Fractions.A word that often scares teachers when it comes to that time of year.Fractions are a very hard concept for students to grasp.The idea of multiplying and dividing things that aren’t a complete whole can be very confusing.It is rather easy to simply teach the students to memorize algorithms but any true math teacher would want to know that the students truly understand the rationale behind the algorithms.
So, here I am in my classroom and it’s time to add fractions with unlike denominators.I currently teach a class of fifth graders ready for the middle school math curriculum, which I am teaching them this year.They know, from being told, that when you add and subtract fractions you only add the numerators.When I gave them a quiz with multiple problems such as 3/5 + 2/7, they all aced it.I wasn’t satisfied, though.I wanted to know if they understood why.So, I asked them to explain to me in writing why, when you add fractions, you only add the numerators.Here are some sample responses: “You only add the numerator because the denominator means how many parts the whole was divided into and the denominators shouldn’t change unless you are finding a common denominators to add the problem.” “You add the numerator and not the denominators because the numerator is the fractional portion out of the whole and it can get larger but the denominator is the whole and it can’t get any larger so it can’t be added.” “When adding fractions you do not add the denominators because you don’t change the parts of the whole.”This activity of simply asking the students to explain their reason took about 5 minutes. It is now very easy to tell from reading the responses who in fact had a true understanding of why they don’t add the denominators and it did not take away from much instructional time and
Writing in math to me is a given.It has never occurred to me that writing doesn’t have a place in math.For me, it is the best way to tell if my students understand the process and concept behind what we are doing in class.Is it a perfect way to get into their minds?No, there are of course, limitations that I will discuss in this paper.Yet in today’s schools, as it should be, deep understanding of mathematical concepts is stressed and to know if a student has that, you need to know the process by which they are solving a problem.And unless each teacher has time to pull their 25 students to the back of the room for a quick interview on how they solved a problem, writing is the next best thing.Writing can help connect math problems to the world.Writing helps students stop and think about the process that they are using to solve a problem.By simply asking why does something work, or how they solved a problem, you can get a glimpse of their thoughts and understanding.It is in their writing that my students can prove to me their knowledge and understanding of what we are doing.
In the book “How To Assess Authentic Learning”, (Burke, 1994) the author mentions the ability to use journals and log entries as pre-assessment, formative assessment and summative assessment.Before you begin a topic, you can have students write what they know about it (pre-assessment).This actually can help you plan instruction better.As you are teaching a topic, you can give them a writing assignment as a formative assessment, to see what they understand midway through a unit.This can be done in many ways and help guide instruction mid-unit.As a summative assessment, a teacher can very well give a writing assignment that will demonstrate what the students learned overall.When I was finally done in my own class teaching addition and subtraction with fractions, I asked my students to prepare a presentation to the 4th graders, explaining the process, since 4th graders are often confused.The writing that came back was wonderful and instead of a few addition problems on a page (which I did as well) I was able to see their reasoning and strategies behind what they were doing.Did they understand common denominators?Did they understand the difference between like and unlike denominators?Yes.I found out after reading and listening to their presentations.I loved being able to hear their thought process; they loved being able to do something a little different in math.
There can be much research found for those in favor of writing in math, and those opposed.In the recent “math wars” over school reform, writing has been a huge part of the argument.Should it be included in our curriculums?Is it valuable enough to make mandatory?Read on and see what you think.
An article “Journal Writing and Math Instruction” by Borasi and Rose, reports on a study that looks into the educational value of writing in math, specifically in journals.Borasi and Rose highlight multiple reasons that writing should be included in any math class.First of all, it helps the student.It helps the student by, in a way, forcing them to think about what they are doing.It forces them to articulate the connections that they may not realize they are making and it helps them reflect on what they are learning.The second major way Borasi and Rose mentioned that journals could help in math is that it truly benefits the teacher, as I have previously mentioned.It helps the teacher in two main ways: It helps them get a better grasp on the student’s understanding of a topic for evaluation purposes and it also helps the teacher reflect on their own teaching practices.A teacher, after reading student’s journals, should be able to tell whether, overall, their teaching of a concept has been successful, or whether there is a need to revisit their approach.The last way that Borasi and Rose say that journals can be helpful is when the teacher gives the student feedback on their writing.They claim that this feedback will help the student-teacher relationship by giving the student more one-to-one correspondence and attention.Wow!After reading this article, I really thought about the widespread benefits of writing in math.I had, truthfully, been selfish in my reasoning and was collecting writing to check for student’s understanding, not even thinking of the major personal benefits it will bring to the student.
