A categorization of the “whys” and “hows” of using history in mathematics education

This week I read Uffe Thomas Jankvist’s A categorization of the “whys” and “hows” of using history in mathematics education.  In this article, Jankvist takes a theoretical approach to the discussion of why and how to use the history of mathematics.  He subdivides the whys into two mindsets:  history as a tool for learning mathematics, and history as a goal of mathematics.  History as a tool is the argument that students should learn the history of mathematics and use this knowledge to help them approach and tackle current issues.  Jankvist cites that history can be a “motivating factor for students in the learning and study” (p. 237). The history as a goal mindset refers to how learning about the history of mathematics can help round out a student’s understanding of what mathematics is, where it came from, and how it developed.   He also divides the hows into three categories based on illumination, the modules, and a history-based approach.  The illumination approach uses history as a “spice added to the mathematics education casserole” (p. 246), splashing tidbits and snippets of history into a student’s learning. The modules approach is a method that teaches history as an entire module, or chapter in math education.  Lastly, the history-based approach is more to do with a pedagogical method, teaching topics in a historically relevant way, such as the way that a concept was discovered, and used.

Jankvist goes into a lot of depth with each of these topics, providing compelling arguments for the inclusion of each.  His section detailing the barriers to integration of history is also well thought out, leaving the reader a good understanding of the issues at stake.

When I reflect back on my own teaching, I certainly fall into a distinct pair of categories.  I teach the history of mathematics to my students as a goal, as a way to fill out their understanding of the topic at hand, and to help give them some historical context as to the importance of our discussion.  I also discuss these historical concepts in ways that Jankvist would call illumination approaches, using them as a way to flavor a topic and bring more depth to it.  Teacher education, of course, is a primary antagonist to the inclusion of history of mathematics in the classroom, but this is true on several levels.  Teachers need to be educated on why its inclusion is beneficial, how it might be included, and finally, the contents of its inclusion.  I would certainly teach more history in my classroom if I knew more history, but I don’t.  What do you think: should teachers spend time researching, and learning about the history of mathematics so they too can include it in their lessons?

Learning Angles through Movement: Critical Actions for Developing Understanding in an Embodied Activity

This week I read Learning Angles through Movement: Critical Actions for Developing Understanding in an Embodied Activity by Smith, King, and Hoyte (2014).  In it, they discuss a fascinating study they undertook with a class of third and fourth grade students and a motion tracking device called Kinect.   They developed a motion-controlled angle task for their students to undertake where they would move their arms to form angles that would then be displayed on a screen.  They concluded that the task was able to help the students develop understanding between the abstract, visual representation of angles, and the physically embodied representation of angles.  Previously, the authors noted that “body-based dynamic angles representations [were] difficult [due to a lack of] visual support” (p. 106).  I think that motion-tracking technology like Kinect has such potential to provide support for learning through embodied moving.  I can imagine students learning about linear equations where they are provided a Cartesian plane, but then must move their arms to create the right slope and intercept. 

This study reminds me of an activity I did with a grade 11 class over the Summer.  We were learning about the hard-to-grasp ambiguous case of the sine law, a topic that students almost always struggle with.  But this year I brought some woodworking tools to school and had students build physical triangles as per a set of instructions.    They noticed that in certain cases their group built a different triangle than other groups.  Our last activity had us constructing a triangle with one swinging arm to represent the ambiguity.  The physicality of the activity, being able to tangibly swing the arm of the ambiguous triangle helped the students connect with this otherwise abstract concept.  

The 50th Issue of For the Learning of Mathematics

This week I am reviewing the 50th edition of the math education journal For the Learning of Mathematics.  The grey cover of the journal has the simple titling of the journal all in lower case, with a large 50 taking the majority of attention on the page.  The table of contents immediately tells the reader that the journal touches on a broad number of topics.  The 11 articles focus on topics from ethnomathematics, word problems (from our own Susan Gerofsky), international viewpoints, what appears to be, at first glance, a creative writing piece (Under the Banyan Tree by Tahta) and more.  There are even two articles that take a more meta-level approach, discussing the journal itself (Reflections on FLM by Higginson, and The "Spirit" of FLM by Lee).

