2009/10/26

How did I discover my passion for mathematics for use in workshop?

I guess I have always had a certain tendency to favour math. As a young child who has not entered an educational system yet, I was very curious about everything; I kept asking questions and seeking definite answers. And I always wanted to see connections in a bigger picture. At the same time though, I was also a creative mind that liked to approach things in my own way. Since I kept seeking definite answers with associations, and I was aware of other approaches, it comes naturally that I found my passion in math – the subject that has explicit answers interrelated in different concepts, but those answers can be found in many different ways.

After the fourth grade, I tested in a mathematical school (eight year gymnasium) for talented children. I continued developing my knowledge and I never had problems understanding new mathematical concepts. Math seemed very easy to me during that time and, naturally, it became one of my favourite subjects. For a period of six years, I never did more work than homework and I always walked out with As after exams.

I spent my last two years of high school studying IB (International Baccalaureate) math Higher and Further Level in another selective school – scholar Red Cross Nordic United World College. It was then when I finally developed my passion for tutoring my peers. Firstly, I had more knowledge than most of my peers and I was happy to share it with them. I did not feel useless; I felt worth when I was doing something beneficial to others. My peers thought I knew more than they did and they expected it to be always like that (psst – nobody has a clue that it changed years ago). But later, as I had to learn totally new concepts in a new language, i.e. English, using a different educational style that was seeking more individual work, I suddenly found myself not knowing everything. I had to re-learn learning again. The subject of mathematics became a challenge rather than an easy ride. I struggled through each lesson maybe even more than my peers did; and I constantly had to invest time in seeking understanding. Thus, I was very glad when my classmates kept asking me for help as long as I was able to help them. I mean, I was not perfect and I could not know everything all the time. But when I did, I could easily anticipate struggles of my peers and offer an explanation to them – the explanation that worked for me. I was aware that I did not know everything immediately. But I knew that, once I finally understood a concept, I could explain where exactly a specific problem in thinking was. I am, maybe, just able to solidify and hold knowledge more easily.

Finally, I entered a college and became a math major. Frankly, I did not choose it in order to get As. In fact, I have not received a solid A on my math exam for a while, while I used to get A+ in my modern dance class…and I have never danced before… Well, I chose mathematics because I wanted to keep developing my passion of seeking definite answers by different ways. My strategy for mathematics has not been performance seeking rather understanding seeking instead. I know that my results have not been perfect and thus I have considered giving up on the idea of math being my domain many times. But I always reminded myself the true reason why I feel comfortable in the field. I like the feeling of satisfaction when, after all of that personal intellectual struggle, I finally find a correct answer.  It is like climbing a mountain in the winter and being able to slide down with joy. I believe that everyone should get a chance to know that sweet feeling, whether they need some push upwards the slope or not. And therefore, I decided to keep helping others with their intellectual struggles in mathematics. I still study mathematics, trying to do my best, and I decided to explore learning and teaching process academically through the Educational Studies department as well.

2009/10/05

Literacy

Literacy is an ability to use a certain skill to which we have been introduce. There are different forms of literacy, mostly grouped as sanctioned or unsanctioned. Unsanctioned uses of literacy happen ´behind the desk´ and they are not approved by a system that manages education. Sanctioned uses of literacy, though, are the ones which an educational system approves and tries to teach. Students in a math class might not understand derivation completely (sanctioned literacy) but they might be skilled enough to use cell phones during a class so that teachers do not see them (unsanctioned literacy). In this short synthesising essay, however, I will be looking at sanctioned forms of literacy presented by educational systems, i.e. reading, writing and comprehension of different material.

One has to go through a learning process to become literate in a certain area. Such ´area´ or domain of knowledge is academically named a discourse group. Whether we are several months old and we try to acquire a skill of talking (become literate in the language discourse group) or we are retired people who try to get comfortable with computers (become literate in the computer science discourse group), we still try to learn new information. Thus, we learn everyday regardless of age: from early infant stages until late phases of our cognitive development. There have been many theories about learning and one of the most significant ones is the Piaget´s theory of cognitive development where “objective knowledge appears as an end result [of development of cognitive ideas] rather than as an initial piece of information” (Ferreiro and Teberosky, page 16). There are, however, many different approaches to literacy. They are each looking at the process of learning differently, especially from the time point of the view. Since learning happens constantly, there is no single literacy type to describe. I consider literacy to be partly shaped by our cultural beliefs. One´s views on the world evolve as his age progresses. Therefore, I will be looking through different non-dogmatic scientific theories of literacy considering a timeline of learning progress, namely Neo-Piagetian theory of literacy Ferreiro and Teberosky, Socio-Cultural theory of literacy by Dyson, Socio-Political theory of literacy by Freire & Macedo and Socio-Historical theory of literacy by Ogbu..

