What’s Right with Writing Math?
A Discussion about Mathematics Instruction
Sunil Santoni-de-Reddy
December 8, 2006
The subject of mathematics is not usually equated with language arts. Generally, people who consider themselves “math people” do not have the same passion for the humanities; the reverse is also often true. When one thinks about math, perhaps what comes to mind are numbers, symbols, and equations. When one thinks about language, presumably what is considered is vocabulary, grammar, and the composition of thoughts and ideas translated into written text. In math, concrete solutions are found whereas language arts subjects usually are based on material that is open for interpretation. Seemingly, then, the teaching of mathematics and ideas associated with the teaching of reading are dissimilar. However, despite the explicit content being from two different worlds (science versus humanities), the acquisition of mathematical knowledge is largely enhanced by principles of reading and writing. Strategies used to foster the positive development of reading in children may also be applied to mathematics instruction. Considering this, math teachers should devote greater attention to incorporating reading and writing into their lessons.
There are many educators currently discussing and researching the implementation of reading and writing strategies in math instruction. Most significantly, traditional ideas about what “literacy” means have come into question. Paul Deane (2004) believes that literacy in the 21st century has taken a new definition and form. Deane suggests that the term is entirely too vague, but cites that “Minimalists define literacy as the basic set of skills required to function on a job—skills that include math and writing as well as reading” (Deane 49). Deane posits that to be literate in something means that an individual has become competent in understanding the characteristics of the thing and the skills needed to put the thing into practice. Thus, people are literate in math when they can perform mathematical operations and explain the processes behind these operations.
Something I value greatly from Deane’s article is his discussion on literacy levels and how the development of one’s reading and mathematical literacy is not, and therefore should not be, bounded by time. The author admits that literacy is often cultivated in the grade school setting where children get “trained” (50) in reading, writing, and arithmetic. Deane does not believe that enough energy is spent on safeguarding one’s literacy development beyond primary and secondary school. Parents, Deane states, should also be “trained” for their own literacy development and because he believes that “literacy is a family, rather than individual skill” (49). Parents should be considered because they have the power to help their children achieve greater heights of educational understanding.
I agree that promoting greater math literacy with parents is necessary and beneficial for children. It is a reality that many parents cannot help their children with math homework, and too often this is a result of their own limited experience with math or because they have forgotten what they learned when they were in school. Thus, in order for parents to serve as a support during homework time, more “parent time” should be spent on conceived literacy activities at school. Just as we did with the literacy bags, I believe that I can assign homework to my students that will ask parents to be involved in some way. I also think it is important to provide overviews for the parents about the things that are being done in class. Minilessons for the parents may not be such a bad idea.
One of the ultimate links between math and reading/writing is communication. Teachers and students alike communicate with each other in the math classroom so that a transfer of knowledge can occur. Patricia Moyer (2000) highlights the “important role that communication plays in helping young children construct mathematical knowledge and form links between their informational notions and the abstract symbolism of mathematical ideas” (Moyer 246). Moyer asserts the importance of introducing mathematical language and concepts to children at an early age. The author admits that math is all around young children but that not enough teachers and parents point out the existence of math. Moyer offers the examples of a child getting half of a cookie or hearing that one object is a shorter distance away from another (247). In a sense, math exists in our surroundings just as environmental print does. We are exposed to numbers and units of measurement, shapes, and fractions, to name a few examples, but more adults should be explaining the math behind everyday objects and occurrences in order for it to become significant to the child.
Moyer cites that there are hundreds of children’s books that develop math concepts but are never used in the classroom. Making these books part of the math or English libraries would help reinforce both subjects. I think the discussion of books with written text and illustrations is important because Moyer acknowledges that diagrams and pictures are often helpful, if not necessary, for students to look at and make sense of with the lesson in mind (248). Moyer also identifies another link between language and numbers:
When language skills are embedded in meaningful contexts, they are easier and more
enjoyable for children to learn. In the same way, numbers and their operations, when
embedded in meaningful real-world contexts, give children the opportunity to make sense
of mathematics and to gain mathematical power (248).
Using real-world examples is helpful when teaching math because many students will understand the material better if it pertains to some aspect of their everyday lives. Additionally, “when students write about math they [place] the subject in a context which makes sense to them” (Baxter 131). Thus, the strategy of putting things into context is another commonality shared by the teaching of reading and the teaching of mathematics.
