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Algebraic Thinking in the Early Grades: What Is It?

Algebraic Thinking in the Early Grades: What Is It?
Carolyn Kieran

 In this article, Kieran tried to give a holistic view of algebra in mathematics. Her main focus is to clarify what does algebra means in early grades. She states that early grade students in solving arithmetic questions are not looking for a meaningful relationship between quantities rather their focus is on doing operations and finding answers. In fact, I believe in early grades students was accustomed to seeing numbers and Dong operations (addition, subtraction, multiplication or division). In these early grades, students believe that mathematics means doing operations. So, the author highlights that this point of view of students must be changed to help them learn and understand algebra. She reveals that student must focus on a relational aspect of the question in addition to performing mathematical operations. They must consider both representing and solving question. They should pay attention to both numbers and letters. They must learn about the important meaning and role of the equal sign in a question.
I agree with Kieran that students must know how they should face with algebraic questions. For example, the equal sign is very important because students are used to putting this sign when they find the answer, not in the first place. For instance, when for a first time, they face a question that already has an equal sign, they usually confused. Moreover, paying attention to words and numbers at the same time is another important issue that students mostly do not know and sometimes they would not like to do at all. One of my friends who was an elementary grades teachers for more than 20 years in Canada told me that her students did not like to read an algebraic question, they just picked up the numbers and applied some random operations to find an answer. She believes that students are willing to find an answer especially when it is recess time so they do not care about the words and only pick the numbers up.  She has found a way to prevent this issue among her students. She wrote all information in words and did not use any number in the question. So students had to read all part of the question to find the number. Meanwhile, they understand the whole questions. I find this solution very interesting. What do you think of this strategy? Do you think it works or students finds a way around it? Do you ever use this strategy?
Kieran (2008) present variety of algebraic definition and characteristics from the point of view of several researchers. For example, she cites “generalization and formalization, the study of structure, the study of functions, relations, and joint variation; and a modeling language” as the most important characteristics of algebra. She argues that “Readers will likely have found more differences than similarities [about algebra definition] among the various papers of the collection.” She finally has ended her paper with a model for school algebra which is presented by Kieran (1996) as “Algebraic thinking in the early grades involves the development of ways of thinking within activities for which letter-symbolic algebra can be used as a tool but which are not exclusive to algebra and which could be engaged in without using any letter-symbolic algebra at all, such as, analyzing relationships between quantities, noticing structure, studying change, generalizing, problem-solving, modeling, justifying, proving, and predicting”.

Prototypes, Metaphors, Metonymies And Imaginative Rationality In High School Mathematics By Norma C. Presmeg

Prototypes, Metaphors, Metonymies And Imaginative Rationality In High School Mathematics By Norma C. Presmeg

Norma starts her article with an interesting question “Why is it that there are such vast individual differences in students' learning of mathematics, even within one classroom?” I believe there are a lot of reasons behind this variation such as student’s mathematical knowledge, students' intelligence, student’s willingness to learn, student’s personal’s issues, etc. However, I assume, Norma’s question has neglected these reasons and only focuses on the same classroom and teacher’s issues. In fact, she asks if everything is the same, how is the student’s learning process?

From Norma’s point of view, student’s understanding of the question or topic directly affects students’ learning. By “understanding the problem”, she means imagining the mathematical subject which can be varied from a shape to an algebraic equation. She highlights that students visualize the problem to comprehend its meaning. A significant portion of the article is allocated to explaining and elaborating of the visualization and imagination. The authors include the view of several researchers to clarify how students might imagine mathematical subjects. From my perspective, Metaphors and Metonymies are the two important kinds of visualization. The author describes the Metaphor using the example of a student who visualizes a Sin graph as the sea waves that goes up and down; it helps her to learn and memorize the subject. In describing the Metonymies, the author uses the Lakoff's (1987, p. 79) definition as “Metonymic usage occurs because of a diagram or image, by its nature, depicts one concrete case. “She also uses the Johnson’s (1987, p. 171) categorization of "synecdoche" and "metonymy proper”, as two kinds of metonymy.

After an excellent discussion of imagination’s types, Norma states that type of visualization isn't mattered and as long as one type helps students to understand a topic, it is good and applicable. However, she had not neglected the teacher’s role in drawing or creating an image in student’s mind. In fact, the author believes that the view of a teacher regarding a mathematical topic affects student’s view and might even change it. I agree with Norma and think that an image that is drawn by a teacher (even an imaginary not a real one) sometimes remains in student's mind forever. For instance, if an equilateral triangle, for the first time, has been introduced to a student as A triangle, he will visualize a triangle in the form of equilateral. However, the role of a teacher in creating an image in student’s mind is entirely related to teacher’s ability to teach, a great teacher who is good at transferring knowledge has this kind of influence on his/her students.

