In Carolyn Kieran’s reaction article on theories of algebraic thinking, she discusses a model with 3 components:
- the generational activities of algebra-rules governing numerical relationships, variables and unknowns, the equal sign and the equation solution.
- The transformational (rule-based) activities-collecting like terms, factoring, expanding, substituting, etc.
- Global, meta-level, mathematical activities-problem solving, modeling, change, relationships
In the examination of different approaches on preparing students for algebra, Kieran goes on to note there is no international consensus on what is algebraic thinking and how best to teach it. She mentions the Chinese curriculum, where its goal is to understand quantitative relationships as well as the Singapore primary mathematics program which focuses on problem solving and relationships. The Korean curriculum focuses on relationships and structure as well as problem solving. The U.S. seems to stand a bit apart, instead focusing on number patterns and generalizations.
Kieran does point out the commonalities between the programs in the development of algebraic thinking: relationships, generalization, justification, problem solving, modeling and structure.
I found Kieran’s article to be insightful and agreed with many points. I found it interesting that many of the curricula in Asia were similar, in that they focus on problem solving and relationships. I have certainly noticed a difference between the mathematical abilities of students who have used the textbook I’m most familiar with, Singapore Math, and students who have come from schools using different texts, such as Math Makes Sense, without such a strong focus on operations and understanding.
I also didn’t consciously realize that there could be students operating with an arithmetic frame of mind and those with an algebraic way of thinking in my classroom. After Kieran’s description of the difference, I feel I could confidently identify most of my students as one or the other.
In several sections of the article, Kieran mentions the importance of the equal sign and that students understand its meaning. It reminded me of an activity another teacher once did with a group of elementary teachers: 11 + 4 = ____ + 3 = _____ + 7 = _____. Most teachers said the answer is 25!
Kieran mention 5 items that should be focused on to succeed in algebra: relations, operations and their inverses, representing and solving problems, numbers and letters together and the meaning of the equal sign.” In your experience, do you think students have a good understanding of these concepts? Or do you find you need to review/re-teach some of them?
I believe student’s understanding of these aspects of algebra is highly related to their age. For instance grade 3 students mostly focus on doing an operation, so words and the relationship between information are not their concern. This lack of attention has been changed by time, and students learn to face with algebraic questions. However, although we expect high school students have a good understanding of algebraic questions, they also have problems facing and solving these questions. Most of them forget about the equal sign in the middle of solving the question, so they could not find the right answer. Although they have learned to solve equation since grade 6, I have noticed most of them have difficulty till grade 10! It is interesting that in the middle of a hard question when they need to solve an equation to find the final answer, they suddenly forgot about how to solve a simple equation!! And I wonder why? So, I believe, in any grade or stage students always need teaching algebra or reminding them basic algebraic rules.
ReplyDelete"I also didn’t consciously realize that there could be students operating with an arithmetic frame of mind and those with an algebraic way of thinking in my classroom. After Kieran’s description of the difference, I feel I could confidently identify most of my students as one or the other. " It is so interesting that you say this, because after reading Amanda's post I started thinking of my students as arithmetic or algebraic-oriented too! I think that training outside of school (Sylvan, Kuman, tutors, how families help with math homework) also has an impact on the type of thinking that is considered 'math' and what is valued in the home. To me transferring between home culture and math culture at school is interesting. (Aikenhead talks about this with science ed. but I think it applies here as well).
ReplyDeleteI did a problem this week that shows how uncomfortable students are with the equal sign.
101 - 102 = 1 (Move one line to make this true...) one solution is to move one line from the equal sign so 101 = 102-1. There was one student who solved it (from 70) and many grade 4 and 5 students had that enlightenment 'aha' face. This tells me that students are seeing a lot of the same at school.
That's a great example of "testing" students' understanding of the equals sign. I think I might try that and see what I find out too!! :)
DeleteI find that many students reach grade 7 without having algebraic thinking and without really understanding what an equal means. Many students think that an equal sign is what you put before your answer. I have had my students work with cuisenaire rods measuring and with weights and balances to try and get them to understand what an equal sign means.
ReplyDeleteI have been reading about Indigenous Yup'ik mathematical knowledge that uses folding in half and proportions to make measurements. The measurements don't have numbers attached to them but are all related. And they are based on body parts. It was suggested that this was a good method for early algebra.