A Categorization of the "Whys" & "Hows" of Using History of Mathematics Education

In this week’s reading, Uffe Jankvist looks into the purposes of teaching mathematics through its history, and how it should be done.  Is the purpose to learn about the history of mathematics or is the purpose to use history to supplement and further enrich the curriculum within lessons?

Those who argued for using history as a tool, rather than a goal, put forth that history engages students and gives math “a more human face.” (p.237)  Students can see that the problems that they are dealing with in mathematics, are the very same problems that mathematical geniuses spent much of their lives tackling.  I have noticed that whenever a subject is seen through the lens of history, my students have become more engaged.  I have found that almost every student enjoys a story, and being read to.  If a teacher can incorporate a story into a lesson, that takes the audience through a series of events that culminate in a problem being solved, or even not solved, I believe that there is a good chance that the students will be more interested in the subject matter. 

When using history as a goal, the author was clear that it was not to simply learn the history of mathematics, but rather understand how mathematics has evolved and developed over time.  I personally prefer using it as a tool, because I would find it very difficult to include it in my curriculum.  Then again, maybe there is a way to expose students to the history and by doing so teach a prescribed learning outcome (PLO) at the same time.

Jankvist discusses three different ways of teaching history of mathematics:

1)   The illumination approach – the teaching is supplemented by historical texts

2)   The module approach – historical units are established (separate from the standard units)

3)   History-based approach – the history is not directly taught, however it influences the possible order in which mathematical units are introduced

Any one of these three approaches may be applied to either teaching history as a tool or as a goal.  I find myself leaning more towards the first approach.  I would like to believe that our curriculum is already taking into account the third approach, however I have noticed that some textbooks would have me believe otherwise.  I find that the second approach separates the history from the math too much.

Using history to teach mathematics is definitely not for everyone.  It may confuse the students more than it will benefit them.  Not all teachers are comfortable or knowledgeable in regards to the history.  It is my belief that it should be a decision made by the individual teacher, and not included in the curriculum necessarily. 

How Multimodality Works In Mathematics Activity

In the article, “How Multimodality Works In Mathematical Activity: Young Children Graphing Motion” written by Francesca Ferrara, different experiences are observed and analyzed to see how they affect mathematical learning.  Specifically, Ferrara looks at perceptual, sensory, and motor experiences.  She noticed that there is often a very concrete connection between the perceptual/sensory neurons and their associated motor neurons so that when you think of solving a problem, you are using the same neurons as when you actually solve it.

To work with these established neural connections, Ferrara created a study where she analyzed how primary students used digital technology (graphing calculators & computer software).  By using the technological tools, the students were stimulating their perceptual/sensory neurons as well as their motor neurons.  The students would capture movement and data would instantaneously be displayed on the calculator or computer.

The benefit of these activities was that the children could view the position-time graph being created as their peer moved in front of the motion-capture device.  Often we are given a graph to analyze, but have not actually witnessed how it was created.  I can definitely see how the students can benefit from experiencing the sensory stimuli and creating a direct association with the graph itself.  It should also be noted that these students started to participate in this study when they were in grade 2.  Position-time graphs are not usually seen in mathematical lessons until later.  After seeing the graph create itself while someone walked past the motion sensor, the students were able to not only make sense of what the graph represented, but they could also in a sense re-create how it was made.

There was a second experiment that also involved the students making an association between the movement of an object and the creation of a graph on a piece of technology.

When the students were asked to explain the graphs, they often used physical movements (recreate the action), and their imagination (“pretend that. . .”).  Due to the numerous ways that the students were able to describe the graph, it was deemed to be multimodal.  I think that this style of learning is an excellent one.  By tapping into the motor as well as the sensory aspects of learning, I believe that there is a greater chance that the student will be able to recall what was learned.   Sometimes, basic physical movements can help students retain knowledge such as basic facts.  For example, one could create specific body movements when reciting their 4 times tables.  This style of learning may not always be possible in a classroom setting, but it should definitely be encouraged.

FLM 33-2 (100th issue)

Upon first glance at the journal I read “An International Journal of Mathematics Education.”  Subsequently I decided to investigate to see how internationally diversified the journal actually was.  Upon looking over the references, I noticed that almost all of the authors were either from England or The Netherlands.  There was the occasional author from Germany, Turkey, or The United States, but a large majority were from England or The Netherlands.  The journal itself is published and printed in Canada.  I would have liked to have seen more articles that were written by Canadian authors, or authors from countries outside of Europe to offer a more internationally holistic perspective.  While five different countries is fairly diversified, they are all classified as first world countries.  I then decided to see where the authors writing the articles were based out of.  From this I saw some greater diversification. 