It is apparent that the learning process starts from before the school time. Neo-Piagetian theory of literacy by Emilia Ferreiro and Ana Teberosky (translated from Spanish by Karen Goodman Castro) applies Piaget´s theory of cognitive development in learning into a literacy context. Ferreiro and Teberosky discuss the basic and most important form of literacy, language literacy – reading and writing. They draw a conclusion that in order to obtain other forms of literacy, reading and writing skills have to be developed. “Case studies have revealed correlations between failure in reading and writing and parallel deficiencies in other domains,” illustrate Ferreiro and Teberosky (Ferreiro and Teberosky, page 19). They believe that children start learning long time before they actually enter any form of a schooling institution, whether it is a preschool or a kindergarten. Children look for patterns and constantly build their own theories about grammar of a native language. They utilize these linguistic ´knowings´ unconsciously in their “daily communication acts” (Ferreiro and Teberosky, page 10). Common are examples of pre-schoolers misusing the “-ed” rule to form a past tense, when they are yet not familiar with irregular verbs. Saying “read-ed” is not a mistake but rather a constructive error leading to additional learning, as long as this kind of a systematic error does not happen too often even after being introduced to a proper form, ie. past tense “read”. Then the learner accommodates information into a proper form. It is also an evidence of understanding patterns and active thinking.

Furthermore, Ferreiro and Teberosky look at schooling process as well and they make a clear distinction of literacy being a two way process of teaching and learning. While the theory of cognitive development from the learner´s point of view is explained by the Piaget´s theory itself, the theory of cognitive development from the teacher´s point of view is evaluated by Ferreiro and Teberosky. In the initial stage of accommodation, a teacher should identify and assess (i.e. correct if necessary) any student´s preconceived ideas. Then a teacher should go through the assimilation phase, where introduction of new schemes while constantly applying prior knowledge happens. For example, my field placement math teacher Mrs. Ringquist keeps warning her students about common misconceptions of math applications as she wants to assure no incorrect prior knowledge is carried along.

Since children have no formal teaching setting before schooling, we could paraphrase Ferreiro´s and Teberosky´s belief of building early theories into a realisation that students learn from their parents by social interaction. The point of social interaction teachings is where Ferreiro´s and Teberosky´s Neo-Piagetian theory and Dyson´s Socio-Cultural theory of literacy intersect.

Anne Haas Dyson builds on Vygotsky´s zone of personal development (which concludes that one can accomplish more with slight help from others, in comparison with trying on his own) and stresses that students need to re-interpret knowledge into their own words. She believes that students as individuals have imaginative universes that help them identifying with information by personalization of knowledge for the purpose of social communication.

Dyson also acknowledges that learning does not happen in a single way or during a single period of time, and so she brings a new dimension of literacy theory happening during the elementary school time. She assumes that students already posses basic skills of reading and writing (have basic literacy in a language) and therefore they can comprehend the content of information, drawing out some cultural information. Active thinking happens in both verbal and written form as Dyson mentions, “In performing a text, an individual makes use of cultural resources (i.e. performance conventions) to take action in her or his social words,” (Dyson, page 15) and similarly writes about the written form, “Readers are not audience members swept into flow of a performance but analytic consumers who compare and contrast text, analyzing and synthesizing the meaning found in each, constructing new texts,” (Dyson, page 16). She therefore classifies literacy as performative – one adds some personal understanding of cultural impacts by partially presenting own comprehension; or communicative – one reinterprets information without changing the content (but still being able to omit information). In both cases, however, the information conveyed is not necessarily the same as the initial one due to own interpretation.

Moreover, Dyson recognizes the importance of a learning environment, i.e. cultural background around. Initially, “the boundaries of the cultural bridge inevitable blur inside the children´s worlds, because, from their points of view, school is not one social sphere” (Dyson, page 18). Then the boundaries get their shape again as children realise power settings. Coming from a homogeneous home setting, children suddenly find themselves in a social world full of differences. They realise the racial and cultural composition (they see a bigger picture), yet they still do not completely understand the relationship of knowledge and power (they do not see outside of the frame). Here, where the literacy development of Dyson´s theory ends, starts a new level of Socio-Political understanding.