Dan Fleming (2004) posits that teaching math in a classroom can become restrictive and that more instruction in other resource areas benefits student appreciation for mathematics. Reflecting on libraries, Fleming admits that most people would characterize the space as one catered towards English and other humanities based classes. Fleming states: “Math is probably the last thing that comes to mind when you thing of the library, but information literacy doesn’t just apply to the written word” (Fleming 43). The author refutes the claim that math cannot be encouraged in the library and goes so far to say that math should sometimes be taught in this space. Libraries are resource locations, and a student’s math skills can be strengthened if this student is asked to conduct research at the library for a class project in math. I believe it will be important for me to incorporate literature about math into my class lessons. Very often math is taught as a series of equations and properties, but students do not get the history behind many of the theorems and concepts that they implement on their homework each night. I have made the decision to incorporate research projects into my curriculum. While my students may not like the idea of writing essays or preparing presentations about the history of math, I believe this will deepen their knowledge of mathematics while also honing their writing skills.
Fleming also believes that “practicing math skills in the library strengthens students’ abilities to apply and understand math” (43). Not only will supplemental material found in the library foster a greater understanding of math principals, but I also believe that teaching a math class in a library would be beneficial because it would expose the students to a new setting in which they can learn and appreciate math. I believe that the more opportunities a student has to engage in subject-related material outside of the classroom, the more this student will enjoy the subject. I think there are many students who do not feel accountable for learning math outside of their school classrooms, and this makes it harder for the students to accept math into their lives. With the implementation of math discussions in the library or in other classes, it is presumed that there is a greater likelihood that students will not shy away from the material but instead will become more comfortable with it.
Fleming speaks to all teachers in his article and stresses the importance of teaching across subjects. He asks teachers outside of the math department to infuse math into the work they assign to students. He suggests that, for research projects and papers, students can be asked to analyze statistics and create graphs or charts about their findings, and that doing these things would show the students that mathematical relationships exist in all books and other literary works. Simple questions such as how many years an author published a book after their birth year are relevant information in a research project about the author but also the questions get the student to practice math. Clearly this is a simple example; it would be more difficult to have students in a calculus class, for example, incorporate their class material into an English report. Students could be asked to create original words problems for the class to solve based on the research they conducted (44). No matter, the idea remains that there needs to be a greater practice of interdisciplinary work in schools.
When thinking about the subject, mathematics can be thought of as a written language with symbols representing different operations just as letters represent different sounds in spoken language. Laura Novick (2004) points out that “numeracy is the mathematical counterpart to literacy, and likewise is a key goal of K through 12 instruction” (Novick 308). With numeracy, students should be able to understand the mathematical principals behind many real world phenomena, and Novick suggests that creating diagrams is one effective method for students to achieve this goal. Diagrams, which Novick states are “among the oldest preserved examples of written mathematics” (307), help students organize their math work and communicate their ideas to other people. I believe this heavily relates to reading and writing: words are used in language to convey feelings, beliefs, and ideas just as diagrams will help a student communicate their understandings of a math problem.
Clements and Sarama (2006) also discuss the importance of visual representations of math in learning the subject. The Agam Program, the authors mention, is based on enhancing visual literacy through the study of geometry shapes. Building children’s visual memory of shapes and pictures will help them understand much of geometry later; math vocabulary is shared at a young age. For example, a parent of a toddler can hold a glass up to the child and trace the glass’ lip while saying “circle.” After the word is repeated many times with different circular objects being traced, the young child will eventually recognize that the shape and word go together (Clements 12). This reminds me of sight words because new vocabulary is introduced to the child until he or she has been exposed long enough to store the information in their working or long term memories.
Most significantly, Novick (2004) addresses six types of mathematical knowledge: implicit knowledge, construction knowledge, similarity knowledge, structural knowledge, metacognitive knowledge, and translational knowledge (Novak 310). Math students approach a word problem, for example, and there are certain pieces of implicit facts that the students will construct meaning from, relate to previous ideas or concepts, and reflect on what they have learned so that they can communicate to others how to best approach the problem. A similar process occurs when students approach a new vocabulary word: they attempt to construct meaning from the word using context clues and using past information about possible analogous words, and ultimately they will translate what they know and learn by using this word in the future.
Numeracy, or “mathematical competence,” as Novick later describes it, requires that students use the appropriate representations of mathematical concepts so that their thought processes can be easily communicated. This also relates to reading and writing since language involves selecting the right words to use in order to express a thought. Students justify their math reasoning or thought processes by using the best diagram or vocabulary to most accurately convey what they are thinking (315). I will require my students to write down all of their steps and explain to me how they got a solution because by reading their words I will get a clearer understanding of their thoughts and math reasoning. Thus, math and language are intertwined.
