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

Gerofsky explores the goals of using world problems in school by using linguistic analysis on the structure of mathematics word problems used in schools. There are usually 3 components seen in a typical word problem: 1. the exposition, 2. some information to solve the problem, and 3. the question. Gerofsky pays special attention to the first component which establishes the characters and places of the story, and argued that it only worked as a statement which is actually irrelevant to the question. The author continued to examine the structure of word problems with the lens of linguistics analysis. It is pointed out that there is a mutual understating over the inconsistency of the tense used in word questions (sometimes, it is all present tense, while in others it is mixed with present and future tense), which might be explained by the philosophy behind word problems: hypothetical solutions to hypothetical situations. Gerofsky argued that most questions could be rewritten in a standardized form: Suppose condition A exists. Then, if conditions B and C held, what would be the answer to D. This leads to an interesting argument that word problems are hardly a fiction genre as word problems don’t present “truth values”. Gerofsky invites us to think of word problems as parables, but also cited contradictory studies about the analogy. According to her, delineating the boundaries of word problems as a genre opens a door to generate more productive discussion over the rationale for using word problems at school.

Stop:

While many teachers hold the belief that word problems help students to learn “adaptive” or “contextual” mathematics, the author invites us to look closely at the elements, structure and variance of world problems. There is no doubt that students have developed an “immune system” to word questions, as they skip the story and quickly search for numbers and key phrases which are useful for plugging in to formulas. With that mindset, students gain a limited understanding of goal setting, planning, and do not develop a positive attitude toward problems.

Question:

Do you prefer “real world” word problems with the three elements mentioned, or abstract problems that cut to the chase?

My Research Interest

I am interested in how technologies have helped students’ problem solving. I have been actively involved in developing and teaching an extra-curricular robotics and coding program, offered in libraries and learning centers. Although I observed a proportion of students in these programs demonstrating significant improvements in engagement, more research is needed to investigate the contributions of technologies to mathematical problem-solving. 

On "Mathematics as Medicine" & "Balancing Equations and Culture"

Article A: Doolittle (Canada): Mathematics as Medicine (in Proceedings of the Canadian Mathematics Education Study Group (CMESG) 2006, pp. 17-25).

Edward Doolittle from the University of Regina is a mathematician and a Mohawk Indian who grew up in the suburbs of Hamilton, Ontario, knowing almost nothing of his indigenous culture before attending the University of Toronto. He was connected with indigenous traditions upon participating in an Indian Health Careers Program, a program designed to help to increase the representation of Aboriginal people in medicine and other health-related careers. From there, he started to seek efficient research methods to help mathematics education for indigenous people.

He is skeptical that quantitative research efforts, such as quantifying or justifying assessments with standardized outcome, can have fruitful results. Doolittle argued that these approaches would not be able to offer adequate solutions as the complexity of the situation seems to be beyond the capacity of any research which tends to offer simple responses to a complex question.

Ethnomathematics is a mean of addressing the interactions between math thinking and cultural beliefs, and the presenter found it far more reflective and respectful to Indigenous traditions of thought. However, he reminded researchers to be aware of the tendency of oversimplifying the inquiry such as saying “The tipi is a cone”. Another two pieces of advice include using culture-appropriate language and words that blend in with the indigenous language when introducing mathematics terms, and respecting indigenous spiritual traditions by trying to view mathematics as a power of medicine from the perspective of it connecting lives and making us better people.

Article B: b) Doolittle & Glanfield (2007): Balancing Equations and Culture: Indigenous Educators Reflect on Mathematics Education. FLM, pp. 27-30.

In this conversation between Doolittle and Florence Glanfield, two aboriginal mathematics educators, several issues relating to perspectives on the value of mathematics were discussed.

1. Doolittle elaborated on a story in which a boy came back to his camp with a strange animal, a horse. The boy showed his people how to make use of and keep a good relationship with the horse. The horse became an asset to the camp and the boy later became the chief. Doolittle raised the question of whether math can be as useful as the horse, or would it rather be a Trojan horse which might actually cause huge loss to the camp.
2. Florence was concerned about the power of mathematics which she felt that she benefitted from, but was not able to be distributed equally to her tribe. She recognized that there was a division and barrier between “the secret mathematics society” and the mass public.
3. Doolittle asked the question of whether mathematics is universally powerful or only has power in western contexts. He was skeptical that many requirements for math in the job market were just arbitrary and did not serve a real purpose.
4. Florence believed many pre-service teachers were not offered opportunities to explore relationships within math. They were “silenced” in their early formal years in expressing their ideas about math notions and relationships.
5. Doolittle found that the aboriginal perception of relationships was not compatible with mainstream relationships presented in math. He was concerned that current prevailing school mathematics might force aboriginal people to give up their natural and innate mathematical understanding of the world.

