Early Algebra and Mathematical Generalization

Early Algebra and Mathematical Generalization

By David W.Carraher, Mara V.Martinez, Analu´cia D.Schliemann

 

 

In this article, the authors investigated how 15 grade 3 students generalized on algebra questions. The questions and exercise were given in a sequence of lessons. The previous 33 lessons were used to introduce some algebra concepts such as variables. The study focused on two table seating capacity questions. In a warm-up class, separated tables were used to introduce a relatively basic expression of p=4t in the practice. The study then moved to the “harder” algebra questions: how many guests could sit when tables are jointed. The authors argued that “many students scan output values of (n) as n increases, conceptualizing the function as a recursive sequence which means “students identify recursion(‘‘keep adding two’’)as the principle that generates successive values in the output column” (p.5).

 

This is actually an interesting article which links to our previous readings in discussion of abstraction and empirical experiences. Bobby’s written response to the question demonstrated that some students were able to identify the pattern and express it in an algebraic manner with several trail-and-errors. In this sense, a good mathematics learning experience should be one which can eventually lead to mathematical generalizations and abstraction; rather than focusing on memorizing multiplication table by rote drills.

However, I found that the authors’ notion on patterns needs further discussion. They argued that “a pattern is not a mathematical object”(p.4) and moving towards pattern will create frustration since it lacks rigorous inference. I appreciate their stance of preferring function over pattern but for younger students from K-3, I found it is more acceptable if I start with pattern identification.

My question is whether or not you are supportive to provide algebra lessons for elementary students as young as grade 3?

 

 

 

" HOW MULTIMODALITY WORKS IN MATHEMATICAL ACTIVITY: YOUNG CHILDREN GRAPHING MOTION by Francesca Ferrara"

• Ferrara, F. (2014). how multimodality works in mathematical activity: Young children graphing motion. International Journal of Science and Mathematics Education, 12(4), 917-939. 

If you have a problem like this in front of you:

3x^2+5x-6=0

You can perceive the solution in different ways by imagining the process in your mind. In order to understand the role that imagination plays in multimodality, the author the descriptive analysis of the work of one child, Benny, who constructs mathematical meanings in an activity involving graphs.

The author acknowledges the complexity in the construction of mathematical meaning. It is claimed that different sensorial modalities become integral parts of students’ cognitive processes.


Drawing upon research on gestures and other bodily activities in mathematics teaching and learning, the author moves beyond the fact that multimodality takes place during the learning, but rather stresses how multimodal cognition occurs in relation to the interplay of perceptual, bodily and imaginary experiences of elementary students. According to the theory of embodiment, students literally utilize their bodies to think, and some recent neuroscientific studies explain that sensory modalities like vision, touch, hearing, and so on are actually integrated with each other and with motor control and planning. The author argues that imagining plays a critical role in any perceptual and motor activity, including in mathematics learning.

This is a very complicated piece for me as I haven’t come across anything like this before. First, it involves an area (multimodality) that I am not familiar with; and secondly, I do not quite comprehend its methodology. The way the author interprets the gestures made by the student Benny is so detailed that it not only enables me to vividly replay the scene but also makes me look into the approach of microanalysis. As an example, “on the one side, pointing with the left hand, he conveys positions on the trajectory, as demonstrated by the head and the torso turned towards the line (by the way, the left hand is the closest one to the space where motion was performed)”.

However, sometimes I have to stop and ask myself how robust the interpretation is. For example, in Benny’s expression that “I go [left hand running t-axis to the left end, body shifting from right to left], I arrived here [left hand pointing to the left end] and this piece came “ The author explains that this happens from the point of view of the child and according to Benny’s experiences (“I went”, “I arrived”) (p. 927). I might consider some other factors involved in the expression, such as the influence of teaching, as in my teaching experience, I notice that students sometimes unconsciously mimic teacher’s instruction style (if teacher uses the expression “let us start here and move there and they mimic “let me start here”) when they are asked to explain their solutions.   

Question:

With the rapid change in digital and Internet technology, other than text books and worksheet, do you anticipate any other learning modes will prevail in schools?

