Tuesday, May 12, 2015

Problematic Behaviors, Classroom Management, and Teaching Math

When math is taught as a social activity, students are communicating their thinking, learning from others, and gaining confidence in their ability to solve problems.  In order to teach math as a social activity, we need to cultivate a learning environment that involves a safe setting for social interaction. Students need to know how to communicate at a low volume so others are not distracted, how to listen to other people's ideas, and how to respond in a respectful manner.  We are actually encouraging our students to talk during class (!), but in a manner that is conducive to learning and problem solving.

Recently I had the pleasure of sitting in on an hour-long professional development session presented by Paul White at Morse Street School.  Mr. White is a Kindergarten teacher and Responsive Classroom trainer, and he gave us a nice overview of Responsive Classroom's four major domains and seven guiding principles.

I had been thinking about the impact classroom management has on math instruction and found the Responsive Classroom (RC) model helped to illustrate the connections to the mathematics learning experience really nicely.

(You can learn more about Responsive Classroom here:

The first of four domains of Responsive Classroom (Engaging Academics) illustrates the cycle of engagement --> positive behaviors --> engagement --> positive behaviors that you may be familiar with.  Excluding emotional outbursts (which can be due to any of a variety of factors), we know that when children are not engaged they are prone to boredom and that can lead to distracting behaviors. Immediately we are faced with a real challenge, because traditionally, math is not the subject every student immediately falls in love with.  Engaging students in math takes practice and careful planning.

Maintaining a positive community is another domain that is directly related to teaching math.  Since anxieties around math begin at an early age, it is crucial that we create a safe climate in our classrooms where students feel comfortable making mistakes in front of their peers.  That means setting guidelines for discourse and practicing those guidelines.

Awareness of developmentally appropriate language and academics is also important in math, and is one of RC's domains.  This is precisely what makes teaching elementary math as complex as teaching high school math, and it is the very part of teaching math that parents often have difficulty understanding.  Perhaps you have heard: "Well, I learned that when you divide fractions, you flip and multiply.  Why are they teaching with all these weird pictures and models?"  Or, "I just don't understand this partial products stuff.  It is so confusing!"  If you are a parent, perhaps you have even felt these sentiments.  We don't remember a lot about our early mathematics experiences, but even back when we were kids, chances are we used manipulatives and models before we learned algorithms.  Those things helped us to understand the quantities we were working with before we simplified them into written equations and solved them with operations and algorithms.  A great deal of research goes into determining what most children at varying grade levels are capable of and ripe for in terms of discovering and exploring mathematical ideas.  As adults, we may remember learning the algorithms, but much of the meat of our mathematics learning happened before that.  And for some of us that did not have as much modeling and practice prior to being taught algorithms, math can be more challenging, even as adults.  To come full circle with the RC domain dedicated to developmental awareness and its connection to classroom management, consider that for those students who missed out on important early experiences with physical models, games involving quantities, pieces, parts, geometric shapes, and opportunities to discover, math is harder and less engaging.  Thus, what we are trying to do with the emphasis on understanding, and experimenting with all these different algorithms, and encouraging our students to share their thinking and solve problems in more ways than one, all that effort is intended to make math a more meaningful and worthwhile experience for children.  When the content is meaningful to the students, it becomes easier to teach and classroom management becomes less challenging.

The four RC domains are engaging academics, positive community, effective management, and developmental awareness.

In addition to the domains, RC also has seven guiding principles, and they are also completely relevant to math instruction at all levels.  Let's look:

1. The social and emotional curriculum is as important as the academic curriculum.

Above, I mentioned the importance of feeling safe in the classroom.  Students must feel comfortable making mistakes, and learn how to critique others without ridiculing their effort.  This is addressed in another post devoted to facilitating math discourse, but let's just summarize by saying that when a student does not feel safe, learning can be significantly stifled.  Speaking personally, if I am in a room of people and I feel like I am the only one who does not understand what is being discussed, I am likely to shut down and not participate in the discussion.  I don't want to burden my peers with my inadequate knowledge!  It is very difficult for children, too, when they are put in that position.  Keeping whole-class classic Q+A style discussions to a minimum and allowing students to work out their confusions in small groups is less stressful for them.  Saying "I don't get this" to one or two peers is much less daunting than admitting it to the whole class.

