Reasoning about the Multiplication and Division of …



Statement of the ProblemAs an elementary school teacher I have encountered certain math concepts that are difficult to teach. By difficult I mean, without the procedure I learned as a child and use as an adult, I am unsure of how to explain the problem to my students.? Two ideas that I understand procedurally, but not conceptually, are multiplying and dividing of fractions.? For example, I was taught that to multiply two fractions you multiply across and then simplify.? Likewise, for division of two fractions, you invert the second fraction and then multiply across.?I find that my teaching of fractions is limited to repeating and remodeling the same procedures over and over again. I don’t know how to engage my students in reasoning around this subject. It is unclear to me how to connect these concepts to my students’ lives and therefore make them worthy of learning. I would like to increase both my own and my students’ understanding of multiplying and dividing fractions through improved instruction, student participation, and class discussion. In order to make these improvements in my classroom instructional practices I need to know what types of conversations I should be having with my students about fractions? I need to know what prior knowledge is necessary for students to have in order to enable them to engage in mathematical reasoning about fractions. I am curious as to what mathematical reasoning looks likes in regards to multiplication and division of fractions? Lastly, I feel that it is important for me to determine how other teachers are approaching multiplication and division of fractions with their students? To begin to examine these questions I have looked at examples of textbooks, read dialogue in which children are reasoning about multiplication and division of fractions, and have researched different literature regarding mathematical reasoning and the teaching of fractions.Review of the LiteratureIn, Multiplication of Fractions: Teaching for Understanding, Cramer and Bezuk (1991) suggest using the Lesh Translation Model, (1979), as a way to increase conceptual understanding of multiplying fractions in students. The Lesh Translational Model suggests 5 modes of representation including: real-world, manipulatives, pictures, spoken symbols, and written symbols. (Cramer and Bezuk, 1991, p. 35) “Relationships between and within these modes of representation are called translations. A translation is the reinterpretation of a concept from one representation mode to another or within a single representational mode.” (Cramer and Bezuk, 1991, p. 35) For example, if students are presented with a story problem this is considered to be a representation of real-world. If the problem is then worked out using a geoboard or counters the problem has been translated from real-world to manipulatives. Cramer and Bezuk (1991) suggest that “organizing instruction around these translations enhances students’ understanding of fractions and improves their ability to transfer and apply this understanding to unfamiliar problems. Below is a diagram of the Lesh Translation Model: Spoken SymbolsWritten SymbolsReal-World SituationsThe Lesh Translation Model (1979)PicturesManipulativesCramer and Bezuk emphasize that even though students can perform the algorithm for multiplying fractions, it doesn’t mean that they have a conceptual understanding of why they do the steps or why the answer makes sense. In this article they suggest asking quantitative questions involving estimation of the answer and estimation of size of answer in comparison to the size of each factor. Students are encouraged to engage in mathematical reasoning by judging the reasonableness of the answer they get. (1991, p. 36) I think that this is a good model to use in both multiplication and division of fractions. Presenting problems in one of the modes and asking kids to translate them to another mode seems like a really effective way to strengthen the conceptual understanding of the problem and to increase student flexibility in solving different problems involving similar concepts. In, Understanding Division of Fractions, Bezuk and Armstrong guide teachers through a short, 5 day unit of real-world division problems that increases in difficulty as the days proceed. Although it is not specifically stated, they use the same method of translating between different modes of representation to build students’ conceptual understanding of dividing fractions. In the form of spoken symbols, they state, “dividing fractions can be viewed as counting the number of parts of a certain size that it takes to cover another part of a certain size.” (1993, p. 43) They then use real-world problems to illustrate this concept to students. To solve the real-world problems they use the manipulative mode of representation. For example, one of the problems they present the children with on the first day is as follows:Paula and her road crew can resurface 1/8 of a kilometer of highway in one day. 1/8 of a kilometer is represented by this strip: __________________ --------------------------__________________ In May, Paula and her road crew resurfaced the ? kilometer section of highway represented below. How many days did it take to finish the work? First try to estimate how many times the 1/8 kilometer strip can be measured out of the ? kilometer section of highway. Then use the strip to check._____________________________________________________________________________ ------------------------- ------------------------- --------------------------- --------------------_____________________________________________________________________________You can think of this problem as ? of a kilometer divided by 1/8 of a kilometer per day is _______.On the fourth day they have evolved to questions such as: List five ‘division of fraction’ problems that equal 3. Describe a pattern that you see in these equations (other than that they are all equal to 3). How would you convince someone else that your problems are equal to 3?You can see that the level of cognitive demand increases as students’ conceptual understanding of dividing fractions increases. The unit that Bezuk and Armstrong present begins as “procedures with connections” as described by Stein and Smith, in that there is a suggested pathway to follow that has a close connection to underlying conceptual ideas. There are multiple representations such as visual diagrams, manipulatives and problem situations. (1993, p. 348) Later in the unit problems develop into “doing mathematics” as described by Stein and Smith. There is no predictable pathway and the problems require students to explore the nature of the mathematical concept. Furthermore, later problems “require students to access relevant knowledge and experiences and make appropriate use of them in working through the task.” (1993, p. 348) They emphasize the importance of having students work collaboratively on the problems and encourage discussion as a mechanism to further strengthen conceptual understanding. Note that the algorithm for dividing fractions is never introduced to students. This short teacher’s guide demonstrates that the Lesh Translation Model is versatile and can be used in teaching mathematical reasoning about multiplying and dividing fractions. These two readings helped me to understand ways in which I can strengthen my students’ conceptual understanding of multiplying and dividing fractions. I did get a couple of ideas about when multiplication and division of fractions might be used in real life, however, would still like to explore this question further. These articles touched a bit on mathematical reasoning by mentioning the idea of judging the reasonableness of answers. However, I feel like the readings provided a pretty structured way of teaching and I am curious to see what it looks like when the students are allowed to explore and openly discuss these concepts among themselves, in a more student-centered atmosphere. I originally wanted to know why the procedures we traditionally teach work, however am now wondering if that is even important. Some of the readings suggest avoiding teaching those procedures until high school! Both Chazan, in Beyond Formulas in Mathematics and Teaching and Lampert, in Teaching Problems and the Problems of Teaching, have ideas about how to create good whole class discussion about mathematics. In Chapter 4, Chazan describes some impressive techniques used by Lampert. According to Chazan, Lampert has a very unique way of approaching teaching mathematics. Instead of assigning problems for students to do after an algorithm has been taught, she assigns the problems before any algorithm has been taught. This forces students to explore and reason their way through the problem. She has found that this allows students to approach math problems in more innovative ways. (Chazan, p. 121) Furthermore, Chazan states that “Lampert portrays school mathematic like the discipline itself, as a living and growing field in which developments occur when people create solutions to problems. She encourages her students to see mathematics as a field in which one makes hypotheses and revises them.” (Chazan, p. 122) This makes so much sense to me but it is completely opposite of how I was taught mathematics and of how I was taught to teach mathematics! Chazan states that we must “avoid categorizing mathematical statements as right or wrong.” It places the teacher as the authority and forces students to take our word for it. It also causes students to devalue their own reasoning and logic. Instead, he cites an example in which two of his students spend a week arguing a misconception (zero is not a number) and he lets them hang on to this argument even though the class disagreed. He discussed the idea that the mathematical community considers zero a number and was able to introduce the idea of accepted views in the field of mathematics. (Chazan, p. 137) Through this process there resulted a lot of mathematical conversation in which the teacher wasn’t the center of instruction. When teaching math, Chazan and Lampert do not focus on obtaining an answer to questions they propose to their students. Instead, they value the rationale for student responses. They want students to discuss the logic and evidence that they use to solve a problem in order to convince their classmates that their responses make sense. (Chazan, p. 127) In, Teaching Problems and the Problems of Teaching, Lampert walks the reader through the steps she takes in engaging students in a conversation about a specific problem she has proposed. There are several things that she does in order to inspire conversation. First and foremost, she aims to teach students that “they are responsible for reasoning through a piece of mathematics.” (2001, p. 159) She practices asking questions in a way that provokes student responses, whether or not they think they have solved the problem. For example, she might ask, “Who has something to say about A?” (2001, p. 145) Another