On "Sociomathemtical Norms, Argumentation , and Autonomy in Mathematics"

SOCIOMATHEMATICAL NORMS, ARGUMENTATION, AND AUTONOMY IN MATHEMATICS

By Erna Yackel And Paul Cobb

In order to make sense of mathematics learning and teaching, the authors put forward a framework to interpret students’ learning from the perspective of sociomathematical norms. Sociomathematical norms, described by the authors, are unique to mathematics, in contrast with social norms which are general across all subject areas. “For example, the understanding that students are expected to explain their solutions and their ways of thinking is a social norm, whereas the understanding of what counts as an acceptable mathematical explanation is a sociomathematical norm” (p. 461). In this paper, the sociomathematical norms of mathematical difference and mathematical sophistication are established. The central idea of sociomathematical norms is that the normative aspect of mathematics discussion in classrooms is co-constructed by the teacher and students, which opposes the notion that mathematics learning is context-free. The author argues that what becomes mathematically normative in a classroom is shaped by the teaching goals, understanding of constraints, and negotiation among the classroom participants.

Further, the author illustrated how different sociomathematical norms regulate mathematical argumentation and provide extensive learning opportunities for both the students and the teacher. Whiles students consistently contribute to the classroom discussion by explaining their solutions, the teacher is offered opportunities in turn to develop notions of what is sophisticated and efficient for the children. The paper also discussed the link between sociomathematics norms and being autonomous learners. The author explained that autonomy takes place when students take responsibility to an extent beyond “being a student”. Such a transition requires awareness of sociomathematical aspects created by students and teachers.  This is also noticed in many inquiry-based approach classes I have observed.

The authors argued that “initially, students' explanations may have a social rather than a mathematical basis.” (p. 467) While I had a very vague conception of the mentioned process, I experienced an enjoyable enlightenment reading this paragraph. One particular example was used for clarifying how the teacher and students interactively constitute what counts as an acceptable explanation and justification. In example 4, a student changed her answers and challenged the mathematical basis for explanations. The teacher explicitly acknowledged that “she changed her answers on the basis of her interpretation of the social situation rather than on mathematical reasoning, the teacher invents a scenario to clarify his expectations for this class.” (p. 468). I sense that this practice of the teacher was surprisingly effective and powerful since it created a concrete case that every student can comprehend and refer to.

Question:

The author distinguishes sociomathematical norms from social norms. How do you think norms in math classrooms differ from norms in other subjects?

Gender and Mathematics: recent development from a Swedish perspective

Gender and Mathematics: recent development from a Swedish perspective

By Gerd Brandell, Gilah Lede, Peter Nystrom

In the past, research attention to gender equity has been mainly drawn to areas in which females appear disadvantaged and underrepresented. However, during the time of this study, more considerable attention has been placed on educational disadvantages faced by boys. Concerns of boys’ disadvantages, even in the fields of math and science which are traditionally perceived as male-dominated, are receiving increasing attention from the media and researchers in Sweden. Mats Björnsson (2005) noted that focusing on girls’ needs in the classroom could somehow be a reason for boys’ declining results in school. As a result, Björnsson suggested looking into pedagogical reform and other factors that influence education among boys. 747 participants were chosen in the questionnaire study, and 24 among them were selected for interviews. For a majority of the investigated items, such identifying which gender enjoys math more, more than half of the students replied with "no difference" between girls and boys. The study also revealed that female students are perceived as hard-working while their male classmates are associated with disrupting class.

A comparison of this Swedish study with Australian data shows that mathematics is also perceived as a mostly male gendered field at the secondary level. However, Australian students were more inclined to view mathematics as a female gendered domain when compared to Swedish students, and female students also enjoyed math more than Swedish students. The finding seems to agree with the assumption described by Elizabeth Fennema and Julia Sherman (1976): the less a female stereotypes mathematics as a male domain, the more likely she would be to study and learn mathematics.

It is quite interesting to notice that female students from different countries perceive math differently. The study did not investigate very deeply into this complexity. I have some teaching experience related to this. For years, I have heard stories that female students struggle more with high school math than boys in China. It is a common impression that math is a male-gendered subject. Interestingly, several students who hated and were scared of math in China found math quite enjoyable when learning in countries as the US and Canada. I would assume that it might be attributed to the less challenging curriculum used in newer schools in the West. This might be true for elementary level math, but I don’t see a huge difference in difficulty and depth between countries when comparing grade 11 text books.

Do you have similar teaching experiences, and what do you think?

On "Teaching Mathematics for Social Justice"

Wagner and Stocker trace their interests in social justice back to their family upbringings and life experiences. Stocker was raised up in an Adlerian family in which punishment and reward are replaced by democratic approaches. On the other hand, Wagner was raised up in a Mennonite family (a Christian denomination) but changed drastically after backpacking around the world and saw disparities and injustices. In the conversation, Wagner and Stocker exchanged ideas over what math educators must do with social justice.

