Asking Mathematical Questions Mathematically
"Put simply, the only point of asking questions is to raise issues about which the teacher needs information or about which the pupils need to think." (Working Inside the Black Box)
“Analyses of the kinds of questions that teachers ask in classrooms have shown that many teachers ask questions that test the ability of pupils to recall facts and procedures (often called lower-level questions), rather than the ability of pupils to apply, synthesis or explain their knowledge (often called higher-level questions).” (Askew and Wiliam).
Main Section
Put simply, the only point of asking questions is to raise issues about which the teacher needs information or about which the pupils need to think.” (Working Inside the Black Box).
The difference between asking a question like “Is 7 a prime number?” and “Can you explain why 7 is a prime number” can be quite powerful. The former often puts learners in a sort of “guess what’s in the teacher’s head” mode. We, as teachers will often collude in this by making conscious or subconscious cues which might prompt learners to give us the right answer. The latter gets rid of the needs for an answer and concentrates the dialogue on reasoning and justification
Other similar types of questions which can prompt learners’ thinking include:
How can we be sure that (this shape is regular)?
How do you find (a fraction of a sum of money)?
Is it always, sometimes or never true that (when you increase the area of a shape you also increase its perimeter)?
If questioning is to lead to dialogue and discussion and a development of ideas and understanding then the ability to listen carefully to a pupil’s response is crucial. Brent Davis makes a distinction between “listening to” (i.e. genuinely listening to what a learner is trying articulate) and “listening for” (i.e. listening out for what you are expecting to hear) as a useful prompt while asking questions.
Another key aspect of the research in this area is the importance of the wait time in between asking a question and expecting answer, asking another pupil, re-phrasing the question or giving up. The average wait time is 1 sec and studies with teachers indicate that by trying to increase this wait time to 3 secs has a measurable difference in the pupils’ responses.
With adults you can afford to be explicit about wait time and who will get the questions. Start a dialogue with the group or with individuals about how they prefer to be questioned and how much time they might need to get articulated.
Probes and Prompts
How long do you wait after asking a question before saying something? Are you sure?
How often do your questions require a single answer?
How often are you genuinely seeking information you do not already know when you ask a question?
What is mathematical about the questions you ask most frequently?
What do you think your learners would say was your most frequent question?
Do you ask more questions to learners who get things right or who get things wrong? What are your motives?
Do you have a strategy for questioning for those learners who are shy or inarticulate, to minimise their stress in the group, but also develop their confidence?
Taking Action
When working on some mathematics for yourself, pay attention to the questions you tend to ask yourself, as well as those asked by others around you: do you ask those kinds of questions when teaching?
Select a particular question or class of questions and try using them frequently, then gradually use less direct, and more indirect prompts until learners take over asking those questions for themselves.
It is a reasonable conjecture that the kinds of questions learners are asked will influence and inform what they think are typical mathematical questions?
Case Studies
Primary
I have been really fortunate to have observed some fantastic discussion and thinking around two questions posed by a couple of teachers. The first, a teacher of a Year 5 class, asked the children whether in the Land of Part Numbers, they would rather be a decimal or a fraction. It was great to listen to their replies which often showed a really good understanding of the topic. Answers included 'a decimal because I could go on forever and ever', 'a fraction because I would be part of a family, if I were a fifth I would have four brothers and sisters.
I heard the same kind of question in a Year 2 class: 'In Shape Land would you rather be a sphere or a cube?' Answers included 'a sphere cos I could roll about', 'a cube because I could be part of wall and have neighbours'.
This type of questioning gives great insight into a child's understanding of concepts and also their ability to use the higher order thinking skills referred to in the article. They also provide good assessment opportunities.
Adult learning
One of the best techiques I observed for questioning young adults, where one was painfully shy and inarticulate, was set up as follows. In a previous tutorial the teacher had explained that she wanted to help A communicate more confidently without putting her under pressure with her peers. They arranged that A would only get a question when the teacher came and physically stood near her, and addressed the question personally. A could relax for the rest of the time and think up answers for herself. The next stage is to make this a two- way process. The teacher seeks to make eye-contact with A during a questioning session. If A returns the glance then that is a signal she is OK to take the question. Needless to say, A's teacher was able to comment on improved confidence and communication skills as well as maths learning at the next tutorial!
Research Sources
Davis, B (1996) Teaching mathematics: toward a sound alternative. New York: Garland Publishing, Inc.
Ref: Recent Research in Mathematics Education 5-16, Askew and Wiliam.
Ref: “Working inside the black box: assessment for learning in the classroom”, Black, Harrison, Lee, Marshall and Wiliam.
Ref: Pedagogy and Practice: Teaching and Learning in Secondary Schools”, Unit 7: Questioning, Ref: DfES 0430-2004 G.
Ref: “Assessment for Learning: Using assessment to raise achievement in mathematics”, section 2: using effective questioning techniques, QCA Publication, QCA/03/1070.
Ref: “Assesment for Learning: Whole school training materials”, Key Stage 3 National Strategy material, Ref: DfES 0043-2004 G, Unit 7: Questioning and dialogue.
Ref: ATM “Questions and prompts for mathematical thinking”
Ref: Guess What’s In My Mind, John Holt (Ruth)
Ref: Janet Ainley
Ref: Sullivan & Clarke
See Also
Categories
Pedagogy