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** Excerpts from A Case Study Of Mathematics Instruction In An Intermediate Elementary Multiage Classroom written by Dr. Kevin O'Connor.
Whole Class Discussion, Using Small Groups and Peer Coaching
Successful Strategies of a Master Teacher
Christine Ortlund, a teacher in the northwest suburbs of Chicago, successfully integrated problem solving and assessment in her multiage fourth and fifth grade classroom using a combination of whole class discussion, small group instruction and peer coaching. Dr. Kevin O'Connor, a principal in another school, interviewed Christine for over 60 hours and observed her classroom for his dissertation study. He has written about Christine's strategies in his dissertation, De-Emphasizing Grade-Level Identities: A Case Study of Mathematics Instruction In An Intermediate Elementary Multiage Classroom. What follows is an account taken from Dr. O'Connor's dissertation describing how the three strategies were effectively implemented.
Snapshot: Reflections on Whole Class Discussion
Christine's concept of whole class activity was not one of a teacher dispensing knowledge to the whole class. Rather, Christine orchestrates discussions among students in the entire class. The whole class discussions I observed appeared to be what are referred to as "congresses," meetings in which ideas about mathematics and mathematics problems are shared and reflections are expressed. The students hear about other students' thinking, encountering a variety of mathematical representations and multiple approaches to get solutions. The types of prompts Christine reported she used in these whole class discussions include:
What's the next step I might take here? What strategies could I list to go back and check my work? What are some different ways I could solve the problem and share those on the board? Was there one strategy that was more confusing for you to use than another? How would you go about solving this one? Make up a story about it.
When Christine asked students these types of questions about their thinking, she was promoting learning that goes beyond the mathematics itself. She was helping the students consider their own thinking related to the thinking of others. This metacognitive perspective, i.e., thinking about their own thinking and activity, encourages teachers to provide times for student to reflect on their activity and learning. Reflecting on one's own thinking can be encouraged as students discuss, draw pictures and make presentations. Teachers can model metacognitive activity by sharing their thoughts with students as they work through a problem, through prompts similar to those mentioned in her above commentary.
Christine tried to put students in positions where they need to encounter other perspectives. She describes how students construct new knowledge by modifying and refining previously established learning and cognitive views, when faced with alternative perspectives that need to be assimilated. Christine encouraged this by asking students to compare and contrast their thinking or approaches to each other:
By comparing mathematical ways of thinking, Christine appeared to believe that focusing on what one student knows and how it relates to what other students know facilitates learning. In articulating their thinking to their peers and trying to get their fellow classmates to understand their thinking, Christine believed that students could potentially discover their own inconsistencies and errors. In restating their views, students report more accurate and efficient solutions, that is, always progressing to better explanations and clarity in thinking.
Snapshot: Using Small Groups
Christine used a variety of grouping methods in her class. She sometimes assigned students to groups and at other times allowed students to select their own groups. She also used groups in a variety of dynamic and interesting learning environments.
Christine modified the student grouping procedures based on the purpose underlying the lessons. For example, she reported:
Maybe we'll 'fishbowl', that is, isolate a pair or small group of students. The other students gather around while this group in the center is working and talking about a mathematical problem. The inside grouping may get to a conflict point about a mathematics problem where they don't know what to do and maybe somebody on the outside might say, 'Why don't you try this' or offer another suggestion. The audience's first role is to be listeners, but then they can offer ideas about various alternatives and approaches. The activity promotes a focus while at the same time allows students to comfortably express their ideas.
We might also do 'Popcorn', an activity in which there are partners or a small group working on a task or discussing a problem. A student audience is observing. Periodically, one of the students in the working group leaves and another person from the audience takes their place; or, someone in the audience says 'popcorn' and takes the place of someone in the working group. It's an approach that assists students in accepting and recognizing each other's talents, approaches and diversity in learning styles and knowledge.
"Fishbowl" and "Popcorn" were types of activities that put students in the role of identifying their own strengths. When they knew they could offer assistance, they were able to take someone's place. The person who left the center could come back in later. The flow and interchange of students usually kept the students alert and ready. Sometimes Christine would have two groups involved simultaneously to allow for more participation by the students.
At the beginning of the year we all come in and I say to the kids, 'We all have three jobs, learner, coach and risk taker. That's who we are now.' I change their roles on them or have them identify which role they are playing as they work in groups on their mathematics problem situations.
