that was published in ESM in 1988 discusses the fact that mathematics is often times taught in a way that is “culture-free” and therefore many students do not see mathematics as something useful.  He focused on the culture of Mozambique, which is located in Africa, to give examples of how the surrounding culture can be used to teach mathematics.  One example he gave was on the axiomatic constructions of Euclidean Geometry.  He looked at the rectangle axiom, and instead of just teaching that axiom, students were asked if their parents created rectangles in their daily life.  The students realized that their parents use their own rectangular axioms to construct the base of a house with bamboo sticks and rope.

I found this article really interesting because I think that in order for most students to actually have an interest in learning math, it needs to be made more meaningful for them.  After having a history of math class in my undergrad degree and also in my masters degree, I have started using it a lot to help humanize the subject.  For example, I have a rap on Pythagoras that I like to share with my students just to explain how a real person created this theorem that we use.  This leads into the fact that other cultures used the Pythagorean theorem before Pythagoras’ group proved it.  This then leads into some historical drama, and we know how kids love some drama. J

Also, I am teaching a “meaning of math” class to two grade five students right now.  We have done many things in this class but some of my favorites are: ancient Chinese multiplication, writing in ancient Egyptian hieroglyphics to discuss place value, recreate the “12 Days of Christmas” song using candy for all of the gifts and seeing how much candy we would actually get in the end, and now we are making a blueprint of the building we have class in so that we can eventually make a 3-D model using clay, toothpicks, and popsicle sticks.  This class has been very rewarding for both me and my students.