What did I discover? What's new to me here?
Certainly of greatest value to me is the new exposure to the very valuable resources on the NCTM website. In particular, I see the tremendous potential and benefit of the Journal articles. Though there are others too. Further, I realize that there are a plethora of ways to 'tear down the barriers' that exist between mathematics and language arts. I am empowered to attempt just a few, at first. And I'd like to challenge any and all Language art teachers to encourage their students to reflect and opine about their math experiences. Finally, I realize that curricula and practice can are informed by multiple foundations simultaneously, sometimes at odds. What will I research further?
Well, as a non-related next practical step, I intend to focus on block scheduling as we are moving core subjects from a non-block to complete block next year. That change will necessarily effect my educational practice and methodology. As a related step I will likely investigate ways to measure the success of this proposal. I've heard great things about the use of portfolios, especially when used in student-led conferencing. Where will I go from here?
I will start slowly but consistently to build up both the variety and frequency of language arts based assessments. Since I teach grades 6-8, I should be able to phase this in over a period of three years, growing that dimension annually.

Hi Serge,
I just wanted to pop on and say well done on your presentation. I thought that your level of self reflection throughout the presentations and your conceptualizing possible future lessons were both well done. I liked your ideas for including writing and language arts in math lessons. I also liked your idea of having students write an autobiography at the start of the year to be a great idea.
Well done,
Jenn

Here is the most recent link to my spicynode. In it I am attempting to organize my thoughts around the myriad of assessment tools possible if allowing language arts into the mathematics classroom. http://www.spicynodes.org/yournodes_view_nodes.html?nodemapNum=0 -Serge

I haven't yet been able to pull it up through TCNJ. I think that they have it, but in hard copy only.
Serge Thanks for responding Serge. I think you've done a terrific job with the limited resources we have here in a 2 wk. course. I'll say this tomorrow in person but it's been such a pleasure to know you. Karen

Karen, I am so very pleased with the journals and methods of research that this project has forced me to find and use. They are excellent. Having re-examined the summaries foundations of cultural eduction I've concluded that the study of mathematics traditionally falls into scholasticism and/or essentialism. Would you agree? Do you see further influences? -Serge

I add a further reference which I found through our e-library subscription at school. Though I am not available to reproduce the article here, I make reference to it.

Journal writing in mathematics: examples from elementary classrooms

Karen, unfortunately it seems the archive articles from Mathematics Teacher that are only easily available until 1997. Beyond that photocopies must be ordered. I am still trying to get around that as the article seems to be exactly what I am looking for. However, I did find the following excellent research articles from Mathematics Teacher or Teaching Mathematics in the Middle School. Since they are available from the NCTM publisher gratis online I don't suppose making them available for reference, temporarily, would be a breech of copyright:

Not directly part of my project but this research sees math as opportunity to promote social justice. Math as a tool for SJ

-Serge

==
==

Hi Serge,

I don't have much to add, but I would like to say that this sounds like a great project. Literacy truly does transcend subject areas and I love the idea of incorporating writing into math. I will continue thinking about the topic, and will get back to you with any other input. In the meantime, though, I look forward to your presentation! -Lori

Karen, thanks for the links. Most enable me to access some articles which I am reading over. I understand the need to reference scholarly journals, and not just professional organizations. -Serge

You've done some good work researching the topic to date. I'd like to see you find research though--not "here's an idea" kind of anecdotal material. Have you tried looking at math journals? Research journals? Here's the NCTM pub: http://nctm.org/eresources/journal_home.asp?journal_id=1

Featured Topic: Writing in Math Class

Teachers incorporate writing in math class to help students reflect on their learning, deepen their understanding of important concepts by explaining and providing examples of those concepts, and make important connections to real-life applications of the math they are learning. Teachers use the writing assignments to assess student understanding of important concepts, student proficiency in explaining and using those concepts and each student's attitude toward learning mathematics. Writing in mathematics is a win-win for both teacher and student. Although it may be difficult to introduce this practice, it is well worth the effort. Look for simple ways to incorporate short writings throughout daily lessons and longer writings over the course of weeks or math units.

More Information on Writing in Math Class

See 10 Big Math Ideas to read why Marilyn Burns emphasizes the importance of writing in math class. Be sure to read #3 and #4.

Math Journals Boost Real Learning offers practical tips on using journals, responding to student's writings and storing journals in the classroom.

Read Math Out Loud for more reflection on the connection between speaking and writing.