Maryann and Gary Manning (1996) add to the discussion of writing in math. The authors clarify that writing in math “is to help students clarify and extend their knowledge in [this] subject area” (106). Writing is an expressive act and enables one to communicate their thoughts onto the page. Baxter et al. (2005) believe that math in the United States is simply seen as arbitrary rules that make little sense because instruction is focused too much on getting an answer than understanding how to reach the answer (Baxter 119). When students approach a math problem for the first time they may feel anxious and not feel that they will be able to figure out the problem. Throughout my math career I have been told—and have told my own students—that the best way to start is to write down what kind of problem it is and what the best starting point for the solution is. Burns and Silbey (2001) agree that when students face difficulty the teacher should ask them questions such as “Tell me what you’re thinking,” or “What kind of math operation is being used in this problem?” The teacher should listen to the students’ responses and then advise the students to write down their ideas (Burns & Silbey 19). Writing down each step is helpful because the student is kept from getting lost in the solution. By writing down what they think is most appropriate, the students use mathematical reasoning and have a basis for justifying their answer at a later time. Should they make an error, they can go back and find the step that they may have made a wrong turn at.
The Mannings suggest the use of “content journals” in math classrooms. They state: “content journals provide a way for students to review or interpret information discussed in class or read in a text” (Manning 106). I believe content journals are helpful because it allows the students to reflect on past notes and also keep a record of things that they have learned or things they have had trouble with. If I were to implement the use of content journals I would have my students write in their respective books for a few minutes at the end of each class period. I would ask them questions about what they learned from the lesson, if there were any interesting examples, and if they had further questions about anything discussed. Perhaps at other times I could supply the students with a question or problem to solve and they would have to solve it in their journals. The authors suggest asking questions of the students including what the students know about the material, what they wish to discover, and what things weren’t discussed. These types of questions remind me of the KWL questioning technique. I would make sure to collect everyone’s journals to review at the end of the week or even daily if the time presented itself. The review of my students’ journals would help me discover who understood the material fully or who was still having trouble. Thus, the journals would serve as an assessment tool.
Written conversations are another method the Mannings include in their article. Writing. This extension of learning involves the students writing down comments about a lesson and then trading their journals with another student’s to make comments on (107). The partners’ questions or comments would reflect how much knowledge of the material they acquired and how much extra thought was put into the lessons by them. Again, with written conversation journals, students can either be given a topic to write about or they can come up with their own. In either case, the purpose of the writing is to discuss the math content presented in the class that day. While I think it would certainly be beneficial to collect the conversations periodically to see what the students were thinking about, I also believe that it would benefit the class for the conversations to be shared sometimes. I think that any questions or concerns that a student has to offer would benefit his or her peers.
Hari Koirala’s (2002) article fully supports much of what has already been discussed. Koirala introduced the article by describing the National Council of Teachers of Mathematics (NCTM) and their standards for math learning. The council “emphasize[s] that students should be able to communicate mathematically, both in written and oral forms, using mathematical vocabulary and notations […] writing in mathematics provides opportunity for students to express their thinking” (217). Again, allowing students to reflect on material covered through writing responses to lessons helps the students understand what they know and what they still need to learn about the material—this closely relates to metacogniton. Additionally, teachers who review written work in math journals not only see what their students are learning and having trouble with, but this also gives the teacher a chance to reflect on his or her teaching. If many students seem confused about a lesson in their journal entries, then teacher should go back and modify their lesson plans. Thus, not only can the teacher evaluate the students’ progress, but the teacher can also engage in self-reflection via indirect evaluations from the students.
After reviewing his own students’ journals upon initial implementation, Koirala observed that “more than 80 percent of the students simply chose to express their beliefs about mathematics. Although they were asked to demonstrate their mathematical understanding in their journals, no such understanding was noted […] the journals were rather brief and did not demonstrate any in-depth reflection from students” (219). I think that perhaps the author’s journal idea failed initially because the students did not know what was being asked of them or perhaps they found writing in their journals boring and a waste of time. In order to avoid something similar happening to me in my classroom, I would communicate to my students the importance of keeping track of where they’ve been and where they are going in math. If students are aware that reflections are important for understanding the material, I think they will be receptive to the idea of spending a short period of time jotting down some notes about the lesson. I think there also needs to be a balance of teacher-prompted questions and free response. If students are left to design their own topics then perhaps it becomes easy to lose track of the purpose of the journals in the first place. The danger in never allowing a free response is that kids may not feel they are given the outlet to admit they do not understand something.
