Why Is Math So...?

Over the course of the summer, educators enjoy the glorious weather but also spend time preparing for the upcoming year.  This often entails reviewing curriculum, aligning assessments to standards, adjusting procedures and reflecting on what was successful from years past.  Educators will often take on the job of creating new, challenging and exciting tasks which they cannot wait to present to students.  And as many educators know, the newly designed work can quickly seem like wasted time spent when the time comes to present it to the students.

The task is distributed, the teacher is excited and the class is bursting with energy.  Unfortunately, the energy is talk about lunch, the school dance or what is happening next weekend.  The teacher will patrol the classroom and quickly find that students do not care about unit cost, cost comparisons and the best deal; they don’t even do the grocery shopping.  The complaint which can be heard throughout the classroom, “Why do you make math so pointless?”

The goal for educators  is certainly not based on turning students off from mathematics.  In fact it is quite the opposite.  There is the never ending hunt and challenge of creating daily lessons which spark enthusiasm, intrigue and excitement in the eyes of every student in the classroom.  However, there is often the roadblock of students who just don’t care about the math.  They have no desire to make efforts to learn about the mathematics and see no point in persevering through useless problems on cross products, linear equations or proportional reasoning.  These are the students who speak their minds, “Why do you make math so boring?”

There are also the cases when a task initially excites students but then something happens and the students give up.  This can happen for some of the reasons noted earlier but also can occur when a task is seen as too easy or too hard. Students will quickly give up on a task when they come to the realization that they have a few days to complete it and really only need one class period. This means plenty of play time.  On the other hand there are the students who just can’t wrap their heads around the task and aren’t quite sure where to start.  In their minds, they shouldn’t bother to waste time on the task.   The complaints that are shared all stem from, “Why do you make math so difficult? Can’t we just do problems from the book?”

For educators these are the dreaded questions which they have worked on diminishing via their summer work.  Now they have come to a place where there is frustration and a feeling of giving up. “Why not assign 25 problems from the textbook?  Students will be happier and I will have saved months of wasted time.”

For these educators, keep in mind the goals of inspiring these young minds and preparing them for the world beyond the classroom.  Accepting the challenge of turning math into a meaningful, exciting, and pertinent subject for all students is the focus.  The question is, “How can this be done?”

First and foremost, remember that not all students are going to love every lesson created.  However, try creating a survey asking student about their interests and hobbies. Ask students to take a multiple intelligence test to determine who is sitting in the room.  Is the class full of interpersonal students who tend to be musical?  Or is the class full of intrapersonal students who are focused on the environment? Use the results to form mathematical lessons around the make-up of the students.  Or to make it even easier, use problems from a math textbook as the foundation.

Take out the math textbook.  Choose a problem which students have shown an interest in the past.  Analyze this problem and determine how it can be changed from a closed, one-answer problem to an open ended problem which allows students to be creative in their solutions. Instead of asking students to determine the area of 12 foot by 13 foot room ask students to design a dream room (or a combination of rooms or even an entire floor plan) and be prepared to discuss the dimensions and make-up of the design.  Leave it open as to how students create the room and share their design.  Some may choose good old paper and pencil while others may turn to technology.  The artists in the classroom are going to add their creative touches and the mathematicians are going to bring out the precision. 

Of course there is still going to be the student who is complaining.  Pull this student aside. Talk with this student and get to the bottom of the complaint.  Work with this student to redesign the task so make it pertinent to them.  Are they interested in a skateboarding?  Design a skateboard park.  Interested in horseback riding?  Design a new arena. Don’t take “it is boring” as an answer.  Make this task one which they have vested interest in.  In the end, make it a point to showcase this student’s work.  Students will see the willingness to make math connected to their lives and the creativity will start flying.

The most challenging aspect of this type of task is student adjustment.  Students are accustomed to being told what to do and how to do it.  By letting go of the reins, they will struggle with less parameters and the openness of the task.  However, the result students who are excited about the mathematics take pride in their work and ownership in the task.  A boring, unrelated and challenging problem can be redesigned to become exciting, pertinent and focused on the individual in a matter or minutes; say hello to summer again.

Modeling Mathematics

A constant ponder amongst educations is why so many students have trouble learning mathematics.  At the start of the year, a common teacher practice is taking a student inventory and it is sad to say that there are responses which include “I can’t do math.”  Whether or not this is true is another topic of discussion but one can say that many students make this statement based on the fact that they struggle with learning mathematics. 

One the best tools for struggling students is to provide concrete and visual representations to supplement the abstract concepts which are taught at the middle level.  Concrete representations may include pattern blocks, unit cubes, 3D models, fraction bars, balance scales, colored counters and other items which students can manipulate during the mathematical process.  Visual representations may include area models, graphs, number lines, tables, and diagrams.  And with classrooms being filled with 21^st century tools, the usage of technology cannot be forgotten when looking for resources for visualization in mathematics.

The goal of exposing students to visualization tools is to support them in developing efficient and effective strategies when working with abstract mathematics.  This is not to say that teachers expose students to one tool and quickly revert to the abstract.  Providing students with a multitude of visuals where they are asked to apply, create and prove their mathematical work over and over develops a strong foundation for the learner. This means asking students to take ownership in the visualization of mathematics and creating deep roots of understanding long before moving onto the abstract usage of mathematical symbols.

