I enjoyed Adler’s article, but at times I understood and then felt that I didn’t understand the ideas of invisibility & visibility, and would love to hear from others to see if I am on the right track.
With respect to understanding mathematics and how the acquisition of the mathematical language can be a hindrance to gaining or showing that understanding through the invisibility/visibility lens, my interpretation:
‘talk’ can be a resourceful tool and the mathematical language does need to be seen (has a purpose, ‘visible’) with respect to both vocabulary and notation that we would like our students to learn; however, there are times when the language should ‘disappear’, become ‘invisible’, especially (at the beginning stages?) of student learning – when they are ‘practicing’ what they are learning, showing their understanding, yet they may not be doing so in the conventional way that math teachers, in general, would like to see/prefer/hope for. This would allow the students to gain an understanding of the concepts in their own words before learning to express in a more conventional manner.
I would really like to hear if I am on the right track, please. Thank you.
Similar to Vivian’s concerns, I too wondered if I have been putting too much emphasis on acquisition of math language both vocabulary (do love those ‘let statements) and notation as a way ensuring understanding of math ideas. Then there was another line that struck me as well – Helen’s concern as to whether or not “saying it [was] actually indicative of understanding, of knowing” (Adler, 1999, p.56). The reason why this line struck a chord for me was because I was in fact a student who could do the math without ever understanding what I was actually doing. It was becoming a teacher that helped me to really understand how and why things ‘worked’ in mathematics. Then a few years ago, I took a math course and discovered that as a student I hadn’t changed – I could still be successful without understanding. I have to say that this was somewhat distressing and made me more determined to ensure that my students did not feel the same way. I tried to ensure this understanding through various real world problems and projects, but I too used techniques that were similar to Vivian’s. I felt that good communication, through proper vocabulary and notation, would help students to gain better mathematical understanding. I understand the confusion that Vivian expressed, as I feel it too. I guess the key, if I understood both readings correctly, is to find balance and to allow students to express themselves initially through different mediums and over time, guide to the more conventional math language.
I guess having said this, who is to say that proper math language is truly ever needed? If students ‘get it’, no matter how it is expressed, isn’t that enough? Food for thought…
"Who is to say that proper math language is truly ever needed? If students ‘get it’, no matter how it is expressed, isn’t that enough?"
ReplyDeleteIt's an interesting thought, LM. I think there is a need for the use of proper math language for clarity purposes. Being clear and concise is part of being a good communicator in mathematics.
I wonder WHEN do we start insisting on the use of proper math language?
We have actually been discussing "What is proper math communication" in out moderated marking group. For one faction the answer has to be laid out traditionally - For a question to find the perimeter of a triangle ie 2(x+3), 2x and (x+4)(2x+4), the student would have to write the addition statement, use the distributive property directly below the brackets, collect like terms and then solve.
ReplyDeleteAnother group suggested that if the work is done anywhere on the page that that is communication is valid - we had answers where (x+4)(2x+4) was expanded using the box method and then brought into the the additiona sentence and others where the work was hard to follow but was mostly correct.
Does the first example preferrably position some students? Is that strictness in write up necessary for later in their education? Will students learn the form on their own if it is only accessed on clarity of meaning?
If we are going to stress "style," I like LM idea to work on meaning in the initial stages and then move to conventions.
Devika,
ReplyDeleteInteresting question to answer isn't it? Difficult too, maybe? If students are able to convey their understanding using their own words, why do we need to push math language? I would say because there is a value to learning math language as well, as with any other language. There are nuances to a language which are important, I believe to understand, and perhaps for others to understand. As you say proper math language could lead to better clarity. Conventions are not necessarily a bad thing - they too have their place in the world.
As for when to insist - that perhaps is even a tougher question. There is a belief that learning languages at an earlier age ensures that students 'get it' and retain it. Should we equate learning the math language as with other languages such as French? If we believe students retain more at an earlier stage then should we insist students learn it earlier? Learning a language takes time and practice.
Are there any right or wrong answers?
Maybe what's right and wrong on any given day depends on where the students are-If students are disengaging from a discussion on form, maybe its time to refocus on the math, and return to form another day. As LM wrote, learning a language takes time and practice - and I would add, with a focus on meaning.
ReplyDelete"Should we equate learning the math language as with other languages such as French?"
ReplyDeleteI think we do! We forget that math is a language in itself and students need time and practice, as you said, to learn how to use it.
I agree, DT, that a focus on meaning is helpful. Vivien mentioned the use of a word wall in her post. I think a word wall that contains the words AND a picture/definition/example is one way to attach meaning to the words we use in mathematics.
I have some pictures of what this looks like from a Grade 7 teacher I work with. I'll post them separately.
i do agree with math being a language on its own and should be mastered like one masters their mother tongue.
ReplyDeleteHow they master it might not be important but being able to express it in a format that can understood by others is important because what is learned if cannot be communicated is of no use.
I believe that if students are able to describe something in their own words, it shows a lot more understanding than if they regurgitate a definition from a book or restate the words that were said in class. But when it comes to marking tests and assignments, I always have a hard time deciding based on what a student wrote whether or not they really understand the idea. I think that reason for this comes down to the ambiguity of the words they use. If they don’t use correct mathematical language, I don’t know if the intended meaning of the word they use us the meaning that properly describes the mathematical concept. Taken one way, the statement might be correct, but taken another way it could be incorrect or incomplete.
ReplyDeleteI agree with you in your first post, Devika, when you mentioned the importance of clarity. Developing mathematical language is not done just so that we can test if our students understand what we have taught them. Having clear language helps students distinguish between similar concepts in their minds, build on prior knowledge, and make connections.
We hope that eventually students will develop new mathematical ideas that they will want to share with others. When that happens they will need the correct mathematical language to help others understand what they are thinking.