technique she uses to guide the conversation is strategically choosing who to call on. She watches their independent work and knows what each has to offer. She can then choose the individual whose information is particularly useful for consideration at that time. Then she can direct the class to consider what the speaker has said. (2001, p. 146) In terms of classroom conversation, Herbel-Eisenmann suggests using a focusing-interaction pattern. This pattern, as Herbel-Eisenmann suggests, serves many purposes including, “allowing the teacher to see more clearly what the students were thinking or requiring the students to make their thinking clear and articulate so that others can understand what they are saying.” (2005, p. 486) This method is less teacher-centered and puts the responsibility on the student to explain their thinking. The teacher’s role is more to ask clarifying questions that they anticipate other students might have. In her planning, Jill Lester tries to make the problems that she presents to the students, “challenging but accessible.” (1996, p. 89) She is patient at the beginning of the year and begins teaching the process of mathematical reasoning by asking, “how they had arrived at the answer”. (1996, p. 90) She frequently provides manipulatives for children to use to validate their answers. Lester gradually moves from a teacher-centered learning environment to a student-centered learning environment. By the first month of school, during one of her math classes, she notes, “It was also clear that they were paying absolutely no attention to me.” (1996, p 100) The kids were interacting with one another and listening closely to each other’s explanations.Chazan struggled with the question, “Why should I learn this?” when teaching Algebra to his middle school students. He believed that the key was to think of Algebra as a way, “to capture the relationships between quantities – where quantities are qualities of experience that you can quantify – that change. When you begin to think that way, functions become things that you see all around you.” (p. 13) He realized that to teach middle school Algebra successfully he would have to make it relevant to his learners. He decided to ditch the text book and create meaningful projects. Chazan realized that he didn’t know half of what went on in his students’ lives. This disconnect was a barrier to creating meaningful Algebra problems for his kids to work on. He tried things like asking them to write stories about measurement. “I have to find ways for them to teach me about them - instead of me making assumptions about their lives.” (p. 14) He also boosted student efficacy by letting them go out and interview local businessmen about “the role that measurement and quantity play in their work.” (p. 14) It turned out that many of the businessmen didn’t even realize that the calculations they do could be written down. The students actually taught their interviewees something through this project. (p. 14) Lester and Chazan have similar strategies to approaching mathematics with their students. Each teacher demonstrates questioning as a strategy to encourage thinking in their students. When a wrong answer is given these teachers allow the students to discuss what makes sense and what does not. Each teacher has made the effort to connect the math that the kids are learning to their lives in some way. I notice that more time is spent on one problem; the goal being to develop and demonstrate a conceptual understanding of the math involved. In each class the student’s show enthusiasm and begin to interact with each other instead of relying on the teacher for all the answers. In chapter 8, Lampert discusses helping children to make connections across lessons. She chooses to uses problem context to connect ideas across lessons. “More pedagogically challenging is the problem of figuring out how to use a context, both to bring students in contact with the connected universe of important mathematical ideas that the problems posed make available and to make the contact that they have with those ideas productive of learning.” (2001, p. 211) In the lessons she describes, she uses time-speed-distance relationships relating to the story, The Voyage of the Mimi, to connect ideas about fractions, division, and ratio over time. By using a story that the kids are reading it makes the math meaningful and “worthy of investigation”. (2001, p. 181) She also encourages students to communicate about mathematical ideas by using graphic representations. Along the way she mentions the difficulty deciding how organize the work of her class without creating a dependence on her. She decided to provide them with a diagram (journey line) for consistency and to make communication about the mathematics easier. Most of her students were creating variations of this diagram in their personal notebooks and were not yet using it to “figure out the relationships in the problem.” She expresses her uncertainty as to how much she should direct her students verses letting them come up with as many variations on the correct diagram as they could think of. The dilemma being that if she directs them entirely she will become the expert but if she lets them go in different directions it will make communication about the mathematics very difficult. (2001, p. 196) Student work is often a tool she uses in this process to help her create a foundation on which future lessons can be built. By looking through their notebooks she can determine where their understanding