Learning mathematics in community accommodating learning styles in a second-grade problem centered classroom.
Specifically focusing on mathematics education, some scholars and practitioners argue that an alternative exists in a problem centered approach where, during a typical day:
the children first attempt ... to solve the educational activities in pairs or, occasionally, groups of three. The teacher move[s] ... from group to group, observing and interacting with the children as they engage... in mathematical activity. The teacher... [might then] call-the class together and orchestrate ... a discussion of the children's solutions. During this phase of the lesson the teacher [does] ... not explicitly evaluate the children's solutions or attempt to steer them to an official solution that she has] ... in mind. Instead, she ask[s] ... questions to clarify an explanation or to help a child reconstruct his or her solution. If, as frequently happen[s] .... the children ma[ke] ... conflicting interpretations or propose ... conflicting answers, she frame[s] ... this as a problem for the children and guide[s] ... their attempts to resolve the conflict. In general, one of her primary responsibilities [is] ... to facilitate mathematical dialogue among children. (Cobb, Wood, Yackel, 1991, p. 26)
Kathleen, the second-grade teacher whose mathematics class is our study's focus, heartily endorses problem-centered education, intuitively asserting:
There have been many gimmicks to try to engage children in mathematical activities, but they do not allow for the child to construct knowledge nor do they encourage sense making. I think if the child finds the importance of their thinking and they are constructing knowledge from within, then learning will take place.
Thus, throughout the 1999-2000 school year, we examined Kathleen's mathematical lessons to determine whether her students grasped the concepts without directly addressing their individual learning styles.
The concept of learning styles has its roots in Howard Gardner's Theory of Multiple Intelligences (Gardner, 1983, 1997), which challenges educators to broaden their conception of children's capabilities and how they might be stimulated. Dealing specifically with how youngsters learn, Dunn (1999), further identifies five categories of learning styles: reaction to classroom environment, children's own emotionality, sociological, preferences for learning, physiological characteristics, and global versus analytical processing. Each category contains several subcategories while assessment exercises and instruments exist to identify individual students' particular style. Flexible scheduling and rotation of classes is necessary to allow for subjects to be taught during individual students' peak energy levels. Research further shows that educators tend to teach in their own learning styles; therefore students should be matched with teachers who have similar learning styles because when educators force children to re ceive information in a style which is not their own, stress, frustration, loss of motivation, and lowered performance results (Dunn, 1999; Dunn & Griggs, 1989).
On the other hand, problem-centered learning involves three elements: tasks, groups, and sharing (Wheatley, 1991). The teacher selects challenging tasks, but students are not shown particular procedures for solving them. Rather they work in homogeneous pairs or small groups to devise their own meaningful solutions. Class members then come together to share their different solution strategies with each other.
The theoretical inspiration for a problem-centered learning environment is constructivism (von Glasersfeld, 1991). Learning occurs as students construct meaning for their experiences, and the learner acts and interacts with the world, actively trying to resolve conflicts while engaging in purposeful activity (Wood & Sellers, 1996). As students actively engage each other they try to resolve personal conflicts or differences between their existing ways of thinking and the aspects of their experiences. Resolution of conflicts takes place during genuine communication among students and teacher. Communication in this sense is not linear, from teacher to student or student to teacher. Instead, it is a circular process involving all learners, whereby students actively share, respond, negotiate, and listen while striving to interpret, for example, the mathematical meanings embedded within an activity.
Just as all scientific theory is negotiated among so-called expert adults, so too can students derive mathematical meanings (Cobb, Wood, & Yackel, 1991). Through the problem-centered approach, children hypothesize and continually revise their constructions with each new individual and group discovery and/or perturbation. In order for students genuinely to communicate and exchange ideas, an interactive relationship of support and mutual respect must be present among all class members. Then the classroom becomes a community in which in-depth interaction takes place to develop mathematical meaning (Wood & Sellers, 1996).
