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”.
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”.
I don’t think that this strategy would teach children to understand the question more thoroughly. Even with the numbers written in words, students still know and use the vocabulary they will be familiar with in problem solving. Students know to look for words like more than, in all, and sum to mean addition; difference, left, less to mean subtraction, per, times, twice to mean multiplication, and divided, out of and percent to mean division. I think that many textbooks use similar vocabulary and the students quickly crack the code! One way around it is in writing word problems that vary the vocabulary and situations. Those are opportunities to challenge students in their thinking. Another way to extend student thinking is by asking students to provide more solutions.
ReplyDeleteTeaching students to look for particular combinations of words or even writing the numbers in words is a short cut and won’t lead to deep understanding of the relationship of the operations. Without a firm understanding of number, pattern and relationship, I think algebraic thinking will be harder for students.
I agree with Nancy. I have been looking at problem posing for my final paper and so I wonder if having students pose algebraic problems could help? I had one student ask me yesterday if he could pose a really complicated problem and have the class solve it. Even in this though, students are practicing arithmetic rather than algebraic thinking. They too, can simply mimic what they have seen in textbooks and in math class.
ReplyDeleteI like Nancy's idea of asking students to provide more than one solution. I find that word problems can be problematic on many levels. I always try to write word problems for my students that involves them and something that we actually did or might have done. It could be interesting to have them write word problems that need to be based on something that really happens.
ReplyDeleteWith algebraic word problems, I think that practice and familiarity make a big difference. But then, are the students mainly tuning out any meaning and just decoding? Probably.