STUNTING GROWTH THROUGH GENERALIZATIONS

POSSIBLE WEEDS TO PULL:  TEACHING

GENERALIZATIONS

There are many generalizations we learn in Mathematics classes as young students, that do not translate, as we learn higher level Math.  One of these generalizations is “You cannot take a bigger number from a smaller number.”

We are taught, as we learn to subtract, to always subtract the smaller number from the larger number.  This statement will not always be true.  As children progress through Mathematics, they learn about interesting new numbers.  These new numbers are called integers.  When integers are introduced, children learn that indeed you can have a number that is less than zero. This is very confusing to children who have been taught that you can’t subtract more than you have to begin with.  As an example, let’s look at the following problem.

Tom has 13 donuts and John has 12.  Tom ate 6 donuts and John ate 15.  How many donuts does John have now?

John ate more donuts than he had.  The confusion is how John could eat more donuts, than he had at the beginning of the problem.  Students, under the Mathematical Practice Standards, are able to discuss the possibilities.  These standards are designed to assist students in the use of their logic and problem solving skills, rather than learning rules and applying them.

I learned about negative numbers at a very early age.  I was unaware of it at the time, however.  I began playing card games with my grandfather, at the age of 4 or 5.  In fact, he “created” a game, just for us to play.  If Grandpa went out before me,  I had to figure out my score.  My score was dependent on the cards played against the cards still remaining in my hand.  Sometimes, I had sufficient “points” on the board to cover the “points” remaining in my hand and sometimes not.  If my hand had more points in it, than I had played, then either I “went in the hole,” on the first hand of the game, or “went” back that number of points.  It was even possible to go in the hole, if my score, from the previous hand was not large enough to consume the points remaining in my hand.

I enjoy playing cards.  There are many games that children can learn to play that
will teach many mathematical principles.  This would allow children to learn more about how numbers work with each other, even if there is no real understanding of the concept at the time.  While playing cards with Grandpa, I didn’t realize I had utilized negative numbers, nor that I could have less than zero.  However, I did understand I had to get enough points to “get out of the hole.”  This is the concept of integers and negative numbers with no real “hard and fast” rule attached to it.

I can also remember, in Algebra II, learning about systems of equations.  The problem we were solving required us to calculate the number of adult and child tickets sold for a play.  As we, the entire class and teacher, worked out the problem, one of the solutions was a negative number.  My teacher said, “You can’t have negative money, so that answer can’t be correct.  The correct answer is the positive one.”  I was stumped.  You certainly can have negative money.  However, I understood her statement.

It is interesting that teachers continue to make these types of statements to their students. As I have gotten older, I have learned that it is more important to be able to problem solve and “work through” a problem in my own way, than to adhere to any “rules” uttered by my teachers.  I have always excelled at math.  I am not as confident or fluent in math theory.  I like numbers and playing with them.  I am generally lost on number theory.  I would have liked a little more theory and problem solving practice, as a child.  Abstract Algebra and Nonlinear Geometry may not have been so tough in college.  I could be wrong, as both classes are just theory and proof, but may be not.  I may have had better problem solving skills.  

The Common Core Standards for Mathematical Principles allow students to learn more problem solving skills and strategies.  It also gives them the ability to “create” their own and argue for their processes.  These skills are necessary to understand number theory.  I wish I had had better training early on.  As a teacher, I believe using cards and other open ended problems will formulate the practice and instruction I would have liked to have had.

GROWING SOMETIMES MEANS STAYING PUT

GROWING SOMETIMES MEANS STAYING PUT

In this week’s portion of In the Moment, the author discusses using the conference for assessment purposes.  One of the greatest tools in a teacher’s “tool box” is the formative assessment.  Formative assessment allows teachers to informally assess student knowledge.  A conference is a great way to accomplish this assessment, as it is immediate and relevant.  During a conference, teachers have the ability to listen to student thought processes and “assess” their thinking and understanding.  

Formative assessment also allows teachers to ensure that their students are not “left behind” nor “stuck in a rut.”  It is important for teachers to form their instruction based upon the information received during this assessment time.  A teacher can use assessed information in a number of ways.  The author specifically mentions five such ways.  These are 1) immediately during the conference; 2) during the closing discussion; 3) creating the next day’s lesson; 4) during future conferences; and 5) planning for anticipated struggles and strategies.