Marilyn Burns, a very well known “math guru” and proponent of school reform, not only believes in writing in math but also has written multiple articles and books claiming so.Her article, “Writing in Math Class?Absolutely!” (Burns, 1995) is a wonderful glimpse into the benefits of writing, as well as ways to get started.She mentions both the benefits to the student of when they clarify and organize their own thoughts; as well as the teacher, to be able to truly assess what the student understands.She states, “Their writing is a window into what they understand, how they approach ideas, what misconceptions they harbor, and how they feel about what they’re discovering.” (Burns, 1995)
For the many proponents I found on the benefits of writing in math, there are also many opponents.These opponents are really opponents of what is now termed “fuzzy math,” another way of saying “reform math.”The term “fuzzy math” is used to include many aspects; a focus on concepts, children exploring; problem solving emphasis and, very often, writing in math.Its opponents claim that fuzzy math takes time away from the “fundamentals.”Too much time is spent exploring and communicating (often writing) that should be spent getting back to the fundamentals.
There are not only many school districts and counties who are debating which direction to go in math, but states as well.In 1994, California decided to adopt a reform-style curriculum; which led to what is considered “math wars.”In an entire state, there were many for the reform style, and those against.Such as Bill Evers, who has headed a California based group called HOLD (Honest Open Logical Debate on math reform), a group against math reform.In the article “The Math Wars – California Battles It Out over Mathematics Education Reform (Part 1)” (Jackson, 1997), Evers group is mentioned to still believe in problem solving, and other aspects of reform math but that too much time is spend on students solving problems on their own, when it could probably be done in a more efficient manner.The article goes on to mention multiple other groups who share many of the feelings as HOLD.The article quotes Richard Schoen, a professor at Stanford University as saying on page 698 “The idea that students should be understanding what they are doing and not simply regurgitating facts and doing things by rote certainly is a very good idea; I agree with it entirely.But that does not imply that you therefore remove the technical side of the subject so that it becomes essentially a descriptive subject.” (Jackson, 1997).It seems opponents of writing in math see the extreme side, as the article goes on to mention how students often do more writing about math than actually doing math.Schoen, on page 698, even says that often students write “flowery essays about how they feel about a problem” (Jackson, 1997).
Where does all
this fit into the changing world of Mathematics?To
find out, I looked at two documents that are considered the guidelines
from which curriculum is written.The
first, the more prominent and well-known document from the National Council
for Teachers of Mathematics and the other, an up and coming document from
Achieve, Inc.
When counties and states look for guidelines on writing curriculum, the National Council for Teachers of Mathematics (NCTM) has been there for them since 1989, putting out three major documents, which helped give direction.Their latest document, The Principles and Standards for School Mathematics 2000 (Standards), is an updated version of their very well known document by the same title in 1995.NCTM considers this document to be something that all teachers and schools should strive for, a goal to be reached.It states a very clear vision of what they think the ideal classroom should be.In the very first chapter, the Standards document has a paragraph that helps you envision their ideas and expectations.It states, “Imagine a classroom, a school, or a school district where…orally, and in writing, students communicate their ideas and results effectively.” (NCTM, p. 3)Right from the third page of the document, it is easy to see that NCTM values writing in math and expects students to be engaged in it.
The way that the Standards document is set up, it states standards (as you can tell by the title) from which teachers and schools should be guiding their instruction.NCTM gives six Content Standards and four Process Standards by which to direct our teaching.The document breaks down each standard into grade level groups and explains how it should be taught at that stage in a student’s academic career.The Communication standard is where writing, for the most part, comes into play.NCTM states communication is a way for students to reflect, share ideas, and understand the meaning of math.When students need to explain and justify their thinking to others, writing helps them organize their thoughts and processes clearly.It is also extremely important, says NCTM, that teachers create a classroom where students will feel open to share their ideas.“Because mathematics is so often conveyed in symbols, oral and written communication about mathematical ideas is not always recognized as an important part of mathematics education.Students do not necessarily talk about mathematics naturally; teachers need to help them learn how to do so.” (NCTM, p. 60)Obviously, NCTM sees the value in writing, and recognizes that it will be up to the teacher to make it happen.Students often do no associate writing with math, and it takes a while to open them up to the idea.
