Tag: math education research

Posted in Math research

Math Education Studies

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Math-ed studies is my  new blog about research in mathematics teaching and learning and teacher learning. It contains studies, books, links, etc about mathematics education. It is actually my nth attempt to organize myself. Evernote is not enough to organize me. With this blog I hope it will be easier for me to trace where I read this or that idea. I hope this will also be useful to you especially in doing a literature review for your research. For obvious reason I cannot provide access to the full paper or books just the abstract and some important ideas from them. Usually these ideas has to do with what I’m currently writing. I have included the title of the journal and publisher of the paper in each post if you want to read the full paper.

To give you an idea what the blog contains, here’s the most recent posts:

Recent Posts

If you have a paper about mathematics teaching and learning and teacher learning published online with public access, just e-mail the link to me. Of course I reserve the right to link it or not in math-ed studies blog.

Posted in Teaching mathematics

Should the historical evolution of math concepts inform teaching?

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Should the history of a math concept inform the way we should teach it? Some camps, especially those that strongly object to the usual axiomatic-deductive style of teaching, advocates the use of a “genetic” teaching model that takes seriously into account the historical roots of mathematical knowledge. Here are some studies that support this approach.

Harper (2007) compared the historical analysis with students’ empirical data and found a parallelism between the evolution of algebraic symbolism and the way students understand the use of letters in school algebra, concluding that “… the sequencing of conceptual acquisition appears to parallel that which is to be detected through the study of the history of mathematics.”

Moreno and Waldegg (1991) found that “… in situations involving the concept of infinity, the student response schemes are similar to the different response schemes given by mathematicians throughout the history of mathematics,…, when faced with the same kind of questions”

However, there are also those who contradicts this conclusions: For example, on solving linear equations, Arcavi argues that,

….solution methods generated throughout history are quite different from the usual methods generated by students. Consequently, we cannot assert that a reason for the study of linear equations is based on or inspired by parallels between history and psychology – these parallels do not seem to exist (Arcavi 2004, p. 26).

Herscovics acknowledges that while obstacles in the nature and evolution of knowledge are in parallel with some of those met by the learner and are associated with his/her cognitive evolution, she also warns that this parallelism should not be taken too literally, since learning environments in the past are significantly different from those of our learners now (Herscovics 1989, p. 82).

In their investigation of the parallelism between historical evolution and students’ conceptions of order in the number line, Thomaidis & Tzanakis (2007) has this to say:

If room is left for genuine problems to help the emergence of the new concepts, motivate students to appreciate their necessity, or formulate their own alternative ideas (as it happened historically), teaching will not be restricted to the presentation of formal constructs in their polished final form, as it is often the case under the additional pressure of factors peculiar to the modern educational system itself, but will help students conceive mathematics as a creative, adventurous human activity.

Like in most issues related to teaching and learning, there is no clear cut answer here,  but it will always pay to know for teachers to have a sense of how specific math concepts evolved in history. It could provide valuable information both in the design of instruction, in anticipating cognitive obstacles and, for making sense of students difficulties in learning the concept. Teachers must also always remember that the evolution of a math concept is always tending towards abstraction. And because definitions of math concepts are already abstractions of those concepts, starting with definition in teaching is a no-no. Read why I think it is bad practice to teach a math concept via its definition.

Posted in Mathematics education

Forms of mathematical knowledge

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Anyone interested to understand how mathematics is learned should at first understand what mathematical knowledge consist of. The book Forms of Mathematical Knowledge: Learning and Teaching with Understanding describes various types of knowledge that are significant for learning and teaching mathematics. It defines, discusses and contrasts various types of knowledge involved inthe learning of mathematics. It also describes ideas about forms of mathematical knowledge that are important for teachers to know and ways of implementing such ideas. The book is a collection of articles/papers from well known mathematics educators and researchers.


Top in the list of forms of knowledge presented in the book is a discussion about intuition and schemata.
While there is no commonly accepted definition, the implicitly accepted property of intuition is that of self-evidence as opposed to logical-analytical endeavor. Now, what is the role of intuition in the learning of mathematics?

While in the early grades teachers are awed by intuitive solutions by our students, those handling higher-level mathematics would find intuitive knowledge to constrain understanding of mathematics. In the book, the author of offered examples of these. He also defined the concept of intuitions and described the contribution, sometimes positive and sometimes negative, of intuitions in the history of science and mathematics and in the teaching process. The author argues that knowledge about intuitive interpretations is crucial to teachers, authors of textbooks and mathematics education researchers alike. The author further argued that intuitions are generally based on structural schemas.

My favorite article in the book is about the description of mathematical knowledge as knowing that, knowing how, knowing why and knowing-to

Knowing why, meant having “various stories in one’s head” about why a mathematical result is so. For example, when partitioning an interval into n subintervals, one might recall that n+1 fenceposts are required to hold up a straight fence of n sections. Knowing why and proof are different — in many cases, the proof doesn’t reveal why. As an example, the author suggested that when primary teachers ask why (-1)(-1)=1, they want images of temperature or depth, not a proof, or even a consistency argument that negative numbers work like positive numbers.

Knowing to means having access to one’s knowledge in the moment — knowing to do something when it’s needed. For example, in evaluating a limit, a student might just know to multiply by a certain quantity divided by itself. This kind of enacted behavior is not the same as writing an essay explaining what one is doing — it often occurs spontaneously in the form of schemas unsupported by reasons, whereas explanations require supported knowledge.

Forms of Mathematical Knowledge: Learning and Teaching with Understanding is a must-read for teachers, educators, and those doing research in mathematics teaching and learning.