Archive for the ‘Classroom Environment’ Category

Parent: I don’t understand why you won’t answer my son’s questions in class.

Teacher: This is the new common core teaching method. Students need to figure things out on their own.

Parent: I am unclear on the way you want my son to complete this homework assignment. Why can’t he just follow the traditional algorithm?

Teacher: We are using the new common core curriculum. This is how common core wants students to do that particular computation.

Parent/Teacher/Student: (with much frustration) &%!@# common core *&!!@($^ math *@##!$%^% stupid…


Parents, if you have had this kind of a conversation with a math teacher recently, I encourage you to challenge that teacher to explain the difference between common core STANDARDS, common core CURRICULUM, and common core TEACHING METHODS. 

The interesting thing is that one of these things is not like the others…STANDARDS! The common core STANDARDS are not like the others because the others don’t even exist!

The common core standards for mathematics include content standards for grades K-12…that is, what content should be taught. The content standards are written in such a way that mathematical ideas and understanding can be developed in a coherent, structured way through what they call “progressions.” For example, the ways in which students are to think about, make sense of, and ultimately compute multiplication problems are developed so that the way students are thinking in 3rd grade are extended to 4th grade, etc.

What the common core standards DO NOT do is tell teachers HOW to teach in order to provide the opportunity for students to master the standards. Nor do the common core standards provide curricular material to support the learning of the content standards.

In addition to the content standards, the common core includes Standards for Mathematical Practices.

  1. Make sense of problems and persevere in solving them.
  2. Reason abstractly and quantitatively.
  3. Construct viable arguments and critique the reasoning of others.
  4. Model with mathematics.
  5. Use appropriate tools strategically.
  6. Attend to precision.
  7. Look for and make use of structure.
  8. Look for and express regularity in repeated reasoning.

Will teachers need to find/create/adapt high quality curricular material to meet these standards? Will teachers have to think about their teaching practice and possibly adjust so that these practices can be developed? YES! 

The question is…so what DOES good teaching and learning look like (whether held to the common core standards or not)? 

I will try to capture the essence of this in the following vignette:

You open the door to the classroom and your initial reaction is shock.  It seems that chaos reigns as you notice students out of their seats as they talk with one another.  Furthermore, the desks are not neatly arranged in rows, but seem to be randomly clustered into small groups as the students use the desk tops for what appears to be some sort of technical equipment set-up.  Where is the teacher?  You are not sure.  Concerned with the outward appearance of this classroom, you enter the classroom to investigate.  You stop at the first group of students and try to figure out what is going on.  Quickly you realize that the topic of their conversations is highly technical.  They are discussing the result of an experiment where two tuning forks of different frequency are struck and the combined frequency is measured.  Would the frequency be the sum of the frequencies of the two tuning forks?  The difference?  Or some combination that will be difficult to determine?  They follow through with the experiment using a data collection device and a graphing calculator and the students begin to analyze the results.  They are discussing ideas and concepts foreign to you as they search for the beat pattern, calculate the period and frequency, determine the equation of the sinusoid which fits over the beat pattern, and ultimately discover the relationship between the frequency of the individual tuning forks and the combined frequency.  You ask one of the students, “why are you doing this?”  Annoyed by the interruption, she responds that they are trying to solve the problem at hand where, in the situation given, an old World War II mine needs to be deactivated.  This can only be accomplished by directing the proper sound frequency at the timing mechanism in the bomb in order to disable it.  You find it refreshing to see students motivated to solve a problem and notice that students are simultaneously using technology, paper and pencil calculations, and teamwork to find the solution to a problem that has obviously engaged the interest of the students.

You move on to the next group of students.  They are equally engaged in what appears to be a different problem.  You first notice that the students are huddled around a graphing calculator screen.  Suddenly, a shout of celebration is heard as students triumphantly give each other “high-fives”.  You get the attention of one of the students and ask what they are doing.  He explains that their team was given data related to the height of waves in the ocean during a particularly dangerous storm.  Their job was to create a scatter plot of the data and determine a sinusoidal regression model for this data so that they could calculate the period, frequency, and height of the waves at the moment in time a small touring vessel was capsized.  Given the manufacturers specifications of this vessel, the students would be able to determine if it was built to withstand such a storm or not.  This would aid in the investigation and eventual lawsuit brought by passengers of the vessel who believed that the vessel’s skipper used poor judgment in venturing out into the storm. 

Impressed by the level of engagement of the students, the high-level discussions that were taking place between students, and the interesting combination of technology and paper-and-pencil calculations that were taking place, you still have one more concern: where is the teacher?

You move on to the next group and find that they are working with tuning forks also.  Peering in to see what this group is discussing, you find that one of the members of the group seems to be asking questions.  “What would you know about the frequency if the period is 0.123 seconds?”  “Do you see a pattern when you consider the different combinations of two tuning forks and their resulting beat pattern?”  “Can you explain the difference between period and frequency?”  You realize that this is the instructor asking probing questions to the students to help them construct their own understanding of the mathematical concepts being explored.

