Response: 'Writing in Math Class Is a Win-Win for Students & Teachers'
The new "question-of-the-week" is:
How do you integrate writing in math classes?
All teachers these days are supposed to be literacy teachers, including math educators. Yes, students need to read word problems to do math, but how do you integrate writing into mathematics instruction?
Today, Dr. Linda Dacey, Sandy Atkins, Andrea Clark, Mike Flynn, ReLeah Cossett Lent, and Shannon Jones share their responses. I also include reader comments. You can listen to a 10-minute conversation I had with Linda, Sandy, Andrea and Mike on my BAM! Radio Show. You can also find a list of, and links to, previous shows here.
You might find this related collection of resources helpful: The Best Resources For Writing In Math Class.
In addition, check out posts that have previously appeared here on Math Instruction.
Response From Dr. Linda Dacey
Dr. Linda Dacey has written numerous professional resources about teaching mathematics. She began her career teaching at the elementary level, taught preservice and inservice teachers for many years, and now writes and consults:
Writing in math class is a win-win for students and teachers. Students have the opportunity to organize, clarify, explain, and share their ideas. Teachers get access to what and how students think, gaining invaluable information they would not get otherwise. So, what are some ways we can engage our students in writing activities that support their learning of math?
Start with quick writes that need not take more than 2-8 minutes and they don't need to be formally evaluated. When introducing a lesson, you might show students a picture, a pattern, or a and ask them to record what they notice, You might show them a graph with no labels, and have them record possible titles for it. At the end of a lesson, you might ask students to write a headline for what they learned or to identify a question they have.
Suggest that students work with partners. They can brainstorm and write together or brainstorm and then spend time reviewing each other's work. Such reviews lead to greater clarity, more precise language, and give students the experience of critiquing the thinking of others.
Offer open-ended problems, which have more than one answer and can be solved in more than one way. Students are much more interested in explaining their thinking in writing, when it is likely to differ from others. They learn to write for themselves, so they can remember their ideas, and are often eager to read what others have written.
Encourage students to write about what and why, perhaps in a two-column format. Such writing helps students keep track of what they are doing and aware of why they are doing it. Writing in this note-like format also helps to prepare students for writing more formal justifications.
Provide prompts that provoke, ones that engage studens in debate, while honing their reasoning skills. Writing a mathematical claim and arguing for that position is the essence of mathematical proof. Possible argument starters include: Is zero a number? Are all prime numbers odd? If you double the y-values in an equation, will the slope double as well?
Use math journals or notebooks on a near daily basis. Students might use them for reflective writing at the end of a class meeting or to record all of their notes; what they notice and wonder about; their work, explanations, and justifications. Students often become attached to their journals and eagerly open them to record their thinking.
Make connections to creative writing to capture students' interest. Students could write a dialogue between a sine graph and a cosine graph or write an advertisement for why the circle is the best shape. They could write a myth about the day numbers were discovered and how it changed the world or a two-voice poem about positive and negative numbers.
What's most important is that writing in math class feels meaningful to students. If students only write so that we can evaluate their learning, they will not see writing as something that has personal meaning to them. However, if students are involved in engaging tasks, they are more eager to write about their ideas, because that thinking is important to them.
Response From Sandy Atkins
Sandy Atkins, owner and Executive Director of Creating AHAs, LLC, is author of Creating a Language-Rich Math Class and Creating Fraction and Decimal AHAs. Sandy is a popular conference speaker and a national and international K-8 math education consultant:
Writing is essential to helping students excel in math
Language (oral and written) is important to help students understand and excel at mathematics. Students who write about math
- access the math to be learned and connect key representations (concrete, pictorial, verbal, and symbolic);
- open a window into their thinking and current understandings; and
- engage in the mathematical practices of constructing viable arguments, critiquing the reasoning of others, and clarifying their own thinking.
Provide access and build connections
- Conceptual translations give meaning to the symbols and build the language of word problems by connecting language and imagery to mathematical symbols. Many of us would read 1½ ÷ ½ as one and a half divided by one-half or 1½ ÷ 3 as one and a half divided by three. Very little imagery accompanies these symbolic translations. However, read 1½ ÷ ½ as, "How many halves in one and a half?" Translate 1½ ÷ 3 as, "One and a half separated into three equal groups" and the symbols come alive. Have students write conceptual translations as a component of "showing their work."
- A split second after giving students a math problem hands are raised for help. Use writing to shift their focus from quickly solving the problem to first understanding the problem. After reading, students write questions they need answered to help them understand the problem. They show a partner the question, ask other pairs for help, or involve the class in finding answers to written questions.
- Observe a classmate model a problem using concrete materials and record their process. For example, to model 64 ÷ 4 (sixty-four separated into 4 equal groups) using base 10 a student observed:
- Made 64 using 6 tens and 4 ones.
- Put one 10 in each group—4 groups.
