By the team of O Círculo da Matemática do Brasil and Professors Bob and Ellen Kaplan

An approach is always difficult to convey, because it requires an appropriation that in most cases only comes through practice. In any case, below is a list of recommendations about the approach presented by Bob and Ellen Kaplan during the training and discussed by the team of teachers from O Círculo da Matemática do Brasil. The recommendations are grouped in eight sets of questions:

Constant asking

  1. Never give away the answers to your students. Do your best for them to arrive to their own answers. Continue asking. The beauty of teaching is to ask your students the right question at the right time. Sometimes make ambiguous questions: half of the way in mathematics is in guessing how to answer an answer that can be answered.
  2. Be patient. Children need time to think and to answer. Do no press for results. Give children time.
  3. The questions must stimulate children to express their ideas. Think: how can I make stimulating questions? There is no recipe, it depends on the context.
  4. Call children by their name. The constant asking acknowledges over all else the individuality of the children. Children like numbers, but the do not like being a number.
  5. The Math Circle has its own ‘choreography of teaching’. Accelerate until reaching a climax and then wait. Use the language:
    • “What do you think of that?”
    • “What a wonderful idea!” (without excessive enthusiasm)
    • “I believe you may be right”
    • “I don’t know either” (even if you know, just a show of empathy)
  6. The asking must promote strategic interaction among the children. When a child gets an answer wrong, ask a colleague what he/she thinks of that answer. Ask also when a child gets it right. Encourage interaction, leave to them the authority of answering in practice.
  7. The questions must be simple. You do not need to use baby talk, but the language must be clear and simple. May times, however, the children will not understand. Never mind, rephrase until what you are saying is clear to all (that is also a way of widening the mathematical vocabulary of your students).

Truly listening

  1. Often teachers have the right attitude for listening but they do not manage to carry it through. For that to happen, write down on the board all the types of answers, including in particular the wrong ones. It is not because the student gave the wrong answer that he/she is less important. All answers must be listened to and written down.
  2. Deconstruct in order to construct. Do not worry overmuch about time and targets (not that they are not important, but they must not be reached merely formally), the focus must be on the quality if the process that you are building.
  3. Be a good listener. Often teaching is mistaken with talking, when listening is as, if not more, important.
  4. Simplify as much as possible complex questions. Translate complex thoughts into simple thoughts. Help your students to understand the problems by giving them an access point to simpler ways of thinking.
  5. Translate the answers into a language that has mathematical equivalent, that way it becomes easier to register the participation of the children in concrete terms and it is a way of including them.
  6. Do not praise children when they get it right. Praise their participation. Praise when they get to the right question. Example: “very good, then, what does this suggest?”

The organization of the blackboard

  1. Give flexibility to your blackboard, keep the main reasoning supported by other parts of the board. Do not divide the blackboard, give it organic functionality.
  2. Do not forget to record the answers by all the children on the board, because the board also belongs to them.
  3. Leave some right answers on the board when you are asking, so that children can answer your questions by looking for them on the board. The board may appear messy but it must have a logic to it.

The error

  1. Making mistakes is good. Wrong answers are very welcome. Record all wrong answers as your starting point to organize your thinking and that of your class towards the right answer.
  2. Use the error pedagogically to organize the thinking of your students. Wildly wrong answers are always a good starting point.
  3. Do not be afraid of making mistakes yourself – in fact, make some intentional mistakes because that can reveal what the students understand and relax the atmosphere.

Strategies for inclusion

  1. If you listen, record the answers, and welcome mistakes, that is going to help a lot in the inclusion of students. But if students are not participating, try to captivate them, make very general questions such as “what is your favorite number?”, “what colour would you like us to use to draw … something on the board?”, and so on.
  2. Respect diversity. Each child must be allowed to advance on his/her own time. But do not tolerate bad behavior, rude children, smart-alecks. Do not promote a competitive atmosphere by allowing impolite behaviors by the children.
  3. Diversify problems and questions. If you realize that a problem is too difficult, try a different problem and return to the difficult one later (the use of the function machine activity with an easy rule is always a good alternative for a break to recover ‘lost’ children).
  4. If the environment is hostile, introduce gentle questions to show that the conversation is not ‘dangerous’, such as ‘How should we call the winner of this race? (on the number line)”.

The end

  1. When you are nearing the end, retrace the logical path you followed to get to the result. Emphasize the important points.
  2. Simply getting to the right answer is not the end, nor the objective, of everything. The secret is in the construction of the process. Play with the result. Do not finish the problem with the right answer, extrapolate, invent, apply the result. Ask of there are other ways of getting to the same result. Try to end on a high note with an open question: “lets imagine what can be done with this result on the next class”.

The nos

  1. The class cannot be a session of copying the blackboard, without due thought. Children must focus on thinking, not on copying.
  2. Do not be a slave of the contents. Merely getting through the material must not be our priority (see point 9). Our priority must be to stimulate the interest of the children in thinking mathematically.

The subtleties

  1. The greater emphasis must be on the promotion of the children’s imagination and of what they are feeling. That is why knowing what are the hard bits for each child and the points of access to mathematics is important.
  2. When to formalize? Whenever you feel that without rigor the discussion will remain too loose – but not when rigor could kill the discussion.
  3. You must judge when you went though a point that was not vivid to all, while at the same time being careful not to be repetitive to the point that it bores everyone.

“Is this too much to remember? Relax. You will enjoy yourself and your students will become your friends for life” Bob and Ellen Kaplan