In order to be an effective math teacher there are four areas that need to be taught in all areas of the math curriculum: computation, procedural skills, conceptual understanding, and problem solving (Geary, Siegler & Stern; Sophian). All four are necessary and are interrelated. Teaching how to add a fraction (computation) without understanding what is happening and why will lead to short term learning. Whereas, if they understand exactly what is happening (conceptual understanding) behind the straight addition the learning curve goes up, and they retain the knowledge over a much longer period. Likewise, what good is learning to add a fraction if you do not know when or how to use it (problem solving). That’s the whole reason we learn math is to be able to solve our own problems throughout our lives, isn’t it. We teach procedural skills whenever there is more than one step to solving a problem. What do we do first, second, and so forth. Like the clown who is juggling, effective teachers are able to juggle and keep going all four of these areas in the mathematical curriculum. What should this mean for you as the math teacher? As you plan each math lesson make sure you are covering each of these areas and that will put you right at the top as an effective math teacher.
Do you remember how math used to be taught? The teacher would stand at the blackboard and demonstrate how to do a problem, then write the page number, turn to the students and tell them to get their books out and do problems 1 through 30, and the lesson was over. I guess you could call this the 2 phase method of teaching math. Effective math research has identified a much better method of teaching called the 3 phase approach (Dixon & Belmont). Phase 1 is to introduce, demonstrate and explain the concept. Phase 2 is where the students try the new concept under the guidance and help of the teacher. Today we call that “guided practice”. Phase 3 students work independently either in class or at home doing homework related to the lesson. Adding that middle phase of the teacher working and guiding the students has made a huge difference in overall understanding of the lesson. Make sure you always put it in your lessons.
As I would read about different types of mathematical research, I would ponder as to what each piece of research would look like in my own math lessons. Systematically I began to take each piece of research that I would find and build them into my own lessons. I developed a huge collection of powerpoints that used at least 11 different pieces of research. In the next few posts I will share the research and how I incorporated it into my lessons.
Today I will focus on informing the students ahead of what they are going to learn. Siegler & Crowley in their study came up with this conclusion: Knowing the goal of problem solving appears to facilitate the development of procedural and conceptual problem-solving competencies. There are many other studies out there that support this same idea. So it is up to the teacher to tell the students what they are going to learn. This can be done with a Title Page and or subtitle pages, an objective, or an essential question. Above are some examples of what I use in my powerpoints to make sure the students are fully aware of what they are about to learn.
A huge push in the latest effective teaching research is student engagement. One way to keep students engaged is through hand signals. Madeline Noonan, a fifth grade teacher, demonstrates several hand signals used throughout her entire school (https://www.teachingchannel.org/videos/classroom-silent-communication-signals). When I was a math coach for Math Matters, they also had a set of signals: don’t understand (moving hand over head), agree (thumbs up), disagree (moving hands across each other back and forth), can’t hear (cupping the ear), etc. When I taught I also added some of my own: sharpen pencil (hold pencil in the air), bathroom (sign language letter B) etc. When we were doing math I might say, “Show me on your fingers the answer.” If we were doing test prep and the answers were A, B, C etc., then I would have them show me the sign language letter for their answer. The great thing about hand signals is the students have quick access to you and you can read what is going on in the classroom, and it all happens silently. However, the most important thing is they can participate and be engaged in the lesson.
Over the years as I tried different teaching methods and strategies, I finally found one where I could quickly check my whole class to see if they understood the concept I just taught. This was with whiteboards and dry erase markers. I had my students divided into groups of four (cooperative learning). After demonstrating, let say, a math problem with my powerpoint lesson, I would then give the students a problem to do on their whiteboards (guided practice). The groups competed and the first team done got an extra point. Each team would get a point when everyone in their group was done and they had 100%. They were to work their problem separately for the first part, and then I told them to check with their other team members to see if they had the same answer. If they didn’t, I had the stronger students help those who were having difficulty redo the problem. They just couldn’t copy the answer. They had to show their work on their whiteboard for the complete problem. I would award top teams an ice cream on Fridays. Was there a little cheating going on in class? Maybe. But I kept a very close eye on the teams and felt there was very little. It was a great self-esteem builder for the smarter kids as well. Plus, one of the best learning methods is to be able to verbalize what you have just learned, so the smart students got smarter. Think also of the many hours I saved of individually helping students who were having trouble. Using individual whiteboards is a win-win situation for everyone.
Are You an Effective Math Teacher?
Here are some identified practices effective teachers use in the classroom when teaching math. Look at the picture and ask yourself how many of these things do you do in a lesson? On the comments below please add what you do as a teacher to make your math lessons more effective?