Insights on Mathematical Writing at EARCOS Teacher’s Conference 2017

 

Blackboard Classroom Symbols Lesson Pie Chart Math
Blackboard Classroom Symbols Lesson Pie Chart Math From Maxpixel CC0 1.0

Writing and Common Core: a Math Made in Heaven?

There are some people who might still argue that mathematics has little to do with writing and that both manners of thinking should be taught as separate disciplines.

These people are most likely not mathematics teachers teaching in schools that follow the Common Core Standards for math. (Based on this article by a physicist-father of two young Common Core math learners/writers, they may not be scientists in the fields that require the most amount of math understanding either.)

A person has only to see the types of word problems that grade 2 students are learning to understand and solve to realise a distinct connection between math and language learning. There is therefore a need to teach math IN CONJUNCTION with language, keeping in mind the functions language must serve in mathematical discourse. For example, in order to effectively EXPLAIN their thinking when discussing math, students need to use cognitive academic language to infer, describe cause and effect, evaluate, and sequence.

As language teachers, a job that especially teachers in international schools must embrace, we try to integrate speaking, listening, reading, writing and presenting as much as possible. We believe that language develops best when these strands are taught in tandem. As such, it is logical that if we want students to skilfully use language to explain their mathematical thinking, they need to have opportunities to write in the course of studying mathematics as well.

Writing in Mathematics: It Can Be Done! at EARCOS Teacher’s Conference 2017

I had the good fortune to attend a session at the EARCOS Teacher’s Conference this year entitled “Writing in Mathematics: It Can Be Done!” led by two proponents of teaching students to learn and write for math, Jessica Balli and Dr. Patrick Callahan of Callahan Consulting, an organisation dedicated to helping schools make solid bridges between math and communication.

Background

The session leaders began with some surprising information about student perceptions of why we as teachers ask them to show and explain their math work; most students in interviews confessed that they believed teachers wanted them to prove they had not simply copied someone else’s answers.

Next they threw some shocking Common Core statistics out:

  • Only 37% of students produce written explanations of math work that show grade level understandings (based on exemplars provided with various CC based scope and sequences like New York Engage);
  • By 8th Grade, in one study, fewer than 4% of students could write a grade level mathematical explanation of an algebraic problem.

Writing for Mathematics as a Genre of Writing

We then began to discuss mathematical writing as a genre unto itself, which immediately perked my attention as I have been thinking along these lines for a few months now since I have begun teaching grade 2 EAL students strategies for comprehending math word problems. Colleagues and I have been developing lessons in which students learn to visualise both the information offered by word problems AND the question(s) they ask us to answer. Questions we have raised in this development are:

  • Why do we present especially younger students with levelled texts during reading instruction, but then expect students to decode and comprehend math word problems that are usually written at a higher level? It seems in order for students to understand math word problems, we have to intensify or modify our reading instruction.
  • Why do we teach students to draw a picture or diagram as a problem solving strategy without making explicit links to the reading comprehension strategy of visualisation?
  • How are pictures that students draw to help them solve a problem different from pictures they might draw to show they understand the information, numerical and otherwise, presented in the problem and the essential question the problem begs us to answer?

Ms. Balli laid out that math writing as a genre should include features and characteristics such as:

  1. precision of language and definitions;
  2. clarity and logic;
  3. the statement of assumptions;
  4. vivid description of quantities;
  5. accurate and logical comparisons.

In hearing this, I thought of how these features combine many language purposes: identifying, sequencing, comparing, explaining, hypothesizing, inquiry/seeking information (when it comes to understanding the question part of the problem), analyzing, and evaluating. One school district in the USA has even described an academic language function for problem solving (p.12) that incorporates many of these language functions.

In order to effectively describe math thinking in written form, students need to be aware of these language functions and they need have practiced listening, speaking, and writing for these purposes. As I listened to Ms. Balli and Dr. Callahan, it seemed to me that I had to keep this goal in mind as I planned instruction for my grade 1 and 2 EAL students. I felt that students also needed to be made aware of how to tie all these language purposes together when writing for mathematics.

Big job…but it is good to see the big picture.

A realisation hit me…at least some part of math class needs to look more like writing class (or vice versa)!

Practical Teaching and Learning Strategies for Mathematical Writing

Ms. Balli then led us through a series of activities she uses in her classes to encourage the development of mathematical writing through analysis, writing about, and discussing of Bongard problems.

(A Bongard problem (examples here) is a type of puzzle, named after the inventor of such puzzles – M.M. Bongard, in which the inquirer tries to identify rules that three pairs of figures on the left side of the puzzle follow by comparing them with three pairs of figures on the right side of the puzzle that do not follow the rule.)

Balli’s students analyze Bongard problems using a template she has devised and write out solutions to them in which thinking is described. She stresses that:

  • these Bongard puzzles do not need to explicitly connect to content being studied in math (remember the objective is for students to gain proficiency with writing about mathematical thinking);
  • the puzzle element of these problems are engaging for students;
  • writing about such problems, which may have more than one solution, is a low stakes activity for her students, a fact that she highlights throughout her lessons. Students have opportunities to go back and revise initial hypotheses about solutions using the template, much as they would revise any piece of writing to increase clarity and logic. This encourages students to be risk-takers in their writing and explanations. She also allows for variations in thinking and teaches students to tolerate and appreciate ideas that are different from their own.

(For more on using Bongard problems to teach math reasoning and writing, check out Jessica Balli’s blog!)

Beyond Bongard…

While Bongard problems may provide these classes with a low stakes introduction and practice of mathematical writing, Balli also has students solve and write about content based problems as well using another template she created that is meant to be used as part of a peer feedback exercise. You can see an example of problems on this template here. Again, in this situation, a problem is posed. Students write about their reasoning behind their solution. Students then share these written solutions with peers, who offer feedback based on a set of discussed criteria and using key sentence frames.

Guidelines for good feedback on math writing include:

  • Be specific;
  • Avoid opinions;
  • Think about what feedback you would find helpful;
  • Giving feedback does not equal being mean;
  • Good feedback pushes our peers to clarify their logic, language, and meaning;
  • Giving good feedback takes practice.

Sentence frames that Balli encourages students to use while writing peer feedback include:

  • I don’t understand…
  • How did you know _____ ?
  • I agree with ______ but…
  • What if you tried…?

Then, as a final step, students use this feedback to revise their initial written reflections.

Final Thoughts

I loved in this session how Callahan and Balli really helped clarify the connection between writing and mathematics. In particular, I think it is useful for teachers to understand that just as good writing instruction involves drafting, peer editing/reviewing, feedback and revision, so should writing instruction in terms of learning to express mathematical understanding. If the purpose of expository writing is to express understanding about a topic clearly, so is the purpose of mathematical writing to express understanding of math concepts and information found in problems and solutions in a way that allows a reader to understand the writer’s train of thought.

The main challenge, like usual, is where to fit everything in? It seems that sometimes the best way, as is often the case, is through integrating and blurring the lines between subjects. Again, we can devote some of our writing time to math writing. We can devote some of our math time to writing.

Lastly, having already discussed Bongard problems with lower Elementary teachers, I think that these problems may be slightly advanced to begin working on with younger students. Perhaps simple math analogies such as can be found in this series might be more appropriate to have students express their understanding through writing.

 

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