The journal itself is formatted with two columns of text and paragraphs barely being indented.  This gives the familiar 'wall of text' feel that so many academic journals take.  Furthermore, in the 47 pages of the journal there are only two pictures (both in the ethnomathematics article), one photocopy of a page of a book, and two mathematical doodles at the ends of articles that don't quite fill up a page. The articles do not have abstracts, which might lend one to more informally read through the journal, rifling through it more casually instead of simply head-hunting particular research interests.

One of the last pages in the articles is a page-long 'Suggestions to Writers' which is very telling of the journal's stylings.  It details that math education should be "interpreted to mean the whole field of human ideas and activities that affect or could affect the learning of mathematics."  This tells the reader that FML has a very broad range of publications which can take many forms, but which all focus on the learning of mathematics.  "It is a place where ideas may be tried out and presented for discussion" tells the reader that FLM is on the forefront of mathematics education, welcoming change and fresh ideas.  It is this last line that, to me, really sells the journal.  I want to read a journal full of articles that hope to shed light on new, and maybe controversial areas within math education.  I want to read articles from a variety of sources and FLM even supports "informal research, especially from the classroom."

The Decline and Rise of Geometry in 20th Century North America

This week I read "The Decline and Rise of Geometry in 20th Century North America" by Walter Whitely, a very interesting read from someone on 'the inside' of research in mathematics and in math education.  He begins by discussing the death of geometry in the mid 20th century.  This is most notably chronicled by the decline of its consideration as an important field.  This had a cascade effect: less research was done on the topic, so less graduate curriculum was taught on the topic, so fewer future math educators were able to teach the topic, which eventually relegated geometry to be a supplemental high school mathematics course.  Whitely recounts geometry's fall from grace in wonderful detail, culminating the decline with a discussion of the nature of mathematics, that the public would believe that "mathematics is essentially about logical intelligence," and that "popular culture sees mathematics as detached from spatial intelligence" (p. 3).  As for the rise of geometry in recent years, Whitely credits much to do with the resurgence with computers. As a resource for learning, computers provide dynamic geometry programs which can be used for teaching, learning, and research.  They also provide an unparalleled visual aspect that is so important for developing understanding.              

Personally, geometry is one of my favorite topics within math.   Finding unseen relationships that can be linked to a visual representation is so satisfying.   Geometry gives a way to very easily provide the context for a problem that students can immediately understand, at least in a visual sense.  This understanding can lead to powerful problem posing, and intuitive problem solving. 

The Experimental Mathematician: The Pleasure of Discovery and the Role of Proof.

A very interesting article this week by Jonathan Borwen called "The Experimental Mathematician: The Pleasure of Discovery and the Role of Proof."

The first paragraph of this article had me hooked.  The premise:  computers are becoming so powerful that proving by "quasi-inductive methods" is becoming a viable way for proof.  Where does this leave the mathematician?  Borwen explores the "relationship between proof (deduction) and experiment (induction)."  He carries on to show a large number of examples where hand-deduction, and computer-induction proved successful methods in helping to illuminate a problem.  

          His discussion of large and small numbers I found to be very engaging.  Statistics relies on the ‘law of large numbers,’ where correlations, connections, and relations are forged between quantities of data that humans could not find (potentially).  As mathematics ventures into more abstract domains, it too ventures into large-number-territory, however, not with the quantity of data, but rather simply because often you are dealing with a very large or hard-to-find number (like the trillionth digit of pi).  In both of these cases, computers offer a reliable way to access these kinds of numbers.  Small numbers, however, is where mathematicians can shine.  Small numbers, Borwen argues, are important numbers of which a computer cannot grasp the significance of.  He cites wonderfully concise proofs of √2’s irrationality.