lies and can tailor her instruction to best meet their needs. Another part of her work in helping students to see the important relationships among mathematical concepts she calls anticipation. She notes the importance of developing the foreknowledge that she needs in order to take full advantage of the opportunities that could arise as students work in a particular problem context. (Lampert, 2001, p. 184) In chapter 9 Lampert discusses connecting the required topics of the curriculum to the student’s work and creating a situation in the classroom in which these topics seem worthwhile of investigation to the students. With experience on her side, she notes that by using a mental checklist of all the topics in the curriculum, she is able to take advantage of opportunities in which these topics might come up through the independent investigations of her students. (2001, p. 213)What she does in her classroom by using problems is allow the students to uncover the math as they approach a problem in which the math is necessary. She documented her student’s work over a period of 6 class periods and found that they “covered” the math in an order completely different from that of a curriculum framework or textbook index. She found that in order to connect the math she couldn’t teach one topic at a time, one after another. Instead, students must be “simultaneously engaged with several topics in each lesson.” (Lampert, 2001, p. 217) Put another way; students are, “moving back and forth between big ideas and the facts and procedures that logically flow from them.” (Lampert, 2001, p. 217)“A real-world problem is one that someone, perhaps even the student, might encounter outside of school. The notion is that students will be attracted to problems that implicitly suggest that mathematics is a useful body of knowledge that allows people to solve problems they face in their lives.” (Chazan, D., Beyond Formulas in Mathematics and Teaching, p. 40)Mode of InquiryIn order to investigate my questions I focused on collecting classroom data from experienced teachers. I looked at both their approach to the mathematics and their approach to encouraging children to become actively involved in mathematical reasoning. First, I collected data in the form of teacher-suggested strategies from different experts on teaching methods. I was observing their approach to the math. From these resources focused on teaching methods I recorded the following: What prior knowledge is suggested?What strategies does the author suggest using when introducing multiplication and division of fractions?Does the author suggest using pictures and/or manipulatives?Second, I looked at actual classroom conversations in which students are engaging in mathematical reasoning. In these samples I am focusing on what mathematical reasoning looks like in a classroom. I am paying particular attention to the role the teacher plays and what types of questions they ask to facilitate mathematical reasoning among the students. ResultsApproaching the Content:SAMPLE 1:Marilyn Burns begins her lesson on multiplying fractions by reviewing six statements about multiplication that the students had created together in a previous lesson.Six statements about multiplication: (Burns, 2003, p. 13) Multiplication is the same as repeated addition when you add the same number again and again.Times means “groups of”.A multiplication problem can be shown as a rectangleYou can reverse the order of the factors and the product stays the same.You can break numbers apart to make multiplying easier.When you multiply two numbers, the product is larger than the factors unless one of the factors is zero or one. (or a fraction smaller than one)She continues this lesson by posing each of the above statements to her students by asking them to think of them in terms of fractions.She begins with statement one and asks, “Let’s think about this statement together with this problem – six times one-half,” (Burns, 2003, p.15) Burns encourages students to, “Talk with your neighbor about how you might make sense of this problem.” (Burns, 2003, p. 15) This leads students to discuss repeated addition. They decided that:6 x ? = ? + ? + ? + ? + ? + ? = 3 To facilitate discussion, Burns asked questions like, “Does it make sense to you?” and “Can you explain what Juanita did?” In regards to statement two Burns asks her students, “Does it make sense to read ‘six times one-half’ as ‘six groups of one-half’?Going on to the 3rd statement she asks: “Do you think we can use rectangles to show multiplication problems when we’re multiplying fractions? Can we draw a rectangle to show six times one-half?” (Burns, 1994, p. 15)She used the rectangle for 6 x 1 to help students visualize what a rectangle might look like for 6 x ?. The students came up with the following rectangle: 6 ??This leads a student to ask the question, “What about if both the numbers are fractions?”Burns replies, “Let’s think about drawing a rectangle for the problem one-half times one-half.” 1 1/2Burns models the drawing of this rectangle when her students are quiet and unresponsive: 1/2 1Burns uses statement 4 to address the language that can be used when discussing multiplication of fractions. She states, “If we think about the times sign as ‘group of’ then one-half times six should be ‘on-half groups of six.’ But that doesn’t sound quite right. It does make sense, however, to say ‘one-half of six’, and leave off the groups part. This sounds better, and it’s still the same idea. What do you think ‘one halve of six’ could mean?” She is able to help her students make sense of the language of fractions and in turn give them another way to think about what it means to multiply fractions. (Burns, 1994, p. 17)With statement number 5 Burns engages students in a discussion about the problem 6 x ? again. “Talk with your neighbor about how you could apply this statement to the problem six times one-half.” She wants them to decide whether you could break apart one of the numbers to help with the multiplication. One student decides that you could break the six into three twos. 6 x ? = (2 x ?) + (2 x ?) + (2 x ?) = 1 + 1 + 1 = 3Another suggests breaking the six into two and four.When the last statement is addressed the students decide it cannot be true for fractions because the answer of 3 is less than the factor 6 above. They modify it to state “…unless one of the factors is zero or one or a fraction smaller than one.” This particular sample illustrates a method used to introduce multiplying fractions to a group of students. The teacher’s questions help to encourage the students to create their own understanding and strategies to solve the problems. She connects this new information to what the students have already come to understand about multiplication and helps them to revise their previous understanding. She models the use of rectangles to illustrate certain problems. She also addresses language used in multiplying so that students will gain a better understanding of what it means to multiply fractions. SAMPLE 2: Burn’s students have a strong understanding of division of whole numbers already. They have created the following statements about division that they revisit during their introduction to division of fractions: (Burns, 1994, p. 76)You can solve a division problem by subtracting.To divide two numbers, a ÷ b, you can think, “How many b's are in a?”You can check a division problem by multiplying.The division sign (÷) means “into groups of.”The quotient tells “how many groups” there are.You can break the dividend apart to make dividing easier.Remainders can be represented as whole numbers or fractions.If you divide a number by itself, the answer is one.If you divide a number by one, the answer is the number itself.If you reverse the order of the dividend and the divisor, and the quotient will change.The first problem that Burns presents to her students is 3 ÷ ?. The first statement is used to solve the problem through repeated addition until zero is reached. The second statement causes Burns to ask her students, “How many ?’s are in 3? One student drew 3 circles, cutting each in ? to illustrate the 6 ?’s within the three circles. The third statement is used to check the answer of 6 that the class had come up with. Through this discussion, one student noticed that doubling the dividend and the divisor made the problem even easier:6 ÷ 1 = 6When the problem 4 ? ÷ ? was proposed to students, one student decided that statement six from the above list would help make the problem easier to solve. She changed 4 ? into 4 + ?.4 ? = 4 + ?4 ÷ ? = 8? ÷ ? = 18 + 1 = 9So, once again, Burns draws on her student’s prior knowledge. She is helping them make connections to both division of whole numbers and their new understanding of multiplication of fractions. SAMPLEX or ÷Prior knowledge usedQuestions posed by teacherStrategies used by studentsPictures or maniputatives used1XSix statements about multiplication of whole numbersDo you think that all of these statements are true when we think about multiplying fractions?Repeated additionBreaking numbers apartRectangles2÷Rules for division of whole numbersMultiplication of FractionsDo you think that all of these statements are true when we think about dividing fractions?Repeated subtractionBreaking apart numbersCirclesI think these two data samples help to answer my question about what type of prior knowledge students need in order to demonstrate mathematical reasoning. They need a deep understanding of both multiplication and division of whole numbers. I think that mathematical reasoning was evident in the way that students drew upon common knowledge that the class had obtained earlier in the year. I think it was also demonstrated in the way that students took risks to use their understanding of multiplication and division to create strategies to solve new types of problems. Students used tools that they were familiar with, such as rectangles and circles, to illustrate their thinking to other students. They also applied familiar strategies such as repeated addition and repeated subtraction to show others that their answer made sense. Students even simplified problems using strategies such as breaking apart numbers, another method that demonstrates their ability to connect their prior knowledge to the new problems they are trying to solve. SAMPLE 3:In, Elementary and Middle School Mathematics, John A. Van de Walle makes several suggestions of the teaching of multiplication and division of fractions.Van de Walle suggests using the concept of equal shares to begin multiplying fractions. He suggests that asking students to first find a fractional part of a whole number is a good place to start. For students who are ready to approach multiplication of two fractions here is a sample problem he has provided and his suggestion on how to think about the multiplication necessary.