The particular classroom under study was part of a low to middle socioeconomic suburban public school, a site for the district's emotionally handicapped program. Of the 18 students, 8 were girls, and 10 were boys: 1 was gifted; 2 had been tested for and subsequently described as learning disabled, and 2 more were on a waiting list to be tested; 3 were emotionally disturbed; and 2 were enrolled in an English as a 'second language program. Ethnically, the class consisted of 2 Latinos, 1 Chinese American, and 15 Euro-Americans. Referring to Dunn and Griggs' (1989) and Dunn's (1999) categorization of learning styles, augmented by Kathleen's assessment, we determined that five students were emotionally responsible while four appeared emotionally low. Two students were global and needed to move around the room continually, and two others preferred to work alone.
The problem-centered mathematics lessons usually lasted one hour, beginning with a short whole group activity, a brief explanation of the problem to solve, paired or (in rare cases when a child found it difficult to communicate with another student) individual collaboration, and whole group discussion. By accommodating select individuals who chose to work alone, Kathleen attended to the sociological factors involved in learning styles (Dunn, 1999). In general, Kathleen's instructional activities emphasized tasks wherein she did not give students specific procedures for finding a solution; instead she encouraged them to complete the tasks in ways that made sense to them (Wheatley, 1991; Wheatley & Reynolds, 1999).
To frame our research, we employed Cobb and Steffe's, (1983) teaching experiment methodology. In this design the researcher can join the teacher as a co-facilitator and, thus, is a participant observer, working with the teacher and students and listening to their interactions and ideas (Steffe & Thompson, 2000). In so doing, two of the researchers observed the mathematics lesson one day each week throughout the school year. They also monitored and questioned students where appropriate, helping to work out solutions to assigned tasks and clarify thinking. The primary author took extensive field notes during each observation period, supplemented by transcribed video recordings of these lessons. She also interviewed Kathleen for approximately 45 minutes after each observed lesson and transcribed those texts.
We divided each of the above data sets (field notes, teacher interviews, and class video tapes) into three categories, those involving: tasks, small group collaboration, and whole class sharing. To ensure trustworthiness or validity, independently, we coded each category within the individual sets and compared resulting themes among categories in each set and then across sets. Employing a constant comparison method we coded continuously throughout the school year, periodically assessing individual results (Strauss & Corbin, 1998). As concepts and relationships emerged, we investigated them further, using these as frames to focus our subsequent classroom observations and interactions.
Analysis of Data
The data for this article derived from preparatory work for and the actual study of a balance task activity. (See Figure 2.) It was intended to help students determine the relationship among numbers, relating specifically to addition and subtraction (Wheatley & Reynolds, 1999). Although we discussed Kathleen's and the students' experiences separately, the classroom in this study (comprising teacher, students, and artifacts) was like others--a self-organizing entity and a complex adaptive system (Varela, Thompson, & Roche, 1993). All elements were interwoven because educator and students negotiated the classroom environment as they attempted to make sense of and respond to each other's actions.
When investigating these relationships we focused on socio mathematical rather than social norms, the latter being the roles played and expected in a complex classroom environment; they could be the same for all subjects, the guide for the whole day, and/or the general behaviors expected within this classroom (Cobb & Yackel, 1996). The socio-mathematical ones were those behaviors that were pertinent only to the students' mathematical activities. In short, a social norm was an explanation of a solution, but the understanding of what counted as an acceptable mathematical explanation was socio-mathematical (Yackel, 2000).
The Teacher's Role in the Students' Activities
Acutely aware of fostering socio-mathematical thinking by building on prior knowledge, Kathleen prepared students for the balance task activity through other activities that developed an understanding of relationships among numbers. For example, a few days prior to the lesson, Kathleen had asked the children to choose any number between 10 and 30 and find as many ways as possible to partition that number into equivalent sets. In the whole group discussion, she began with, "As I went around I noticed many of you chose 20 and 30. Let's talk about your ideas:' One student said that he had, "10 plus 10 plus 10 equals 30:' and showed his solution. Kathleen then asked if anyone else had come up with three 10's, and several students said, "Yes:' She queried, "Is there another way?" A couple of students demonstrated how they came up with 2 sets of 15. Kathleen then elicited more responses by asking, "Are there any other ways?" One student said, "6 sets of 5' and another said "5 sets of 6." Another then observed that those two solutions were the same. So Kathleen asked the class, "Are they the same?" The students began discussing and working with their manipulatives to discover the answer and decided that there were different solutions but that the combination of numbers worked because they, as the children all agreed, were "reversed."