Secondly, I looked over the articles themselves to see if there were any noticeable connections between them.  The articles seemed to discuss vastly different topics.  I wonder how the editors decided to group them together.  Does the image on the cover allude to some overarching theme?  The articles do seem to be all very similar in length (6-8 pages).  I also found it interesting that occasionally there would be an “Editor’s Note” at the end of the article.  They seem to be used when the editor wishes to add a contribution to the article.  I wonder if they need the permission from the author to do this.  While it is intended to aid in the understanding of the article, it may also take away from the point that the author is trying to make.

Lastly, upon visiting the website I noticed that the journal’s intent is to bring forth potential ideas for discussions.  Perhaps this is the reason for a variety of article topics.  If one of the articles in particular  receives an abundance of feedback, perhaps other authors will pursue the field further.

Is There A Geometric Imperative?

In this article, Dick Tahta discusses how geometry largely “lies in imagery rather than in the words that accompany or describe such imagery.” (p.20)  He does this by breaking down the geometric abilities of people, which are already present in students when they first start school: imagining, construing, and figuring.  It is difficult to fully understand imagination as it presents itself differently amongst every person.  In addition to this, it is near impossible to visualize what someone is imagining, because the moment the image is described in words, the image changes as the words can mean different things for different people.  Construing is defined by Tahta as either “seeing what is drawn” or “saying what is seen” (p.20)  He goes on to discuss how depending on who you are and how much two-dimensional imagery you have seen, a 2D picture may look like many different things, depending on shading, or perspective.  When I read this, I immediately thought of M.C. Escher and his puzzling imagery of never-ending staircases, the hand drawing the hand picture.  These are classic examples of how subtle nuances in imagery can elicit different visualizations.  The last geometric ability, figuring, is defined as “drawing what is seen” (p.20)  Tahta goes on to discuss how many students may be able to visualize numerous shapes, both two dimensional and three dimensional in their head, however if asked to draw the image they will often do so with great difficulty.  For example, a cube before the age of seven or eight is often drawn as the net of a cube (6 squares).  Most often, a student is taught how to draw three dimensional shapes, or two dimensional shapes from a different perspective, by someone else.  Tahta also makes some connections between understanding geometry and the human vestibular system.  Through interactions of our semi-circular canals and the objects in the world that we see and manipulate, a sense of space and perspective is gained.  Multiple perspectives can be achieved through sharing with others. 

In response to this week’s focus question, I must admit that I do not feel that I have enough of a geometry knowledge-base to bring it in whenever it is relevant in mathematical lessons.  This does not mean that I do not bring it in, in other subject areas such as art, or even P.E.  I have gained an appreciation now for students who may have different perspectives in geometry due to their past experiences, and I will attempt to honour those different perspectives whenever possible.

Learning from Learners

I was immediately drawn to this article within the first sentence as the author stated that they had been teaching grade 5 for the last ten years.  While I haven’t quite taught for ten years yet, I have primarily taught grade 5.

The purpose of the study was to analyze how grade 5 students problem-solved, with respect to a proof.  The particular proof that they were asked to do was, “Find all the squares (in a four by four grid as a figure).  Can you prove you have found them all?” (p.3) There were also extensions to this question, such as finding how many squares were in a 5 by 5 grid, or 10 by 10, or even 60 by 60.

The author, Vicky Zack, posed possible solutions to her students and asked them what they thought of them.  What I like about this strategy is that it encourages students to question a possible solution, and they can do it without potentially hurting any other students’ feelings.  Occasionally, I notice these types of questions asked of students from mathematical textbooks.  They are usually at the very end of a lesson, as they are thought of as being the “challenge” question.  I wonder what would happen if these types of questions were asked right from the start.  Would students become immediately engaged, trying to disprove or prove a theory?  Or would they be more or less apathetic towards the problem, with no prior knowledge of the subject matter?

Zack found that my analyzing the students’ counter-arguments she gained a higher understanding of their mathematical thinking.  In order to prove or dis-prove a potential solution, often the students had to extend and apply the proof to test it.  In this way, the students began to understand the problem on a much deeper level.  Zack also noted that her students would display sophisticated reasoning, but would often not vocalize it.  She found it useful to give an alternative voice to their reasoning by putting it in current mathematical terminology.  In this attempt, Zack connected the problem-solving abilities of her students with the mathematical curriculum.

If I were to do this activity with my class, I would be tempted to perform it at the beginning of a patterns unit, and again at the end of the unit to see how the students’ thinking changed (or remained the same) over the period of time.