As we have discussed, during the conference, the teacher elicits information from her students and provides a nudge in the correct direction or to encourage the students toward their goal.  This nudge is an immediate response to student

thought and the information obtained through this assessment process.  Sometimes, this simple nudge is all that is necessary to clear up any confusion or misconceptions. The remaining four strategies are for future planning purposes.

The second planning strategy is to focus the lesson’s closing discussion on any misconceptions discovered during the conferencing.  These discussions could be “student” led, where the teacher asks certain groups to share their work and strategies.  This sharing will allow the students to discuss, debate and justify their work.  During these times, the other students, who may have struggled, are able to listen to the strategies used by their classmates and possibly learn some more efficient strategy to complete the work.

The teacher could simply explain the situation to the class and allow the class to discuss, debate and justify the strategy presented.  This also allows the students to revise their thinking or explain more thoroughly how they reached their answer.

The information gained through the conference process could be used to plan the next lesson.  In planning for the next day, the teacher reflects on the struggles and challenges observed, during the day’s lesson.  It may be necessary to provide the students with more time to fully comprehend the current concept.  If so, the teacher may provide the students with a similar problem or the same problem with different numbers to work through the next day.  This way, no student is left behind and the class has additional time to learn the concept.

The conference also allows teachers to learn more about their students’ learning trajectory.  It is important for a teacher to know how a student thinks, processes information, and when to push or step back.  A teacher, armed with this information, is equipped to reach the student.  It is important to understand your students.  Some are quiet and reserved.  It may be due to one student “running the show” and not allowing the others to contribute.  It could be that a particular student only speaks up when he or she disagrees with the rest.  The conference is a wonderful time to gain insight into group dynamics and rectify any “roughshod treatment.”

Lastly, the teacher can use the conference information to anticipate possible confusion, struggle, or challenge the next unit might bring.  Some students will struggle learning new material.  The conference allows a teacher to discern what challenges a student may or may not have experienced in the past and anticipate any new challenges the student may face with the new material.  If a child struggles with multiplication, it is reasonable to assume that the student may struggle with division.  Similarly, these struggles could be found in fractions, as well.  It is important for teachers to understand and anticipate these potential challenges for his or her students.

In order to be completely prepared for any possible challenges, a teacher needs to keep records of his or her interactions and thoughts from the conferences.  It is only through careful reflection and notes that a teacher will be able to have a complete picture of his or her students.  Documentation is very important.  It allows teachers to understand their students.  It also provides ample evidence for any necessary differentiation or support that may be needed for a student.

In keeping records of these discussions, so that planning reflects student needs, a teacher should take the opportunity to reflect following a conference and record more specific thoughts later.  I always felt on the spot, when the teacher took notes while having a discussion with me.  I felt as though I may have said the wrong thing.  It can be intimidating when a teacher takes notes during discussions with children.  Also, it could appear to the students, as if the teacher was not paying attention to them nor interested in what they have to say.  It would be best to have the conference, jot a couple of thoughts, away from the students and move on to the next group.  More detailed notes can be made at a later time.  

I believe teachers need to understand their students and to provide a safe environment where all ideas are welcome.  It is important for students to feel valued and respected.  Our classrooms need to foster an environment where there are no “wrong” ideas; where all children feel safe and comfortable.  Implementing conferences will allow those students that are more introverted and less likely to speak in a larger setting the opportunity to “blossom”.

SENDING LIGHT TO THE GARDEN: WHEN CONFERENCES GO WRONG

SENDING LIGHT TO THE GARDEN:

WHEN CONFERENCES GO WRONG

Our students are like all plants.  They need good soil, watering and plenty of sunlight to thrive.  Our goals as teachers are to provide the water and sunlight.  Their minds are the good soil.  When we provide “water”, we nurture their thinking and help them discover all they can be.  However, there are times when we need to “send in the light.”  Plants don’t survive on “water” alone, they need plenty of sunlight to stretch their stems, leaves and petals.  Our students need the same.

There are times when our conferences can go wrong.  It is during these times, we need to “send in the light” for our students.  There are three instances where conferences can get “off track” and no growth is happening.  At those times, it can be as though we are banging our heads against a brick wall, pulling teeth, etc.  Sometimes, we are just speaking a foreign language.  The language of math and not the language of our students.  

The first instance of unproductive conferencing is funneling.  Funneling occurs when a teacher is looking for a specific answer and does not receive it.  The teacher continues to question the students, each question is narrowed until the only answer remaining is the teacher’s desired one.