2. How children learn is as important as what they learn

When Mr. White shared this guiding principle, I simply wrote in my notes, "SMPs!"  This is precisely why the Standards for Mathematical Practice were written into the Common Core.  The SMPs were designed as a guideline for us to give students a wide variety of mathematical learning experiences.  Post them in your room somewhere if it helps, because if you can manage to incorporate the SMPs into your instruction, it is one way of differentiating!  And one of my very favorite things that Everyday Math has done in their newest edition is divide the Standards for Mathematical Practice into more specific Goals for Mathematical Practice, essentially showing you where these different learning experiences lie within and throughout the curriculum.   So yes! How children learn math is as important as what they learn in math.  And how does this impact classroom management? By giving your students all these different types of learning opportunities, you are helping to address their learning needs, preferences and styles, and also addressing their weaknesses.  This means not only will students be performing more varied tasks in class and solving more interesting problems, but they will also be developing greater understanding which yields greater engagement.

3. Cognitive growth occurs through social interaction

This guiding principle is heavily discussed in the post devoted to Mathematical Discourse, but I'd just like to throw out there that nowhere is this statement more true than in mathematics.  Much of mathematics is a language, and languages are meant to be communicated.  Students learn and grow by sharing their thinking with one another and having their conjectures confirmed or critiqued.  The traditional 20-problem independent homework assignment is standard for engraving procedural sequences in one's brain, but there is not much use for it beyond that.  It takes too long to go over 20 problems in class; students who have mastered the procedures are bored during that time; students who did not perform the procedures correctly must now revise the 20 questions (and re-wire their brains from the incorrect procedure sequence they practiced the night before).   Today we see more deep problem solving being sent home, and fewer sets of problems to practice procedures with.  With one or two problems to try at home, the work can be revisited in much shorter time in class, and mistakes are learned from without becoming so consequential.  Moreover, time spent in class can be devoted to sharing strategies, exploring new concepts, and analyzing each others ideas and work.  Cognitive growth in math definitely occurs through social interaction, as math is a social activity.

4. To be successful academically and socially, children need to learn a set of social and emotional skills that include cooperation, assertiveness, responsibility, empathy and self-control  

In order for children to work on tasks together, whether it is in pairs or small groups, they need guidance, instruction, encouragement and practice with the communication skills required for cooperative problem solving.  For example, listening is a skill some need more practice at than others, but it is essential for working together.  Knowing how to listen to another's ideas, without interrupting, and how to ask questions about those ideas, re-state those ideas, and/or clarify those ideas is necessary before one person can endorse or criticize the ideas of another.  Many adults have problems with this!  In many ways, children are at an advantage.  There is less history, less experience, a.k.a. "baggage," and that allows children to learn how to share different ideas and strategies with a common goal.  In other words, children and adults often have a hard time sharing opposing ideas without getting impatient and competitive.  But adults have a lifetime of being impatient, under-appreciated, self-conscious, shy, dominant, overconfident or overly anxious that can make this kind of collaboration difficult.  For children, there is more room for learning how to collaborate effectively, even debate, with the understanding that everybody wins when a solution is reached.

Communication skills for collaborative problem solving should be introduced at the start of every school year, practiced, and re-taught at various times throughout the year.

5. Knowing the children we teach-- individually, culturally and developmentally-- is as important as knowing the content we teach.

This comes from formative assessment, which includes the observations you make as a teacher, notes you take about every student, direct communication with students, and data from multiple summative and cumulative assessments.   It also comes from communication with parents and other teachers, but that is covered in the next guiding principle.  Knowing the children we teach through the various ways of formatively assessing informs our instruction.  And in math, this means knowing more than who is strong and who is weak, who has test anxiety and who has trouble writing.  Math includes a whole spectrum of different types of problem solving.  You might learn that one of your students has a special love of algebraic patterns, while another has specific difficulty with angles and geometric properties.  And we all are familiar with the dreaded "does not work well with others" label... You may discover that a student who previously owned that label actually works very well with others in certain types of problem solving scenarios.

6. Knowing the families of the children we teach is as important as knowing the children we teach.

Communicating regularly with parents about math is more important now than ever, mostly because from their perspective, much of the math we are teaching their children just seems weird.  There is greater emphasis on understanding concepts than ever before, and that often looks funny on paper.  Whether it is a homework assignment or a unit assessment, parents have a lot of questions about what their children are doing in math class.  Combine that with all the information parents can share with you about their son or daughter, and y'all have an awful lot to talk about.