This article reminds me of Poly’s suggestion in solving a new mathematical question in his book “how to solve it.” He proposes to imagine or draw a picture of the questions to have a better understanding of a mathematical problem. Although I think, it is more applicable in solving geometric problems; Norma highlights that it is even practical for an algebraic problem. This part of Norma’s article is interesting for me as I never thought it is possible to imagine an algebraic subject. Now that I am writing this reflex, I am also thinking how it is possible to imagine numbers or operations? And I remember a film that we have watched in our last class, the teacher knock on the wall to show the increasing or decreasing the numbers. Although he did not draw a number line, I think, his action will remain in his student’s mind as some hidden number line.

How about you, do you ever think of imagining an algebraic subject? Or did you ever draw a picture (imaginary picture) in your student's mind to help them memorizing or learning an algebraic topic?

BELIEFS AND NORMS IN THE MATHEMATICS CLASSROOM

Beliefs And Norms In The Mathematics Classroom By Erna Yackel And Chris Rasmussen

The primary purpose of this study is to examine the possibility of changing student’s expectations of the mathematics classroom. The “beliefs about one’s role, others’ roles, and the general nature of mathematical activity in school and specifically mathematical beliefs and values.” Students usually believe that mathematics classroom means listening to a teacher who teaches a mathematical subject, taking the notes and solving the questions via formulas. In such a class, students can ask the questions and teacher must explicitly answer these questions. Besides, student’s answer to mathematical questions must be clarified by the teacher. So, it means that there is not any mathematical discussion or debate in these classrooms. The authors put emphasis on the social norm of the classroom and state that it is different in the context of math. In fact “social norms are regularities in interaction patterns that regulate the social interactions in the classroom. By contrast, socio-mathematical norms refer to regularities in the interaction patterns that relate specifically to the fact that the class is a mathematics class.”

This study attempts to investigate to what extend changing teaching methods might change students’ beliefs of the regular mathematical classroom. In this study, a university level mathematics class with the subject of differential equations was tested. In this classroom, instead of focusing on solving several equations, the instructor presents real examples and asks students to solve them using differential equation. However, students must explain and discuss their solutions to prove the answer. The point of view of these students, regarding this method, was registered via a journal writing. The instructor could not directly respond to the student’s questions rather leads them by asking more questions.

At first most of the students did not understand the new method, and even some of them feel the class is not useful and practical. They could not understand how discussing a mathematical equation can help learning the abstract mathematics or how it is possible to use words to describe a mathematical equation. However, after several sessions and observing and contributing to the discussion, students get the main point of the method and classroom. They finally learned how to communicate to come up with the right solution. It is interesting that students realize there might be several solutions to get the same answer. They understand the role of the instructor as a facilitator to lead them toward the right solution instead of offering the right answer.

Honestly, I had the same beliefs of these students that mathematics is numbers, signs, and symbols and it is not rational to prove it by words. From my point of view, the best and shorter way of solving a mathematical question is using numbers to find the answer. However, reading this article open a new view for me that discussing a mathematical question in the classroom and involving all student in solving this question is a new method of education. In fact, it helps students to remember the answer because they have found it via an extended discussion or even debates. Some students might not be good at numbers and formula, but they might be good at making the rational explanation, so it is an excellent opportunity for these students to learn math. I do not have many experiences of teaching math by interaction method, however, whenever I have started teaching new topics by making a conversation and asking questions, students gradually understand the lesson. And I realized that students learned and memorized these lessons better than the regular teaching methods. In this process, students feel that they are in charge and they are discovering something new. They did not understand that I am pushing them towards the answer or new idea. They accompany me to find the right answer or learn the new topic. From my point of view, this feeling makes a pleasure that students hardly forget it.

What do you think of involving students in teaching process? Do you have any experience doing this? What is your student's belief of mathematics classroom?

Snips And Snails And Puppy Dogs' Tails: Genderism And Mathematics Education By Indigo Esmonde

Snips And Snails And Puppy Dogs' Tails: Genderism And Mathematics Education By Indigo Esmonde

In this article Esmonde mainly focuses on genderism and its role in education. Although the title of article promises to explore educating mathematics in the context of genderism, there is not much about mathematics education in particular. She starts his article by emphasizing on boy’s issues in learning mathematics and highlights to what extent “the troubles of boys have not been a major of focus” in the past studies. She represents several boy’s educational problems that have been revealed by Dr. Spence as the director of Toronto district school board, such as underachievement when compared to girls, over-representation in learning support program, anti-learning culture, and disruptive and violent behaviors in schools. From Spence’s point of view, the reasons behind these problematic issues are the lack of role model in boy’s life or having female teachers who do not know the true nature and needs of boys. Esmonde believes that Spence’s solutions to overcome these issues are not appropriate and might make more gender issues.