Standing back to observe, as a whole, the classroom setting that initially brought disgust (nearly), you now feel joy (nearly) as you realize the high engagement of students as they study difficult math (and physics) concepts.  But the students seem to be enjoying it.  Could this really be a math classroom?  What curriculum development and implementation steps were employed by the instructor in this classroom to get to this point? 

The National Council of Teachers of Mathematics recently released a book (Principles to Actions: Ensuring Mathematical Success for All) describing the Mathematics Teaching Practices that “provide a framework for strengthening the teaching and learning of mathematics:

  1. Establish mathematics goals to focus learning.
  2. Implement tasks that promote reasoning and problem solving.
  3. Use and connect mathematical representations.
  4. Facilitate meaningful mathematical discourse.
  5. Pose purposeful questions.
  6. Build procedural fluency from conceptual understanding.
  7. Support productive struggle in learning mathematics.
  8. Elicit and use evidence of student learning.

Much support and patience is required as mathematics education is improved in the USA. Let’s all commit to making sense of the problem of mathematics education and then persevere in solving it!



The development of an effective class culture begins on the first day of class but may take several class periods to develop. Students are typically uncomfortable working with people they don’t know. Team building activities are excellent ways to break down the barriers and begin to build relationships. During the first several class periods, students should work with as many different classmates in teams as possible. This can be achieved by assigning students to teams by their favorite foods, birthdates, number of siblings, shoe size, and so on. Once the day’s teams are assigned, conduct a 10-minute class activity with the intent to get students talking to one another in a discussion that may not be related to mathematics. By allowing students to interact on a level playing field in terms of knowledge and experience, students learn to share ideas, debate, challenge, and argue rather than worry about what they will look like to their classmates in terms of their mathematical “know-how”.

The goal of these team building activities is to build a culture that will transfer to mathematical learning. Time spent developing an effective classroom culture through team building pays dividends throughout the course. The deep learning that occurs when students are engaged in an empowered learning environment prevents having to re-teach material that would not have been learned in a less engaging, instructor-centered environment. An added benefit of the team building activities is that students learn how to work effectively in a team, negotiate conflict, and communicate clearly.

The physical setup of the classroom helps to define the classroom culture. Ideally, furniture should be arranged to allow for student-to-student interaction. The following diagram shows one classroom layout that facilitates student interaction and group work.This setup also facilitates the student use of hand-held whiteboards. These whiteboards make it easy to display student work for presentation and discussion. Although small whiteboards may be purchased commercially, they can also be hand-crafted from large sheets of the whiteboard material purchase at a local hardware store. A sketch of sample whiteboard is shown in the following figure.

Mathematics classrooms where students sit passively, working in isolation to copy notes, should be a thing of the past. Instead, the classroom should be alive with collaborative problem solving activity. Current professional standards (AMATYC and MAA) suggest that an active classroom learning environment should be the norm. Too often instructors and students spend much of their time focusing on basic computational skills rather than engaging in mathematically rich problem-solving experiences. Worthwhile mathematical thinking occurs when students struggle and learn to overcome initial frustrations to make sense of powerful ideas while solving realistic problems. Developing this type of thinking requires a classroom environment where students are comfortable sharing their correct or incorrect ideas in the public forum of a classroom setting. However, such an environment does not occur naturally: it must be developed by the instructor. The ideal classroom learning environment is characterized by the following attributes.

  • is positive and cooperative
  • piques student interest and provides the opportunity to enjoy learning
  • creates a strong personal sense of motivation and responsibility
  • encourages relevant questions and discussions
  • helps students learn that making mistakes is a natural and acceptable part of the learning process
  • develops student confidence in making sense of mathematics
  • produces an understanding that correctness is determined by the logic of mathematics, not by the instructor,  a “smart” student, or the back-of-the-book

In my not too distant teaching past, I became frustrated with the lack of student involvement and their willingness to “take chances” by providing answers to problems or to even speak up at all. I would develop what I thought was an outstanding lesson or activity that required students to think deeply and tackle some of the big ideas of mathematics only to be disappointed by how little they would willingly participate.

I lamented the STUDENT’S’ lack of effort to engage and even came to resent THEM for this! “How dare they not appreciate how hard I worked to make the lesson exciting and well-planned? What was THEIR problem?” I became so upset that I spoke to a colleague (co-author Scott Adamson) about the challenges I was facing and we came to a startling conclusion….maybe it wasn’t the STUDENTS’ fault….maybe it was the INSTRUCTORS’ fault and that just so happened to be ME! I would need to change (scary thought) and “force” them to behave how I wanted them to in class. It was eye-opening but I learned that they actually had to be taught how to act in my classroom.

Within just a few minutes I came to realize that if I wanted students to feel comfortable interacting with others they most likely did not know I had to teach them how and create a comfortable environment for them to branch out in.  I learned that even more frightening for students than just talking to others they don’t know is the notion of asking them to actually attempt to solve a mathematics problem in the presence of others who might think they are “dumb” if they are wrong.

In my next posts, I will begin to share some of the techniques I have used over the years that have made a huge difference in getting students to collaborate, hypothesize, challenge, and debate their own and others’ work in a constructive and exciting way!