“You have ? of a pizza left. If you give 1/3 of the leftover pizza to your brother, how much of a whole pizza will your brother get?” (Van de Walle, 1990, p. 323) He remarks that the problem is 1/3 of 3 things. “The focus remains on the number of unit parts in all, and then the size of the parts determines the number of wholes.” (Van de Walle, 1990, p. 323) Another example of this is as follows: (Van de Walle, 1990, p. 323)? x 4/5? of 4 things is 3 things ? x 4/5 = 3/5 In more difficult problems such as the following:The zookeeper had a huge bottle of the animals’ favorite liquid treat, Zoo Cola. The monkey drank 1/5 of the bottle. The zebra drank 2/3 of what was left. How much of the bottle of Zoo Cola did the zebra drink?Van de Walle maintains the importance of the different roles that the top and bottom number play. “The top number counting and the bottom number naming what is counted play an important role.” (Van de Walle, 1990, p. 324) He notes that in the problem above you are finding thirds of four things, the 4 fifths of the cola that are left.One interesting problem that he uses in his book is as follows:3/5 x 2/3One strategy he suggests is counters. (Van de Walle, 1990, Figure 17.8, p. 324)2/3 is 10 counters1/5 of 10 is 2 counters3/5 of 10 counters is 6 counters 3/5 x 2/3 6/15 or 2/5Another strategy he suggests is using the Communitive Property.Switching the problem around to read 2/3 of 3/5 is easier to immediately visualize using the same method he suggests earlier. Finding 2 of the 3 things is simple. Van de Walle does suggest introducing a task that could perhaps lead students to the traditional algorithm used for multiplying fractions. By using grids students can illustrate problems such as 3/5 x ? as follows: (Van de Walle, 1990, p. 325) By first drawing all the lines in one direction students are representing the ?. Next, students can be instructed to divide that region into fifths. Then, by extending the lines it becomes evident what fractional part each little square is of the whole. He suggests that the teacher should avoid pushing the students to formalize any rule. Let students notice on their own that the number or rows and columns are actually the two numerators and the two denominators. SAMPLE 4: Van de Walle also suggests certain strategies for introducing division of fractions.In the problem,Cassie has 5 ? yards of ribbon to make three bows for birthday packages. How much ribbon should she use for each bow if she wants to use the same length of ribbon for each?Van de Walle suggest thinking about 5 ? as 21 fourths. Then it is easy to divide 21 by 3 to determine that there will be 7/4 yard of ribbon for each bow. In more difficult problems in which the numbers must be split apart as follows, Van de Walle has another strategy he suggests.Mark has 1 ? hours to finish his three household chores. If he divides his time evenly, how many hours can he give to each?Because the 5 fourths Mark has for his chores doesn’t split evenly into 3 parts, this problem is more difficult. For this type of problem he suggests using pictures such as number lines and counters to illustrate dividing each of the fourths into three equal parts, creating twelfths. Because there are a total of 15 twelfths there is obviously 5/12 for each chore. There are two algorithms associated with division of fractions. One is of finding the common denominator and the other is the algorithm for the actual division.To help students develop the common denominator algorithm he suggests using pie pieces and fraction strips to model converting each fraction into the same fractional part. To help students develop the invert and multiply algorithm he uses the following example:A small pail can be filled to 7/8 full using 2/3 of a gallon of water. How much will the pail hold if filled completely? (Van de Walle, 1990, p. 330)The task is to find the whole. A full pail is 8/8. The water in the pail is 7 of the 8 parts needed to fill the pail. Therefore, dividing the water by 7 and multiplying by 8 solves the problem or fills the pail. (Van de Walle, 1990, p. 330)Sample #X or ÷Prior Knowledge NeededStrategies Suggested to use in teachingPictures or manipulatives used3XMultiplication of whole numbersFinding fractions of a wholeThe focus should remain on the number of unit parts in all.Use the word “of” in place of “times”.Use the Communitive Property if it makes the problem easier.Use pictures to illustrate the problem RectanglesCountersGrids4÷Division with whole numbers2 meanings of division: partition and measurementStart with whole-number divisors.Think about mixed numbers as fractional parts. Keep in mind that the question for partition problems is “How much is one?”CountersNumber-linesAgain, I think that this resource does emphasize the need for a deep understanding of multiplication and division of whole numbers. Therefore, it is important that the teacher is connecting these new ideas to what the students have already learned. Van de Walle stresses the importance of choosing the numbers used in story problems carefully so that they work to illustrate the concept that you are trying to teach in that particular lesson. He uses a lot of drawing and illustrations and suggests the use of counters in working to create a common denominator. I think that his methods could be applied in a classroom in which the goal was to encourage mathematical reasoning.Observing Mathematical Reasoning in the ClassroomSAMPLE 1:In, Keeping Out Right Answers, Marty Schnepp, a once traditional teacher of mathematics, visits Chazan and Bethel’s classroom to witness a new way of teaching math. “Kids in her class were thinking, and finding ways to represent that thinking…These kids who