Esmonde highlights although there are several studies regarding the genderism in education since the notion and definition of gender and sex is not clearly defined, the issues are not addressed well. So, in the rest of her article, she tries to clarify the differences between sex and gender in the view of biology and social terms. She believes there are many parameters that are involved in forming a person character such as culture, society, and personal beliefs, so, it is not so easy to point or judge a person as a man or woman, male or female. She argues people’s behaviors are influenced by the social situation that people are involved. For instance, a manager wears a suit and tie at his work but in the park, he just wears a jean and shirt. She believes, nowadays, gender is not limited to male and female, so, another gender such queer people, or gender non-conforming people must be considered in mathematics education. She put emphasis on mathematical questions that only address male and female and the way people interpret questions means there was not any other gender. From her point of view, neglecting these people in the educational system is a mistake that makes gender issues.

My expectation from this article was reading boy’s issues in mathematics education and understanding the reasons behind these problems and finding at least some solutions. However, I have found this article advocating for involving queer gender people in mathematics education and several issues that have been arisen from this lack of attention. I do not contradict the point of view of Esmonde but I still do not understand the necessity of involving another gender in mathematics educations. Also, I do not see the relevance of this involvement with the boy’s issues in mathematics educations. What is your point of view regarding the involvement of queer people, or gender non-conforming people or another gender in mathematics educations?

As a part of this article, the author states that to overcome issues related to boys and girls in the school, it might be a good idea to de-gender the classroom. As a person who comes from de-gender schools, I am not agreed with this solution. Separating boys and girls might solve some banal problems but my experiences show that it arises several problematic issues. I believe boys and girls have different nature and needs and it is not reasonable to treat them similarly, however, they need to study with each to have this opportunity to know their nature, abilities or even weakness. A combination of boys and girls in a class creates a real life situation that men and women work, live and interact with each other. Lack of this opportunity makes a mystery of woman for a man and vice versa. When they were separated in early age, the desire of togetherness makes a big deal when they are teenager or adult. This desire usually surrounds their mind and affects their every decision. This situation usually happens in a society that relationship between men and women is forbidden. So, young boys and girls take so much time to plan to be with each other and seeing each other secretly. Aside from this issues, when these men and women start working with each other they will face a variety of issues because they have not had this opportunity to know their nature and now in a workplace, it would be too late to know each other. What is your point of view regarding separating boys and girls in the school? Do you believe, nowadays, issues related to boys need more attention and consideration and girl’s issues have had enough attention?

Talking About Teaching Mathematics for Social Justice by David Stocker, David Wagner

Talking About Teaching Mathematics for Social Justice

David Stocker, David Wagner

In this article Stocker and Wagner express their opinion regarding the involvement of mathematics education in understanding social justice. They believe that similar to other scientists, the mathematician also “contribute to the shaping of this world”. Stocker states that his interest in social justice came from his experiences as a family member. He has learned “people who are encouraged, feel capable and appreciated and as a result will tend to act in a connected, cooperative way”. “The family dynamic is organized democratically and behaviorist practices (such as reward and punishment) are replaced with authentic listening, choice, and real opportunities to shape the environment”. Wagner also believes that his point of view regarding social justice came from his experiences in the family. He has learned about Mennonites history and their sacrifices to achieve social justice to make a better life for the human. However, he has thought his achievement in his life is only because of his “hardworking and superiority” but his experiences in Swaziland showed him the influence of social issues on people life and to what extent it might prevent capable people to promote.  

To contribute mathematics in social justice, Stocker believes that teacher can propose questions related to social issues (unions, youth crime, economic systems, defense spending, and etc.) to bright a light in student’s head and lead them to understand their society issues. For instance, he asked his students to draw two graphs in one paper and compare them together, one, the amount of money has been spent for advertisement and the other the amount of money is needed to eradicate hunger in the world. The results were shocking for students, so, Stocker believes that this shock becomes a basis for action in the future.

In their discussion, Stocker puts emphasis on a very interesting point. He argues that usually in each classroom there are some students who are dealing with social issues such as domestic abuse, racial profiling, and certainly poverty. He states that “not confronting these issues directly, I worry that we take students who are disempowered, to begin with and silence them further.” This point of view was a big stop for me. It is really true if we will not talk about the issues that surround our society, children who are involved, never have the courage or even opportunity to talk about their problems. They always ashamed of their situation and will not dare to discuss them. I think gradually these children will believe they are a part of these issues and feel guilty. This feeling might lead them to act as a villain. So it cannot be so surprising that children who live in such a neighborhood often are involved in crime and became a criminal. I think classroom can be a safe and secure environment that children express their feeling and talking about their issues. However, it needs welcoming atmosphere from teachers to insure students that they can express themselves. What is your point of view as a teacher to make this environment for students?

