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.

MEASURING THE MARIGOLDS

MEASURING THE MARIGOLDS

https://www.youtube.com/watch?v=fXi3bjKowJU

Measurement is a multidimensional topic.  It includes telling time, determining elapsed time, perimeter, area and the degrees of angles.  While some of these topics seem to fall into other categories, each is a measure of something.

TIME IS A MEASUREMENT

Most of us do not consider time to be a measurement.  However, without the separation of the day into hours, hours into minutes and minutes into seconds, we would have no idea where we were at any given point during the day.  This is a measurement of our activities.  We determine the amount of time or more aptly we measure the amount of time it should or does take to complete certain tasks.  Remember that old Algebra problem: “Two trains leave the station at 9:00 a.m., one traveling south at 55 mph and the other traveling north at 65 mph.  How long will it take for the trains to meet?”  This problem measures the elapsed time of each train to a meeting point at an imaginary location.  

In  the Third grade and into the Fourth grade, students are given word problems to solve involving elapsed time.  The problems are not as advanced as the train problem.  There are three possible types of problems that students could be asked to solve.  They could be required to determine how long a flight took, when a flight took off, or when a flight landed.  In the first type of problem, the problem would provide the take off time and landing time.  The second type would contain the landing time and the length of the flight.  Lastly, the third problem type would provide the take off time, along with the length of the flight.    As with earlier word problems, the students would be doing addition and subtraction to solve these problems.

Determining elapsed time is just one of the measurements related to time.  The other is the division of our lives.  This division is the way we measure our days.  The calendar divides our year into months, weeks and days.  Our clocks divide our days into hours, minutes and seconds.  As adults, most of us are constantly looking at our watches or “watching the clock”.  These are the measurements that drive our lives.  The calendar and the clock are our first introduction into measurements.  As we grow older, we begin to apply these divisions to other things.  We begin to look at how long, wide, high, or heavy something is.  These divisions allow us to make sense of our world.

THE INCHWORM AND UNIT BLOCKS

When students begin to look at length, height, and width, they are required to begin with measuring the length of objects based upon a specified unit.  In Kindergarten, a child might be asked to compare a paper circle to the number of pennies it is across.  This is the beginning of measurement.  The students are tasked with measuring objects by the number of a specified “item” it is long/wide/high.  Once this type of measuring is understood, students progress to using rulers and measuring to the nearest inch, which begins in about second grade.  

By third grade, students are again measuring area and perimeters with a unit block or item.  The beginning concept of area is how much of an item is “covered” by a shape.  In other words, if I place a piece of paper on my desk I will have covered an area of 93,5 square inches.  Students would use a unit cube, which is 1" by 1" and count the number of squares contained in that piece of paper.  In the 4th and 5th grades, students are expected to use the formulas for the area of most regular polygons.  (Rectangle–A= l * w. Triangle–A= (b * h)/2, and so on.)  Also, in the 4th and 5th grades, students begin to experiment with volume and learn the formulas for those measurements, as well.  The most important thing for students to remember when calculating these figures is to appropriately label their answers.  93.5 is the correct calculation for the area of a piece of 8.5" x 11" paper.  However, without the appropriate labeling, the area could be square feet, inches or yards.  It is important for students to label their work with the appropriate measurement unit.

MEASURING ANGLES

In the 4th grade, most students begin to measure angles.  I remember buying “pre-filled” school boxes and measuring tools, as a child.  These always had a protractor included.  I never knew what a protractor was for.  Today, students use the protractor in the 4th grade to not only measure angles to the nearest degree, but to also draw these angles as accurately as possible.  I was intrigued, while observing a 4th grade math class this semester, watching the teacher and the students using a protractor to draw and measure angles.  It was great.  I always had to have a protractor in my supplies, but I can never remember using it.  To get the opportunity now was fantastic.  The understanding of the students on how to create the two rays and draw the angles was very impressive.  I learned so much that day.

