Notice what happens in your classroom when you write something incorrect on the board…

…Or when you admit that you don’t know the answer to a problem… When this happens, the energy level in your classroom tends to escalate. Students sit up in their seats. Now there is something worth thinking about. Even the teacher doesn’t know the answer!

More often than not, teachers are expected to know the answers. Consequently, students do not feel a need to check their work, or even to figure out the answers themselves. Students know that if they wait long enough, their teachers will tell them how to solve the problem. This results in classrooms where the teacher is the mathematical authority in the room and, as such, does most of the maths.

Within the contexts discussed in our RME materials, often students will have as much knowledge to bring to the problem as the teacher. Students will likely see the problems in novel, if informal, ways. The role of the teacher is to make space for a diversity of student approaches. By remaining neutral, the teacher nurtures a classroom culture where students feel comfortable contributing their ideas. They model good listening skills for the students and they hand over responsibility for thinking and reasoning to the students.

Here are some ways to practice remaining neutral:

  • Facial expressions: By genuinely listening and concentrating on what a student is saying, the teacher models what they want the students to do — listen carefully to others’ contributions. The teacher also sends the message that their role is not to judge or decide if what the student is saying is correct. When the teacher raises their eyebrows or looks puzzled, they model the effort to understand others’ ideas. They also return the onus of ‘making sense’ back to the class.
  • Repeating what a student has said: Inevitably, many students will talk to the teacher when they offer a response. The neutral teacher’s role is to repeat verbatim what the student has said. Alternatively, they may ask a different student to “Say what he/she said” in their own words. This will promote a listening culture, while helping teachers resist the temptation to dismiss solutions that don’t fit with what they had in their head.
  • Being appreciative of student contributions: Instead of appraising the contribution itself, neutral teachers respond with comments such as “Thank you.” / “Okay.” / “That’s interesting.” All these comments let the student know the teacher is aware of the contribution, without giving an indication as to the correctness (or otherwise) of the solution. The aim is for students feeling free to try out strategies and solutions, without fear of being ‘wrong’.
  • Standing away from the board: The move away from the front of the classroom can be a revelation. By standing behind learners, or to the side, the teacher  is in a position to casually hear student conversations. When students are invited to bring their maths to the front, they start to own that space much more than if the teacher is there beside them. This also encourages students to project their voice when explaining solutions, so that the teacher at the back of the room can hear. This will make what they say audible to the rest of the class. The space at the front becomes a more neutral zone, as opposed to the sole domain of the teacher.

Remaining neutral encourages more contributions to a problem.

When the teacher remains neutral, multiple ways of looking at the same problem can be discussed and explored. This leads students to justify their answers and helps them to develop a more connected view of mathematics. Students who are used to RME classrooms don’t expect to stop thinking once someone has come up with ‘the right answer’.

Take, for example, these slides from Sorting It Out (D1). They invite students to develop a variety of strategies which will eventually lead them to a deep understanding of what we mean by ‘average’. The first slide elicits a variety of strategies:

  • Guessing a value and checking by totalling the values and dividing by how many there are
  • Recognising that the total needs to be 25+25 (for 2 throws) or (25 +25+25) for three throws and working back from there
  • Comparing how far scores are from 25 and balancing the gaps below with the gaps above

The second encourages a honing down of strategies, and students develop the argument that Kelly does not qualify, with various strategies:

  • Re-distributing: 9 from the 35 goes to the 16 to make 25. 3 from the 28 goes to the 20 plus from 1 the 35 goes to the 20. This gives scores of 25 24 25 25 25.
  • Balancing: Comparing with 25 gives -9 -5  0  +3  +10  = -1 below 25
  • Totalling and dividing: 124 divided by 5 is 24 r 4

Remaining neutral while using these slides will encourage students to explain their ideas to the rest of the class. It should also enable them to make new connections between various approaches to the concept of average.