Kathleen followed up, posing, "Does this make sense?" A student called out, "15 plus 20 equals 30," and the teacher asked if anyone had anything to say about that solution. One youngster said, "I don't think 15 plus 20 equals 30" and began counting the tiles she had been using to help her find a solution earlier while asking the first student if he would explain his solution using the tiles. As they both counted, a third student realized that some tiles had been counted twice. In the process, they were able to make sense of each other's thinking and obtain a correct solution.
Meanwhile, Kathleen recorded on the chalkboard the various solutions as they were presented. She began the next lesson by reminding the students they had talked about sets of 30 and asked if anyone wanted to make a comment. She asked them again about the "reversal" (drawing on their knowledge and words) they had noticed the day before. The students then realized that there were more "reversals," and in so noting Kathleen continued to enrich her own understanding of their individual and collective mind sets. This progressively would enable her to devise developmentally appropriate lessons throughout the year.
Opening Activity: Double Ten Frames
The balance task activity was one. To introduce it Kathleen presented the "double 10 frames" concept. (See Figure 1.) She selected this exercise because it promised to foster mathematical thinking and encourage students to relate those ideas to the day's task. She displayed Figure 1 for approximately three seconds and then asked students to share what they saw. Kathleen encouraged them to share different ways of seeing the dots by asking questions such as, "How many did you see, and how were they arranged?" 'How many sets did you see?" "Did you see a pattern?" One student noticed 4 dots missing on each (the empty spaces in each frame) so he knew it was 6 and 6, which makes 12. Thus, he thought of the 6 as 10 minus 4. Another student saw 4 rows of 3 and knew that when added together they equaled 12. When asked about Kathleen's purpose in using an opening activity like this, she responded:
I think it is a way to get kids to get their minds engaged to be thoughtful--the engagement of ideas, the conversation that comes about because of them. They (opening activities) are a good way to warm up to what [we] will be doing for the rest of the lesson-getting them to talk to each other.
After the opening activity, Kathleen introduced the task at hand. (See Figure 2.) Having listened to the students in small group interactions and during whole class sharing time, (the session first discussed in this section) Kathleen was able to select an activity to enhance their mathematical thinking and sense making. Rather than finding a different task for each student she always chose one for the entire class that all the children could complete in a variety of ways, depending on each one's level of sophistication. She displayed the task on the overhead projector, asked the students what they knew about balances, and was careful not to impose particular analysis methods on them.
The children's discussion centered on the meaning of the word "balance" with the students noticing that each was level. Kathleen then told them that their goal for the day was to find numbers that would make each balance level, where one side was equal to the other. Then she asked if there were any questions. There were none, so the students went to work finding solutions for the balances. Primarily, they negotiated what was intended in the assigned task, "finding numbers that make it balance," as several children noted, rather than establishing procedures to use in finding these numbers. Kathleen commented to us that she attempted to take herself "out of that position where they [students] are looking to me for knowing how to do it."
Small Group Collaboration
While students worked on the assigned task, Kathleen walked around observing their work, taking notes, and asking questions as she learned about the mathematical thinking of individual students. She did not look for right or wrong answers but for a variety of strategies being used by the students. As stated and demonstrated, Kathleen's intention was to be aware of methods that students used so she could be prepared to orchestrate the whole class discussion by discussing all of them. Explaining to us her routine she said:
As I walk around, I ask lots of questions about how did you see that and get them to talk. I will ask the partner [within a given pair of students] if that makes sense to them and/or pull in another couple sitting close by. I am trying to see what they are making sense of as well as disturb their thinking or facilitate them in taking another step.
In so doing, Kathleen deftly monitored and encouraged each student's participation. She was almost always able to understand their thinking processes and determine when assistance was necessary.
Sometimes she helped youngsters develop mathematical meanings and aided making sense of or explaining solutions. Usually, her facilitation came in the form of posing provocative questions or initiating dialogue. Other times she encouraged children to work cooperatively or to listen to each other's explanations. Kathleen's role in her students' small group collaborations or, with the case of two students, individual deliberations supported each one's own construction of mathematical ideas.
