The definition of what effective mathematics teaching looks like very much depends on what purpose we assign to teaching mathematics. A classroom where the primary objective is to teach students a specific set of mathematical skills for them to use later will look much different than a classroom where the primary objective is to teach students how to think mathematically, although there is obviously overlap between those two classrooms. For a good description of the type of classroom which achieves the first goal but fails at the second goal, see When Good Teaching Leads to Bad Results by Alan Schoenfeld.
I will describe a classroom where the primary purpose of the classroom is to encourage mathematical reasoning, with a secondary benefit of students practicing mathematics skills they have developed.
What are the students doing?

Students are engaged in the standards for math practice.
The Common Core Standards for Mathematical Practice, which are similar in many ways to the NCTM Process Standards, are a useful tool for understanding the types of activities students should be engaged in within a mathematics classroom.
In order to really do mathematics, students need opportunities to problem solving, to use mathematics they know to model processes, and to do all of this in the sociacultural contexts of their classrooms. Both of these sets of process standards do an excellent job of defining what it means to do mathematics, but are flexible enough to allow for a variety of different activities to qualify.

Spending significant time solving rich mathematics problems.
Routine problems with limited opportunity for investigation might be acceptable for students to use to practice skills they have learned, but they do not have the breadth necessary to allow students to do the inquiry necessary to learn mathematical reasoning. A significant amount of time in the mathematics classroom investigating, postulating, formulating, deciding, and analyzing mathematical situations is necessary if the habits of mind required for mathematical reasoning are ever going to be adopted by students.
For excellent examples of rich mathematical tasks (some of which are used for assessment of understanding, and others are used more to prompt student thinking) see some of the web sites linked here.

Students talk to each other about math.
While there is definitely value in students spending at least some of their time thinking independently, there is tremendous value in students having opportunities to discuss mathematical ideas and problems with each other. The first is that it is through the repeated access to different linguistic and representative variations on an idea that we come to more than a superficial understanding of that idea. If I say words, you hear the words, and you might even think you can assign some meaning to those words, but it is only when you hear other variations on the formulation of the ideas behind the words, and see other representations (often physical or pictoral) of the ideas represented by those words, that you can come to a full understanding of the concept. For more information on the dangers inherent in a “linguisticonly” understanding of concepts, see Richard Feynman on Education in Brazil.
There are other benefits of students talking to each about mathematics. One benefit is the person who describes their solution is either likely to see flaws in their reasoning (or at least receive feedback on those flaws) or in the articulation process of their reasoning, come to a better understanding of the concept. Another benefit is that instead of just one person in the room able to give feedback to students, every student in the room becomes a resource for each other. Finally, someone who has just learned a concept, and more importantly recently moved past their own flawed models of that concept, is often more able to explain the concept as compared to someone who learned the concept long ago, and no longer remembers their struggles with it.

Students have the opportunity to revisit and reflect on mathematics they have learned.
Human memory is limited. Essentially, our mind trims information from it that is not used frequently (or possibly archives it so that it is difficult to access).
In a highly effective mathematics classroom, concepts students have learned are revisited, often as embedded practice within the current unit of study. For example, when students create graphs of linear functions by plotting points, they are also practicing plotting points. The skill they practice is used within the context of current problem solving. Note that this practice is only really effective once students have mastered the concept as practice without understanding leads to student confusion.
What is the teacher doing?

The teacher uses formative assessment practices on a daily basis within their classroom.
Aside from engendering the opportunities for “what students do” as described above, an effective teacher gathers evidence of their student’s learning in a systematic way. Formative assessment is a process through which a teacher assesses their students, and then uses this information to inform their teaching. It acts as a feedback loop within the cycle of teaching and learning. If we consider what is to be learned as being like a vast wilderness, then the curriculum the teacher follows is a map through that wilderness, and formative assessment is the process they use when checking their compass so as not to get lost.
There are three basic indicators teachers can use to collect formative assessment information; what their students write, what their students say, and the body language students use that indicates how they feel. All of these are important markers for teachers.
Unpacking formative assessment is not the goal of this blog post. For more information, I recommend reading Dylan Wiliam’s Embedded Formative assessment as a good starting place.