          Concluding with one thought of “there are different versions of proof or rigor” was a rewarding way to finish the article, as Borwen points out that historical proofs have since been reneged and renewed on the basis that the standards of rigor change over time.  Computer-assisted proofs may offer a current ‘standard of rigor,’ but if it will stay the test of time is to be determined.

Mathematics in the Streets and in the Schools

This week I read a very interesting article by Carraher, Carraher, and Schliemann titled "Mathematics in the Streets and in Schools."  This article explains how street vendors in Brazil do mental arithmetic to calculate the prices of goods, and to calculate change.  The catch is, is that these street vendors are children, who's average age is around 11.  The researchers set out to determine the children's abilities to do similar calculations that they would on the streets but in a formal setting.   The researchers found that, across the board, these children were able to do calculations better in the informal setting of street vending than they were when the same or similar questions were posed to them in formal settings.  

These are very interesting results, and they give rise to several interesting discussions surrounding the topic.  In North America we tend to heavily favor first learning procedures in abstract settings before applying them.  One motive for this is that there are fewer factors to take into consideration when solving a problem; they are streamlined for a particular task.  This research, however, shows that this may not be the best method. Having never been taught in this particular way, these Brazilian children are now able to solve problems in a variety of methods depending on the context. 

In the end, I am not surprised with the results from this study.  Children practicing to do math in a specific context will inevitably do better than their peers who practice in a different context.  The flexibility with which the Brazilian children answer problems, however, is something that I think that we should strive to teach to our pupils.

Problem Posing in Mathematics Education

In Brown and Walter's article "Problem Posing in Mathematics Education" they discuss the recurrent themes that should be a part of a student’s mathematical experiences if “learning mathematics is to be viewed as an act of liberation.”  They call these themes sensitivities, and in this article in particular, shy away from the mathematical content, and instead focus on the pedagogical issues.  Each of these sensitivities revolves around the idea that mathematics is, at its essence, a “problem-solving activity.” Filled with examples and anecdotes from their team-teaching experiences, this article was an engaging look at how problem-posing can motivate and involve students, giving them a voice in an, otherwise externally-imposed, curriculum. 

            In their discussion of the efforts that go into creating an atmosphere of problem-posers, they describe three layers: context, posing, and solving.  Typically, a teacher would provide a problem given a particular context, and the students would be asked to solve it. In problem posing, a context, or scenario, is provided for which a problem is to be posed.  This gives the students creative reign over what kind of question they might ask of the scenario.  Brown and Walter hope that these individual efforts, creative endeavors, and opportunities to explore problems of their own choosing will result in students who take more responsibility for their learning. 

            I once taught an extracurricular problem-solving course in mathematics.  The engagement and creativity that students exhibited was truly impressive.  Looking back on my choice of problems, I think that it was their open-ended nature that was a major contributor to the high levels of engagement.  Giving students the choice of how they might attempt the problem was reflected in their range of answers and opened up the door for inspired learning and creative discourse: students defended their own methods and learnt from their classmates.  Open-ended problems could be a good gateway to developing the independence and confidence needed for students to pose their own problems.  

"On the Dual Nature of Mathematical Conceptions"

In the article “On the Dual Nature of Mathematical Conceptions: Reflections on Processes and Objects as Different Sides of the Same Coin” Anna Sfard discusses the dichotomies present in mathemactical concepts, contrasting them with such words as structural and operational, abstract and algorithmic, and declarative and procedural. Sfard focuses on the structural/operational dichotomy and attempts to find the bounds of these two modes, to find where they begin to overlap.  She goes on to rescind  her use, and academia’s use, of the word dichotomy claiming that, in fact, since these two terms are “inseparable, facets of the same thing” it should be called a duality instead as they complement each other.