Check out Writing in Mathematics for some suggested ideas and prompts that promote different kinds of writing to make sense of mathematics.

My Life in Math Class describes a beginning of the year writing assignment Brenda Dyck created that requires students to write a math autobiography to meet specified criteria.

And, I found the following research:

Writing was developed by the Sumerians approximately 5,000 years ago. At the same time, the Sumerians developed some written notation for mathematics. Writing and mathematics are brain tools--they are powerful aids to the human mind.

The abilities to use both written language and mathematics are so useful to people that these are "basics" in our formal educational system. Students study and practice the "three Rs" year after year in K-12 education and even on into higher education as they work to develop contemporary and more advanced knowledge and skills (expertise) in these areas.

Our math education system pays some attention to the idea that math is a language. For example, many math teachers have their students do journaling on the math learning experiences and their math use experiences. Some math teachers make use of cooperative learning--an environment that encourages students to communicate mathematical ideas. Some math assessment instruments require that students explain what it is they are doing as they solve the math problems in the assessment.

There has been a great deal of research on the teaching and learning of reading and writing in one's first, natural language. In addition, there has been a great deal of research on the learning of a second language. It seems likely that some of the research findings and practical implementations of these findings would be applicable to teaching and learning of mathematics.

Marzano, Robert J. (September 2005) Preliminary Report on the 2004–05 Evaluation Study of the ASCD Program for Building Academic Vocabulary. Accessed 11/30/05:

Mathematics as a Language

Crannell, Annalisa. Writing in Mathematics [Online]. Accessed 1/26/02:
. This '94 content is "old." Still useful? Perhaps not as much as some of the more recent, but the idea still supports the use of writing in math. -Serge

Crannell gives writing assignments in the calculus classes she teaches at a university level. Her Website includes a 1994 booklet A Guide to Writing in Mathematics Classes. Quoting from the first part of that booklet:

For most of your life so far, the only kind of writing you've done in math classes has been on homeworks and tests, and for most of your life you've explained your work to people that know more mathematics than you do (that is, to your teachers). But soon, this will change.Now that you are taking Calculus, you know far more mathematics than the average American has ever learned - indeed, you know more mathematics than most college graduates remember. With each additional mathematics course you take, you further distance yourself from the average person on the street. You may feel like the mathematics you can do is simple and obvious (doesn't everybody know what a function is?), but you can be sure that other people find it bewilderingly complex. It becomes increasingly important, therefore, that you can explain what you're doing to others that might be interested: your parents, your boss, the media.

Nor are mathematics and writing far-removed from one another. Professional mathematicians spend most of their time writing: communicating with colleagues, applying for grants, publishing papers, writing memos and syllabi. Writing well is extremely important to mathematicians, since poor writers have a hard time getting published, getting attention from the Deans, and obtaining funding. It is ironic but true that most mathematicians spend more time writing than they spend doing math.

But most of all, one of the simplest reasons for writing in a math class is that writing helps you to learn mathematics better. By explaining a difficult concept to other people, you end up explaining it to yourself.

Language and the Learning of Mathematics [Online]. Accessed 1/26/02: http://www.mathematicallycorrect.com/allen4.htm. A speech delivered at the NCTM Annual Meeting Chicago, April 1988 by Frank B. Allen, Emeritus Professor of Mathematics Elmhurst College. Quoting from the paper:

This brings me to my major thesis that natural language, gradually expanded to include symbolism and logic, is the key to both the learning of mathematics and its effective application to problem situations. And above all, the use of appropriate language is the key to making mathematics intelligible. Indeed, in a very real sense, mathematics is a language. Proficiency in this language can be acquired only by long and carefully supervised experience in using it in situations involving argument and proof.

However, the language of Mathematics does not consist of formulas alone. The definitions and terms are verbalized often acquiring a meaning different from the customary one. Many students are inclined to hold this against mathematics. For example, one may wonder whether 0 is a number. As the argument goes, it is not, because when one says, I watched a number of movies, one does not mean 0 as a possibility. 1 is an unlikely candidate either. But do not forget that ambiguities exist in plain English (the number's number is one of them) and in other sciences as well. A a matter of fact, mathematical language is by far more accurate than any other one may think of. Do not forget also that every science and a human activity field has its own lingo and a word usage in many instances much different from that one may be more comfortable with.

The Language of Mathematics [Online]. Accessed 1/26/02: http://www.math.montana.edu/~umsfwest/.
This Website is based on a book by Warren Esty and a course at Montana State University by the same name. The first quote given below is from the Website, and the second is from the Warren Esty book.

