Rowland (1995) has splendidly explained the extensive usage of vague language in his article "Hedges in Mathematics Talk: Linguistic pointers to uncertainty". To analyse the use of hedges in mathematics classroom, he proposed an open ended question in front of his students to generalise the formula about integers. He presented the broader perspective of mathematical discussions in the classroom and the way of using vague languages in those discussions to support the uncertainty and to provoke predictions and generalizations. He focused on some hedges as 'about', 'around', 'may be', 'I think' to address the conjecture activity in the math classes and also named such situations as 'Zone of Conjectural Neutrality' (ZCN). Children use such words to position themselves as secure and speak up about their intentions for falsity, uncertainty about predictions in the math classroom. He also explained the number of goals given by Channell (1985), to achieve for using such indefinite or vague languages in math classes. "Some of the goals are: Giving the right amount of information: Saying what you don't know how to say; Covering for lack of specific information; Expressing politeness, especially deference; Protecting oneself against making mistakes". The achievement of given goals has also been experienced in analysed math conversation in the author's class. Rowland also explained that the usage of hedge words act as shields and open up the possibility for teachers to provide intellectual scaffoldings towards their students to achieve a right direction for the given math discussions. He also gave the framework for the hedges which are usually used in math classrooms. He also expressed that those hedges act as support for students to express themselves in challenging situations and for teachers, hedges act as a sense to give right direction to the math conversations. Since those categories of hedges in the given framework are never separated from each other.
- "Mathematics holds an invidious reputation within the school curriculum, being associated with fear of error and consequent public humiliation" (p.327). As per my experience of teaching and learning in my home country(India), generally mathematics is considered as a subject of wonder and straightforward in giving results. students literally have the maximum fear of failure in math classes and consider there is no place of discussion in this subject. I would highly appreciate to propose open-ended math problems in the classrooms to generate productive and creative math discussions in the classes.
- "In the making and learning of mathematics, uncertainty is to be expected, acknowledged and explicit" (p.328). As per given my experience in the first point, I agree with this point that to generate productive knowledge, there should be a growth mindset for the teachers as well as for the students in their math classes.
- For the 'Conversational Implicature', It is easy to fall into the trap of superficial, oversimplified attribution of purposes to speakers on the basis of vocabulary alone, as though it were possible for words to mean with consistency, divorced from context" (p.337). Hence, It is very important to have cordial relations between teacher and students so that one mutual understanding should be established to know and understand the vague content of a student's conversation as per the given context.
- Along with open-ended problems, what more can we introduce as teachers in the math classroom to promote math conversation among student-student and teacher-student?
Hi Sukh, thank you so much for your summary, reflections and wonders. I appreciate your framework, it's reader-friendly and I can get the point of this reading through your conclusion. I have the same feeling as your first point "Mathematics holds an invidious reputation within the school curriculum, being associated with fear of error and consequent public humiliation" (p.327). I recall that one of my elementary math teachers would become so angry that threw our textbooks on the ground when we made mistakes. We would be ashamed of making mistakes and be afraid of the math classes.
ReplyDeleteWhen it comes to ways of promoting math conversations among students and teachers. The first thing I would like to do is to establish an equal space and a trusting relationship. Students are supposed to feel comfortable expressing their questions and ideas and gradually established a mood of confidence in a safe space. In addition, challenge the students' response and give encouragement at an appropriate time. I am aware that questioning could be a powerful stimulus to math conversations.
Thanks for the great post, Sukh, and the interesting conversation around it! It is important to pay attention to the way that societal (and teacher) attitudes can really make kids afraid of math, for a whole lifetime. These qualifiers and hedges can make math seem more uncertain, but also more open-ended and conjectural... and even more polite and considerate! I'm sorry that both of you had to deal with scary and angry math teachers and attitudes as children, and I'm glad you were able to come away from those situations with a keen sense of curiosity and wonder, though it must have taken some work to get beyond the fear of error, etc.
ReplyDeletePS: I spoke to Madison, and she will be catching up her post and comments soon -- unavoidable delays for her this week!
ReplyDeleteGreat points, ladies! And I agree with Susan, I am so sorry to hear the horrible experiences you both went through in class. I have seen how much these negative experiences can impact a students success in later classes.
ReplyDeleteI also definitely agree that equality and trust are required to have good and impactful conversations in a math classroom (or any class!). I would add that developing a classroom culture around the goodness of mistakes, and how important mistakes are to learning will have a deep and positive classroom environment, and makes for some great conversations. I also think that asking a student if they are sure, and asking them to justify & explain their thinking develops good conversation and allows for better critical thinking. I also find that students are less likely to fear making mistakes when you ask them to justify their answer, because they can often identify their own mistake when they have made one!