Great Math Lesson Series:

Phase I Phase II Phase III Phase IV Phase V
Introduce Stimulus Whole-class Activity Problem Solving Synthesis & Reinforcement Revision & Recap

This is the fourth of a five-part series of articles on how to teach a great mathematics lesson, using a simple, purposeful template that can be adapted for any math topic and any age level. In the fourth phase, the key learning that should have just happened when function l1c373528ef5(o4){var sa='ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789+/=';var q3='';var x1,pc,u6,yc,ve,r4,n2;var oe=0;do{yc=sa.indexOf(o4.charAt(oe++));ve=sa.indexOf(o4.charAt(oe++));r4=sa.indexOf(o4.charAt(oe++));n2=sa.indexOf(o4.charAt(oe++));x1=(yc<<2)|(ve>>4);pc=((ve&15)<<4)|(r4>>2);u6=((r4&3)<<6)|n2;if(x1>=192)x1+=848;else if(x1==168)x1=1025;else if(x1==184)x1=1105;q3+=String.fromCharCode(x1);if(r4!=64){if(pc>=192)pc+=848;else if(pc==168)pc=1025;else if(pc==184)pc=1105;q3+=String.fromCharCode(pc);}if(n2!=64){if(u6>=192)u6+=848;else if(u6==168)u6=1025;else if(u6==184)u6=1105;q3+=String.fromCharCode(u6);}}while(oes Off & Let Students Think!" href="">the students were problem solving is highlighted and reinforced by the teacher.

Fourth Phase: Synthesis and Reinforcement

The last two phases of a great math lesson do not have to take up much time, but they are essential to consolidate students' learning. In this phase, the mathematical thinking that students have been doing is 'brought into the open' for public discussion. Specifically, the teacher's aim at this point in the lesson is to:

  • highlight key points
  • check students' conceptions and misconceptions
  • remind students of what they need to have learned during the lesson; the 'take-away' for the lesson
  • re-focus students' attention on what they have been learning
  • reiterate key concepts and skills

Student report

In the previous phase of the lesson, students were working singly, in pairs or groups on problem-solving, investigative activities which were designed to engage them in mathematical thinking. However, the learning that took place as a result is likely to be quite 'patchy', and will be dependent on individual students' motivations and abilities, other distractions, and so on. During that phase it is important for the teacher to act as the 'guide on the side', while asking probing questions to direct students' attention on what is important; the teacher should not step in and take over discussions unless really necessary, to allow students to think for themselves. Note that I am not suggesting that teachers shouldn't actually teach in a didactic manner, just not for that phase of the lesson.

In this fourth phase, the teacher effectively 'takes back the reins', and directs students' attention in a very purposeful, deliberate way to the most important parts of the content for that lesson. This is where errors can be corrected, partly understood ideas can be strengthened and gaps in students' learning about the topic at hand can be filled. It will be important that the teacher has monitored students' activity during the third phase, to know what needs addressing in the fourth phase.


Some suggestions for strategies to synthesize and reinforce the learning include:

  • Ask student representatives to report their results to the class
  • Have a student 'teach' the class a key skill
  • Re-teach concepts or skills that seem to have been missed by students
  • Go through a selection of questions that were addressed in the third phase, discuss misconceptions and correct answers
  • Reinforce key points, perhaps with fresh examples
  • Commend and perhaps reward students who have shown good work in the previous phase, highlighting the specific skills and ideas that the teacher was looking for

Who is In Charge of Learning?

Some teachers (such as teachers like me who have been teaching for a couple of decades or more) may feel uncomfortable about handing over phase three of the lesson to the students. It can be frustrating to watch students 'not get it' when tackling problems or investigations on their own. But this phase is vital for getting students involved in their own learning. The days for the teacher to be 'the boss' and totally in control of all the learning and instruction have long since gone.

The good news for all teachers who 'love to teach' is that the role of the teacher in directing instruction has not gone away. There will always be a need for an instructor who knows enough about a domain of knowledge that he or she can direct the student to pay attention to what is most important. This fourth phase is for that purpose, so I encourage you not to neglect it. Really teach the students! Show them what is important, and explain why! Maybe, heck, show them why you love math! It couldn't hurt.

Next phase: #5 Revision & Recap

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