Teaching in the Twenty-first century demands teachers to be highly proficient in all key learning areas but particularly in mathematics where planning, preparing and implementing a lesson involves many aspects. Margaret S Smith and Mary Kay Stein, authors of the book 5 Practices believes in order to advance mathematic thinking and reasoning skills, teachers need to orchestrate productive, whole class discussion through the five practises; anticipate, monitor, select, sequence and connect. These five practices not only depend on cognitively demanding tasks with well defined goals and multiple student responses, but also on teachers’ understanding of their students current mathematic thinking and practices.
After carefully selecting a rich task a teacher must anticipate how a student will approach the problem. They must be aware of the common misconceptions that could arise. They must identify the mathematics and find all possible solutions and the only way to do this is to actually do the task themselves. By doing this teachers are better able to pre-empt misconceptions, making them more prepared with enabling and extending prompts and probing questions. They must ask themselves the same question they ask their students ‘Can you do it another way?’ Planning with a partner is particularly helpful for this as each person will bring their own solutions to the table, which can then be discussed.
During the activity a teacher should monitor student thinking. Observing and identifying key strategies and by being well prepared and having anticipated the various responses the teacher is well equipped with effective questions and prompts that would support, consolidate and extend their students as they steer them toward the learning intention of the lesson. Offering this support could help students who are on the verge of implementing an important strategy achieve this, as well as assist students who are challenged at starting the problem. Teachers are encouraged to walk around taking notes of student discussions and capturing images of materials used when possible to make selections of work to be shared during reflection.
Once students have had an opportunity to explore and the teacher has monitored, a selection of appropriate work samples beginnings. It is important to know the students thinking in advance and the mathematics they want their students to unpack. The selection should be mapped out to show a range of strategies as well as misconceptions not for ‘show and tell’ but to ensure that mathematical ideas are discussed, allowing ideas to be illustrated and highlighted. The teacher must be in control of this selection process, which is planned and implemented with the learning intention in mind.
Sequencing student work samples should also be well thought out and purposeful. There are many different ways to do this. The easy to understand strategies before the more complicated ones or the most common strategies and the most common misconceptions or related strategies first followed by contrasting strategies or ‘scaling up’ practises demonstrating varying degrees of complexity in problem solving are all useful ways to sequence in order to facilitate discussion and to explicitly teach particular concepts. Both selecting and sequencing are done ‘on the run’ demanding teachers to think critically and analytically while managing, monitoring and orchestrating the lesson.
Finally it is important to draw connections between mathematical ideas through different strategies and help students to notice that the same ideas can be embedded in two different strategies. Allow them to decide which would be the most efficient strategy, challenge them to think if the numbers/ratio etc changes would one strategy be more efficient than another? They should be given time to reflect and revise their thinking. Further questions can be posed to advance the maths used and to set up the learning for the next day, which is then added to the Success Criteria.
In conclusion, the ideal mathematics lesson is a cycle that consists of detailed planning, strategic implementing with intentional navigating that will then lend itself to further planning that will continually advance students’ problem solving skills and develop growth mindsets. I believe, as a twenty-first century teacher adopting the five practices is paramount to nurturing innovative problem solvers.