Jointly with Anne Teppo, Warren Esty published an article in the Mathematics Teacher (Nov. 1992, 616-618) entitled "Grade assignment based on progressive improvement" which was reprinted in the NCTM's Emphasis on Assessment. and posted on the web by the Eisenhower National Clearinghouse for Mathematics and Science Education. In a language course, you can expect continual improvement. This article discusses why grading should not be based on averages of unit-exam scores and how a course like "The Language of Mathematics" can be graded.----Mathematical results are expressed in a foreign language. Like other languages, it has its own grammar, syntax, vocabulary, word order, synonyms, negations, conventions, idioms, abbreviations, sentence structure, and paragraph structure. It has certain language features unparalleled in other languages, such as representation (for example, when "x" is a dummy variable it may represent any real number or any numerical expression). The language also includes a large component of logic. The Language of Mathematics emphasizes all these features of the language (Esty, 1992).

The Language of Mathematics [Online]. Accessed 1/26/02: http://www.chemistrycoach.com/language.htm.
This Website contains a number of quotations that relate to the topic of mathematics as a language. Here are two examples:

Ordinary language is totally unsuited for expressing what physics really asserts, since the words of everyday life are not sufficiently abstract. Only mathematics and mathematical logic can say as little as the physicist means to say. Bertrand Russell, (1872-1970) The Scientific Outlook, 1931.---The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. Eugene Paul Wigner, (1902-1995): The Unreasonable Effectiveness of Mathematics in the Natural Sciences.

Using the developmental model of incorporating Writing into the Mathematics Classroom http://www2.ups.edu/community/tofu/lev2/journaling/writemath.htm -Serge
(Not relevant to your project, but this research project was conducted by one of my math professors at my undergrad college. It's neat that his work is helpful for your project! -Alycia)

Certainly of greatest value to me is the new exposure to the very valuable resources on the NCTM website. In particular, I see the tremendous potential and benefit of the Journal articles. Though there are others too. Further, I realize that there are a plethora of ways to 'tear down the barriers' that exist between mathematics and language arts. I am empowered to attempt just a few, at first. And I'd like to challenge any and all Language art teachers to encourage their students to reflect and opine about their math experiences. Finally, I realize that curricula and practice can are informed by multiple foundations simultaneously, sometimes at odds.

What will I research further?

Well, as a non-related next practical step, I intend to focus on block scheduling as we are moving core subjects from a non-block to complete block next year. That change will necessarily effect my educational practice and methodology. As a related step I will likely investigate ways to measure the success of this proposal. I've heard great things about the use of portfolios, especially when used in student-led conferencing.

Where will I go from here?

I will start slowly but consistently to build up both the variety and frequency of language arts based assessments. Since I teach grades 6-8, I should be able to phase this in over a period of three years, growing that dimension annually.

Hi Serge,

I just wanted to pop on and say well done on your presentation. I thought that your level of self reflection throughout the presentations and your conceptualizing possible future lessons were both well done. I liked your ideas for including writing and language arts in math lessons. I also liked your idea of having students write an autobiography at the start of the year to be a great idea.

Well done,

Jenn

Here is the most recent link to my spicynode. In it I am attempting to organize my thoughts around the myriad of assessment tools possible if allowing language arts into the mathematics classroom.

http://www.spicynodes.org/yournodes_view_nodes.html?nodemapNum=0

-Serge

Serge, your spicynode is a good start at plotting your course for the project. I can see you've found a good deal of information about writing in math. Were you able to access any scholarly research? This is, again, a 20 year old discussion but Peggy McIntosh is a good source. Can you locate this journal in TCNJ's database:http://eric.ed.gov:80/ERICWebPortal/search/detailmini.jsp?_nfpb=true&_&ERICExtSearch_SearchValue_0=EJ436632&ERICExtSearch_SearchType_0=no&accno=EJ436632

I haven't yet been able to pull it up through TCNJ. I think that they have it, but in hard copy only.

Serge

Thanks for responding Serge. I think you've done a terrific job with the limited resources we have here in a 2 wk. course. I'll say this tomorrow in person but it's been such a pleasure to know you. KarenKaren,

I am so very pleased with the journals and methods of research that this project has forced me to find and use. They are excellent.

Having re-examined the summaries foundations of cultural eduction I've concluded that the study of mathematics traditionally falls into scholasticism and/or essentialism. Would you agree? Do you see further influences?

