Response to "Integrating history of mathematics in the classroom: An analytic survey"

In Tzanakis and Arcavi’s chapter titled “Integrating History of Mathematics in the Classroom: An Analytic Survey” an argument is given as to why history of mathematics should be integrated in mathematics education.  They do this by listing objections to this integration and then counter these objections by exploring ways in which this integration is beneficial.  They then give an explanation for how this integration can actually happen, and give many applicable examples from a variety of resources to close the chapter.

It is only fair for me to disclose that I am a huge advocate of incorporating history of mathematics into mathematics education.  I do not believe that history should be the main focus, but I think there is great value in using history as an engagement tool.  That being said, I have taken two history of mathematics classes (one in my undergrad degree at the University of Georgia and one at UBC in my masters program), and there have been people in both of those classes who do not agree with using history of mathematics in their classrooms.  Two of the main concerns they mention are that there are no resources for teachers to use and there is already such a time constraint to teach the prescribed material, so how could they possibly incorporate outside material.  I think that these concerns are primarily due to lack of expertise in the area of history of mathematics.  This reading mentioned how one of the objections against history of mathematics was just that – teachers lack expertise in the history of mathematics, and that this stems from the lack of appropriate teacher education programs.  What might help this issue is making a history of mathematics course a requirement.

Another thing that is mentioned several times in this reading is the nature of mathematics.  The authors say that by incorporating history of mathematics into the classroom, student can see that real humans discovered and struggled with the same concepts many years ago.  This is an interesting topic for me because we have talked a lot about the nature of mathematics in our Monday night class.

Lastly, I would like to share a link to a youtube video.  https://www.youtube.com/watch?v=mZHYE8lOIe4 This was for a project I did in the history of math class I took in my undergrad.  Just thought I should share! :)

Response to “How Multimodality Works in Mathematical Activity”

The main goal of Ferrara’s article titled “How Multimodality Works in Mathematical Activity” is to “further elaborate on the ways multimodal aspects are exploited by learners thinking about mathematics by considering as relevant, not the fact that multimodality manifests itself, but how it happens”.  She tries to do this by looking at how one child learns to graph motion through his perceptual, bodily and imaginary experiences.  The setting of the study was at a primary school in the northwest part of Italy.  The study worked with the same students from grade 2 to grade 5 by having them interact with two different tools to interpret graphs related to given movements and to check movements associated to given graphs.

The way in which one student, Benny, described a graph was a main focus of this study.  He was asked to walk out a certain path using a specific graphing tool, which then created a graph of his movements in a position-time graph.  He had a recollection of moments of his own motions, then imagination of qualities of motion in relation to qualities on the graph, and then he could interpret its shape.  This shows how he was able to enter into a temporary possible world so that he could pretend something was happening in order to explain and understand something else.  Also, at a different point in the study, he was able to communicate this process to the teacher and his peers, which is in the social context of knowledge construction. 

When I was first reading this article, I really struggled with the meaning of multimodality.  I understood after reading the example about Benny that it means using imagination and physical action to conceptualize mathematical concepts.  I think teachers try to create this environment for students by primarily using manipulatives.  In this article, the manipulative was the graphing tool.  By experiencing the movements physically and seeing how the movements created the graph, Benny was able to understand position-time graphs.

I like how this article used graphing technology for younger students because I would have thought to use such a technology more in the high school years.  This has made me reconsider how valuable a technology or manipulative is for any age.  Also, in the article they used a lot of story telling to describe scenarios.  For example, they used Mister x and Mister z who were two secret agents who worked in Movilandia, but were invisible there.  However, Mister x and Mister z could communicate with the students through Cartesiolandia, which is a special language that describes the movements performed.  I love how Cartesiolandia sounds like the “Cartesian” for Cartesian plane where we use coordinate points to describe positions on the graph.  Here the teachers are basically doing the same thing, but are making it more accessible and fun for the students to engage with.  I had a precalculus teacher who did a similar thing to describe concepts in trigonometry.  Even though I don’t remember the specific stories today, I remember loving math because my teacher made is accessible for me to engage with by making it fun.

Comments of FLM 1-1 (the first issue of the journal)

Table of Contents:

The first article in the table of contents is titled “About Geometry” and the second article is titled “The Foundations of Geometry”.  While the rest of the articles appear to be about different topics in mathematics education, I have to wonder if all of them are viewed through a geometrical lens.  What I mean by this is that I wonder if all of the articles are framed around the subject of teaching geometry.   For example, one of the other articles is titled “The Stereotyped Nature of School Word Problems”, and I wonder if this article discusses geometrical word problems or just something about geometry in general.