Teachers must build a classroom environment where students want to talk about mathematics and have a growth mindset.
Developing a positive and productive classroom culture is a critical component of effective teaching. Students must feel that their contributions to the classroom matter, and that they feel safe to make mistakes. Making mistakes, and learning from those mistakes is an important part of learning. The goal of mathematics classrooms should not be to prevent students from making mistakes, but to treat mistakes as opportunities for everyone to learn and to grow.

An effective teacher uses questioning technique carefully and thoughtfully.
Teachers ask a lot of questions. For example, in one classroom observation I did this year, a teacher asked 170 questions in a 40 minute period, which averages out to about 1 question every 14 seconds. Given that many teachers ask a great number of questions each class, improvements in questioning technique are therefore likely to improve overall teacher effectiveness, perhaps even dramatically.
Good questions prompt students to think. Teachers with effective questioning technique do two things well; they have a set of generic questions prepared they can ask students to prompt their thinking which they use frequently enough that students begin to ask these questions of themselves before even talking to the teacher, and they actively listen to, and clarify their understanding of, student reasoning before responding.

An effective teacher learns about the linguistic and cultural backgrounds of their students and adjusts accordingly^{1}.
Mathematics is a cultural activity. Therefore, as mathematics teachers, we are not only teachers of mathematics, but also teachers of the sociocultural norms of mathematics. In order to do this effectively, we need to understand our students at a more than superficial level. A cautionary note here: This is an area where it is easy to fall prey to cognitive bias and judgemental attitudes. Teachers need to make their best effort to objectively understand their students’ cultures and their linguistic understandings and then make sense of how their students’ backgrounds impact what is effective for their students.

An effective teacher uses technology to focus students on mathematical reasoning.
Classroom technology, in an effective mathematics classroom, is used to support student’s mathematical reasoning. Rote practice exercises, even if administered via technology, do little to help students develop their reasoning skills, and because they lack context, have limited ability to help students develop connections between different areas of mathematics.
Imagine a classroom where students are looking for connections between different forms of a quadratic function. They could plot these functions using pencil and paper, and then look for connections, but during the time students would take to draw the functions, they would lose track of the goal of the graphing. Every time we ask students to do another task in preparation for mathematical study, they lose active cognitive resources to keep track of the overall purpose of the task. Instead, in an effective classroom, the teacher would give students access to a graphing calculator or graphing software, and students would be able to focus on seeing connections between graphs, instead of creating the graphs.
What else would you add to this description of an effective mathematics classroom?
Reference:
1. Suggestion offered by Ilana Horn. See this tweet.
Kelly says:
David,
Thank you for your article. I think you make several very good points, teacher uses formative assessment (formal and informal), students discuss the mathematics, mistakes are viewed as opportunity to iterate, technology is used effectively, etc. I would add to the list of teacher actions that the teacher is transparent. What I mean by this is that the teacher must be a voice in the conversation about a challenging math task. The teacher needs to verbalize and model his/her thinking. I believe most importantly, the teacher must be brave enough to take on challenging math task when they are unsure of the necessary process or outcome. Make it acceptable to say, “I’m not sure how to do this. Perhaps we can try ____________ strategy and see what we can learn from it.” I firmly believe that this is the road to student actions 13 in your list above.
In my classroom experiences these were the most powerful ways to get students interested in math and to intiate really great conversations about the math. Dan Meyers speaks of the math serving the conversation. I beleive being transparent is the first step to getting the ball rolling in that direction.
April 2, 2014 — 12:18 pm
Mike Lawler says:
Saw your request for your favorite blog post, not so easy to choose one, but I did enjoy thinking through this one:
http://mikesmathpage.wordpress.com/2014/02/18/stevenstrogatzandmathwriting/
April 2, 2014 — 12:44 pm