They both believed a balanced approach towards math from the lens of aboriginal culture. Mind, body, emotion and spirituality can be most promising to help both aboriginal people and math society.

Reflection: :

A sample used to illustrate the complexity of mathematics in indigenous society  is the following,

Q: If he gets four dollars a day, how many is he going to have in two days?

A: Six.

Although the answer appears wrong at first glance, there are more subtle ethnic implications behind the answer. For example, if a labourer were to work for two days and was told that they were to receive 4 dollars each day, but only ended up receiving 6 dollars in total, this question would be treated as “real math” and the answer would end up being a reflection on reality. One reason that “street math” is not a major chapter of our current curricula is that “street math” is more of a personal or group interpretation of a phenomenon with minimum abstraction and lack of ability of generalization. Of course this is against some of the perception of math and its value. Since we have limited the term of math to the western context, I kind of feel some new names should be introduced to recognize the different “original math” base on its culture roots.   

Question: 

In teaching current school math curricula, have you experienced any hesitation, resistance or conflict due to cultural differencecs?    

On " Models and Maps from the Marshall Islands" By Marcia Ascher

Models and Maps from the Marshall Islands:

A Case in Ethnomathematics

The researcher investigated how stick charts/maps were made and used by the Marshall Islanders to navigate the Pacific Ocean. Stick charts were widely used by the Marshallese for navigation before World War Two. They were usually made from palm ribs which were tied up to form an abstract geometric composition. Two types of charts are introduced in the study: Mattang and Rebbelith. Mattangs were used in the training of navigators and were constructed with identical standardized models. The Rebbelith charts are used as maps of the region, with the same lines and curves seen in Mattang. Unlike modern maps, they were not "images" of the landscape, but rather contained abstract shapes of the land and water.

Wiki: Marshall Islanders 

The study focuses mainly on the mathematical ideas of modeling and mapping embodied in these charts/maps. According to the writer, these maps and charts are analogical and visual representations of the original space, which require scientific knowledge and mathematics skills in establishing and formulating these spatial relationships. “Maps, therefore, must be viewed broadly and, with a broad view, we can better appreciate them as products of mathematical abstractions.”

Stop 1. The charts had to be studied and memorized prior to a voyage and they were not available during a trip. Wait a second, memorizing a map? It must have been a more challenging task than memorizing times-table, which including its practice in elementary school has been debated over for many years. Such a pedagogical practice can only be understood in its cultural context. As the writer explains, due to the limited land and water dominated environment, knowledge and techniques were highly prized and well kept in a way that the “textbook” had to be preserved on land. The charts that represented major ocean swell patterns and silent tides are considered the most valuable assets in the community.

Stop 2: It is also interesting to note that when navigators choose their successor "knowledge carriers”, women were not excluded. This is against my understanding that women were now and then under-presented or mistreated in most education systems in male-dominated cultures. 

Question: The public does not commonly say that mathematics and culture are connected. Do you have experience of teaching mathematics in a culture-rich activity?

Reflect on "On the Foundations of Mathematics Education" by Bill Higginson

William Higginson (1980) presented a model to explain the foundational disciplines which constitute modern mathematics education:

1.       “The psychological dimension of mathematics education is mainly concerned with the way in which the individual attempts to learn mathematics.”

2.       “The social-cultural dimension deals with the influence of groups of individuals and their creations on this experience. “

3.       The philosophical dimension deals with the concept of mathematics knowledge

4.       And of course, mathematics dimension itself

This model of mathematics education is simplified into a tetrahedral shape, referred to as MAPS, with each side representing one constituent discipline:

M-Mathematics, A-Philosophy (arbitrary?), P-Psychology, S-Sociology

Reflection:  

The analogous model does have its appeal as it gives a more vivid presentation of dynamics and interactions than a plain text description. However, although with good intentions, the author overlooked the complexity of the contributing disciplines when he tried to express that mathematics education is a complex phenomenon. Indeed, I tend to think each of these building elements as equally sophisticated as the main topic of education itself. Inspired by the author, I would rather express mathematics education as a four-pronged model where each dimension of learning is equally complex as the subject itself. At the centre, we can find the core concept of “mathematics education” which is greatly affected by the four substituents, each of which pulls it into its respective corner.

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Question: 

The author claimed that that the four foundational disciplines  are not only necessary, but also sufficient, to determine the nature of mathematics education. Given the growing complexity of mathematics education, do you see any new constituent dimensions emerging after 30 years of the publication?  

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Foundations of mathematics education