Whole Class Sharing
Unlike the traditional lecture-oriented classroom where children wait for the teacher's answers or explanations, the class discussion gave the teacher and students opportunities to listen to many children's explanations and elicit responses from others. Kathleen encouraged the exchange of ideas among all the students. She expected them to explain and justify their solutions during this time while, as in the other settings, maintaining respect for each other's ideas. For instance, she asked questions such as, "Does anyone else agree with this idea--why or why not?" "Did anyone solve it in a different way?" or "Does anyone have a question or comment about this?" She expected the students to listen to one another, make sense of the ideas, and ask questions of those who were presenting their solutions. Thus, Kathleen modeled appropriate types of inquiries that the students could ask one another during the whole class discussion, deepening their mathematical thinking.
Kathleen's actions encouraged true mathematical dialogue among her students. They were not just giving correct answers to procedures but also offering explanations for and clarifications of the strategies involved in finding solutions. This helped students sharpen their thinking and make conceptual connections from one problem to another. The conversations also taught students to develop a language for expressing mathematical ideas more clearly. With Kathleen's patient insistence, as the children attended to each others' ideas and expressed their own, they learned to listen, paraphrase, question, and interpret. The classroom goal was to make sense of tasks, negotiate meaning, resolve any conflicting ideas, and strengthen students' reasoning abilities. In the process, they learned mathematics through communication and to communicate mathematically (NCTM, 2000).
Kathleen was aware of her important role in determining the extent and quality of students' mathematical understanding and knowledge. However, she did not perform this duty through the traditional means of clearly explaining procedures and providing students with adequate time to practice, but by promoting curiosity and intellectual autonomy (Kamii, 1982). She enhanced learning through interaction in a risk-free environment, encouraging students to take responsibility for making sense of the various mathematical tasks. To this end Kathleen expected and encouraged respect for each other's right to speak and made allowances for the youngsters' mistakes.
In order to develop this type of atmosphere, Kathleen and her students negotiated and renegotiated the classroom's, socio-mathematical norms. From the tile manipulatives example, the students were expected to give an answer as well as be able to explain their solutions mathematically and understand each other's solutions. When the youngsters disagreed about an answer, Kathleen expected them to resolve the issue mathematically in their own way. The negotiation of these norms was an ongoing process, occurring with each task or perturbation, because Kathleen was more interested in the students' reasoning strategies than in managing their behavior.
Because Kathleen was interested in the students' thinking and developing autonomy she chose tasks that led to conflict and discussion.
In order to promote group understanding of individual children's thought processes she used their own words when clarifying, asking a question, or repeating so that all could hear (Cobb, Wood, & Yackel, 1991). In this way the youngsters heard their own original solutions or responses and reflected and reorganized their thoughts so others could better understand each one's perspective (NCTM, 2000).
Reflection, therefore, became a way for students to strengthen their mathematical reasoning abilities and develop autonomy. The children felt free to disagree with each other and not feel threatened when other students disagreed with them because the end result was individual and collective understanding. For example, in the double ten frames opening activity students discussed an answer to an addition task in which half the class had one and the other half had a different answer. Kathleen asked several students from both groups to explain and demonstrate their solutions. After several minutes of explanations, she realized that the groups had reached a deadlock. She then asked them to get with their partners, use the white boards, think about their solutions, and try to develop explanations that would help the other groups understand their thought processes as well as the mathematics involved in this particular task. Thus, their cognitive processes and solutions became objects which could be examined. They w ere thinking about their own thinking. Giving the students time to reflect in this way helped them to become aware of their methods and options, which promoted internalized meaning (Wheatley, 1992).
As the students' responsive interactions (regarding the balance task in Figure 2) indicated, Kathleen's teaching went well beyond encouraging youngsters to follow given procedures, encouraging their appreciation of the mathematical relationships involved in adding and subtracting (Wheatley & Reynolds, 1999). Kathleen's goal was not to complete every problem, but to develop students' additive reasoning (Smith, 1997). Children completed the tasks in their own way, using whatever made sense to them. A variety of manipulatives were always available; as a result students developed personal problem solving strategies. As earlier suggested, the children later used them consistently in various problem solving tasks throughout the year.
