A Linguistic and Narrative View of Word Problem in Mathematics Education

“A Linguistic and Narrative View of Word Problem in Mathematics Education” by Susan Gerofsky discusses the importance and value of mathematical word problem in facing the real life situation. It describes mathematical world problem as a linguistic genre and its pragmatic structure then compares word problems genre and others literary genre. Gerofsky endeavors to explain to what extent word problems are applicable and useful for students. She states the point of view of several scholars who believes word problem in mathematics textbooks do not represent the real world problem because most of them are not the real story. She raises the example of people who are good at solving mathematical problems in the real life but cannot solve some of the word problems of the mathematics textbook.

the author categorizes the word problem in three groups of “set up”, “information” and “questions”, then explains to what extent “set up” part is relevant and helpful in finding “information” and solving the “question”.  She believes “too much attention to the story [set up] will distract students from the translation task at hand, leading them to consider “extraneous” factors from the story rather than concentrating on extracting variables and operations from” the text. I am completely agreed with the author that most set up part of word problem distract students from the main part of the question. Students take much time to understand the setup part of the word problem and sometimes they would loose the track of information especially with some extraneous data. In solving the word problem, some of my students, ask why these questions are so long and full of non-useful information, how these part of questions (set up) is relevant to the answer and solution? Set up part frustrates my ELL students who have issues understanding English language. They usually give up answering the question when they face a long word problem. When I ask them (sometimes force them) to read the question, they emphasize that we cannot understand it, we cannot solve it, it is too complicated. However, after reading the questions and carefully translating it, they mostly solve it easily. They laugh and say “it was easy!”

In Gerofsky’s article, there are some reasons behind the importance of the setup parts. Some scholars believe that without setup part, students cannot see the relevant of this question to the real life situations. This part helps them to imagine the real situation that they can use their mathematical knowledge. However, the critics of this viewpoint state that these parts are fictions and do not represent the real situation or even if show a real life, solving the problem relating these situations are not helpful for students. For instance, they bring up the example of a word problem that asks about the time of filling a tub or the amount of water that can be filled in the tub. They wonder how calculating filling time or volume of a tub help students? I think this kind of discussion shows the importance of setup part of a word problem. It means by modifying the setup part of these question, most of these deficiencies will be disappeared. Instead of telling a story about a tub and asking its volume, we can tell a story of a factory that fills the water bottle and asks the volume of water that is needed for each bottle or filling time. So, the question would be practical and useful. Gerofsky believes asking students to propose a word problem will show to what extent students have learned from those word problems that they solved. Studies reveal that most of the students propose questions exactly similar to what they already solved or faced in the school. They usually did not make new question based on their real life situation. It shows that students did not see any real life situation involved in the word problems. They just see the setup part as extraneous words of the question.

These issues highlight the weaknesses of the word problem, however, I believe it does not mean word problems are not required or applicable or useful. I believe we need to understand these issues and address them carefully. My question is what are the issues of current word problem? What are your suggestions to improve word problems?  Do you remember any word problem that was not applicable? And finally what is your point of view regarding the word problem? (Is it useful or not or..)

my research interest

I am interested in researching on Problem-solving and language deficiency among second language students, mainly who recently migrated to Canada. The important reason that I am interested in this topic is that most of my students are second language learner who has difficulty in understanding mathematics especially world problems. I hope the results of this study brings a good understanding of these students and help them to overcome their problems. 

Doolittle and Glanfield

Edward Doolittle in “Mathematics as Medicine” gave a brief biography of his life and his Indigenous root. He states “One of my major life goals is to resolve the apparent incompatibility between the two aspects of my identity, being a mathematician and being an Indigenous person.” In his journey to know his culture, he understood that it is not simple to extract some mathematical concept from Indigenous culture and simply teach them to students, rather deep understanding and consideration are required. He believes mathematics is “an essentially simple way of thinking” so we can “pull mathematics into Indigenous culture rather than how mathematics might be pushed onto Indigenous people”. He points out several examples that show the relationship between Indigenous culture and mathematics. For instance, he describes the Medicine Wheel as a tool that Indigenous people have been used in their life, he described that Medicine Wheels “were used not to divide and analyze, but as “maps” of processes of ceremony, thanksgiving, timekeeping, and communication.” As I understand from Doolittle explanation, elders Indigenous have a spirit with a special power and supremacy to feel and understand things that ordinary people might not sense. He even brings up some related examples. He indicates that “we can weigh and measure and test, but true complexity cannot be handled by simple means”. I personally believe in scientific proof of phenomena and events, so, providing some examples and signs might look like justification. However, if I could be agreed with these power, I do not understand how Doolittle connects this power to mathematics learning. Although I am unfamiliar with Indigenous' culture, I think people’s belief is rooted in their culture and tradition. Since a culture might have hundreds of years of history its interpretation and understanding take time and efforts. Now, my question is what do you think of these kinds of powers that Doolittle mentioned in their elders?