I teach Sunday School to the high school students at my church.  When we discuss a scripture passage, I generally have a specific answer in mind.  I find myself changing my questions, until I get the answer I want.  I have never thought of it as funneling, but it is true, my questions become narrower, until only my answer remains.  I have often told my students that there is no correct answer, but there is generally an answer I am looking for.

A second indication that a conference has gone of the rails could result in the dreaded “blank stare.”  These deer-in-the-headlights, eyes glazed over looks, occur when students do not comprehend the material nor the teacher’s questions.  The students seem to respond with vague, almost cautious and noncommittal answers; answers, such as, “Ummmmm;” “Errr;” “Okay.”  These indicators show that the students have not understood the material, have a mistaken reasoning or misunderstood the teacher.

The third indicator of a unproductive conference is the “pushback.”  A student will push back when he or she feels that the teacher does not understand his or her reasoning.  In a conference where the students feel misunderstood, the students protest the teacher’s nudge.  A pushback occurs when the students are performing the task and have a clear understanding of the material and the teacher is trying to correct their work.  The author emphasizes that students have a strong desire to understand the material and to be understood.  It is in those times that students will pushback and “fight” for their position.

In the book, In the Moment, there is an example of a pushback.  The students are sharing 45 crackers among 6 friends.  The students have parceled out six groups of crackers.  However, the teacher believes the students have confused the number of friends with the number of crackers.  One student explains that the group knows that 6 * 6 = 36, so their first sharing is 6 to each friend.  Afterwards, they will share the “leftovers.” This example shows that the students have not confused the assignment and are working to solve the problem, based upon their knowledge.  Once they have divided the initial 36 crackers, they will deal with the 9 crackers that remain.

Once the teacher is aware that there is a mis-communication, he or she has many choices to remedy it.  The author describes three.  The first is to “anchor to the task.”  Anchoring to the task occurs when the teacher refocuses the conference and, in essence, starts the conference over.  The teacher can ask the students to retell the problem, act it out, visualize the problem through manipulatives, and connect it to their current work.  These activities will allow all parties to begin anew and help to alleviate any lingering confusion of the students.  This strategy is best used when the conference results in a blank stare or pushback.

The second strategy a teacher can employ, when putting the train back on the track, is to elicit more information.  By asking additional questions and restating student answers to ensure everyone is on the same page, the teacher is able to alleviate some of the students’ confusion, along with his or her own.  This strategy is best used when there is confusion after funneling a desired response and receiving student pushback.  The teacher is able to state student reasoning through more probing questions and remove any lingering confusion.

The third strategy is to “anchor to student work.”  The purpose of the conference is to ensure that students are reasoning through the assigned task.  If confusion arises between the students and teacher, the parties can look at the work performed by the students to discover where any misconceptions or faulty reasoning on the students’ part has occurred.  The teacher can point out a specific portion of the work and ask probing questions and reiterate his or her understanding of the student’s process.

As I read these obstacles, I thought about how I ask questions, not only to students, but in other areas of my life, where I teach.  I discovered that my primary form of gaining student information is through funneling.  I have received many a blank stare.  After receiving the stare, rather than start at the beginning or restate student ideas, I begin to restate and narrow my questions.  I like having some new strategies to rely on. 

I think my favorite strategy is to elicit more information.  It is important to remember that students have a strong desire to be understood.  This desire is almost as strong as their desire to please the adults in their lives.  By asking more probing questions and eliciting more information, teachers and students can find themselves, again, on the same page and not at logger heads.  It is important to be sure that all parties have a clear understanding of what the students are doing; what the teacher has requested; and what the assignment really is.  Once these three items are understood by all parties, the “sun begins to shine” and our students begin to grow.  Just like flowers in a garden.

 

TEACHERS BEING "SEEDS" PART THREE

TEACHERS BEING "SEEDS" PART THREE

On March 19, I attended my third Twitter Chat hosted by the Ohio Council of Teachers of Mathematics.  This chat was on Math Warm-Ups.  A warm-up can be anything a teacher uses to start the math portion of the day.  It could be an open question, a journal entry, anything.

The first question asked about whether math warm-ups or lesson starters were used.  A warm-up, in my mind, would be a transition from one content area to the other, so that students are ready to talk math. I have since learned that a warm-up is a review of the previous math topic.  A lesson starter is a way to introduce the topic that will be discussed that day.  Whatever you call them, these are great ways to start a math class.  It allows the students to focus on the day’s question/routine and begin to think in a math direction.