Perseverance

With the transition to the Common Core and a recent session at the IMPACT Center at Plymouth State University, perseverance has come into the fore front of my teaching and planning.  What does it mean for students to persevere in the math classroom? How can we teach this to students?  How can our lessons, activities, instruction and curriculum encompass this idea of perseverance?

The quick and easy answer here is problem solving.  Ask teachers how to challenge students and the likely response will include problem solving.  Check in with directors of mathematical instruction and surely problem solving will arise as a focus or solution in one way or another.  However, problem solving does not appear to be a simple answer for developing and promoting perseverance in our students.  In reality problem solving is what makes our students throw their hands in the air. One could say problem solving is the anti-perseverance.

Engaging students in the problem solving process can be described as exploring, examining, applying, testing, reflecting and developing conjectures. Students are given mathematical problem upon problem with the dream that consistent exposure will develop student understanding and mathematical skills needed for the real-world. The emphasis becomes cooperative learning, active participation and investigations that build new knowledge. Again, where does the habit of perseverance come into play? It is expected but how is it taught?

Perseverance is commitment, hard work, patience and endurance. Perseverance is a willingness to try and try again.  Check Webster’s online dictionary and you will see perseverance defined as a “continued effort to do or achieve something despite difficulties, failure, or opposition: the action or condition or an instance of persevering.”  How do we incorporate this into our mathematical classrooms?  We expect it and want students to develop this as one of their very own habits of mind but as teachers we need to teach this.

There is not a quick and simple answer to teaching perseverance.  It is not going to magically happen in one class or even in one year. As educators we can only model, expose and encourage this habit regularly and regularly.  Here are some suggestions on how to develop perseverance:

• Give students non-routine problems which may appear impossible.
• Let students wonder and inquire for extended times. Refrain from giving out the correct answer.  Prolong the awe and wonderment for classes, days and even weeks.
• Allow students to share observations and build off one another’s reasoning.  Let students support and argue the findings of another. 
• Ask questions and expect students to prove themselves.  Expect convincing arguments.
• Let students experience failure.  It is okay for students to take a risk and find out that was the wrong route to take.
• Work with students and show them that you don’t have all the answers.  Model the process of learning together.

Perseverance….the habit of mind which teachers are expected to teach must persevere to perfect.

The Common Core State Standards are Coming: What’s Your Problem?

            Lingering in the air are the Common Core State Standards.  Administrators, teachers, specialists, and entire districts are in a state of wonder and insecurity.  How are we going to transition from the GLEs to the CCSS with ease?  How are we going to transition to the CCSS without leaving gaps in student understanding? How can teachers efficiently and effectively transition to the CCSS without affecting student learning? 

            The CCSS stress the mathematical habits of mind, the inclusion of mathematical practices and mathematical processes.

·      Make sense of problems and persevere

·      Attend to precision

·      Reason abstractly and quantitatively

·      Construct viable arguments and critique the reasoning of others

·      Model with mathematics

·      Use appropriate tolls strategically

·      Look for and make use of structure

·      Look for and express regularity in repeated reasoning

           Embedded in our current practices are these very same habits, practices and processes.  Currently we push students beyond the everyday “drill and kill” into the higher levels of thought through modes of instruction, classroom activities, applications and assessment.  When we narrow down where and when we are asking students to perform beyond the basic skills we can certainly look at one’s mathematical competence in problem solving.

            Assessing students’ problem solving abilities gives educators a peek inside the mathematical mind of the learner.  Given a rich problem, students can show the mathematical knowledge, skill, understanding and any misconceptions they may have. To proficiently problem solve students must have a deeper knowledge base than the application of algorithms and procedural knowledge. When given a rich problem, students must have an understanding of the mathematics; how to justify, apply, explain, use appropriate tools and engage in the mathematical practices.  The key here is providing rich problems where students are given the opportunity apply the skills and habits we need to assess; the proficiencies and practices proposed by the CCSS.

             Look through those file cabinets, memory sticks and internet resources for your favorite problem solving task. Adapt this problem to make it one which is asks students to apply pattern building, conjecturing, generalizations and mathematical justifications.  Ask students to explain what they are thinking, how it can be solved or modeled another way, how they know that something is true or if something always works.  Give students the opportunity to show their skills in adaptive reasoning, conceptual understanding, productive disposition, strategic competence and procedural fluency. You have made this problem your own and adaptable to the CCSS. Connect it to a standard and you are one step closer to a smooth transition. This is your “problem.”

The CCSS are not asking educational systems to get rid of what works.  In reading a draft of the Content Specifications with Content Mapping for the Summative Assessment of the CCSS, there are four claims which put the goals of the CCSS in perspective.

·      Explain and apply mathematical concepts and procedures with precisions and fluency

·      Construct viable arguments to support their reasoning and critique the reasoning of others

·      Analyze complex, real-world scenarios and use mathematical models to interpret and solve problems

·      Frame and solve a range of complex problems in pure and applied mathematics

The CCSS are asking educators to continue working with what they know works with a clearer national focus to develop a cohesion that has been lacking in the mathematical arena. This is the time for educators to reflect on their instruction and determine what connections can be made to the CCSS.  Start with problem solving. No need to reinvent the wheel; just refine it. Think to yourself…The Common Core State Standards are coming. What’s my problem?