Small Group Collaboration
Students demonstrated that the construction of knowledge was not only based on the immediate situation but what they had experienced mathematically prior to this activity. (The particular balance task being solved is in parenthesis with "f' denoting the balance point.) For example, solving (6 6 /_____) Ken said, "Six and six are twelve; I remember from the ten frames?' (Recall that the class opener that day had been a double ten frame with six dots in each frame.) Therefore, Ken referred to this earlier activity in explaining his solution to the balance task. In addition, resolving (9 7/_____) Melinda stated, "I remember the other sheet." She picked up the double 10 frame overhead master and demonstrated with the pennies, arranging 9 pennies in the left 10 frame and 7 in the right then moving one from the right 10 frame and putting it in the space of the left 10 frame, making 10 and 6. (See Figure 3.) This also illustrated the use of a "malting 10's" strategy. So both of these students found solutions based o n prior experience with the ten frames activity.
Other students found that using their knowledge about doubles was a good strategy for solving tasks. Working the problem (6 6/_____) Joey stated, "I remember from my doubles--twelve." Ben added, "All my problems are doubles because I know all my doubles; like 30 and 30 equals 60." Contemplating (7_____ / 14) Rex replied, "Seven goes in the blank because it is a double." And regarding (______ 7 / 15) Kristy commented, "It's eight, because seven and seven are fourteen and fifteen is one more." Moreover, Dan was able to use the double seven for finding a solution to (_____/8 6) . He added one to seven and took one away from seven, stating, "It is like seven and seven. I took one from seven and added that one to [the other] seven so six and eight equals fourteen." He was able to convert the double seven appropriately to find the solution. Another student, Missy, solved a balance task (13 ______ / 24) by looking for a multiple often. She explained, "I made it thirteen plus seven equals twenty and another 4 to get 24; so it's eleven-7 plus 4." She was able to make a 10 and solved the task easily, as the other students also built on their prior knowledge about doubles to understand and solve the balance tasks (Wheatley & Reynolds, 1999).
Whole Class Sharing
Kathleen did not specifically "teach" any particular strategies, but rather provided opportunities for students to share and use them to solve particular tasks. She encouraged students to make sense of each others' solutions during whole class discussion and challenge each other by disagreeing with solutions or asking questions of clarification. Therefore, they revised their thinking through negotiation and argumentation. For clarification and understanding, Kathleen incorporated explanations at the end of each activity. She did not, however, make presentations as an authority figure, telling students they had right or wrong answers or grading their assigned tasks. Instead, she asked if anyone disagreed, had a different answer, or had a question.
The following discussion is instructive: it took place as students presented their thinking concerning the balance task (6 _____ / 12). Joey said, "6 plus 5 equals 12." "1 don't remember how we did it." Aaron explained, "That is 11 because 5 plus 6 is 11, but 7 and 5 equals 12." Then Kim said, "6 and 12 equals 16. 1 did it on my calculator and got 16." At that point Kathleen asked if anyone had anything to say. Several students disagreed with Kim's solution, indicating that "6 and 6 equals 12." So Kathleen asked: "How can you prove that?" Dan responded, "Because there is 6 here and 12 here (pointing to the 6 and 12 in the balance task) so that means 6 goes in the empty one." Others in the class responded that 6 and 6 was one of their doubles and that they had it on the 10 frames earlier. The teacher then asked Kim if she agreed and she shook her head (yes) and changed her equation from 6 + 12 = 16 to 6 + 6 = 12. Kim had originally constructed the problem as 6 + 12 rather than as 6 + _____ = 12 which is how mo st of the other students had thought about it. So, in this situation the students took responsibility for helping each other through mathematical discussion.