Christine told the students that they would be working with many different students. Sometimes they would be learning from someone else. Sometimes they would teach.
Sometimes, Christine would put certain students together who needed assistance with the development of a particular skill. Then she met with them as they are worked to develop and master that skill. She said that this type of group was usually a short-term situation. The group stayed together until they were proficient with the skill. She also had longer-term grouping that usually lasted three to five or more class periods. The purpose of longer-term groupings was to help students learn how to work with each other for an extended period to reaching a goal of completion, usually on a substantive problem-solving task. For most of the long-term group situations, Christine assigned the members to the group. She made the assignment based on the mix of personalities and development of the members. She made the groups heterogeneous in nature so that those more mathematically developed students would have the opportunity to assist other students who are just learning concepts:
I want them to appreciate learning styles and differences beyond age. I don't think they see age first. They see other things. They know who is good at things and they know who struggles. I keep trying to mix them up. It's never one person is struggling all the time. At some point kids see each other succeeding and open to learning from each other.
Christine often changed the composition of the groups as lessons evolved. She had a concept in mind of who should work together, but made adjustments as she realized students were not working well together, were too easily distracted or were not challenging each other.
In multiage, older students can offer the voice of experience to younger students. This time of year [May], I want the older kids [fifth graders] to get the new fifth graders [this year's fourth grade students] ready. It is neat to hear older kids talk about their first year in the class and what it is like in the second year. It's powerful. ... In multiage, a 4th grader can see what a 5th grade or older student is like and it isn't so scary. This is a very reassuring experience for kids.
Christine looked closely at social as well as academic factors when she formed groups. Her purpose was to heighten the students' acceptance of differences and to lessen social issues and concerns that can occur. She planned the group so that she could get certain students together for social and emotional as well academic reasons.
Grouping helped to dissipate some of the social issues that students encountered. Each grouping gave students different opportunities for leadership roles. When they weren't in a group that they selected, they knew that there would be other opportunities and that the present group arrangement was short term. This helped to make the grouping arrangements more acceptable. Since Christine wanted more variety in student groupings and did not feel that she had to be the one to always create them, there were times she gave students opportunities to be responsible for creating their own groupings. This freed her of the time it would take to juggle students' names on charts to construct groupings that would work at optimum levels. After whole class discussions, students formed self-selected groups often based on how they heard other students were approaching a problem.
The groupings needed to allow for the changes that occurred with students' changing self-perceptions and attitudes about each other, their evolving interests and their developing academic development. Because of these ongoing changes, the make up and sizes of groups were fluid and changeable. For example, during one period of time students were grouped with regards to fraction work:
Currently, I have three groups of math for this class: One - Equivalent fractions, abstract thinkers who come up with rules on how to strategize. Two - Fractions as part of the whole. They need practice that is more concrete. Three - Geoboards and beans. Lots of manipulatives. The groups are not based on overall ability, but rather students' understanding of a concept as per assessments. Groupings in fractions will change as the concepts are presented. Grouping for geometry may be completely different. I get the information from short assessments or other activities that we do to determine the groups.
Snapshot: Peer Coaching
Christine established a culture of acceptance of the diversity of individual's talents and learning among her students. With this culture established, Christine was able to reduce the risk involved in students being labeled with the identifiers of "Novice", "Practitioner", "Apprentice" and "Expert".
An apprenticeship atmosphere was created in Christine's classroom when students who were achieving a high level of proficiency in a particular concept were recognized by other students as sources of help and information by those who were striving to learn that same or similar concept. The student who wanted to learn sought out the help of one who was more proficient, much in the same manner of the "master - apprenticeship" culture of feudal times and even currently in fields such as carpentry, electricians, and tool-and-die makers. Christine worked with students to enable them to understand how the apprenticeship model works, so that they would be more inclined to assist each other and seek help from each other.
Christine learned about these classifications from the Exemplars rubric, which became a tool that Christine used and adapted to unify the classroom experience. It enabled her to use consistent language and performance criteria across various aspects of the learning experience in her class including instruction, assessment, observations and commentary.
Using the language of the rubric, she evaluated the many kinds of mathematical products (writing, drawings, group work, dialogue, conversation) produced by the students, while encouraging students to identify their own level of understanding. She said, "Rubrics give every student a chance to determine their level. Determine where they are. We should do more with kids so that they can identify where they are at in their own learning."