Another note about communicating with parents.. Year after year I encounter adults who insist that math was just never their cup of tea, and they expect their children to experience the same difficulties in or apathies toward math in school.  This can be a real road block to learning, especially as the child gets older and nurtures the image of him or herself as mathematically hopeless.  One cannot blame them for having this attitude; it is entirely likely these parents were not given enough opportunities to explore, play with, discover, and eventually appreciate math as children.  But it is important for us to to what we can to share with them that their child is a young mathematician, and that we are doing everything we can to help them understand, enjoy and excel at mathematics.

7. How we, the adults at school, work together is as important as our individual competence.  Lasting change begins with the adult community.

Just as math can (and should, in my humble opinion!) be considered a social activity, teaching math is in the same category.  I am so lucky to be able to see so many great teaching styles and techniques as I make my way from school to school, but as far as exposing one teacher's great work to another, the best I can do is tell you about it or write to you about it.  Talking to each other about teaching math and visiting each other's classrooms are the best ways to collaborate and learn from each other. Hopefully we can make more time to do just that in the future.  The teaching that I witness is too good not to share.  As a math strategist, I look forward to teaching with you, learning from you, and passing along the good teaching practices I witness far and wide.

Teacher Talk vs Turn-and-Talk: Some Ideas for Increasing Mathematical Discourse

In a book published by Marilyn Burns' Math Solutions, called Classroom Discussions in Math*, the authors outline four steps toward productive math talk in the classroom:

1.  Helping individual students clarify and share their own thoughts

2.  Helping students orient to the thinking of others

3.  Helping students deepen their own reasoning

4.  Helping students engage with the reasoning of others

Students often need to gather their own thoughts before even beginning to clarify them.  In an earlier post, we explored the importance of providing think time for students before they engage in discourse (either with each other or when sharing out).   Using that think time is an excellent way to help students gather, clarify and share their own thoughts.

In the first couple minutes of the Teacher Channel video below, you can see a nice example of a sequence that encourages productive math talk.  The teacher asks students to observe and think, then turn-and-talk, then share:


The student that is first asked to share what patterns he sees seems relaxed, confident and not intimidated.  He was given an opportunity to look at the math, to think about it, and then to share his ideas with a peer and listen to his peer share his ideas (turn and talk), all before being asked to share anything with the whole group.  Also, students are all seated on the floor, where they have become accustomed to communicating about math as a group.

I have seen some terrific student discourse happening in classrooms I visit.  It is late in the school year and students in many classrooms are developing stellar talents for communicating their mathematical thinking.  Especially in the K-2 grades, where the new Everyday Math curriculum requires more collaborative thinking and sharing of mathematical thinking than ever before, students are getting used to sharing not just their solutions to problems, but also how they solved those problems.  I have even seen students volunteering to restate others' strategies, sharing new strategies, and critiquing each other's strategies.  I notice more and more students leaning over their neighbor's desk, walking him or her through a problem to help find an error.   They are not just learning how to communicate their mathematical thinking, but they are becoming very good at it too.

Once upon a time, in middle school and high school, collaborative learning opportunities used to be a rare thing in the math classroom.  Desks were mostly in rows, and looking at your neighbor's strategies, or sharing your strategies with your neighbor, could have landed you in the principal's office for cheating.   Teaching usually involved a lot of explaining and not much discovery.  It was not all bad, at least for those students who already had discovered an appreciation for math and had confidence.

A more traditional style of teaching math, at least ideally, went something like this:

First:  The teacher talks the class through a new math procedure in front of the class.

Then:  The teacher gives the class an example to work through together, with guidance from the teacher.

Last:  The students work on a series of problems, similar to the above work, independently.

This approach is commonly referred to as the "I do, we do, you do" approach, and it is a logical way to teach someone how to complete a procedural task.  In high school, I was taught how to graph a linear equation that way.

Even as we experiment with different classroom techniques, employing a workshop model in the classroom, facilitating group and partner work, using centers, and incorporating movement into our lessons, the tendency to want to simply demonstrate how something is done remains dominant in most of us.  "My students are clearly confused about area and perimeter.  I know the distinctions between area and perimeter," we think to ourselves, "so I will just show them so they know too," and we launch into a mini-lecture on the topic, complete with nice diagrams and a laser pointer.

There are two main dangers that linger when this happens.

The first danger is that in order for even a very brief mini-lecture to be effective, you need complete student engagement.  As soon as you have to re-direct just one student to face the front or keep his or her hands to his or her self, there is a disruption to your presentation of the topic.  If it happens twice, it becomes a burden for those that are listening to remain focused.  And this is assuming you began with 100% attentiveness.  In the event your students are not all that interested in what you have to say from the start, your mini-lecture is doomed... The more behavioral disruptions the more you lose student engagement, the longer the mini-lecture becomes and the period of time you expect your students to remain focused on you.