Thus, through collaboration the children employed a variety of thinking strategies to contemplate large numbers--breaking them into smaller parts, relating to smaller units, and using sets and doubles. When numerous students finished their tasks they presented each other with several challenge problems that they, personally, had created. The youngsters were excited and wanted to do the extra work. For instance, Bill's challenge problem was: 30 + 30. Don said, "that was an easy problem," so Kathleen asked him what he thought might be a challenge problem, to which he replied: "Fifty plus fifteen. I added ten and then I added five, equals sixty-five." Here Don thought of 15 as 10 and 5, adding 10 to 50 to get 60 then another 5 to get 65. Another student posed: "What if you had 50 dogs and then 52 more dogs?" Dan answered, "Fifty is like five and one-hundred is like ten so it is one hundred-two. Fifty and fifty-four dogs (creating his own problem to illustrate this idea) equals one hundred-four because fifty is l ike five and ten is like one hundred." Then another student suggested the following: thirty kittens and seventy puppies. Ben responded, "It would be 100 because 7 (pause), 8, 9, 10 equals 100," holding up three fingers and touching each finger as he counted, "8,9, 10." Here the youngsters used their sense of the number system's structure to relate previous knowledge of number combinations to 10's and 100's.
Other students also made contributions which led class members to think creatively. For example, Don asked, "How many balls if I have 4 sets of 15?" And Helen responded, "I know! 60! 15, 15, 15, 15," holding up four fingers and touching them as she counted. She then grasped two of the four fingers together--then the other two as she said, "30 and 30 are 60." Helen was able to think of 15 as a unit, double it, and then double that answer (30) to get her final answer.
After snack time, another youngster challenged the class with the following problem: "What is 36 plus 36?" During mathematics time Richard, Joey, and Zoe decided to work on this challenge. Richard had written, "36" and "36" on one side of a blank balance sheet and stated, "36 + 36 = 81." And Joey replied, "No," Richard said, "Remember the snack problem, 36 + 36 = 81?" Joey picked up the tub of unifix cubes, making 3 groups of ten and 6 one's. He did this twice. Then he proceeded to count, "30, 40, 50, 60, ... 1, 2, 3, ... 72." Zoe then found a 100's board, located the square with 36 on it and proceeded to count on it by 1's to get 72. Both students used the manipulatives as a way to explain the reasoning they had already cognitively developed. In this way manipulatives facilitated the learning of mathematics and gave meaning to their activity.
Irrespective of the classroom diversity, all of the students took responsibility for malting sense of the problem in their own ways and debated among themselves about correct solutions. For example, in the small group discussion, Joey (identified as learning disabled) used his knowledge of doubles to solve the problem correctly. Moreover, the whole group process fostered cognitive mathematical development and autonomous learning, as the Principles and Standards for School Mathematics champion (NCTM, 2000). For instance, with Kathleen's direction and the students' help, Kim, labeled as emotionally disturbed, was able to work through her initial comment that 6 + 12 = 16 to understand that 6 + 6 equaled 12. Not only Kim, but all of the students seemed to find their own voices and were never belittled for having done so. Either voluntarily or through mild prompting, each was willing to go beyond the problem at hand to create additional ones dealing with the current concept-problem.
Many children also used different manipulatives to help think through given problems. Thus, while representing a singular lesson plan, in some respects, the problem-centered approach proved flexible enough to accommodate a wide variety of students: bringing them together in collaborative ways; helping them individually and collectively make sense of addition and subtraction; and nudging them to take responsibility for their own learning.
By creating an environment wherein youngsters were able to develop their own thinking processes, exchanging and debating, Kathleen's students collectively and individually negotiated socio-mathematical norms. In so doing, the problem-centered approach accommodated an average, highly diverse group of second-grade youngsters' varying learning styles and abilities. The interchanges displayed self-confidence and a willingness to make sense of their mathematics along with the other students. It is exactly this type of climate which gives rise to scientific advancement in any number of fields, promoting the advancement of human knowledge (Cobb, Wood, & Yackel, 1991).
Although the classroom social norms were not a focus of this study, Kathleen and her students created a democratic climate wherein they all came to appreciate each other's similarities and differences.
For society as a whole, the potential by-product of this extends far beyond children's increased mathematical thinking skills (Darling Hammond, 1998). Each child, to varying degrees, entered the mind of another- seeing beyond his or her own way of reasoning. The implication for improved human relations that may result from this discovery are legion.
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|Publication:||Focus on Learning Problems in Mathematics|
|Date:||Jan 1, 2003|
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