Teaching Mathematics in Rich, Complex, and Realistic Contexts: Part 2
Strategies to Use in the Classroom
In the first blog post of this series, I talked about the need for students to develop critical thinking and literacy skills as a part of teaching mathematics within complex, realistic contexts. In this piece, I suggest strategies to use in the classroom to support students in developing these skills and share some resources I have found helpful.
Establish a safe and positive learning environment: A positive learning environment is one in which the students and instructor take collective responsibility for supporting the learning of all and in which the contributions of each individual are valued. In such an environment, students feel safe to admit when they are struggling and can better understand that struggle is a part of the learning process.
Normalize the struggle: Many students assume that reading and writing are easy for other people. Instructors can help students understand that they are learning complex skills, which many people find difficult, and that they will improve. It is also important for students to understand that these skills are empowering—they need to see how these skills will help them in their daily lives.
Scaffold activities and materials to develop skills over time: Establish learning goals for reading and writing and make a realistic evaluation of the skills students have when entering the course. Plan learning activities that will help students gradually build skills to attain the goals. For example, if the goal is for students to read and understand a lengthy news article with complex data, begin with short paragraphs and build up in length and complexity. The same method applies to writing. Scaffolding can also be used to gradually incorporate independent work. Start with class discussions and small group work to support students with reading and writing tasks. Then move students to independent practice for tasks of the same complexity. As more difficult tasks are introduced, return to collaborative activities.
Set clear expectations: Students often do not understand what it means to write a good response using quantitative information. Instructors should provide brief and explicit guidance on expectations. For example, the Dana Center’s Foundations of Mathematical Reasoning course is designed for students who are not college ready in mathematics. Often this population is also not college ready in reading and writing. We provide four basic Writing Principles for students to use throughout the course.
Principle 1 – If the problem has words, so should the answer!
Principle 2 – Each answer should be in a complete sentence that stands on its own, which means that the relevant information from the problem should be in the answer. The readers should understand what you are trying to say even if they have not read the question or writing prompt. Relevant information includes:
- Information about context
- Quantitative information
Principle 3 – If you use tables or graphs in your response, be sure they are clearly and thoroughly labeled.
Principle 4 – Let the reader know if you are making any assumptions.
Note that the principles are written simply and are not highly technical. The students receive a two-page resource that explains why writing is incorporated into the math course, provides guidance about how to get started with a writing prompt and explains each principle.
Use modeling to illustrate processes and examples: Modeling is a critical instructional strategy for reading and writing. Instructors can use this technique during class discussions, and materials can provide modeling during individual work.
One modeling strategy for reading is a “think-aloud” strategy. The instructor models reading for the students by explaining her internal thought processes when she reads. For example, if reading a credit card disclosure form, the instructor might say, “As a I skim this form, I notice that certain words and phrases are bolded. That tells me that those are particularly important.” For a writing example, an instructor can present a poorly written answer and ask the students to help improve it to better communicate information and issues.
Modeling in Materials
The Writing Principles Resource from Foundations of Mathematical Reasoning described above models each principle with examples of poorly and well-written responses. Principle #2 is illustrated with the following examples based on a prompt taken from a lesson.
Refer back to question 8 from Student Page 2.A: What are some factors you think may have led to this change in doubling times?
Insufficient response: Improved health care, better food.
Good response: The world’s population has increased rapidly. This increase may be due to factors such as improved health care, better food supplies, and clean water.
The Dana Center also uses materials to model writing is through multiple-choice questions in which students select the statement that is best written. The following example is from an early class activity. Each response is factually correct so students have to select an answer based on how well the information is communicated.
Which of the following statements best describes the change in doubling times before 1800 AD?
a) The doubling times generally decreased over time.
b) Before 1800 AD, estimated population doubling times decreased from 2,000 years to 1,000 years.
c) The doubling times decreased from 2,000 to 1,000.
This question is followed by a prompt in which students write a statement in response to a prompt. The students can refer back to the previous question for a model. The instructor notes suggest the instructor call out this strategy in the class discussion so that students will become more aware of it and use it in the future.
Provide opportunities for practice and feedback: Students must practice their skills regularly, and they need feedback about how to improve. Feedback is perhaps one of the most challenging aspects for math instructors because it is time consuming and requires training to provide effective feedback. Rubrics are a useful tool for instructors as an efficient way to give informative feedback and for students help them understand expectations.
Include reading and writing in assessment: The old adage that “what gets assessed gets valued” is all too true. If an instructor incorporates reading and writing into activities but bases student grades solely on traditional calculation-based tests, the students will focus their time and attention on those calculation skills. This is not an either–or situation. Instructors can create a grading system that values all of the skills.
I do not mean to imply that incorporating literacy skills into math classes is simple. I do want to communicate that it is possible. A first step can be to look for support within your institution. Language arts instructors can help math instructors think through how to implement these strategies. There are resources online or through professional associations that offer additional support. For example, the National Numeracy Network provides teaching resources for activities and assessments. The Network also has a highly active listserv whose members are highly responsive to questions and requests for help. The Association of American Colleges and Universities’ Quantitative Literacy VALUE Rubric is useful both to jumpstart conversations about expectations and to use as an assessment tool. The Chicago Guide to Writing about Numbers by Jane E. Miller is an excellent resource for instructors who want to delve more deeply into effective communication with quantitative information.
One last word of advice: Start small, especially if you are working alone with little support. Set reasonable goals based on the resources you have and gradually integrate more as you build a repertoire of skills and materials. The reward will come when students start telling you how they found ways to use math in their lives!
About the Author
Amy Getz is the Strategic Implementation Lead for the Higher Education Team at the Charles A. Dana Center. Her work focuses on the reform of developmental and gateway mathematics programs at community colleges and four-year institutions through working with state and regional systems. Amy led the team that developed and implemented the New Mathways Project in Texas and is currently leading work to expand the NMP to new states. Amy also led the development of the Quantway™ curriculum in partnership with the Carnegie Foundation for the Advancement of Teaching. Before joining the Dana Center, she taught mathematics for 20 years in high school and college. As the founding director of the Freshman Mathematics Program at Fort Lewis College in Durango, Colorado, Amy taught developmental and freshman-level math and led curriculum redesign that resulted in significant improvements in student success in both developmental and college-level math courses. She also led initiatives to provide professional learning services to K–12 math educators.