Mathematical Discourse

Mathematical Discourse

            Mathematical discussion in the classroom is not a new idea or concept in education. Educators utilize this as one of the most powerful tools in the development and assessment of student learning. How is the level of mathematical discourse in your classroom?

Connolly and Smith (2002) studied classroom discussion and “the way it should be.” They found student and teacher viewpoints differed in terms of the teacher’s role within small group discussion, competition versus collaboration, familiarity with the discussion topic and comfort within small group settings. As a result of their studies, Connolly and Smith stated the importance of teachers “talking about talk.”  Students not only benefit from knowing the teachers expectations for discussions but also benefit from the definition of norms pertaining to who gets to speak, in what manner, about what, and in what format. 

            Cohen (1992) addressed the value of focusing on the concepts and the type of interaction desired by the teacher. Teachers should not base the value of a discussion solely on the frequency of the student interactions. Instead, teachers must turn their focus on the fact that if students are not taught differently, they will tend to operate at the most concrete level. Exposure to deliberate skill building or motivation techniques can enrich what is happening in the classroom. This means creating tasks which are resource and goal interdependent, individual and group rewards, role assignments and constructive controversy; tasks which require teacher direction. The teacher’s role should be delegating authority to the students and fostering conceptual and elaborate discussions by teaching specific and relevant behaviors.  

Discussing the various roles individual students can take and assigning students these various roles can have a direct affect on the quality of the mathematical discussions. Defining the components of quality discussions and the roles that individuals play within discussions can clear student misconceptions and provide the necessary tools needed to meet teacher expectations. Providing students with role assignments and on-task tools are simple ways to increase student responsibility, bring ownership to the individuals, increase participation and enhance the quality of the discussion.

          Engaging in a quality mathematical discussion is as a complex task. Despite the exposure during the elementary years, teachers must have an awareness of student dependence on teachers as models for communicating with peers and take an active role in establishing expectations. Focus on how students contribute to mathematical discussions over time and the role of the teacher in this process.  Focus on the student’s ability to construct useful or useable concepts to take part in the society of the classroom. Pay special attention to the elements of student discourse including patterns of discourse, levels of response, teacher and student roles, norms and classroom practices. Bring the level of mathematical discourse in the classroom to a new level.

        

Somethin' Fishy

This is one of my favorite activities to do with students.  Modify it for by using any candy, item or special day.

Playground Designs

Playground Designs

Provide groups of students with the CurrentPlayground Design.

Have students take turns identifying the fractional parts of each area of the playground. Direct students to record the fractional areas onto their individual Current Playground Design. 

Note:  Students may ask whether or not the fractions should be written as a fraction of the section or of the whole playground.  This is an opportune time to lead a discussion. Allow students to discuss their question and come to a consensus as a group or as a class based on this discussion.

Give each group a copy of the FuturePlayground Design Guidelines (see attached). Have groups of students follow the Future Playground Design Guidelines to create the new playground design onto graph paper.  Although students are working as a group, each student should create a new playground design.

Note:  Students will be asking many clarifying questions.  Allow the students to ask the whole class for clarification. Allow students to discuss their views and understanding.  You may decide the whole class will need to come to a consensus or allow individual groups to develop their own.  However, students should be prepared to discuss support their reasoning in their written explanations in addition to clarifying any assumptions they have made.

Students should then follow-up the design process by individually responding to the questions found at the bottom of the FuturePlayground Design Guidelines.

Display the completed playground designs and rationales. 

Have students examine the solutions of the groups. As students are looking at the works of their peers, ask them to provide feedback to their peers in the form of guiding questions.  In other words, students should be writing questions that guide the playground designer to consider the accuracy of their work based on a given guideline.   

Have students review the questions asked by their peers. Ask students to reflect on the playground development process including the feedback received by their peers. 

Based on the Playground Problem developed by the Virginia Department of Education 2004

Mathematics Standards of Learning Enhanced Scope and Sequence. (2004). Retrieved from http://www.doe.virginia.gov

Inspiration

“There are many ways to organize curricula. The challenge, now rarely met, is to avoid those that distort mathematics and turn off students.” - Steen, 2007

            Those individuals who attended NCTM Denver in April 2013 could not have felt anything less than inspired.  Inspired by the energy, enthusiasm, research and knowledge shared by the professionals who swarmed the city.  In the hotels, restaurants, walkways, buses and throughout the convention center, there was a constant hum of those who were energized by the mathematics present at this annual meeting and exposition.

                  In one session, Picture Yourself Having Fun at Math, Mary A. Robertson shared how photography can be used to incorporate real-world situations into the math classroom. The use of pictures can be used to reinforce concepts involving geometric shapes, areas, volumes, similar figures, transformations and so much more. I found myself reflecting on how the simple task of incorporating photography into the math classroom can be used inspire a student to look at mathematics through a different lens (which happened to be one of the hashtags used throughout the week).

Who has heard of mARTh?   The basic idea of mARTh is to connect mathematical concepts in a visual, kinesthetic way to make math fun, hands-on and beautiful. The presenter noted the goal of mARTh is to use creative expression to connect students to mathematical concepts.  This is a teacher whose goal is to help students make a personal, physical and visual connection with mathematics. 