In Kindergarten through Second Grade, NCTM states that teachers really need to guide and walk students through learning how to share their ideas and explain answers. At this stage, teachers also need to help students learn to listen to each other’s ideas carefully to see if they make sense.This is where students will begin to learn what is acceptable as far as explaining and justifying their ideas.In these grades, writing is just beginning to be developed in all other content areas as well, so it is a good place for students to be introduced to the idea of writing in math.This will make it less of a struggle in the later grades.
In grades 3-5, students should begin to use writing as a way to understand and think of strategies they are using to solve problems.Writing should help them make sense of the process they are using to solve a problem.This is the stage in which students should really be justifying and explaining their ideas openly to others (including the teacher.)By sharing their ideas through writing, teachers will be able to understand the process the student is taking, and the students will be able to think about whether their answer and process was reasonable.When students at this stage share their ideas, it will hopefully give the students a look into other people’s thinking and maybe even give them another strategy to use.When I was in the middle of my fraction unit, I had students using fraction pieces to show subtraction with unlike denominators.When students were sharing how they solved a problem, most had shown how they converted the denominators to be the same, then subtracted.One student explained that he simply put the two fractions one under the other, and saw what fraction was needed to fill in the gap.If you could only have seen the faces on many students.A light went on.A light that said, “Oh, there are other ways to solve this problem.”It is here that students not only listen to each other, but also hopefully gain insight into other ideas besides their own.NCTM says that teachers at this level need to expect the writing to be rather clear and, even in the beginning, give it back to the students for revisions.Teachers at this stage will be able to get a very good assessment of what the child understands, and will hopefully see their understanding build on itself and make connections in the world of math.
As NCTM continues on the grade continuum, it explains what students should be able to communicate in grades 6-8.Now that students can, hopefully, explain their thinking through writing, they will now be expected to compare their thoughts to others, but also to use their own writing to think about different strategies.“Is the strategy efficient?” is an example of a question a student in this age group should be able to ask themselves.It is in this section that I found a wonderful summation, really, of what I love about writing in math.On page 272 of the NCTM’s Standards document, it states “Teachers can use oral and written communication in mathematics to give students opportunities to – think through problems, formulate explanations, try out new vocabulary or notation, experiment with forms of argumentation, justify conjectures, critique justifications, reflect on their own understanding and on the ideas of others.”Beautifully stated.
It is in grades 9-12 that NCTM expects students to be making the connections between the various aspects in math, through their writing.Hopefully, students will be using proof from one strategy to help solve, prove or disprove another.By connecting these mathematical ideas, they are building a true understanding of the interconnections in math.Their understanding of math should now become prevalent in other content areas as well.
Through its obvious stress on communication and writing in math, NCTM has made it clear that writing in math, not only is important, but also should be part of everyday math across all grade levels.
Achieve, Inc. is a nonprofit organization brought together by a common goal of raising the expectations we have of our children in schools.In 1995, the Third International Mathematics and Science Study (TIMMS) compared students in the United States to students in other countries at the same ages as far as abilities and understanding in math.Overall U.S. student achievement in math did not compare well with students from other countries.The gap was even more apparent for middle school students.AchieveInc., after reading the TIMMS study, decided that we as a country need to raise our expectations for students in the middle grades (6-8.)Achieve, along with 10 states, collaborated to form the Mathematics Achievement Partnership (MAP), which brought together an advisory panel to create a draft document called “Foundations for Success:Mathematics for the Middle Grades” (Foundations) that would identify key skills students will need to become successful as they move onto high school and then a career.Foundations has an appendix of skills that would need to be taught in the elementary grades and then what would be taught following these expectations in high school.As noted in its introduction, any child growing up in the 21st century will need a truly deep understanding of the fundamentals of math in almost any future career they may choose.That is what they hope Foundations will do.It is quickly becoming a challenger, so to speak, for the NCTM’s Standards.Although Foundations mentions NCTM as a “roadmap” for improving our schools with similarities, it also mentions distinctions.Foundations claims to have a stronger focus on the fundamentals of math with higher expectations in the same areas.Foundations has a clear emphasis on algebra, geometry, data analysis, and a deep understanding of number concepts; but not a very clear emphasis on any processes (including writing) of how to achieve them.