On the other hand, “Balancing Equations And Culture: Indigenous Educators Reflect on Mathematics Education” is mainly a conversation between Doolittle and Glanfield. This conversation has clarified many parts of “Mathematics as Medicine” by Doolittle for me. I have to admit that I did not understand well Doolittle’s paper till I read this article. Doolittle and Glanfield have discussed their opinion regarding mathematics education from the perspective of Indigenous people. The main point of their conversation was, how mathematics education can help people to achieve peace, power and good mind in their life. This point of view reminds me of my ancestors who believed in having good thoughts, words, and behavior in their life. I am wondering, how mathematics knowledge can help people to achieve these goals? Or how my ancestors involved these beliefs in mathematics as a part of their life. As I have observed in Persepolis (an ancient city in Iran, back to 2500 years ago) the statues represented several patterns of respect and love among people, a combination of mathematical knowledge and peace. The stone tablets that are discovered from this city shows how my ancestors fairly calculated the salary of workers; who built the Persepolis; which is a true prove of mathematical understanding and human right. Another important matter that Doolittle and Glanfield discussed was forcing people to learn mathematics and to what extent its effect people’s life. Now my questions are, is learning mathematics necessary for everybody? In high education level, should each major be involved to the mathematics courses? Is it fair for some people to be failed in their study just because of lack of mathematical knowledge, even mathematics is not  the main part of their training?

On Culture, Geometrical Thinking And Mathematics Education

On Culture, Geometrical Thinking And Mathematics Education

By Paulus Gerdes

In this article, Paulus Gerdes, investigates the mathematics education in the Third World Countries. He discovers, in these countries, learning new mathematics (come from developed countries) has been difficult for students because they could not make a connection between its content and their real life. Gerdes comes up with the idea of making a curriculum based on student’s culture and tradition. However, he knows making curriculum based on a culture that has been crushed (from whatever reason) is difficult and sensitive. So, he faces with this question how we can retrieve the traditions “when probably many of them have been - as a consequence of slavery, of colonialism... - wiped out.” He proposes to look to the geometrical forms and patterns of traditional objects like baskets, mats, pots, houses, fish traps, etc. He believes this exploration reveals the mathematical knowledge that has been used in creating these objects.

Gerdes’ exploration shows many examples of using mathematics in the real life. The first discovery was using the characteristics of the rectangle in making rectangular based house. Then, Gerdes finds out how artisans in the north of Mozambique make a funnel. He shows this method can be employed to create an equilateral triangle, regular pyramid or even regular octagon. Afterward, he investigates the southern parts of Mozambique and notes that people fasten the top of a basket by pulling a little lassoo around a square-woven button. Gerdes carefully considers these patterns and realizes this is a right triangle surrounded by 3 squares which it represents the Pythagorean Theorem. Then he discovers the formula of the internal angel of a polygon or sum of the first n odd numbers from the pattern in regular hexagonal holes that has been used in making the “Litenga” and “Lema”.

Gerdes’s study is very interesting for me, as I never though connecting culture to mathematics education can be important. I always though wiping out a culture has the influence on people behavior, social study, history or even geometry but not mathematics! It made me think of my own culture and tradition and its hidden mathematical knowledge that I never noticed. I remember visiting Persepolis, an ancient city in Iran from 2500 years ago (Figure 1), the architecture of buildings and the shapes of statues exhibits the geometrical knowledge of my ancestors. I always knew these structures are unique but never thought of its hidden mathematical knowledge. Moreover, the design and patterns of ancient Iranian carpet (Figure 2) also suggest to what extent my ancestors have employed their mathematical knowledge in their real life. However, in my country, teachers barely refer to our traditional mathematics to teach us new mathematics. Now, I understand it was easier to learn not only mathematics but also other science if we were referred to our ancestor’s work and knowledge as most of us have samples in our home.

Figure 1: A view of Persepolis

Figure 2: Iranian Rug (carpet)

The important question is how we can connect to our traditional mathematics? How we can understand to what extent we have improved our ancestor’s knowledge or simply forget them and borrow new mathematics from other colonies or countries?

short review on "The Reasonable Ineffectiveness of Research in Mathematics Education by JEREMY "

The Reasonable Ineffectiveness of Research in Mathematics Education 

by JEREMY KILPATRICK (1980)

Kilpatrick, (1981) have written this article trying to follow Hamming’s (1980) strategy in "The Unreasonable Effectiveness of Mathematics”. He starts his article with this question: "Why is research in mathematics education so ineffective?" then claims most of this ineffectiveness of researches in mathematics are perceived rather than real but these are reasonable. To prove his claims, author, brings up the idea of the effectiveness of research in education, explores why research in mathematics education is ineffective, puts emphasis on the differences between mathematics and research in mathematics education, finds reasons for why it is not easy to implement research in mathematics and finally proposes some suggestions to make research in mathematics more effectively.

To find to what extent research in education is effective, Author investigates the content of a thick book; “Impact of research on education: some case studies” edited by Suppes (1978); and the argument of Scriven’s (1980) about this book. Based on Scriven’s (1980) explanations, although Suppes’s (1978) book mentions several case studies to prove the benefits of research in education, at the end, it is not clear how these researches are beneficial.  Kilpatrick believes that the major difference between Scriven’s (1980) and Suppes (1978) opinion is their usage of “effect” or “benefit” of research in education. These researches have effects on education but studies do not show these are beneficial.

