1. Do Now - every day, class starts with a Do Now, so students start doing math the moment they come in the door (best case scenario, in reality they still need some prodding to focus on the task). The Do Now task always recaps something the students learned the previous lesson, and serves as both retrieval practice, feedback for students on their level of understanding and skill and opportunity to correct misunderstandings.
What I like the most about the Do Now task is that it can often bridge the previous lesson to the current one.
For example, a recent Do Now task asked
"What are the x-intercepts of f(x)=x^2+3x+2? Discuss in pairs."The previous lessons, students had solved quadratic equations. This current lesson, the objective was to graph quadratics from standard form. The Do Now bridged the lessons by giving students the opportunity to help each other apply their understanding of equation-solving to graphing.
2. Traffic-lights - these are my cheap-o "clickers". They are just laminated cards red on one side and green on the other. I tried them a month ago and loved them immediately, as do my students. Every lesson, during the Do Now, I hand out these cards. Students use them throughout the lesson to signal their understanding or need for further clarifications. When I explain something, I typically ask students a specific question such as "Are you able to explain every step of this solution to a classmate who is absent today?". If a student holds up red, I ask what step(s) need further clarification, and then ask a student who holds up green to explain those steps. I find that when students are made to take a stand like this, students who would otherwise pretend to understand do speak up. Also, during individual work, students turn their cards red side up if they want my attention.
3. Weekly homework-checks - this could be just looking through students notes during the Do Now, or even a mini-quiz integrated in the Do Now, or a much larger hand-in assignment. In whatever way I do it, I need to check student homework not because I'm so very interested in whether they did it, but because students are much more likely to do it if I check it. In my psychology teaching, every time I've done weekly homework checks, student test results have risen dramatically. Jury is still out on whether the same improvement is seen in mathematics, but I'm hopeful. I'm still working on a system to check homework comprehensively yet quickly.
4. Investigations, where appropriate - I've tended to overdo or underdo investigations in class. Now, I ask myself before each lesson whether the goals for that lesson are reachable by investigation within the time constraints of that lesson. Next lesson, the goal is for students to understand and apply function transformations. This is superbly appropriate for an investigation task, and most of the lesson will be devoted to it. Investigations are great, because of too many reasons to list here. However, investigations are time-consuming and don't always lead to the intended understanding and skill, so they need to be balanced with more structured teacher-led demonstrations.
5. Direct instruction, where appropriate - some goals, such as applying trigonometric relations or log rules, are more skill- than understanding-based and require lots of practice. In such cases, I prefer a structured teacher-led approach in which I or we together solve an exercise, then students practice individually or in small groups, and then the process repeats. This can also work well with less teacher scaffolding and more group-work, such as examining and evaluating solutions of problems.
6. Problem solving, where appropriate - sometimes, when solutions require many different steps, such as when students graph functions from first and second derivatives, or use derivatives in optimization problems, I prefer to put students in groups and have them figure out the solution to a problem themselves, before writing a structured summary on the board. I find that this type of scaffolding, students helping each other, helps students feel confidence and ownership of the solution method. If instead I was to simply present such a complex multi-step solution, students would be more likely to feel intimidated and to try to memorize all the steps instead of trust that they can construct a solution based on their understanding of relevant concepts.
I'd like to integrate WCYDWT style problems, but so far I have never felt that I have the time to do so.
7. Closing summary - OK, I'm not actually doing this one every time or even most of the time, but I'd really like to! When I do have the presence of mind to recognize that 5 minutes remain, I like to ask students to summarize either orally or in writing what they have learned during today's class. Sometimes, rarely, I use exit slips as well.
So - this is my math teaching in a nutshell. Lots of room for improvement still (and I hope I'll always feel that way!), but at least and at last there is some stability with strategies that promote quick and efficient feedback, confidence, understanding, mastery and fun.
As always, I welcome all feedback.