I am going to confess right now that I am not a middle school teacher and do not claim to have any expertise in those grade levels.What I do have is teaching experience, as well as experience writing curriculums.While I see many expectations as far as skills are concerned in Foundations, I do not see any clear place where the process to compute those skills is emphasized.At the beginning of the Foundations document, it states that this curriculum stresses reasoning and deep understanding, but does not go far beyond that.Nowhere did I see that writing should occur (or oral explanations, or any other type of communicating about how an answer was solved.)
The Foundation document is broken up into the four main content areas.For each content area, there are lists of what students should understand by the end of the “unit.”Continually, I saw “Students should understand _______.”How will a teacher be able to assess true understanding?Yes, there are the normal tests and continual assessment mentioned in the document.But without having the student explain their thinking, it is hard to assess true understanding.When looking through the Foundations document, there were sample problems in each content area.The majority of questions had a numerical answer with no explanation.For example, in the Number Section, the first problem states:“There are two comets, A and B.A comes close enough to the earth to be observed every 76 years.B Comes close enough to the earth to be observed every 12 years. If we can observe both of the comets from the earth this year, how many years will it be before we can again see them both in the same year.” (p. 22) On page 70 the question is asked, “Decide where this angle is exactly a right angle, a little more, or a little less.” These are both great tasks and questions, which involve higher-level thinking.Imagine how much more you would be able to assess if at the end of each question you simply said, “Explain your thinking.”Instead of just a numerical or short answer, you would also have the process by which the student arrived at that answer.There are many variations of levels of thinking the student could be going through to get an answer. Yet if you only saw numerical answer, it would be conceived as either “right” or “wrong” with no in-between. In looking through the entire document, I saw only 3 problems that asked “Why” or “Show that” which would both invoke a more detailed answer from the student.I truly enjoyed seeing those 3 problems.
Writing is a way to promote both reasoning and deep understanding as previously mentioned.The sample questions definitely promote higher-level thinking.To be able to solve these problems, students will need to have a clear understanding of the basics.By asking the students to explain their process of solving the problem in writing, you can both assess their way of thinking, and also catch any students who may be struggling with the concept.When a child is asked to explain how they got an answer, after reading it, a teacher can very clearly see their level of understanding on the topic.
Does Achieve not want to emphasize writing?It’s hard to tell. The document did not clearly mention writing, although it is a draft, and a small draft at that.With the mentioning of reasoning, writing would almost have to be involved…yet it was not specified in the introduction or in almost all of the sample questions.It is left unclear how reasoning is to be taught and/or assessed.
I don’t want this to sound like a boxing match.Yet, since many educators are now looking at Achieve’s Foundations document closer, it seems there needs to be a comparison between the two.Obviously, for my interests, I will compare them with respect to how they emphasize writing in math.
When looking at the two documents, there are some very clear similarities.They both stress higher-level thinking.Sample problems in both show much more than the rote math you often see in books.They both have high expectations for students, although one claims to have higher expectations.And, knowing that the 2000 Standards was just recently released as was the Foundations document, they both are looking for a change in math, a change for the better, a change from the “old school” math.
There is also a very distinct and clear difference between the two documents in the way they stress writing in math.NCTM’s Standards document not only stresses the importance at all grade levels through the communication standard, but also even claims it to be something that should be in an ideal classroom.Achieve’s Foundations document, however, barely mentions writing in math.It mentions reasoning as an important component of their document.Is that the same thing?It can be, but not necessarily.If I were to look at the Foundations document, I would not think I needed to include writing in my daily math.Not only that, if I were to think about it, I would see the omission as a sign of de-emphasizing writing as a toll to promote higher-level understanding.In their defense, Foundations is in draft form.It is about ¼ the size of the already revised Standards document.Will the final draft of Foundations include a mentioning of writing?One can only hope.
As far as Achieve and NCTM are concerned, the suggestions are a little different.For NCTM, I applaud how they have laid out what is expected for teachers at each grade level as far as writing and communication is concerned (although it has, to Achieve’s defense, had much more time to develop.)The only suggestion I would have would be to include more student samples of writing at each grade level so that teachers can get a glimpse into what they should expect.