The author points out that people; who think about the effectiveness of research in education; can distinguish between pure and applied research. There are two models to explain the differences between pure and applied research. Hierarchical and complementarity models. To illustrate the Hierarchical model, Greeno (1978) proposes “the pipeline model".  This model explains similar to crude oil that gets pumped out of the ground, the fundamental knowledge and theories have been gotten from the basic research. Then like crude oil that is shipped to be transformed to the useable forms, fundamental knowledge and theories have been sent to apply research settings where it is transformed into something more useful. Finally as the product of transformed crude oil such as gasoline has been sent to filling stations where customers can actually use it, the results of applied research have been sent to the developers who use knowledge in making products for schools and students. In complementarity model “the two types of research are seen as complementary, each with its own domain and its own agenda, and equal (supposedly) in status. When one sees basic research characterized as "conclusion- oriented" and applied research as "decision-oriented," one can be fairly sure these types of research are being viewed as separate but equally valid - for their own domain of relevance.” The Adherents of the complementarity model believe that the differences between two types of research are decreased dramatically in recent years because 1) theory-building has been prescriptive, not descriptive and 2) research methodologies have new capabilities and improvement. The author proposes a third model (lens model) and explains “a study may be basic or applied depending upon the lens you use in reading a report of it. The same study can be either basic or applied, depending on who is doing the labeling and for what purpose. In fact, one can only ask what connection the research study appears to have to theory and what connection to practice. Both of these, to a large extent, are in the eye of the beholder.”

 Kilpatrick believes that “One reason [behind lack of effectiveness in mathematics education] may be that, “despite what appears to be a flood of research in our field, we actually have very little in the way of research to go on in drawing implications for practice”. Besides, he investigates several studies to show; do we have enough data to answer the important questions in mathematics education? The results of his investigation reveal that there is very little trustworthy data representing the facts of the educating process. Also, he discovers that lack of true community for mathematics education research is another reason behind ineffectiveness of research in mathematics education. Moreover, he also finds out that most of the researches in mathematics education have been done by researchers who are not recognized as researchers in mathematics education. So it is not surprisingly if these researchs are not “arise from common concerns, shared knowledge, and mutual interaction”. Kilpatrick also highlights that there are two additional reasons, however, that appear most compelling of all: (1) our lack of attention to theory, and (2) our failure to involve teachers as participants in our research.

Kilpatrick explains why the lack of attention to the theory such a serious problem? “I contend that it is only through a theoretical context that empirical research procedures and findings can be applied. Each empirical research study in mathematics education deals with a unique, limited, multi-dimensional situation, and any attempt to link the situation considered in the study with one's own "practical" situation requires an act of extrapolation.” Kilpatrick reviews several related studies in mathematics education and understands that these researches do not affect practice in mathematics education. Too many mathematics educators think teachers need to know the results of research. Kilpatrick, 1981 highlighted that what mathematics teachers do every day comes close to being research; it is just not quite so deliberate, systematic, or reflective. So teachers can borrow three things from researchers: their procedures, their data, and their constructs (Alan Bishop [1977]).

The author put emphasis on the differences between mathematics and research in mathematics education. In this regard, he uses Hamming’s [1980] explanations (four reasons) of the effectiveness of mathematics. Based on Hamming, first, mathematics has been highly creative in inventing tools and the phenomena we see arise from the tools we use, however, research in mathematics education does not have these tools. Second, we select the kind of mathematics to use and when the mathematics we have, does not work, we invent something new. In research in mathematics education, we have not shown the same ingenuity in adapting our tools to our problems. Third, science, in fact, answers comparatively few questions. Perhaps one reason for the perceived ineffectiveness of research in mathematics education is that too much has been expected of it. Fourth, Hamming contends that “the evolution of man has provided the model for mathematics by selecting for "the ability to create and follow long chains of close reasoning" [p. 89]. However, the history of research in mathematics education is much shorter, and evolution has not had time to select for a research attitude”.  Kilpatrick adds that “applying educational research to mathematics teaching practice is not an engineering problem like applying mathematics to a practical situation. “Doing something about education means doing something about people such as teachers, students, parents, politicians and people are just not that manipulable. They are what they are and do not become new people to suit any new ideas we might have” (Kristol [1973]).

Kilpatrick proposes some suggestions to make research in mathematics more effective. He believes researchers should: (1) “develop a stronger sense of community, which would include practicing teachers as collaborators in research; (2) create their own theoretical constructs for viewing their work; and (3) recognize the limits of their domain as well as its complexity”.

My stops and question

The first stop was the title of the paper. I suppose every research has some merits so how it is possible that research in mathematics might be ineffective??

Facing this word was another stop for me, extrapolation…what does author mean with extrapolation? How it might help us in this paper?

This is another stop for me when the author stated that “however, the results are the least important (for teacher) aspect of a research study”.  As I always though the results is an important part of a study that helps teachers.

My Questions

Since author talked about 1979 and 1980’s researches, I am wondering to what extent the situation has been changed after almost 36 years?

Hi there,

This is malihe and this is my new blog.