Week 2 Reading
Richard Barwell (2005): Ambiguity in the Mathematics Classroom, Language and Education, 19:2, 117-125 http://dx.doi.org/10.1080/09500780508668667
Introduction
1.
The Formal Model
- Views mathematical
language as precise and unambiguous
- Sees ambiguity as a
problem to be eliminated
- Assumes words have fixed,
universal meanings
- Emphasizes
teacher-controlled transmission of vocabulary
2.
The Discursive Model
- Sees mathematics learning
as a social, interactive process
- Recognizes that meaning is
flexible and context-dependent
- Views ambiguity as a
valuable learning resource
The
author supports the discursive model, arguing that informal language can foster
the development of sophisticated mathematical ideas. Using a class on shapes
and dimensions as an example, the author demonstrates how ambiguity can be
generative for mathematical thinking. The examples show students probing and
exploring linguistic possibilities, highlighting the potential of ambiguity to
enhance understanding and creativity in mathematics.
Reflection
Here are the two quotes that I found
meaningful and interesting from this article.
“Meanings, rather than being absolute, are
relative, flexible, and thus amenable to development, deepening, and increasing
complexity. It is this flexibility that makes possible the development of
participants’ mathematical ideas, practices, and language, as well as the
shared meanings of the class.” (P124)
“Rather, students’ development of the use
of mathematical discourse is intertwined with their development of mathematical
thinking. Ambiguity acts as an important resource for students and teachers,
serving as a means of articulating between thinking and discourse.” (p125)
I chose this quote because I completely
agree with it. I believe that the register of mathematics not only serves as a
universally agreed definition but also acts as a daily tool for people to
communicate mathematical concepts and express their unique ideas. Sometimes,
especially for beginners, it is helpful to allow meanings to remain flexible
rather than fixed. This flexibility enables people to better express their
mathematical ideas.
On the other hand, the scope of the quotes
I chose can be limited. For example, precision in vocabulary is crucial for
high-level mathematical work (aligning with the idea from the formal model),
such as writing a thesis. Allowing young students to use mathematical
vocabulary in their own way can help them gain a comprehensive understanding of
these terms. As mentioned in the Week 1 reading, since the process of learning
mathematics is also a linguistic learning process, people can only truly learn
by speaking and applying new words. If we impose strict limitations from the
beginning, we might create an unfriendly learning environment, slowing down the
process of understanding and preventing students from recognizing the nuances
between different applications.
Being imprecise is the same as making
mistakes, which is an important and necessary part of the learning process.
Students can only gain a comprehensive understanding once they have tried a few
times and explored the different potential meanings of terminology in various
contexts.
In both of the quotes that I chose, I see
the word “development.” As we mentioned in class, language is developing all
the time. I think we can treat ambiguity as a short-term or temporary
development as well. It is developed under a single case to help an individual
better express the idea that they want. How can we say it is a problem that has
to be eliminated?
In conclusion, as mathematics educators, I
think we should draw the good parts and learn from both the formal model and
the discursive model. We should balance precision and flexibility while
teaching vocabulary in classes. I agree with the statement that Richard Barwell
made in this article. However, I also think the statement “Mathematics is
widely seen as a precise language with which to describe aspects of the world”
is also true. We should let our students know that they can freely use the
vocabulary because there are no strict rules; vocabularies are just tools to
help you better express your ideas. It is a bad idea to set a list of rules for
using a tool. However, mathematics is also a precise and worldwide subject that
can bring people around the world together. In order to do that, you have to
know the definition of the vocabulary clearly so you can use it more freely
and lead a meaningful discussion.
Question
How do you think the concept of
ambiguity in mathematical language can be applied in your own teaching or
learning experiences? Can you provide an example where embracing ambiguity has
either helped or hindered understanding in a mathematical context?
This is a really good question, Lee! From my experience so far (which is limited), I've found that it really depends for me when ambiguity can be applied. I think for me, the BC curriculum is focused more on curricular competencies now instead of content, so if I'm assessing a competency then it leaves a bit more room for ambiguity. For example, perhaps there is a minor error in specific vocabulary used in a justification, but if they have demonstrated their thinking in other ways, chose an appropriate strategy, and the process is logical which lead to an accurate result, it's hard for me to say that they don't understand something. I would agree that formal models are more necessary at higher levels of math, but in all honesty, most of what we experience mathematically in real life is so context dependent that maybe we should lean more into the ambiguity.
ReplyDeleteIn response to your question, and to be honest, I do think that ambiguity in mathematics has its place. Allowing students to explore concepts and use their own words to describe mathematical ideas can deepen their understanding. It helps them connect with the material in a way that feels personal and meaningful, rather than relying only on formal terminology. That said, I wonder how practical this approach is in a classroom setting.
ReplyDeleteGiving students the time to develop their own language for mathematical concepts sounds great in theory, but it can be very time-consuming. As teachers, we often face tight schedules and curriculum demands, and we don’t always have the luxury of spending extra time on every concept.
One activity I’ve found helpful is "Which One Doesn’t Belong." This activity presents students with four objects, shapes, or numbers and asks them to explain which one doesn’t fit and why. The beauty of this exercise is that there isn’t one right answer—it’s all about how students reason and explain their thinking. This activity can be used to introduce and reinforce the use of mathematical language pressure while building on reasoning and communication skills.
Hi, Lee, thank you for your thoughtful reflection! Your ideas are very insightful and have encouraged me to think more deeply about the ambiguity of mathematical language.
ReplyDeleteI agree with your point that "being imprecise is the same as making mistakes," and this is a natural part of the learning process. I believe it’s not necessary to focus too much on precision in mathematical language, especially at the early stages of learning. It is normal for students to explore concepts in their own way when they first encounter them, and this process can actually enhance their understanding. I also agree with your statement that both formal models and discursive models play equally important roles. As educators, it’s important to strike a balance between these approaches to support students’ learning effectively.
Regarding the question, I think ambiguity can sometimes benefit students by giving them opportunities to explore mathematical ideas in a way that makes sense to them. It also allows teachers to better understand students’ thought processes and learning progress. For example, when teaching students who are less interested in math, I often use examples to introduce concepts instead of providing precise definitions right away. This approach not only grabs their attention but also makes the concepts easier to understand. For instance, when introducing the graph of a quadratic function, I might ask students to draw the path of a basketball when shooting and explain that this shape resembles the graph of a quadratic function. Although this explanation isn’t completely precise, as there are downward quadratic function, it helps students relate to the concept.