The second question asked the teachers to identify their favorite warm-ups/math routines.  In class this semester, we started with a warm-up.  I really enjoyed them.  I was able to think about math in another way, than just solving an equation.  Some of the warm-ups listed were Splat, Esti-Mystery, Open Questions/Numberless Equations, and math talks.   I had heard of some of them before, but not all.  I really enjoyed Splat.  Splat helps with “subitizing”, which is helpful for the primary grades.  One of my favorites is What Doesn’t Belong.  There are four pictures.  Any one of the four could be chosen as not belonging.  It allows the children to look at the pictures and determine which doesn’t belong and why.  These pictures allow for some rich conversation and allow the students to use the mathematical practice standards.  It is important for students to learn to reason.  This routine allows them to “argue” their point and critique others.

It is important for teachers to allow students to think without leading.  While certain aspects of math need to be taught and explained, students need to be allowed to think through problems and learn to reason.  These are necessary skills for later in life and higher level mathematics.

The use of the math warm-ups gives students the opportunity to think and ask questions.  These are skills that are needed as we learn to reason.  The purpose of the warm-up would not be for an answer, but for the exposure and practice.

GROWING THROUGH OPEN QUESTIONS

GROWING THROUGH OPEN QUESTIONS

Many math-minded people, me included, have difficulty with open questions.  Open questions have no true solution.  We silently scream to the math gurus, “It is supposed to have a solution!”  We have a need to find the answer.  It is why the concept of an open question is very foreign and often times scary for math teachers.

I recently watched a Building Math Minds video called, “Why Teaching with Open Questions Makes all the Difference.”  The presenter was Marian Small.  In this video, she explains why open questions are perfect for math instruction.  I must admit, I agree.

She starts with giving some examples of an open question.  “________ is 4 times as much as ______________.  What could go in the blanks?”  Once her students have had an opportunity to fill in the blanks, she asks questions about the numbers, not asking the students to reveal the numbers.  Some of the questions she asked for this example were: Which number is larger?  Could the numbers have been equal?  Could the first number have been 20?  Could the second number have been 20?  How about 21 for the second number?  The answers to these questions are all yes.  If the numbers are not equal (0), then the second number is larger than the first and first number is always 1/4 of the first.  In this question, the students are able to choose any number for the first spot and then calculate second.  Another concept learned through this exercise is that if the student uses a whole number, then the second number will be even.  

One of the greatest points she made is that open questions have built in differentiation, as the students answer the questions based on his or her own abilities.  If one student is advanced and chooses 1/4 as his starting number, the second number will be 1.  If a child starts at 1, his second number will be 4.  Each is correct and each is within the ability of the student.  Differentiation becomes necessary, when the teacher fills in one of the numbers.   Further, once the teacher chooses a number, it is no longer an open question, but an equation to be solved.

Another benefit of the open question is the opportunity for rich conversation.  Math talks are a wonderful way for students to learn and grow.  A student that can logically explain why he or she reached an answer is learning number sense and not just operation sense.  Number sense is more than working out problems.  It is knowing that 111 is 100 + 10 + 1, 100 is 10 10's, 10 is 10 1's and 1 is 1.  These concepts are as necessary for students to develop, as it is for them to learn to read.  Reading begins with learning sight words.  Math begins with learning the numbers, place value and number sense.  These concepts are the building blocks for greater success in advanced level courses.

Finally, she discussed how she creates her open questions.  Each question is created by first reading the standard, then thinking about how it could be used in an open question.  She chooses words that allow for wiggle room, such as “a little more,” “a little less,” “close”, or “almost”.  These words do not have actual numerical equivalents and therefore allow for more answers to be correct.

An open question allows students to show what they have learned and that they can use their knowledge.  It also allows each student to answer, as he or she is able.  The built in differentiation element allows for multitude of answers and rich conversation for all students in the classroom.  A problem without an answer is a wonderful thing.  Who knew?

BLOOMING THROUGH CONFERENCING PART ONE: ELICITING AND INTERPRETING

BLOOMING THROUGH CONFERENCING

PART ONE: ELICITING AND INTERPRETING

As we discussed last time, math conferencing is great way to discover how your students think; what thought processes they use; how they view mathematics; and how they interpret numbers.  There are five steps in the conference process: 1) Being attentive to what they say; 2) Eliciting their thoughts; 3) Interpreting their thoughts; 4) Nudging them to a deeper understanding; and 5) Repeat, as necessary.