POSSIBLE “WEEDS” TO PULL

        As with any math topic, there may be some “weeds” to be pulled, during measurement lessons.   The first possible weed is a confusion between area and perimeter.  These topics are introduced in 3rd grade and honed in 4th and 5th grades.  The perimeter is the distance around a shape.  Another way to think of the perimeter is to imagine putting a fence around your yard.  The perimeter is how much fencing you would need to buy to fence in your yard.  It can also be thought of the distance you walk around your residential block.  Explaining perimeter to our students in this manner will give them a real world “picture”.   In explaining area, it is also important to give students a real world frame of reference, so they are able to visualize it.  A teacher could use the yard analogy for area, as well.  However, rather than putting a fence around it, you are concerned with amount of grass needed to cover the inside of the fence.  A better analogy would be the surface of a table.  If I want to cover my desk with a cloth, but have no over hang.  The amount of space on my desk covered by the cloth is the area.  A table top of any kind will give them a frame of reference for area, as fencing the yard does for perimeter.

Each new topic in math has its own little “weeds” to pull.  Being able to provide real world examples to our students can help bridge the gap between our reality and theory we are teaching.

TEACHERS BEING “SEEDS” PART TWO

TEACHERS BEING “SEEDS” PART TWO

On February 20, 2020, I attended another Twitter chat with the Ohio Council of Teachers of Mathematics.  The title of the chat was “Diving into Discourse.”  As with the other chats, this chat was a half hour and consisted of five questions.

The first question dealt with defining Math Discourse.  There were a great number of responses to the question.  The answers ranged from simplistic to very detailed.  However, each one boiled down to discourse being the language of Math and being able to determine whether or not your students are learning the materials you are teaching.  I believe that true discourse is the teacher and the students being able to talk about numbers, their mathematical concepts and how each applies to real life situations.

The second question’s topic was on the key factors needed for honest and open discourse.  The responses for this question were very similar.  In order for discourse to be effective, the teachers need to be sure they provide a safe zone for their students to discuss mathematical principals without judgment or fear of ridicule.  Only when students feel free to make mistakes will students feel free to share their strategies and approaches to problem solving.  Students also need to know we care about them.

Question 3 dealt with the necessity of teachers completing the tasks, prior to the “lesson”.  It is important for teachers to be prepared for the lessons they are teaching.  In some ways, it is most important with a Math lesson.  For some reason, Math is feared by students.  For a teacher to complete the task prior to giving it to students, allows the teachers to consider the possible strategies their students might choose to complete the task.  I also think it is important for students to know that we can actually solve the problems we ask our students to solve.  It is comparable to a lawyer not asking a witness a question that you do not know the answers to.  Lawyers don’t like surprise answers during trials.  It is the same with teachers.  If we don’t plan ahead and prepare, we are like a lawyer that didn’t prepare for court.

The last question asked which online resources we preferred to obtain the tasks we use.  In the class that I am taking this semester, our teacher begins each class with a “math routine.”  I have discovered some great online resources with these routines.  Splat, Esti-mystery, “What doesn’t belong”–it is a book with four pictures on each page and the students determine  what doesn’t belong.  I have really enjoyed these routines and can’t wait for others.

I really enjoyed reading the responses from everyone who participated in the chat.  I also enjoyed the topic.  However, I had more difficulty with this chat than the last.  I don’t remember having to place the # in my responses on the first chat, but lost my responses that did not contain the # on the second one.  While I enjoy participating in the chats, I personally find Facebook chats more user friendly.  However, I will continue to participate in these chats.  I have learned much from all who participate.  I can’t wait to learn more.  I am really looking forward to the chat with COSI in April. 

WHAT “SHAPES” YOUR GARDEN?

WHAT “SHAPES” YOUR GARDEN?

This week, we turn our thoughts towards geometry.  Geometry is the study of points, lines, rays, segments, shapes.  The study of the geometry is divided into two distinct types of geometry, plane geometry and solid geometry.  Plane geometry contains everything that is not a three-dimensional shape.  Solid geometry contains only three-dimensional shapes.