The only article that gives me any insight about a particular age group is the one titled “The Multiplication Table:  To Be Memorized or Mastered?”  Since multiplication is taught at the elementary level, this title leads me to believe that this article will discuss younger children.  All of the other articles do not give any insight about age groups though.

The Articles:

Most of the articles have some sort of illustration, chart, or graph, and most of them range in length from four pages to nine pages (including references).  It is hard to say a general statement about if there are a lot of references cited about the journal as a whole because some articles have at least a whole column of references and others have no references.  All of the articles are in English. 

The article I find most interesting is titled “Two Cubes”.  I find it interesting because it is only one page long with most of the page being filled by an image of two cubes overlapping.  There is a short paragraph and a caption for the image, but no other writing.  It is strange to me that this can even be considered an article.

There are three articles out of nine total that have subheadings.  One of them has the typical subheadings I would think of where all of the information is chunked into about seven different sections, which seem to be in sequential order of explaining a thought/concept.  The other two articles have very few subheadings in them, which makes me wonder why subheadings were even used at all for these particular articles.  For example, one of these articles only has the subheading of “Postscript” and nothing else.  It seems strange to me to only have one subheading.


The Issue as a Whole:
The front cover has an image of a tessellation, and there is an explanation about this particular image on the table of contents page.  Maybe they used an image of a tessellation since the issue might be framed about geometry?  The back cover has a space where annotations can be made about the information in the articles.

The material on the inside of the front cover summarizes the purpose of the journal.  It also gives information about when the journal is published and how to get a subscription.  On the inside of the back cover there is suggestions to writers who might submit an article for publishing.

There is no material between the articles, which I find interesting because I am use to reading journals where each article takes up an exact amount of pages.  This journal is not like that.  An article might take up four and a half pages, and another one might start where that one finishes.  Maybe they are interested in conserving paper or only want the journal to be a certain length each time?

I am not sure what “author identifications” means.  I do not see a particular spot in the journal where the authors are identified in any other way other than listing their names.  By looking at their names, they all seem to be from North America. 

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.

Response to "A Linguistic and Narrative View of Word Problems in Mathematics Education"

In Susan’s article titled “A Linguistic and Narrative View of Word Problems in Mathematics Education”, she looks at the pragmatic structure of word problems to try to find the “unspoken assumptions underlying its use and nature as a medium of instruction”.  She writes how word problems are typically structured using three components: the set-up component to establish the characters and location of the story, an information component that lists the needed information to solve the problem, and lastly, the question component.  This part was very interesting to me because this is how I actually make up word problems, and yet if someone were to ask me to describe the process I take in making word problems I probably would not describe it this way.  I would say that I use real-life scenarios that ask students to use the mathematics we just learned to problem solve.

This brings me to the next interesting part from Susan’s article about looking at word problems using linguistic and metalinguistic verb tense.  While I don’t quite understand what these terms mean (there are definitions of them within the article), an example she gave really brought home to me the point about word problems not really making sense at times.  The example is “A truck leaves town at 10:00am travelling at 90km/h.  A car leaves town at 11:00am travelling at 110km/h in the same direction as the truck.  At about what time will the car pass the truck?”  She goes on to explain how “a truck leaves” and “a car leaves” are linguistic present tense and “the car will pass the truck” is linguistic future tense.  She explains how the tenses used in word problems are often self-contradictory and how this takes away from the truth-value in word problems, which is the last thing I will discuss.

Susan gives a word problem and then rewords it with the information in parentheses added.  The word problem is this: Every year (but it has never happened), Stella (there is no Stella) rents a craft table at a local fun fair (which does not exist).  She has a deal for anyone who buys more than one sweater (we know this to be false).  She reduces the price of each additional sweater (and there are no sweaters) … The problem continues on in this same way.  She makes note how the truth value of the word problem doesn’t actually change when this extra information is added in. 

I personally have never really struggled with solving word problems, but this article brought to my attention how flawed word problems are.  It is no wonder people struggle with solving word problems.  Their tenses are contradictory and there is no truth in their statements.  This made me question the purpose of word problems.  I understand that we use them to try to make the math seem more realistic and to show students where they can use the learned material in everyday life, but if the word problems are bogus, are we really benefiting our students?