would have to write, or make tables and graphs to represent ideas, they were able to draw the mathematics out of situations…discussions led to new inquiries, and ideas came from students.” (Chazan, p. 15) He decided to adopt their methods and to use their materials in both of his Algebra classes.One example of mathematical reasoning in Schnepp’s class was when they were presented with the problem: 4x-5 Write the rule in wordsMake a table of six inputs and outputsFind the input(s) that make zeroDescribe all inputs that make the output negativeWhen kids began sharing their answers for part (a) one student wrote, “Take the input. Multiply it by 4 and subtract 5.” When Schnepp asked his class, “Anyone have anything at all different?” Another student spoke up and said, “I have ‘Take 4. Multiply it by the input and subtract it by 5’” (Chazan, p. 16) Schnepp pushed his students to continue thinking about these two statements by asking, “What’s different about that?” This led the kids into a discussion about when order matters. The students began thinking about subtraction, addition and then division and discussed whether order matters in each of those situations. They are examining concepts such as the commutative property without memorizing a definition. “…students think through and explain their solutions instead of seeking or trying to recollect the "right" answer or method.” (NCTM Reasoning and Proof Grades 3-5, Cobb et al. 1988) The other thing that this example illustrates is how important it is to articulate mathematical descriptions accurately. This became a lesson in the language of math and what difference one word can make. This is a good example of mathematical reasoning because the kids are comparing two different answers and are working to evaluate which one makes sense. They are using discussion to clarify a mathematical concept. They are learning about how to talk about math.SAMPLE 2:One example of children doing math reasoning in Jill Lester’s class was when they were working on the following problem: If each of 15 6th graders and each of 25 second graders need paintbrushes, how many shall I buy? Lester accepts the answers 52 and 40 and asks the kids, “Are there any answers here that bother you? One child offers that 52 is too big because, “There are only three 10s and two 5s.” (Lester, 1996, p.99) Another agrees, “There’s one 10 in 15 and there are two in 25. That can’t be 52.” (Lester, 1996, p.99) By accepting all of the proposed answers, she encouraged kids to, “assess their answers for reasonableness”. (Lester, 1996, p. 96) She frequently asks, “Are there any answers here that bother you?” (Lester, 1996, p. 99) This is a great way to provoke healthy debate without pointing the finger at any individual. Kids in her classroom did not seem to mind when others challenged their answer. It had simply become an accepted process as students continued to build their understanding of math.She allowed the kids to then work in small groups. Some groups proved the answer was 40 in several ways. They shared their justification for their answer with the whole group by stating such ideas as, “We took the two 5s – one from the 15 and one from the 25. We added them together. That made 10. 20 is two 10s. There’s a 10 from the 15. That made 40. (Lester, 1996, p. 100) She challenges her second graders to think about, “how it was possible that three seemingly unrelated solution processes could produce the same answer”. (Lester, 1996, p. 92) This is a good example of mathematical reasoning because both kids are considering the reasonableness of the answer. They are also solving the problem in multiple ways and providing justification to the group for their answers.It is one month into the school year and Lester makes several observations of her 2nd grade students as she watches their ability to reason about math develop. The children present right and wrong answers, all of which are written on the board by Lester. Students have learned not to be upset when their answer is disputed by a peer. Groups are working together to solve this problem in several different ways. They are dying to share their ideas with their classmates. The kids are no longer simply being quiet while others share to wait their turn to share. Now they are listening to each other and are commenting on each other’s responses. Lester sums up their progress by stating, “The children had listened to one another in order to figure out ideas and how they related to their own solutions. And they had validated each other’s solutions without looking to me for direction or support.” (Lester, 1996, p. 102)The kids were interacting with one another. They were responding directly to the comments other students were making. Lester makes the observation, “…no one looked to me for recognition.” (Lester, 1996, p. 100) This means that they had moved past appeal to authority, as Carpenter, Franke, Levi (2003), describe it. They were exuding confidence and the energy in the room was high. Lester observes the kids while they volunteer to be called on, “Jack is bouncing up and down in his place. His voice almost too loud. Keith is up on his knees and pulling on his earlobe” (Lester, 1996, p. 100) Even the quietest kids are participating. “Everyone seemed to be involved in the process.” (Lester, 1996, p. 101) Conclusions and LimitationsFrom this research several things have become clear to me with regards to mathematical reasoning about the multiplication and division of fractions. Most importantly, I have determined that