-Serge

I add a further reference which I found through our e-library subscription at school. Though I am not available to reproduce the article here, I make reference to it.

## Journal writing in mathematics: examples from elementary classrooms

Cohen, RinaOrbit01-01-2000

Byline: Cohen, Rina

Volume: 31

Number: 3

ISSN: 00304433

Publication Date: 01-01-2000

Page: 45

Type: Periodical

Language: English

Karen, unfortunately it seems the archive articles from Mathematics Teacher that are only easily available until 1997. Beyond that photocopies must be ordered. I am still trying to get around that as the article seems to be exactly what I am looking for. However, I did find the following excellent research articles from Mathematics Teacher or Teaching Mathematics in the Middle School. Since they are available from the NCTM publisher gratis online I don't suppose making them available for reference, temporarily, would be a breech of copyright:

Journal Writing in Math

Not directly part of my project but this research sees math as opportunity to promote social justice.

Math as a tool for SJ

-Serge

==

==

## Hi Serge,

I don't have much to add, but I would like to say that this sounds like a great project. Literacy truly does transcend subject areas and I love the idea of incorporating writing into math. I will continue thinking about the topic, and will get back to you with any other input. In the meantime, though, I look forward to your presentation! -LoriHello, class. My project is to create a proposal to use writing in the mathematics classroom. Your comments are very welcome. As a first step I've investigated types of writing, expository, narrative etc..... And I am just starting the search for publications which support the practice. Mathwire (mathwire.com) summarizes what I've intended to do in the classroom for some time. The conclusion of the article has references that point me in further directions for research. Declaration: the following is not my writing, but does re-express my ideas. Serge, do you have access to the middle level math journal (

Mathematics Teaching in theMiddle School) NCTM publishes (through TCNJ lib?) Take a look, e.g., at this recent issue's cover:http://nctm.org/eresources/toc.asp?journal_id=3&Issue_id=976

Another issue with a topic you might be interested in:

http://nctm.org/eresources/article_summary.asp?URI=MTMS2010-12-311a&from=B

Another re: "math identities"

http://nctm.org/eresources/article_summary.asp?URI=MTMS2010-11-224a&from=B

Another re: students writing word problems:

http://nctm.org/eresources/article_summary.asp?URI=MTMS2010-08-27a&from=B

Another on language in math...

http://nctm.org/eresources/article_summary.asp?URI=MTMS2010-05-516a&from=B

Karen, thanks for the links. Most enable me to access some articles which I am reading over. I understand the need to reference scholarly journals, and not just professional organizations. -Serge

You've done some good work researching the topic to date. I'd like to see you find research though--not "here's an idea" kind of anecdotal material. Have you tried looking at math journals? Research journals? Here's the NCTM pub:

http://nctm.org/eresources/journal_home.asp?journal_id=1

## Featured Topic: Writing in Math Class

Teachers incorporate writing in math class to help students reflect on their learning, deepen their understanding of important concepts by explaining and providing examples of those concepts, and make important connections to real-life applications of the math they are learning. Teachers use the writing assignments to assess student understanding of important concepts, student proficiency in explaining and using those concepts and each student's attitude toward learning mathematics. Writing in mathematics is a win-win for both teacher and student. Although it may be difficult to introduce this practice, it is well worth the effort. Look for simple ways to incorporate short writings throughout daily lessons and longer writings over the course of weeks or math units.## More Information on Writing in Math Class

And, I found the following research:

The abilities to use both written language and mathematics are so useful to people that these are "basics" in our formal educational system. Students study and practice the "three Rs" year after year in K-12 education and even on into higher education as they work to develop contemporary and more advanced knowledge and skills (expertise) in these areas.

Our math education system pays some attention to the idea that math is a language. For example, many math teachers have their students do journaling on the math learning experiences and their math use experiences. Some math teachers make use of cooperative learning--an environment that encourages students to communicate mathematical ideas. Some math assessment instruments require that students explain what it is they are doing as they solve the math problems in the assessment.

There has been a great deal of research on the teaching and learning of reading and writing in one's first, natural language. In addition, there has been a great deal of research on the learning of a second language. It seems likely that some of the research findings and practical implementations of these findings would be applicable to teaching and learning of mathematics.

## Mathematics as a Language

Crannell, Annalisa. Writing in Mathematics [Online]. Accessed 1/26/02:.

This '94 content is "old." Still useful?Perhaps not as much as some of the more recent, but the idea still supports the use of writing in math.