In today’s episode, we are going to look at how our stance can be either productive or unproductive to student growth and understanding.  A teacher’s stance is very important in how receptive students are during a conference.  The author explains that a teacher’s stance is how we appear to our students.  It is not just the tones of voice we use and our words, but also our body language.  If we appear “closed off”, our students will not be receptive.  In contrast, if we appear open and curious, our students will be more receptive.

There are three unproductive stances that teachers emulate.  A teacher can show a managerial stance, a completion stance, or a “fix-it” stance.  Each will not give the teacher a full understanding of student thought processes.

In a managerial stance, the teacher is more interested in maintaining order.  While classroom management is necessary and each teacher wants his or her students to be on task, a managerial stance does not promote the most welcoming forms of communication.  In the managerial stance, the teacher is looking for students that are not on task, rather than how the students are working to answer the problem at hand.  Ensuring that your students remain on task or attend to their work can be accomplished with gentle nudging during the conference.  A teacher exhibiting a managerial stance will often appear defensive.  This does not help a child’s ability to think nor communicate.  It will make the conference very unproductive.

In a completion stance, the teacher’s main concern is arriving at the correct answer, rather than student thought process.  A teacher in this stance will nudge her students to the correct answer.  This stance stifles the students’ ability to creatively work through the problem and discover their own methods and strategies.  It is through this creation process that students learn to problem solve.  Problem solving is a skill that students need to have.  This skill will be used throughout one’s life.

For instance, a family has three children, each child attends a different school and needs picked up a different times.  Which child should be picked up first.  As parents (or adults), we need to be able to determine that it only makes sense to pick up the child that is released the earliest first.  If, as teachers, we are only concerned with the right answer to an equation, our students will not have the skills necessary to solve the problems that arise in our lives.  It is important for our students to explore all the alternatives and create their own solution strategies, so they will be able to do so later on.

In a “fix-it” stance, the teacher is ensuring that his or her students are following the pathway he or she wishes them to follow.  As with the completion stance, this stance does not allow for student created strategies and solutions.  The teacher leads the students through the solution process that he or she wishes to be used, rather than allowing his or her students to explore multiple pathways of their own creation.  If a mistake in thinking has occurred, the teacher is able to gently nudge away from the error without nudging them to a certain process through questions.  This approach allows the students to discover their mistakes and self-correct, rather than being led down a specific path.

When a teacher approaches a conference with an open stance, students are more likely to respond in kind.  An open stance allows the teacher to gently nudge the students toward a solution.  The students are more receptive, as the teacher is open and curious about their work, rather than “judging” it.  When using an open stance, the teacher needs to display genuine curiosity over the students’ thoughts, processes and solution strategies.  We need to ask questions that elicit an eager, rather than reluctant response.  Questions that show an interest in the students’ work and how they are accomplishing their tasks.  It is important to remember the destination isn’t the important thing in mathematics, it is the journey.  While we still need to learn that 2 + 2 = 4, the strategies that students use to arrive at the answer is more important.  Teachers need to foster student creativity through their open curiosity.  

When our students feel comfortable discussing their thoughts with us, we can then nudge their thinking in areas that will assist them to the solution.  It is important that we don’t do this in a fashion that has our students feeling as though we dragged them there, kicking and screaming, but gently and in their own time and manner.  If a student is comfortable using base ten blocks to reach solutions, that is completely acceptable.  Being open to student needs and self-created strategies allows students to explore math and learn it, rather than memorize it.  It also allows them to feel comfortable with math and not fearful of it.

During a visit to a fourth grade math class, I had the opportunity to wander among the students and observe their work.  The class was completing a group of task cards as a break from testing.  One student was trying to discover the perimeter of an unknown polygon.  The problem asked for the amount of fencing needed to fence in a pasture.  The problem contained two measurements.  In order to assist her, I asked some probing questions, so she would be able to discover the shape of the pasture and the process to discover the perimeter.  The classroom had all of the information she would need to complete the task.  However, she was confused by the problem.

I asked, “What shape do you think the pasture is?”

“I don’t know,” she answered.

“How many measurements do you have?”

“Two.”

“Okay, let’s look at some of the definitions on the wall?  Do you think the pasture would be a quadrilateral?  How about parallelogram?”

“It could be a quadrilateral and parallelogram.?”

“Could it be a square?”

“No, the numbers are different.”

“How about a rectangle?”