Socio-Political theory of literacy by Paulo Freire and Donaldo Macedo takes a part of children´s learning mostly during the high school and college education. At this age, children can not only comprehend information, they can also think over it and apply somewhere else, if necessary. This point of transferrable skills is my own input of Freire´s and Macedo´s theory. Though, they stress the point of allowing critical reading in order to build an exemplary democratic society. Socio-Political theory of literacy advocates for literacy as a voice to change social conditions. Since Freire and Macedo identify themselves with Arowitz´s description of literacy as “the ability of individuals and groups to locate themselves in history” (Freire and Macedo, page 12), they conclude that literacy at this stage is not about reading the words any longer, but literacy is about understanding the political context of information presented and power schemes that are influential in that period of time. We say that the aim is to become politically conscious.

Political consciousness is then described as the understanding of one´s place in society defined by voice and identity, and the awareness of power and inequalities. Thus, they warn that students should be self critical and always aware of possible chauvinism in pedagogy. I come from Slovakia, former Czechoslovakia, which used to a communist state. Although I do not remember the communistic era whatsoever, I got an idea of biased hegemonic practice supported nationwide from history books depicting the Western culture as the enemy and the evil. As far as the voice is concerned, Freire and Macedo deduct that it is necessary to be “linking the pedagogy of student voice to a project of possibility that allows students to affirm and celebrate their interplay of different voices and experience while at the same time recognising that such voices must always be interrogated for the various ontological, epistemological, and ethical and political interests they represent” (Freire and Macedo, page 20).

The view about unintentional institutionalised discrimination is shared also in the Socio-Historical theory of literacy by John U. Ogbu. He believes, for example, that typical standardised tests favour dominant cultures because they were created from within it. Ogbu recognizes literacy as a part of dominant structure and raises awareness of different cultures having different values. He characterises a plural society to have voluntary and involuntary minorities which respond to literacy issues differently. For voluntary minorities such as “immigrant minority youths, the collective orientation appears to be toward making good grades; social pressure from the community, family, and peer groups support this. Individuals the threatened with criticism and peer isolation are those who do not achieve academically” (Ogbu, page 162). On the other hand, for involuntary minorities such as descendants of colonised people, the collective orientation “militates against academic success” due to “less community and family pressure” (Ogbu, Page 162). It is important to note that Ogbu´s consequences of historical legacy, defining community and culture, are carried throughout all stages of literacy development described above.

In conclusion, it is visible that all four theories correlate literacy to learning process in one way or another. We can also say that the development of literacy copies the development of learning stages. For learning, young kids first imitate sounds, than repeat words, try to recognize patterns, learn rules and exceptions, start to see a bigger picture of the whole box (i.e. cultural influences) and lastly they see beyond the box as well (i.e. political influences). Going from one stage to the other one, however, does not mean that they replace the stage before (and similarly, their order is not strictly dogmatic). For example, a child, who has learned that the verb “read” is irregular and thus its past tense is “read” and not “read-ed”, still remembers that there was a pattern of adding “-ed” to a verb in order to make a past tense. Thence she will be able to make the past tense of “remember” (“remember-ed”). This all happens while children intellectually develop in a specific family background or a community that influences them either noticeably or silently.

Similarly, the literacy types develop from Neo-Piagetian by Ferreiro and Teberosky, through Socio-Cultural by Dyson to Socio-Political by Freire & Macedo. The factor that may unconsciously influence them all is the Socio-Historical theory of literacy by Ogbu.

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RESOURCES:

Ferreiro E., Teberosky A.: Literacy Before Schooling. Portsmouth: Heinemann (1982)

Dyson A. H.: Social Worlds of Children Learning to Write in an Urban Primary School. New York, Teachers College (Columbia University): Teachers College Press (1993)

Freire P., Macedo D.: Literacy – Reading the Word & the World. New York: Bergin & Garvey (1987)

Ogbu J. U.: Minority Status and Literacy in Comparative Perspective from Literacy – An Overview by Fourteen Experts. New York, The Noonday Press: Hill and Wang (1991)