Koirala adds that journals are important because they illicit participation from every student in the class. It is often the case that many students become marginalized in class, either by choice or because not enough attention is paid to them by their teacher. If a student does not participate regularly in class and the teacher has a hard time making him or her feel comfortable to speak in front of a peer group, then there is greater value placed on the journals because the silent student is still able to communicate with the teacher. This is not to say that the teacher should stop making attempts to draw the student out, but more importantly it is beneficial for both student and teacher to have an open dialog. Ultimately, then, there are many reasons why journals are useful tools in the classroom, and using them in the math classroom is just as relevant and necessary as using them in an English classroom.
Mary Burns (2004) is another educator who has written about writing in math. Although she admits that when she started teaching math and writing were “like oil and water [… she] can no longer imagine teaching math without making writing an integral aspect of students’ learning” (Burns 30). The author discusses four primary uses of writing in the math classroom. First, Burns discusses how math journals and logs provide a chronological record of student learning experiences. Second, she highlights how, in solving math problems, students are asked to show many strategies and write reflections on how they went about their solution. Next, Burns says that teachers should encourage and require the explaining of mathematical ideas through assigned essays on math concepts or asking students to write a “What I Learned” essay after a given unit. Last, the author describes how students should be asked to write about the learning process (30-2). This last suggestion is based on metacognition and it is what we are asked to do in class all the time, to reflect on what we are learning and how we are learning it.
I believe that any or all of the four strategies for incorporating writing into the math classroom could be implemented successfully by most teachers. Thinking ahead, I know that I will require my students to explain the steps to their math solutions on homework and test papers so that I can follow along with their thought processes and give them partial credit for displaying that they knew how to approach the problem. I also think that I will have students maintain math journals that I could collect and review on a frequent basis. Since it will be difficult for me to know what each one of my students is thinking or feeling about the material being taught, the journals will help me reach some level of understanding my students better. I believe it is important for the reader to give feedback and so I will pledge to reflect on their entries and make comments in their journals to continue the conversation. I wonder how much time this will require of my schedule; I worry that I will not have time to review everyone’s journals every day or even every week. Perhaps I could review the journals on an alternating schedule so that each week I am reviewing a fraction of the total class journals. Last, while I believe doing research on math concepts is important I cannot imagine myself assigning too many in-depth reports. I suppose I can have different activities that will incorporate research but will ask the students to display the information differently each time—completing essays, posters, and PowerPoint are some of the options.
The discussion of various writing exercises in math activities is also discussed by Brandenburg (2002) who believes that students become literate in mathematics because writing aids in comprehension. Writing, and reflecting on what’s written, trains students to “pinpoint any confusion, compare and contrast methods, and ultimately deepen their understanding and retention” (Brandenburg 68). The author also discusses math portfolios, but the method for constructing a portfolio differs from the type of portfolios we have discussed in class. Brandenburg’s portfolio assignment asks students to write about each math unit from the semester and explain the unit while demonstrating knowledge of what the unit entails mathematically. Our alternate approach to using portfolios is to have students submit their best or favorite pieces of work from the units to include in their individual portfolio. I believe there are good things included in each approach: in the first, students can look back on what they have done in class and how they have grown or learned since the date of the assignment. The second method asks students to select the most significant items of a unit to study from. I believe both methods can be infused for use in the math classroom: students can submit pieces of work after each unit and at the close of the semester can go back and reflect on the units using their work.
Ryan et al. (1996) introduce readers to the “Write Now” math and science program that incorporates writing into opening classroom activities. As it is described, students come to class and are expected to answer a question that the teacher has written on the blackboard. The teacher-chosen math problem should be based on material covered in the previous day’s class. Students answer the question by writing down their steps and indicating the methods that they learned the day before (78). The activity asks students to recall information and prior knowledge and put the methods into their own words. More interestingly, students are asked to read their written solutions out loud but they must read exactly what they have written in their explanations. This stipulation hones students’ writing and communication skills. The students need to determine what the best method for solving a math problem is and subsequently write direction on how to solve the problem concisely and in a may that the rest of the class will understand. The students learn, form this activity, that mathematical information can be conveyed to others but that effort must be put into explaining the methods as clearly and as simply as possible so as not to bring about confusion in the reader.
In closing, there are many similarities between mathematics and reading or writing instruction. Most common among all three is a belief in student reflection through journal writing. Writing allows one to discover what is known and what is still desired to know. Math vocabulary is learned by the student to help them make sense of larger math concepts. Ultimately, numeracy and language literacy should be fostered in the classroom and encouraged for all ages to maintain.
References
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