- Traded 2 tens for 20 ones.
- Put 5 ones in each group then 1 more 1 in each group.
This process parallels finding partial quotients and long division.
Open windows into thinking
- Have students write to a student who was absent or students a grade or two below about the day's math concept. I find that their explanations are more detailed and include pictures when writing to students two grades below. After all, "they need the help."
- Getting a variety of answers for the same problem is common. Instead of leading a discussion toward the correct answer, have students write about the answers it cannot be and why. Another option is to choose an incorrect answer and have students write about what that student did not understand or the mistake that was made. This increases the cognitive demand on students and requires them to critique the reasoning of others.
Critique reasoning and construct viable arguments
- Listening to what classmates say and critiquing their thinking is an important mathematical practice. When disagreements occur, have students write questions to help classmates change their minds. We initially need to negotiate the difference between a question and a statement. However, I'm no longer surprised that at first the only question they think of is, "Would you please change your mind?"
- When doing math, the defense of a solution should be the final goal. Instead of describing the process used, have students use persuasive writing (proof) to convince their peers that their solution is correct. A prompt I use is, "The answer is _____. I know because..."
Each of these techniques makes writing an invaluable component of excellence in math.
Response From Andrea Clark
Andrea Clark is a 5th grade teacher at a small, independent school in Austin, Texas. She has taught 2nd-5th grades at both private and public schools. She blogs about technology and teaching life at https://teachingwithmrsclark.wordpress.com, and she posts at edtech and books on Twitter @andrea_m_clark:
Writing in math hit me over the head at the NCTM Annual Meeting this past March. I went to the Elementary Mathematics Writing Task Force's session on writing and Zak Champagne's session on formative assessments back-to-back, and it all fell into place: Include writing in your formative assessments! I already do formative assessments at the end of each chapter. If I replaced these five-question quizzes with a more robust word problem, the students could write about it. So I tried it.
My students wrote about 2-3 times a week with these formative assessments. They solved one word problem and wrote about their process. I read what they did and why they did it, and I learned whether or not they understand the concept. It took longer and required additional instruction, but I had a much better idea of their grasp of the concept through their writing. This was a good start, but I thought I could do more.
This year, guided by Smokey and Elaine Daniels's book The Best Kept Teaching Secret, I want my students to write every day in math. Before they start solving a problem, they can write about what they know and their first steps. Once they solve a problem, they can write about the whole process and how they know their answer is correct. At the beginning of class, they can write about their homework: what they need help with and what they feel confident about. At the end of class, they can write about what they learned and what questions they have. Their summative assessments are going to have a written component so they can reflect on the unit as a whole: what they accomplished and what they are still thinking about. I'm keeping my writing-based formative assessments too.
If you are wanting to include writing in your math class, do it! Your students will need modeling of what a good mathematical written response looks like, but once they get it, you will be amazed at the richness of their thinking and the depth of their reasoning.
Response From Mike Flynn
Mike Flynn is the director of Mathematics Leadership Programs at Mount Holyoke College where he runs the Master of Arts in Mathematics Teaching program. He is also the author of Beyond Answers: Exploring Mathematical Practices with Young Children. He tweets at @MikeFlynn55:
Writing in mathematics classrooms can take many forms and serve a variety of purposes. One form that intrigues me lately is the use of contextualization as a strategy for sense-making. As teachers try to build students' conceptual understanding of abstract ideas, they can encourage students to create contexts that help them visualize elements like the relationships between values or the actions of operations. Creating real-life scenarios of abstract concepts can make the mathematics feel more intuitive and natural when done correctly.
Consider a student's strategy for solving 371 - 257 where they add 3 to the 257 to make 260. The new problem 371 - 260 is easier to solve mentally and results in a difference of 111. However, students who first try this strategy often struggle when deciding what to do with the 3 the added to the original problem. Should they add it to the 111 or subtract it? Here's where having students create contexts can be really powerful.
We can contextualize this situation by saying a school got a shipment of 371 iPads and they planned to give one to each of the 257 students in the class. In this scenario, we are trying to figure out how many iPads the school would have leftover after every student received an iPad. When we changed the 257 to 260, we altered the story by adding students to the school who were not supposed to be there. That means we gave away 3 extra iPads so we must add the 3 back to the 111. In essence, we are returning the extra iPads we gave away.
In the above example, the story context helps students see the relationship between the values in the problem and structure of subtraction. In this case, we took away more than we were supposed to so we needed to add that back at the end. The context serves as a mechanism to help students understand that when you add an amount to the subtrahend (the amount being removed) in a subtraction situation, the difference will be smaller. As one of my former 2nd-graders said when exploring this idea, "When you take away more, you end up with less."
Teachers can encourage students to write contexts to describe the mathematical concepts they are exploring as a way to help them deepen their understanding of abstract ideas. When students contextualize the mathematics, they provide a familiar frame of reference from which they can make sense these ideas. This form of writing can be very powerful in math class and can make a big difference in students' understanding.