It is important for a student’s first introduction to geometry be within the context of his or her environment.  The ability to observe the world around us is important in being able to categorize and classify the different types of shapes.  Most children are able to look at a window and recognize that it is shaped like a rectangle or square.  These initial categorizations allow students to build the necessary observation skills that allow students to piece together definitions of the shapes.  These early definitions are refined with each passing year, until the student’s definition of a square is regular rectangle.  The understanding of this definition will not come together without constant observation and refinement.  

In order to refine a definition, a student must be able to observe the shapes in different settings and sort the shapes based on his or her own interpretations of the defining attributes of the shape.  A squares defining attributes are many.  All squares are polygons, quadrilaterals, parallelograms, rhombuses, and rectangles.  However, as we all know, not all polygons, quadrilaterals, parallelograms, rhombuses and rectangles are squares.  Squares are a special 4 sided polygon that has alternate sides being parallel, all line segments form a 90° angle and are all sides are equal. Only by observation will a student be able to understand these relationships between the various classifications to which a square belongs and its special designation as a square.

It is sometimes difficult for children to make the connection between the shapes, as shown above.  They have been taught from an early age that there is a difference between squares and rectangles, that in fact, the two are distinctly different shapes. Only to discover later that the square is a special type of rectangle and not a distinctly different shape after all.  To further their confusion, we begin to introduce other types of shapes that contain the same attributes that squares and rectangles have.   These newer classifications of shapes are not exactly distinct, as their old friends are actually special subsets of the “new” shapes.

I am not advocating that polygon, quadrilateral and parallelogram be introduced prior to the introduction of square and rectangle.  However, I am advocating that these shapes be introduced sparingly in the earlier grades.  Being exposed to these shapes and their names will help assuage some of the confusion later.

The best way to discover the relationships between the shapes and refine the working definitions of the shapes is to sort the shapes.  It is important that the sorts be open with some guidance from the teacher.  These sorts allow the students to determine the rules and make observations of the similarities between the different shapes.  The teacher’s involvement is to direct each sort’s parameters.  The parameters contained on page 126 of Making Sense of Mathematics for Teaching Grades 3-5, by Juli K. Dixon, Edward C. Nolan, Thomasena M. Tobias, and Guy Barmoha are a great example of the teacher’s involvement in the sorting process.   These parameters allow the students to begin with a sort of curved and non curved shapes, so that each sorting loop has distinct and separate shapes contained within.  One of  the instructions asks the students to create 2 new rules, so that there are some shapes outside the loops, some inside one of the two loops and some that are in both loops. In order to succeed, a student will need to look at what makes each shape the same and what makes them different.  Finally, the students are tasked with creating another set of two new rules.  These rules are to sort the shapes  so that some shapes are outside of both loops, some shapes inside the larger loop and some shapes inside embedded the smaller loop. The rules must be such that the student is able to find shapes that while a bit different have the same attributes as another.  These sorts will allow the students to make the connections outlined above that squares, while the shape will always remain a square, it is also a polygon, quadrilateral, parallelogram, and rectangle, as well.

You might be surprised by the additional shapes you can find, while you wander around.  There are shapes everywhere, in tile floors, patio paving stones.  Each of these can make beautiful pictures.  I use many items such as these as inspiration for my quilts.

The Inspiration

The Quilt

Inspiration for future quilts.

Let's Add Multi-Digit Numbers

I really enjoyed making this video.  It took 3 takes and each time I felt more confident.  Hope you enjoy it.


I decided to demonstrate addition of three digit numbers.  I used the based 10 block manipulatives to demonstrate adding 367 + 256 .


It is important that children understand the nature of the numbers they work with.  They also need to have numbers remain within the place value that they begin.  I hope this video helps.

https://photos.app.goo.gl/YMZYrUszEhQQeyC5A