Another interesting stop for me was Wagner point of view that providing meaningless contexts for the mathematical application can be considered as a low-level social abuse for students. I am totally agreed with this point of view. Teachers are responsible for teaching meaningful lesson, to clarify mathematical topics, to answer directly and correctly, to spend quality time with students. Parents always trust a teacher to take care of their children, to put children’s interest in front of everything, to teach them anything they need, so, if a teacher disregards this trust, I believe he/she abuse social trust. What is your point of view of this low-level social abuse?

In the view of social issues that are raised by the act of capitalism, Wagner suggests that rich capitalists and businessman don't really think about the effects their actions have on others, so they plan based on their benefits and interest. Although in general I am agreed with Wagner, however, I have read and heard the opinion of some of these people who believe they are doing so much good for others, so, a little bit bad consequences of their action can be forgettable!! They believe they make the job for hundreds of people and bring comfort for many families and although the salary of workers might not be high, at least they have a job to manage their family. Or they spend a part of their profit for the people of poor countries or educating foster children. From my point of view, these are arguments that are worthy of attention and consideration. There are many important parameters that are involved in these situations. Those people who are the victim of this system cannot be ignored, at the same time those poor people that are benefited from this system cannot be ignored. So, "an extreme illness cannot be cured with a moderate medicine." (Latner, 2005). What is your point of view regarding this complicated issue?

Using Two Languages When Learning Mathematics By Judit Moschkovich

Using Two Languages When Learning Mathematics By Judit Moschkovich

In this article, Moschkovic presents a good explanation of monolinguals and bilinguals in the context of mathematical education and social issues.  She explains the point of view of several researchers in terms of bilingual and monolingual definitions. It was very interesting that some researchers believe that if a person can speak, read and write fluently in two languages, she/he is bilingual, however, others believe that the level of fluently in speaking language is not important in bilingualism. Moschkovic points out two important advantages of bilingualism for children; “enhanced ability to selectively attend to information and inhibit misleading cues” and good translation ability. As a mother of two bilingual kids, I am so happy to know the advantages of bilingualism because my kids have faced many difficulties learning two languages at the same time. The problem is that in our home we are speaking mixed English and Persian, so my younger daughter has difficulties learning one specific language. Although she mostly speaks English, she uses many Persian words among her sentences. On the other hand, the structure of English and Persian languages are different, so it takes time for her to fully understand the structure. For instance, we do not use “her” or “him” for pointing to male and female in our language, so my daughter keeps making mistakes. She has to switch from one language to another to express herself.  

Another interesting point of this article is, the definition of code mixing and code switching in bilingualism. When I am teaching mathematics most of the time I have to use code mixing to explain a particular subject to those bilingual students who have difficulty in understanding mathematics in English language. In fact, I explain a new topic in Persian but at the same time use all of the mathematical terms in English and if students did not understand a term in English, I will translate it to the Persian. Zentella, (1981) believes that “code switching is not an ad hoc mixture but subject to formal constraints and that for some communities it is precisely the ability to switch that distinguishes fluent bilinguals” and similarly in mathematics education Vald´es-Fallis, (1978) states that code switching is not a reflection of a low level of proficiency in a language or the inability to recall a word”. However, based on my experiences, most of the code mixing and code switching in the mathematic classroom is because of the lack of English language knowledge. This inconsistency highlights that the definition of bilingualism must be considered carefully. As I understand most of the researchers who talked about bilingual, they mean those people who fluently speak two languages, therefore, they put emphasis on the other reasons that cause code switching. For instance, Moschkovic describes several situations in mathematics education that students choose to use their first or second language or use code mixing or code switching. She reveals, it mostly depends on the mathematical aspects of the situation, mathematical topic, being in the classroom or in private session, and the past experiences with mathematics instruction in each language. On the other hand, Moschkovic explains that the pattern of social language speaking of bilingual children is not similar to their mathematical language speaking pattern. In the classroom, bilingual children speak as they are spoken to, however, in society, they speak based on the norm of the society. In this step, I was a little bit confused because those children who have trouble speaking English [but still are considered as bilingual] cannot speak or respond in the class as they are spoken to, they usually cannot understand their teacher, or cannot ask their questions, or answer their teacher’s questions. So, again, I believe, Moschkovic refers to those bilingual students that speak fluently in both languages. Finally, Moschkovic concludes that most bilingual students prefer to switch from one language to another to explain a question or describe a solution. She believes it does not mean these students have deficiency in one language but they just feel comfortable to use their first or second language. Her research shows that most of these students use formal classroom language to talk about the mathematical terms but use their first language in description and explanations.    

Now, my questions are: what are your experiences with bilingual students in the classroom, do they use their first language as only a comfort zone for description and explanations or lack of their second language knowledge? To what extent lack of English language knowledge affects their mathematics learning?