-Serge

Crannell gives writing assignments in the calculus classes she teaches at a university level. Her Website includes a 1994 booklet

A Guide to Writing in Mathematics Classes.Quoting from the first part of that booklet:- For most of your life so far, the only kind of writing you've done in math classes has been on homeworks and tests, and for most of your life you've explained your work to people that know more mathematics than you do (that is, to your teachers). But soon, this will change.Now that you are taking Calculus, you know far more mathematics than the average American has ever learned - indeed, you know more mathematics than most college graduates remember. With each additional mathematics course you take, you further distance yourself from the average person on the street. You may feel like the mathematics you can do is simple and obvious (doesn't everybody know what a function is?), but you can be sure that other people find it bewilderingly complex. It becomes increasingly important, therefore, that you can explain what you're doing to others that might be interested: your parents, your boss, the media.
- Nor are mathematics and writing far-removed from one another. Professional mathematicians spend most of their time writing: communicating with colleagues, applying for grants, publishing papers, writing memos and syllabi. Writing well is extremely important to mathematicians, since poor writers have a hard time getting published, getting attention from the Deans, and obtaining funding. It is ironic but true that most mathematicians spend more time writing than they spend doing math.
- But most of all, one of the simplest reasons for writing in a math class is that writing helps you to learn mathematics better. By explaining a difficult concept to other people, you end up explaining it to yourself.

Language and the Learning of Mathematics [Online]. Accessed 1/26/02: http://www.mathematicallycorrect.com/allen4.htm. A speech delivered at the NCTM Annual Meeting Chicago, April 1988 by Frank B. Allen, Emeritus Professor of Mathematics Elmhurst College. Quoting from the paper:- This brings me to my major thesis that natural language, gradually expanded to include symbolism and logic, is the key to both the learning of mathematics and its effective application to problem situations.

Mathematics as a Language [Online]. Accessed 1/26/02: http://www.cut-the-knot.com/language/. Quoting from the Website:And above all, the use of appropriate language is the key to making mathematics intelligible. Indeed, in a very real sense, mathematics is a language.Proficiency in this language can be acquired only by long and carefully supervised experience in using it in situations involving argument and proof.- However, the language of Mathematics does not consist of formulas alone. The definitions and terms are verbalized often acquiring a meaning different from the customary one. Many students are inclined to hold this against mathematics. For example, one may wonder whether 0 is a number. As the argument goes, it is not, because when one says, I watched a number of movies, one does not mean 0 as a possibility. 1 is an unlikely candidate either. But do not forget that ambiguities exist in plain English (the number's number is one of them) and in other sciences as well. A a matter of fact, mathematical language is by far more accurate than any other one may think of. Do not forget also that every science and a human activity field has its own lingo and a word usage in many instances much different from that one may be more comfortable with.

The Language of Mathematics [Online]. Accessed 1/26/02: http://www.math.montana.edu/~umsfwest/.This Website is based on a book by Warren Esty and a course at Montana State University by the same name. The first quote given below is from the Website, and the second is from the Warren Esty book.

- Jointly with Anne Teppo, Warren Esty published an article in the Mathematics Teacher (Nov. 1992, 616-618) entitled "Grade assignment based on progressive improvement" which was reprinted in the NCTM's Emphasis on Assessment. and posted on the web by the Eisenhower National Clearinghouse for Mathematics and Science Education. In a language course, you can expect continual improvement. This article discusses why grading should not be based on averages of unit-exam scores and how a course like "The Language of Mathematics" can be graded.----Mathematical results are expressed in a foreign language. Like other languages, it has its own grammar, syntax, vocabulary, word order, synonyms, negations, conventions, idioms, abbreviations, sentence structure, and paragraph structure. It has certain language features unparalleled in other languages, such as representation (for example, when "x" is a dummy variable it may represent any real number or any numerical expression). The language also includes a large component of logic. The Language of Mathematics emphasizes all these features of the language (Esty, 1992).

The Language of Mathematics [Online]. Accessed 1/26/02: http://www.chemistrycoach.com/language.htm.This Website contains a number of quotations that relate to the topic of mathematics as a language. Here are two examples:

Using the developmental model of incorporating Writing into the Mathematics Classroom

http://www2.ups.edu/community/tofu/lev2/journaling/writemath.htm

-Serge

(Not relevant to your project, but this research project was conducted by one of my math professors at my undergrad college. It's neat that his work is helpful for your project! -Alycia)