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.

Mathematics in the Streets & the Schools

Carraher et al discuss in this article how children with a formal school education differ from children without a formal school education, in the ways that they solve problems.  To illustrate the scenario, we are taken to the rural streets of Recife, a city in Brazil with a population of 1.5 million people.  Most of the citizens either work in the informal sector, or have a family member that works in it to support the family income.  This informal (non-school-based) way of problem-solving is driven by “four pressing needs. . .finding a home, acquiring work papers, getting a job, and providing for immediate survival.” (p.240)  When I consider what drives me in my everyday activities, I suppose that I’m not far off from what drives a citizen of Recife.  We all are looking for a job, a home, and a means of survival.

I found it interesting when Carraher mentioned that it is not uncommon for children who are 8 or 9 years of age to help with the transactions of their parents’ line of work.  I have a student in my class who is 10 years old and often helps at the family restaurant working the till.  We currently are just finishing up our unit on fractions in our class, and recently this student was having difficulty converting ¾ into a decimal.  When I put the questions in terms of money, he was able to solve the problem.

A study was performed where children who worked on the streets were tested in two different scenarios.  One scenario involved solving problems that related to their line of work and were all oral.  The other scenario involved solving problems in a more formal school-based setting.  Some of these formal questions were given context, while others weren’t.  There seemed to be a direct correlation between problems that were solve with context versus without context.  The children found having context much easier to solve.  Interestingly, we tend to teach algorithms by themselves before we teach how they can be applied to word problems.  These children show that the opposite can be achieved as well.  It should be noted that if the children had to solve larger multiplication questions, their chunking strategies may not have been as effective.

A Linguistic and Narrative View of Word Problems in Mathematics Education

In this article, Gerofsky describes how mathematical word problems can fit under the umbrella of a linguistic genre.  She uses “methods from pragmatics [the study of language in use in particular contexts], discourse analysis [the study of extended stretches of spoken or written text in terms of utterances and their relationships], and genre studies [the study of ‘text types’ in terms of their linguistic and contextual features].” (p.37)

A typical word problem, or story problem, has a few defining characteristics.  It usually introduces the problem with some erroneous background information which may or may not be useful, then it gives some values, and clues as to which operations/algorithms you will need to use, and finally it will ask a question.  In my experience as a classroom teacher, children need to consistently practice word problems.  A give my class a word problem everyday and we go over it as a class looking for these key features mentioned above.  It is imperative that a student knows which information is relevant and which information can be ignored.  Often, students will skip to the question at the end, and then look back at all the numbers, and then combine all of those numbers into one or two operations that have recently been discussed in class.  I find that being able to weed out the unnecessary information and truly understand what is being asked is the key to word problems. 

Radatz (1984) mentions that students in their elementary years, and who are knew to problem solving, tend to focus more on the story, while older secondary school students will immediately look for the math in the problem, rather than be concerned about the story.

Gerofsky also points out that often there can be confusion when multiple tenses are used in word problems.  Some languages do not have the same tense structures as English.  This would make it very difficult for a student coming from another country, or even one from Canada but perhaps has parents who speak another language at home, to translate the problem in their head while they are reading it. 

Another interesting point that is brought up is how most story problems are fictitious.  Sometimes they are based on truths, but use words to make it seem fictitious, such as “If the Lions Gate Bridge was 32 m above sea level. . .”  If we are talking about a factual object, then we might as well use facts, and not use words such as ‘if’.  If a story problem is meant to be fictitious, then it should not be based on real things, and should actually be a story. 

In her summary, Gerofsky questions the use of word problems in childhood mathematics programs, and wonders if there are new ways to pose them.  In essence, we should be critical of the lessons that we learned as children, and be careful to not blindly pass them on to the next generation.