Christine also adapted Exemplars to fit with the expectations of her textbook assessment:
I merged ideas from Exemplars rubrics and Chicago Math assessments to create a rubric for the class. I can evaluate their understanding and the strategies from these rubrics through sharing my interpretations with them, engaging them in dialogue and working with them to create visual representations of their understanding. These activities help them to understand themselves as well.
Christine used the levels of rubric as descriptors for the students' apprenticeship roles. She found that students willingly accepted the Novice, Apprentice, Practitioner and Expert roles associated with this rubric in part because the labels continually changed for individuals with different learning activities:
If I give everyone an opportunity to be a teacher and a learner, then there's not that stigma that a certain student always knows the answer or is always the learner or a person always knows more. As I keep grouping by different formats, there are always situations in which a child knows something that another one does not.
During one of my observations in Christine's class, she introduced a group activity for measuring the distance between cities using a scale of miles. Before doing this activity, Christine asked the students to decide for themselves whether they were Novices or Apprentices. Once the students classified themselves based on their self-assessment of what they knew, she had them get into groups of four. In most cases, two Novices and Apprentices were in each group. As they continued the work on the scale of miles project, they referred to each other with terms "Novice", or "Apprentice."
As I walked around, I witnessed the manner in which the Apprentices were coaching the Novices. Since the roles were clearly defined, the Novices did not have to pretend they knew the information or how to do the activity. It was acceptable for them not to know it. This acknowledgment made it easier for the Apprentice to do the coaching.
As students operated side-by-side, a struggling student could seek help from the student who was recognized as having the concept, regardless of age or grade. Sometimes, it was older students who had the knowledge. There were also situations in which younger students became the apprentices. Christine sometimes created situations in which particular students became coaches by meeting with a specific group of students herself:
In a small group, teacher-led situation, especially with kids at the younger developmental level, I will present a new task or assignment that they can learn and teach others, especially the older ones. It's their [the youngers'] job with a particular game or activity. I give it to them purposefully to go teach older students. It shatters barriers. Sometimes, the activity is just some of the 'warm ups' or ice breakers that these kids do. The activity puts them in a leadership role. On the other hand, I can have them work toward solving a math problem together. Then they take what they learned back to older students. It helps break the age and ability barrier because I give them these experiences from which to work.
Another illustration from Christine's class shows how the classroom culture she had established nurtured collaboration as well as appreciation for diversity among students. In the session that I observed, the students were working on the application of multiplication skills using a problem scenario which asked the students to determine how many stacks of dollar bills could fit into a suitcase. They were given the measurements of the stacks and the dimensions inside of the suitcase. In one group of four, two pairs of students worked, verified their answers among the four and then the group double-checked with the students in a different group. At this point, they were satisfied that they had correctly answered the question. This allowed Christine to work with other students uninterrupted.
Christine also worked to help the students learn how to be good teachers when assigned the role of coach:
In working together, I talk a lot with the students about wait time, giving your learner time instead of actually telling him or her the answer or asking someone else. Maybe what question they could ask. They're good at that by fifth grade.
Christine also helped students to accept different levels of accomplishment with new concepts or tasks:
There's a way for the students who don't understand a concept right away, to say, 'It's OK, I'm a Novice and know I will get it later.' It's just a new approach. It's not different than starting anything new.
Christine's students at an Apprentice level knew that in time, with continued experiences, the Novices with whom they worked would eventually understand. The Apprentices also knew that as they learned more about a particular concept, they would become Practitioners and Experts. She believed that this acceptance of varied roles of academic development also increased social acceptance and motivation.
Christine's use of coaching relationships reflected the way she accepted the range of developmental levels and the diverse abilities among the students as an essential instructional strategy. She put students in a variety of roles that changed depending on the mathematical lessons and individual students' circumstances. Christine's implementation of coaching provided for multiple teaching-learning situations simultaneously, unlike whole class teacher-directed instruction.
Her planning and management were further enhanced by the use of the Exemplars assessment tool. With the support that Exemplars provided, she assessed and recognized the differences in students' mathematical development and accepted and supported each individual in their differences of mathematical ability and social development. Her continual monitoring of student development provided the information she needed to constantly nudge students to new conflicts and higher levels of problem solving, yet remain within their own "zone of proximal development" consistent with the views of learning she put into practice based on the theories of Piaget and Vygotsky.
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