Response From ReLeah Cossett Lent
ReLeah Cossett Lent is an international consultant and author of several books and articles on literacy and leadership. Her latest book, This is Disciplinary Literacy: Reading, Writing, Thinking and Doing. . .Content Area by Content Area (2016, Corwin), addresses learning in all disciplines:
A large high school math department was instructed to integrate writing into their math lessons but they were hesitant, explaining that mathematical "words" often take the form of numerical symbols, and sentences may look more like equations than prose. As a consultant hired by the district to support content-area teachers in incorporating reading and writing into their curriculum, I found this knowledgeable department less than enthusiastic about our collaboration—and I could hardly blame them. They had a lot to cover and little time in which to cover it.
Our work together led to important understandings for me as well as for teachers willing to take the time to think about how writing and math might be more like bread and butter than oil and water.
Writing is thinking—but that thinking doesn't always look the same. In math, students must organize and clarify information while planning for, reflecting on, and revisiting possible solutions to problems. Interestingly, these skills are exactly the ones required in writing. Often, the very process of writing clarifies thinking to such an extent that students experience a mathematical epiphany that brings everything together. Writing can become a critical tool for teachers to use in helping students unlock understandings.
One of the best writing activities we developed together led to students' increased mathematical reasoning as well as collaborative problem solving. Students were placed in groups of four and every group was given a different, challenging word problem. Each student was provided with a copy of the group's problem and allowed time to read through it. Then, students were instructed to write about how they might begin to solve the problem until the timer went off (about 2 minutes). The teacher then said "pass," and students passed their writing to the student to their right. Students were given time to read and respond to their neighbor's post and continued to be pass and respond until each problem/solution ended up with the original writer. Students then read their original post and the three responses, discussed together how they might solve the problem, and then shared with the class their problem and possible solution.
When we looked at the students' papers, we found that almost all of them began with some variation of "I don't really know how to solve this problem" with a few offering tentative suggestions. The final posts showed evolving mathematical reasoning and some had even solved the problem. This single activity demonstrated to all of us the power of writing, especially in math.
In what other ways can students use writing as a tool for mathematical reasoning? Take a look at some ideas the teachers in this department are now incorporating writing into their lessons.
- Reflections on learning or confusion in learning logs
- Interviews where students practice asking others questions to clarify their own understanding
- Essays about how to use math in real world situations
- Exit slips that ask students to explain, justify, describe, estimate or analyze
- Silent, written discussions where students defend their process or answers
- Blogs to create interdisciplinary connections
Response From Shannon Jones
Shannon Jones is an 11-year elementary school educator. She has spent all of that time teaching math to 3rd, 4th, and 5th graders. She has worked as a Math Resource Teacher, a school wide math Professional Development Lead Teacher, and this past school year taught an advanced math course to 4th graders. She lives in Maryland with her husband, who is also an educator, and their two little girls:
It is now commonplace on standardized math assessments for students to respond to prompts such as, "explain how you found your answer" or " explain how you used ______ to solve the problem." Writing can reveal understanding and misconceptions to guide our instruction, so how can we incorporate more writing in the math classroom?
In the beginning the school year it helps to write sentence frames for your students when asking them to explain their thinking in writing. This helps set the tone for how you want math writing to be composed. It also helps to provide students with a word bank of key math terms that they should consider incorporating in their written response. As the school year progresses eliminate the word bank, instead encourage students to brainstorm which vocabulary terms should go in the word bank.
Another strategy for adding writing into the math classroom is having students write their own math word problems.This is effective for evaluating a student's level of understanding about an operation. This writing strategy can reveal important math tactical issues, for example, if a student's division word problem has a larger divisor than dividend.
Writing can also be utilized at the beginning and end of units to assess understanding and retention. A teacher can begin a unit with students writing everything they know about fractions and conclude the unit with the same prompt.
Students writing about a strategy they are using to play an engaging math game is another effective way to integrate writing. Their words can reveal if they are being purposeful in their moves or just guessing.
Following a writing prompt, student's names can be removed and students can evaluate their peer's written responses using a student or teacher created rubric. Student created rubrics often lead to improved written responses.
Writing can also be integrated into math class as a closing activity where students can respond to prompts such as "What was easy today? What was difficult about today's lesson?" or "What do you understand now that you didn't understand at the beginning of class?"
Responses From Readers
every single day! Ss must explain their #math thinking with words, math symbols & expressions, & drawings.—Nicole Fahey (@Nikcal88) October 12, 2017
Got another: explain how you solved the problem. Write as if you're explaining it to your little brother. Pushes tons of rigor!—Piers Blyth (@PiersBlyth) October 12, 2017
Thanks to Linda, Sandy, Andrea, Mike, ReLeah and Shannon, and to readers, for their contributions!
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