“Yes.”

“How would you find the perimeter?”

“You add the sides together.”

It was interesting to watch the “wheels” turn, as she and I explored the possible shapes the pasture could be.  I will admit that in that moment, I was unsure whether I was leading her down my chosen path or allowing her to discover her path.  I can honestly say that I am still unsure whether I was more concerned with the answer or her discovery.  As math has always been my forte, solutions have always been my goal.  I need to remember to be open and curious, so my students will discover their own pathways.  It is only when we allow our students the freedom to explore, that they bloom and flourish.

WATERING OUR SEEDS THROUGH CONFERENCING

WATERING OUR SEEDS THROUGH CONFERENCING

When I was in school, I remember being pulled aside for reading group, each day.  It was the time that we met and read with the teacher.  The members of the group were established based upon our reading abilities.  We were all in the same reading book and we all read the same thing.  In these groups, we were given so much time to read a certain passage and then discuss it.  I remember struggling with it, because I didn’t read as quickly as the others.  I was never finished on time.  It wasn’t that I couldn’t read every word, it was that I read much more slowly than the others. (Still do) This representation of a reading group morphing into a math group intrigues me.  How does that happen?

In her book, In the Moment, Jen Munson poses the idea of conferencing with our students during math work.  These conferences could be conducted in much the same manner as reading groups were in the 1970's/80's.  The teacher could split the children up into groups.  These groups would work together on certain tasks and the teacher would ask questions on their work.

However, I do not believe these are the types of conferences the author has in mind.  In one of the examples in the book, two classrooms are involved in a shared “write-the-room” activity.  There are pictures of famous people, along with school personnel hanging in both classrooms.  Each picture has the height of the person depicted.  Some of the heights are in inches only (77 inches), while others are in feet and inches (6' 5").  The task is for the students to convert each height into the other.  (If in inches, then feet and inches.)  The teachers move among the students and interact with each pair.  These interactions are the conferences.

In these chapters, it is apparent that how I learned to complete math, is not the only way to teach children today.  I don’t really remember having to explain why I solved a problem a certain way.  We were instructed in the standard algorithm and those were the only strategies we had.  Today, students are given the freedom to look at math in a way I could have only dreamed of while growing up.  Students are asked to reach solutions to problems in ways that makes sense to them, rather than just following the teacher’s directives.  Students today are thinking for themselves, which I have always thought was the teacher’s role.

Many of my generation and those of my parents’ generation are confused by partial sums, products, and quotients.  It makes little sense to them, as they were not given the opportunity to truly understand what place value meant.  Today, students are looking at numbers and their place values and creating their own strategies to complete problems.  The teacher’s role in this type of learning is to assist the students in refining their thought processes and explaining why they chose a certain path, rather than “drilling” facts into their heads.

The book states that an effective conference has the teacher listening to the students, as they work and observing their work product.  This work product could either written or demonstrated through the use of manipulatives.  Once the teacher has some understanding of the students’ thought processes, the teacher then asks some questions to elicit clarification of those thoughts.  A good conference will also include the teacher nudging the students for a deeper understanding of their work.  If a pair are only discussing the possible solution for a problem, then the teacher might nudge the pair to consider how they would explain their work to another student or construct a model that shows their work.  In these instances, the teacher never tells the students how to dig deeper, but gives them ideas to think about, so they remain the owners of their work and processes. It is important for students to understand what they are doing and for them to own their work.  Student ownership of the work is important, I think, as it not only builds their self-esteem, but also helps them “teach” others.  Finally, owning their work strengthens their problem solving capabilities.  Being able to problem solve is important throughout life.

It is important to remember to provide rich tasks for the students to solve.  These are the types of tasks that lend themselves to effective conferencing.  If the task is too rigid, there is not a lot of room for student creativity in solving the problem.  The tasks need to be open-ended and allow for student exploration.  The method and thinking are the important elements in the process, not the answer.  While, 2 + 2 will always equal 4, some children will reach that answer in ways that others do not.  Some will raise two fingers on each hand, others will use two red chips and two yellow chips, while still others will use a number line or draw a picture.  There are different ways to reach the same answer.  Allowing the students to explore these ways, gives them the courage to try more difficult problems and stretch their mental thinking toward greater solutions.

In these conferences, it is my job to nudge them into a deeper understanding of their thoughts and help them explain those thoughts to others.  Sharing our ideas and learning with others helps us grow and could help a friend learn in the process.