The second danger is just as significant as the first, but more simple:  Your students may not connect with what you are saying.  Any number of them might not understand your way of explaining the concept.  Students learn math in a myriad of ways, and your explanation, algorithm or procedure may be lost on one or more of them.

Since I know (from my own experience) how dominant this tendency to want to explain things to the class is, I am not condemning the practice, but I do offer a suggestion; monitor yourself very carefully when slipping into "explain mode."  Know that if your explanation lasts more more than a minute, you have likely lost at least a couple of your students.  Also, if it happens more than a couple times during a math class,  your students may be missing out on more valuable learning and discovery time.

Here is a neat way to think about the learning experience as it pertains to mathematics;

Think of learning math and problem solving as a social experience.

It is like learning a language.  If you have ever had the experience of trying to learn a foreign language, perhaps you can relate.  You took Spanish I, and got an A+.  Como Estas? Muy Bien, gracias!  Spanish is easy!  You took Spanish II, and it was harder, but you learned a lot!  Spanish III?  Oh my, accents, novellas, dictations, I sure am getting good at Spanish!  Spanish IV made you a master!  I must be fluent now! I feel so confident!  You took a spring break vacation to Cancun and it was: What are they saying? How do I respond?? Ayudame!  Any foreign language teacher of professor will tell you, you don't become fluent until you immerse yourself in the language.

Luckily, we do not need to send our students to Mexico or Spain to immerse them in mathematics. We need to give them opportunities to problem-solve with each other and construct their own solutions together.  I used to tell my middle school students scientists and mathematicians rarely make great discoveries in isolation.  NASA does not usually send a solo astronaut into space to work on its most precious and complex projects.  Problems are sometimes solved independently, but mathematicians and other specialists often work in teams to solve the world's most important problems.

Solving problems in pairs and teams can allow students the opportunity to reach greater heights with their thinking, and celebrate greater victories.  Even a classroom open response experience is better off spent tackled with a partner or two.  A confident solution with a consensus should warrant a high-five!  Eureka!  Math is fun.

As a social activity, math is about sharing, about experimenting, about collaborating, about persevering and about finding solutions.  Every part of that problem solving process can be a stimulating, engaging, enjoyable experience.

Here are some ways some teachers in RSU 5 are increasing that excellent mathematical discourse that is at the heart of a superior learning experience in the classroom:

  • Incorporate turn-and-talk opportunities into every delivery of whole-class instruction, so your students become accustomed to sharing their thinking with each other regularly.
  • Get into the habit of incorporating whole-class sharing and whole-class instruction in a part of the room where students can be seated on the floor, close together and close to you.  Turn-and-talks are effective in this setting too.
  • Practice specific protocols for communicating in groups, and have students do it every day.  For example, speaking at an indoor volume, always remain quiet when a group member is speaking, identify and use specific polite words when questioning or critiquing someone else's thinking (like Can you clarify that for me? and I think you made a mistake.  Can I show you?)
  • Always be looking for opportunities for students to explore new ideas and new concepts, especially when you are tempted to explain stuff to them.  Think: How could I get them to work this out without me showing them how to do it?
  • Assign roles to group members, such as presenter, note-taker, ambassador, fact-checker, editor, time-keeper, equalizer (making sure everybody is contributing), etc.  
  • Promote a strictly safe environment for sharing in class to promote confidence and reduce anxiety when sharing.  Encourage everyone to take each other seriously so nobody is ever laughed at for giving a wrong answer or making a mistake.  This includes the promoting of making "brave mistakes" as the essential element of achieving success! 
  • Arrange desks in ways that allow students to easily communicate with each other.  Tables are best for this, as they can sit across or next to or diagonal from each other and communicate, while having some distance from other groups.  
Do you have other ideas or tips for creating an environment for collaborative learning? Please let me know and I will add them to the above bullets!

*The book mentioned at the top, Classroom Discussions in Math, was published in 2013 by Math Solutions of Sausalito, CA, and was written by Nancy Anderson, Catherine O'Connor, and Suzanne Chapin.

Recently published (Fall 2015) articles on Mathematical Discourse:

Orchestrating Mathematical Discourse to Enhance Student Learning

Establishing a Culture of Collaborative Learning

Tips for Launching an Inquiry Based Classroom

Learning Is Loud