through creating an environment in which children are taught to reason about math, a teacher can make any subject in math reasonable. So, while I started by focusing on the multiplication and division of fractions, I have decided that my biggest gains in understanding have been in the area of making math, in general, reasonable. Mathematical reasoning in school is imperative in teaching students that math is logical, can be tested, and can be proved. If students are engaged in mathematical reasoning in school they are making connections between the math they are doing in school and real life. When students are reasoning about math they are using their prior knowledge to try to solve a new problem. They are being innovative and using creativity. They can demonstrate why their strategies and answers make sense to other students in the classroom. They aren’t asking the teacher whether an answer is right or wrong. Students are using language that is mathematical and familiar to others in the class in discussions about math. With regards to the multiplication and division of fractions, students need to have prior knowledge of the multiplication and division of whole numbers. Real-world problems help to create sense of “worthy-of-learning” among the students. Using manipulatives and number lines help students to visualize the math that is happening. Choosing alternative language and familiarizing students with the different way people can say “times” will help over all understanding to evolve as well. To support mathematical reasoning about the multiplication and division of fractions, a teacher must approach math in a non-traditional way. It is important to allow students to work on new problems without first teaching the algorithm. They must force students to use their own creativity and understanding of math. Problem-based mathematics which connects the math from lesson to lesson over time and which connects the math to real world problems can help engage students and can help them to make important connections to the math that they are doing. Providing different ways of illustrating the mathematics is important; use of diagrams, manipulatives, and pictures can help students to gain a stronger understanding about math. A classroom culture is important in facilitating mathematical reasoning. One that is student centered, allows both independent and group work, encourages mathematical discourse between students and requires that ideas be explained is most supportive of that goal. The discussions that students engage in should not be teacher-centered but the teacher must be there to ask students for clarification when needed. Student responses should be focused on explaining their logic and why the math they did makes sense. The teacher is not after a right answer but instead sound reasoning and evidence presented by students. The responsibility to reason through problems is placed on the students. A classroom culture that is flexible in allowing students to discover math through a natural process instead of rigidly planned curriculum is also important. I realize that the biggest limitation to my research is lack of classroom access. I would have found it beneficial to gather my own data throughout the year as my students reasoned their way through math. I think it would have been helpful to experience some of the benefits to the methods that the literature on mathematical reasoning suggests. Next StepsNow that I have learned about what it takes to make math reasonable I think there are several “next steps” I could take. One of the most interesting ideas I came across was the idea of a problem based curriculum. The connections across lessons and between the math and real life seem very valuable in the effort to make math reasonable. This leads me to ask the following questions: How can teachers who are bound by a structured curriculum and regular, standardized assessments create and maintain a problem based curriculum?Have other teachers found ways to create a problem based curriculum in which the multiplication and division of fractions is covered? What resources are available for a problem based curriculum in which the multiplication and division of fractions is the mathematical focus?Resources:Burns, Marilyn (1944). Teaching Arithmetic: Lessons for Multiplying and Dividing Fractions, Grades 5-6. Sausalito, California: Math Solutions Publications, c2003. Bezuk N.S., Armstrong B. E., Understanding division of fractions.?(1993). The Mathematics Teacher,?86(1),?43.? Retrieved October 22, 2009, from Research Library Core. (Document ID:?5244277).Chazan, D. (year) Keeping out right answers. In H. Featherstone (Ed). Changing Minds. Michigan Extension Service: Michigan Department of Education.Cramer, Kathleen,?Bezuk, Nadine.?(1991). Multiplication of Fractions: Teaching for Understanding.?The Arithmetic Teacher,?39(3),?34.? Retrieved October 22, 2009, from Education Module. (Document ID:?1862133).Herbel-Eisenmann, Beth and M. Lynn Breyfogle. "Questioning Our Patterns ofQuestioning." Mathematics Teaching in the Middle School 10 (May2005): 284- 289.Lampert, M. (2001). Teaching Problems and the Problems of Teaching. Yale University Press: New Haven and London.Lester, J. (1996). In Schifter, D. (Ed): What's happening in math class?Vol. 1: Envisioning new practices through teacher narratives. New York: Teachers College PressVan de Walle, J. (1990). Elementary and Middle School Mathematics. New York : Longman.? ................
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