            In another session, Making Cents of CCSS, Doug Tyson and Jason Molesk addressed ways to make inferences and justify conclusions from sample surveys, experiments and observational studies through spinning pennies and simulations.  The presenters shared ways to lead students in a statistical significance test in a way that non-stats teachers can implement…even at the middle school level. Did you know there are pennies from the 1960’s that will land nearly 100% of the time on heads when spun on its side? 

David Masunaga’s Geometry on a Shoestring Budget was described as “the most profound, interactive and dynamic activities that don’t require expensive technologies” and that is exactly what it was.  Masunaga kept the audience captivated and yearning for more with cheap and nontraditional geometric manipulatives that could be used to reason and prove various geometric concepts.  Every person in the room was engaged and inspired by Masunaga; the power of one.

Jo Boaler made a valid and strong point in Using Research to Make a Difference where she clearly noted that producing research knowledge is not enough to make changes in the math classroom.   How do K-8 Teachers Change Their Practices after Learning More Mathematics?  shed light on the aspects of teaching practices connecting a teacher’s knowledge and beliefs which led directly to Ritual:  A Category for Understanding Persistent Practices in Math Education; a theoretical study on the persistence of practices in math classrooms which contributed to a theory of rituals in math education. 

            With so much happening in Denver it is impossible to share every ounce of awe and amazement one experienced throughout the week.  During the opening session, The Power of Just One Teacher, Mayim Bialik shared a mission which all teachers should consider encompassing into their rituals, beliefs, practices and everyday practices: to inspire students to pursue STEM education.  This is not to say that we should expect every student make the maths and sciences the end all of education but as educators we should make it our mission to ignite a spark in every student.  As educators we have the immense and immeasurable power to inspire our students to develop a love for mathematics in some way, shape or form.  Every lesson, activity, assessment and mathematical discussion keep the common core and the mathematical practices in mind.  However, do not forget to inspire, engage and help students to develop an appreciation for mathematics that permeates beyond the classroom; you have the power to make an inspirational difference.

I look forward to another inspirational NCTM conference in New Orleans.

I Can't Do Math!

The dreaded yet famous quote “I can’t do math!” continues to haunt math teachers.  Is this a true statement? Are there students out there who just can’t learn mathematics?

Teachers of mathematics are expected to have a strong hold on mathematical content as a well as the pedagogy needed to support the various learners in the classroom. They have been trained in Bloom Taxonomy, Gardner’s Intelligences, special education and classroom differentiation.  The challenge is designing instruction to bring the learner to a level beyond rote memorization and regurgitation to a level of inquiry and conceptual understanding. This requires the teacher of mathematics to consider the way in which a student’s mind is activated.

Based on How the BrainLearns Mathematics by David A. Sousa (2008) there are several areas to consider. 

·      Make math meaningful.  If a teacher cannot answer the question, “When am I ever going to use this?” in a way which is meaningful to the students, then the teacher should consider why the concept is being taught. Learning is stored in long term memory when it has meaning. Teachers who find themselves frustrated with a classroom that cannot remember a process from one day to the next are teachers who should focus on establishing meaning for students.

·       Make math emotional. Middle school students are emotional beings.  They are quick to share their opinions and feel strongly in what they believe. A good teacher uses this to his/her advantage.  Take a common objective for the day and make it exciting.  Sparking interest and emotion is yet another way into the long term memory area of the complex brain of a middle level student; good teachers know this and use it to their advantage.  

·       Timing is everything.  The first ten minutes of class, after student attention has been gained, is a peak time for learning.  The good teacher uses these ten minutes to teach new material knowing the brain is absorbing all the information being presented.  Good teaching during these ten minutes of prime learning time is a preventative to “I don’t remember what we did yesterday.”

·       Use downtime to practice. After the first ten minutes of processing information the brain reaches it potential and starts on a downward trend for retention.  The brain is essentially a sponge which cannot hold any more. A good teacher gives students time to utilize and process the new mathematics after peak learning.  This utilization and processing period is the brains way of storing information into the long term memory bank; exactly what teachers wish for.

·       Closure is a mathematical seal.  Given a processing break, the brain begins to rejuvenate.  Spending the last 20 minutes of class to bring closure to a lesson is another good use of student learning.  A good teacher knows this is the last chance to make a meaningful connection to seal the mathematical objectives.  Stress the key aspects of the class and make the last ten minutes the grand finale.

·       Ten is a magical number.  The working memory of a middle level student works best in ten minute chunks.  After ten minutes of the same activity or instructional mode, the mind of the middle school student veers off.  Boredom, daydreaming and distractibility all set in.  A good math teacher knows to change up the activity, instruction and mode of instruction. This is the time to use Blooms and Gardner.

·       Make room for high level mathematics. The brain recognizes and stores patterns; patterns of processes, skills and knowledge.  Teachers at all levels have techniques to the speed of pattern recognition; flash cards, tips to break down word problems, acronyms, rhymes, songs, etc.  These techniques support the brain in long term memory storage; making room for deductive reasoning and high level mathematics.  A good math teacher develops lessons which go beyond the rote performance of skills and knowledge, knowing the brain is well prepared for mathematical reasoning and deduction.

Everyone has a brain.  Everyone can do math.  The key is fine tuning the methods in which mathematics is presented; methods which maximize learning for all minds and eliminate those who believe “I can’t do math!”