When Achieve, Inc. moves from a draft to a final copy, I hope a few things will change.First, I hope they include some sort of writing/communicating component in their document.A teacher and/or educator has so many things thrown at them that when a curriculum or document is handed to them, they complete it as is…often with simply too much on their plate to add more to it.If Achieve thinks that writing is important, they need to mention it in their document.A step further would be to not only mention it, but also be very clear about what is expected and should be happening in the classroom as well as samples of writing at each grade level.
Now all the debating, reading and researching hits home.What should we, the teachers and educators, who see the children from day to day, do about writing in math?As a Math Specialist this year, I am finding that many teachers simply do not know what to write about, or how to go about getting writing started during math.If students have not been exposed to writing in math, it will be rather obvious when you first ask them to write.I hope that there is support for teachers who need help.There are MANY resources available, of which I will mention my favorite that can help an educator ease their way into the benefits of writing.
Marilyn Burns, a well-known math guru, has written a good chunk of those many resources.One of her more popular books, titled “Writing in Math Class” (Burns, 1995) is a wonderful guide for any teacher interested in including writing in math class.The book explains why writing is important, tips and suggestions for instruction, and types of writing assignments that can be used.Hopefully, somewhere over the rainbow, teachers will read this article and want to get started tomorrow.Here are five ways that Burns suggests to include writing in your everyday math.The first type of writing she discusses is the use of journals and logs.These will become the students “ongoing notes.”The writing is open and flexible in its content, with a few guidelines.She gives ideas such as writing about what they learning in math class that day, writing about something that confuses them in math or even reflections on a particular lesson.In the earlier grades, prompts will be more necessary.After continued use, students should become very comfortable with the journals.“Solving Math Problems” is the next way Burns suggests we use writing.If a student writes about the process of solving a problem, it helps them develop the more strategies by making them think through their own process in order to clarify it to others.From my own personal experience, sharing after students have written about how they solved a problem can lead to wonderful discussions about multiple strategies that were used.
“Explaining mathematical ideas” is the next item Burns mentions as a way to introduce writing.This is when you ask students to write what they know about a specific idea or concept in math.It can be as structured as “Tell me what you know about division,” or “Explain why you invert and multiply when dividing fractions.” (Good luck getting an answer for the second one.)Again, personal experience has taught me that this is a wonderful way, when beginning a unit, to assess what the students already know.It can save everyone time if they already know something you were about to start teaching!“Creative Writing and Math” comes next.If a student is more confident in their writing than their math, or if you feel you have been giving your students the same prompt over and over, it’s time to bring in a creative writing assignment.This is when students write stories about something you are working on, etc.This can truly bring out the creative spirit you may not see with other writing prompts in math.The last way that Burns mentions as a way to include writing in math really includes everything else: “General Writing Assignments.”This can include anything from writing about why math is important, to writing a letter to welcome a guest who is coming in math class, or write a thank you note to a partner who helped you out.The five ways that Burns mentioned in her book are certainly not the only ways you can include writing in your math; although they include any type I’ve ever done.They are, also, wonderful guides to help you get started, help you feel more comfortable in your job as an educator and, most importantly, to help your children become more articulate mathematicians.
Writing needs to be included in math instruction.So much can be gained!The students become better learners after writing in a variety of ways.They can share what they feel, clarify their thoughts, look deeper into their own processes, and feel like they are progressing, learning, and becoming better mathematicians.
As a teacher, having students write in your math class is invaluable.It gives you insight into a student’s thinking, which you wouldn’t have otherwise.It can help you become a better teacher by guiding your instruction and truly understanding where you students are.Please make sure to create (it will take a lot of work) an environment in which students feel comfortable sharing any ideas they may have about a topic.Expose your student to the various ways of writing in math.The five that were mentioned in Marilyn Burns’s book are a great place to start.Students need change every now and then.It will take time; but the time spent will be well worth it.If you haven’t already done so, introducing writing in math will take longer when you first start.Setting up the correct environment will take a while.Introduce the idea, as NCTM has done, in a continuum.If you have already been writing in your math class; don’t stop!Keep going and expand what you’ve started. Once you start writing in math, you will never want to turn back.