Principles of peace and social justice

Stocker describes 3 basic principles regarding peace and social justice:

- non-violent approaches to conflict

- democratic decision-making processes

- the elimination of barriers to social, economic, and political inclusion based on race, class, gender, ethnicity, religion or ability

Wagner negotiates the definition of social justice by pointing that much violence is wrought by people with good intentions, and therefore the elimination of structural barriers as an “ends-based” vision should be practiced with caution as “others would be involved in when addressing violence”. He prefers to focus on processes rather than end goals when integrating social justice in teaching mathematics. Stocker refutes this sentiment by noting that it is “extremely competent” businessmen who are maximizing profit for shareholders without sparing thought for “good intentions”.

Exposing children to the real world

In arguments over whether children should be shielded from exposure to social justice and inequities such as race and gender, Stocker claims that children are potential future makers for the world and it is the teachers’ responsibility to direct the eyes of children to these real world issues. Wagner agrees that children should see the negative and broken aspects of the world but he questions the intention of dragging children into the battle of social injustice. Further, Wagner concerns that it is a subtle form of “social abuse” for teachers to force their political and social agendas onto children. 

Balance of teaching

Wagner discusses the need for balance between teaching actual mathematics and the social justice component to mathematics. Stocker notes that there is no semblance of balance in current curricula, comparing the “balance” to the “balance” between an elephant and a pea. In reality, Stocker is not concerned about the balance, but rather to the many teachers who use the concept of “balance” as an excuse to avoid bringing social justice teaching to the classroom. Wagner and Stocker both agree that perspective teaching must also be included to bring balance to teaching mathematics, and emphasized the importance of the social function of play in learning.

Stop:

It has been a struggle for me to balance these elements of mathematics education in the classroom. On one hand, I want my students to know what their ideal world should be by downplaying the severity of issues around us. On the other hand, I know that some issues such as pollution, poverty, and inequity in education itself will only get worse by ignoring them. According to Wagner, the world is always changing, and a “better world” is an end goal in itself, meaning that people working with good intentions towards this end goal have the ability to cause more damage than they fix. I share the belief that implies that each generation has their own contributions and responsibilities; adults who are interested in social justice should take responsibility to “fix it”. 

When enacting social justice in classrooms, how do you address the question and justify of whose responsibility is it (everyone’s or adults’) to fix these issues and make the world a better place?

On " The Linguistic Challenges of Mathematics Teaching and Learning: A research Review" by Mary Schleppegrell (2007)

The article synthesizes research from the mid-1980s to 2005 in the fields of linguistics and mathematics education.  The article highlights several linguistic challenges of math learning:

1.       Multi-semioticity: the construction of the meaning of math concepts is drawn from multiple sources including symbols, oral language, written language and visual representation, order, position, etc.

2.       Specific grammar features in math language: One feature is that the math language uses noun-dense phrases such as “the volume of a rectangular prism with sides 8, 10, and 12cm”.

3.       “Mathematics is highly technical, with characteristic patterns of vocabulary and grammar” (p. 142). An example was used to illustrate the idea. “George has twice as less money than Tina” is considered technically incorrect, in comparison with a newspaper report “traffic in Sydney during peak hours is nine times slower than in Melbourne”.

4.       Precision of mathematical conjunctions to link logical elements: words such as if, when, then are used in precise ways in developing theorems and proofs and they represent different ideas compared to use in everyday life such as in “if I got an A, I would be very happy”.    

I appreciate the notion that a key challenge in math teaching is to help students move form informal discourse to formal discourse which is necessary for disciplinary and interdisciplinary learning in school. Especially with the inquiry-based approach to encourage students to connect math concepts with everyday life, students are comfortable bringing their “everyday” language to participate in the activities. In addition to teaching formal technical terms and demonstrating the language used in mathematical discussion, the article suggests to explicitly teach the language and bring self-regulated awareness of “formal” math talk in classrooms. It is a very insightful article as it casts doubt on the common understanding of the independence of effective math learning over mathematics language.      

Question:

As the author indicated, most teachers have recognized that technical vocabulary is a challenge, but many do not pay enough attention to the grammatical patterns. How should we offer help in the classroom to ELL learners who are new to both “everyday” English and “mathematics” English?

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.

  .

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?  

welcome to Math 550

Foundations of mathematics education