HEDGES IN MATHEMATICS TALK: LINGUISTIC POINTERS TO UNCERTAINTY
Introduction
The researcher examines how children aged
10-12 use language to express uncertainty during mathematics tasks, focusing on
linguistic strategies called hedges that indicate doubt or hesitation.
The researcher categorized key hedgewords
into two main categories:
- Shields: In mathematical discourse,
these hedges present a mathematical assertion as a conjecture, implicitly
inviting comments. Shields can be further divided into:
- Plausibility Shields: These suggest a position or belief while indicating some doubt about its fulfillment or evidential support. Examples: I think, maybe, probably
- Attribution Shields: These attribute some degree or quality of knowledge to a third party. Examples: According to somebody
- Approximators: Unlike Shields,
these hedges are found within the proposition itself, modifying it to make
it more vague rather than commenting on it. Approximators can be further
divided into:
- Rounders: Common in measurements and quantitative data. Examples: About, around, approximately
- Adaptors: These words or phrases add vagueness to nouns. Examples: A little bit, somewhat, fairly
The research analyzed transcripts from
mathematical interviews and found that children use hedges as protective
linguistic strategies to avoid potential criticism or embarrassment and to
navigate uncertainty in mathematical reasoning. The study suggests that
uncertainty should be seen as a valid and productive state in learning,
encouraging more exploration and conjecture.
Stops:
“One could go further, and insist that uncertainty is a productive state, and a necessary precondition for learning. For once we believe we "know", we are no longer open to the possibility of further knowing. When mathematics is coming into being in the awareness of an individual, uncertainty is to be anticipated and expected.” (p328)
“The willingness of schoolchildren to expose their thinking will depend on whether or not teacher and pupil share a belief or explicit agreement that they are working in a 'conjecturing atmosphere'.” (p333)
Whilst truth and falsity may be decided in
the ZCN (zone of conjectural Neutrality), a person may articulate a proposition
without necessarily being committed to its truth. In such a cognitive and
affective milieu, it is the proposition that is on trial, not the person. (p350)
Reflection:
The article highlights a common issue in STEM education: the fear of making mistakes or taking risks. This avoidance is prevalent across various fields but is particularly significant in science and math, where precision and accuracy are emphasized.
This fear can be problematic because many pedagogical approaches emphasize exploration, risk-taking, and making mistakes as essential for deep understanding. For me, these steps are crucial for achieving the desired learning outcomes by the end of each course.
Allowing uncertainty and the use of hedges
in responses is essential. As the article states, uncertainty is a productive
state and a necessary precondition for learning. It encourages beginners to
shape their ideas and reason through answers, and it also allows those with
some knowledge to explore further and justify their answers, promoting
continued learning and growth. This is why I chose quote 1: once we believe we
know and are certain about an idea, we risk stopping there and missing other
possibilities. This applies not only to our students but also to ourselves as
educators.
Furthermore, these linguistic markers
reveal the cognitive processes involved in mathematical thinking and challenge
the traditional view that places a high premium on being right. The key ideas
from the research align with my experience and the importance of creating a
safe learning environment.
I would like to end my reflection with an
example from page 351 of the article, which shows how hedges and uncertainty
shape students' ideas and provide scaffolding:
Question:
Please share some thoughts regarding those
hedges. How can you value this kind of uncertainty or use it to direct your students
in your teaching field?
Thanks for your summary of this very dense article, Lee! I really appreciated your reflection as well and thought the example you chose from the text was very effective. Personally, I think hedges are very valuable especially in my context as a middle school math teacher. I can't even begin to think about how many times students were more willing to engage in genuine math conversations when they didn't have a fear of being wrong. I consider this to be an area that more STEM educators should encourage. The STEM fields in general involve asking and answering questions about how the world works and then communicating the learning in some way. Part of asking questions lies in that uncertainty. It's ok to make mistakes, as long as we use them to learn, and I believe these hedges are a great way for us to invite and work in that realm of uncertainty to build a stronger understanding.
ReplyDeleteIn my experience teaching mathematics to Grade 7 students, I have noticed that many of them become overly focused on finding the correct final answer. This obsession can create anxiety and frustration, making them hesitant to attempt problems that seem difficult. When students fear making mistakes, they often avoid challenges altogether, which prevents them from developing problem-solving skills and mathematical confidence.
ReplyDeleteOne strategy I have found helpful is encouraging estimation. Estimation allows students to think about what the answer could be without the pressure of getting it exactly right. It creates a safe space where mistakes are part of the learning process rather than something to fear. By estimating first, students can make reasonable guesses and use them as a guide to work toward a final solution. In some cases, reaching the exact answer is not even necessary—what matters more is the process of thinking, reasoning, and problem-solving. Another useful approach is using hedges—words or phrases that soften statements and reduce the pressure of finding a single correct answer. For example, instead of saying, “What is the answer?” I might ask, “What could the answer be?” or “What do you think makes sense here?” Hedges help students feel more comfortable taking risks and exploring possibilities without fear of failure.
I also believe that educators, whether intentionally or unintentionally, send signals about what is most important in math. If we emphasize only correct answers, students may feel that their thinking and efforts are not valued unless they arrive at the “right” solution. Instead, we should highlight the importance of reasoning, strategy, and persistence. By shifting the focus from perfection to exploration and using hedges to encourage flexibility, we can help students build confidence, take risks, and develop a deeper understanding of mathematics.
Hi, Lee. Thank you for your thoughtful reflection. I completely agree that uncertainty in language is a productive part of the learning process. For one thing, it allows learners who are afraid of making mistakes to express their thoughts with less pressure. For another thing, it introduces uncertainty in knowledge, encouraging deeper exploration and understanding.
ReplyDeleteSince students often use uncertain language when explaining their thoughts due to a lack of confidence or fear of making mistakes, it is important that we support and encourage them. Instead of focusing solely on correctness, we should acknowledge their efforts and help them build confidence in their reasoning. Even when their answers are not entirely correct, we can still appreciate their thinking process and guide them toward deeper learning. By fostering a 'growth mindset', we can help students recognize the value of mistakes and see them as opportunities for growth rather than setbacks.
Very interesting discussion! I'm particularly interested in the idea that precision and certainty are mostly emphasized in STEM subjects, and less in arts and humanities subject areas (from Lee's discussion). I'm also very interested in Shar's mention of the ways that we as teachers might deliberately hedge our questions to encourage students to feel safe in conjecturing (and I love Tim Rowland's 'Zone of Conjectural Neutrality'!) One question about asking students to estimate: how do you avoid having them start by calculating a precise answer, and then rounding it to artificially create an 'estimate' (that isn't really an estimate)?
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