A very interesting example of this notion in action revolves around the concept of number.  There exists a one-to-one structural relationship between the number of objects in a set and the words ‘one,’ ‘two,’ ‘three,’ but when a child is asked to determine how many in a set, they will repeat the process of counting, showing the operational side of the concept. So number represents both operation, (of measuring how many) and it represents a structure (of representing an amount).

In what order should these two different faces of a concept be learnt?  Or should it be strived to be simultaneous?  Near the end of the article Sfard makes me stop as she mentions that one of the ‘vexing issues’ facing math education is the apparent trade-off between profiency and comprehension, between “promoting the smooth performance of mathematical procedure and developing understanding of how and why that procedure works and what it means.”  What do you think?  Is this an intrinsic issue in learning mathematics, or is it extrinsic of learning math, introduced by ways of how we approach math education?

My personal pedagogic ideology

In Herbal-Eisenmann's paper "Examining the Voice of  Mathematics Textbook" she discusses how a teachers's classroom discourse can limit student learning experiences, and how the language used in textbooks "manifests particular ideological views in mathematics education."  Reading how Herbal-Eisenmann dissected the language of textbooks and how that positions the locus of authority was fascinating and gave rise to a wealth of questions in my own practice.   In particular interest were the use of imperatives in giving exact directing instructions to the student: “make a graph,”  “draw a straight line,” “explain your reasoning.”  Also of interest was the use of pronouns and the absence of the author’s voice, almost as if the textbook were an omniciescent power, a decree from an unknown.

I have been teaching high school mathematics for the last five years in the setting of a small classroom (with 5-10 students) or one-on-one.  Working in a highly supported school environment I realize, after reading this article, that I never assign my students to read through a textbook.  If they have troubles with course material then I will direct them to the wealth of resources we have on hand, namely my colleagues who's job it is to support students with exactly this issue.  This article makes me question my own discourse with my students, and how this is similar or different to what Herbal-Eisenmann describes in her paper.  Being permitted a wealth of time to work with students, I often take a slower, more exploratory approach to learning.  My use of pronouns would differ, obviously, from a textbook, as I am a very real voice in the students learning, not a non-corporeal textbook, and I try to focus on asking ‘how’ questions instead of directing students with imperatives. This personal pedagogic ideology, however, as Herbal-Eisenmann suggests, may certainly be limiting students’ learning in some aspects.  What do you think?

Building from the roof downwards, or the floor upwards?

The introduction to Bill Higginson's "On the Foundations of Mathematics Education" begins with reference to Gulliver's travels, remarking on the far-reaching societies that Gulliver encounters on his voyages. A quoted passage discusses how a math educator attempted to teach his pupils by giving to them a wafer to eat, upon which the theories they were to learn were written. From this passage, I imagine that Higginson will begin to discuss how math education is often conducted en masse, in a one-size-fits-all style, where the teacher is the keeper of the knowledge to be transferred to the pupil. The teacher will offer up their wisdom for the students to nourish themselves on, with hopes that they will be able to digest the offering. Higginson, I imagine, will discuss how math education treats the students who find these offering “too nauseous” to be able to digest, and will offer alternative ways of approaching math education instead of this top-down approach, where the teacher is the purveyor of wisdom.

On further reading, Higginson reveals his thesis statement: “We will not begin to make significant progress in dealing with [the difficulties of learning mathematics] until we more fully acknowledge the foundations of our discipline.” Higginson’s goal in this article is to broach a model of mathematics education called the MAPS-tetrahedral model, where MAPS stands for mathematics, philosophy, psychology, and sociology. He hopes that developing this model as the foundation for math education will help educators better understand their subject, and approach its teaching. Higginson concludes the article in a fascinating, humbling way, claiming that even if this model is shown to be “incomplete, logically flawed, or of very limited use” its ulterior motive, regarding the value it might contribute to help stimulate discussion between educators concerning the foundations of math education, will remain.

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