By Brigitte Lahme, Professor, Sonoma State University
Every university instructor would be thrilled if their students came to their mathematics classes with the ability to make viable arguments and to critique the reasoning of others; if their inclination were
- to persevere through difficult problems,
- to look for and make use of mathematical structures, and
- to strategically use tools in their mathematical toolbox.
But how do students develop these mathematical practices? The foundation is laid during a student’s 13 years of mathematics classes in K-12 – learning from their teachers and engaging in mathematics with their peers. The eight Mathematical Practice Standards that are an integral part of the Common Core State Standards (CCSS) for Mathematics, have elevated the importance and visibility of productive mathematical habits of mind in K-12 education. It is now an expectation and not a bonus. But are teachers equipped to help their students develop the practices until they become habits? Do teachers even have productive mathematical habits of minds themselves?
We actually know quite a bit about pre-service teachers’ habits of mind from research (Karen King: Because I love mathematics, Mathfest 2012, Madison). For example, pre-service teachers who hold mathematics degrees have an inclination to first state rules (Floden & Maniketti, 2005). They are not in the habit to seek meaning, which is such an important mathematical habit of mind. We can think of habits as acquired actions that we have practiced so much, that we eventually do them without thinking. At first, they are deliberately chosen but at some point they become automatic.
This has important implications for teaching at the university level, especially for pre-service teachers. Many professors and policy makers assume that completing a major in mathematics builds some kind of maturity. Undergraduate courses should be an opportunity to further refine productive mathematical habits of mind. Instead, this coursework often appears to reinforce unproductive habits of mind for engaging in mathematical practice. So I think we college/university faculty should take a serious look at what we are doing in our classes—not just in specific classes for future teachers, but in all our math classes. Mathematics faculty have a tendency to assign responsibility for K-12 math teacher quality to math education courses. But let’s think about that for a moment. In California, future high school teachers take 4 credit hours of math methods courses in their credential program. If they are lucky, they take at most a handful of courses as part of a math major specifically designed for future teachers, maybe 6 more credit hours. And they complete about 40 credit hours of mathematics content courses that are part of the normal mathematics degree programs. If they don’t learn productive mathematical habits of mind from their professors in their 10 or more college math courses, then who is responsible for this?
This is our responsibility and our opportunity! Pre-service teachers come to college with already formed ideas of what mathematics is and how the game of mathematics is played. They have already developed mathematical habits of mind—for good or for bad. It is up to us to help them replace unhelpful habits and develop productive habits, and we have approximately 4 years to do it.
When we are trying to change habits and practices, we often focus on directly changing actions and we hope this will lead to better results. In this case, we want teachers to change their teaching practice so that all students will develop productive mathematical habits of mind. But actions are affected by beliefs and beliefs are based on experiences. So it would be much more productive for us to provide pre-service teachers (and all students) with a series of compelling and positive experiences to change their beliefs. This, in turn, will lead to more coherent, consistent, grounded, and therefore stable results.
In my work with in-service teachers around transitioning to the CCSS, we have explored a variety of productive pedagogical ideas that provide students with experiences where they engage in mathematical practices. I have adopted several into my college classroom to better prepare my students for their work as teachers but also because I think this is simply good teaching for everybody. I’ll give two examples that focus on “Make a viable argument and critique the reasoning of others”.
In many of our courses, students write proofs; this is a mathematician’s idea of a viable argument. How do students learn how to write a proof? What are characteristics of a good proof? How do you critique other people’s arguments? On the first day of my combinatorics and graph theory class we worked on the following problem:
Students first collaborated on the problem in groups of 3—4. After students solved the problem, they made a poster to explain how they found their solution and how they knew that they had found all solutions. We then did a gallery walk: With a stack of sticky-notes in hand, students studied each poster. They asked questions about parts that they did not understand and they made suggestions when they found something that could be improved. They also pointed out aspects of the posters they found helpful in understanding the argument.
(sample posters with sticky note feedback)
Next, students went back to their own posters and studied the feedback they had received. They discussed revisions, and for homework each student individually wrote up an improved version of their proofs.
Before we finished the class, we had a discussion about the purpose of this activity. Students were surprised about the variety of proofs they had seen. After reading each other’s solutions, they were able to decide if there were gaps in arguments and describe what made a proof easy to read. They saw that there are a variety of ways to structure the argument, that a complete proof is not necessarily a good proof, and that a “proof by example” is not a proof but could possibly be revised into a general proof. They recognized the value of their peers’ feedback; and that they did not need the instructor to validate their proofs—rather, they possessed the mathematical authority to do so themselves.
You may ask: Our students write proofs and have to show their work all the time, why is this activity useful? In this case, it set the tone for the semester, and it made expectations clear to the students. Aside from seeing that they would be expected to actively work with their peers in class, they also experienced giving feedback and then using feedback to revise their work. They learned that an important goal of mathematics is communicating solutions, not just getting answers, and for the future teachers in the room, they saw a pedagogical structure they can use at any grade level and in any subject.
I do variations of the gallery walk in most of my classes a few times each semester. It works with modeling problems in calculus just as well as with proofs in real analysis.
Every instructor knows the following situation very well: Students have done a task. You assess it. There are major gaps. What do you do? You could
- Re-teach the topic or do more examples.
- Offer review sessions or office hours for students with gaps and work with them separately.
- Ignore the gap, go on, and hope the students will pick the content up later.
I want to describe another option: re-engage the students with the task and the concepts, using their responses to move everybody forward.
While learning how to write proofs involving the algebra of sets in my “Intro to Proofs” class, students did the following standard problem on a homework assignment: Given sets A and B, prove that A U (B – A) = A U B. While grading the homework, I found myself writing the same comments over and over again: “Pick a point,” “double set inclusion,” etc. I decided to use the proofs that students had written as the basis for the next day’s activity. To prepare, I compiled a collection of students’ proofs. In class, I handed out copies of these proofs to pairs of students. I asked them to discuss:
- What is good about each proof?
- Are there actual mistakes? Gaps?
- What makes a proof easy to understand? Hard to understand?
- Fill in gaps, correct mistakes.
Then we had a whole class discussion, keeping track on a document camera of changes students suggested.
Why was this activity better than just going over the proof again on the board or doing a similar problem, which would certainly have been faster?
By using a compilation of actual student work, students were invested in the exercise from the start. They already had engaged with this problem, so even if they had not written a perfect proof, they had a basis to build on. The examples I chose included good and bad features of proofs. The contrast and repetition allowed the students to transfer ideas from one to the other. The setup of the activity allowed students at every level to engage and benefit. One of my top students told me after the class that he had learned a lot about reading and critiquing others’ work. Finally, by contrasting several proofs, we had an excellent discussion about the structure of proofs, not just small details.
Research is compelling that students learn more from making and then confronting mistakes than from avoiding them (Boaler, 2016). My goal as a teacher is shifting from providing clear explanations so students don’t make mistakes, to creating situations, which are likely to produce important mistakes, and then helping the entire class confront and learn from those mistakes. Re-engagement lessons are a great method for this confrontation.
This is just one example of a re-engagement lesson. David Foster from the Silicon Valley Math Project contrasts re-teaching and re-engagement:
|Teach the unit again.||Revisit student thinking.|
|Address basic skills that are missing.||Address conceptual understanding.|
|Do the same or similar problems over.||Examine task from different perspective.|
|Practice more to make sure student learn the procedures||Critique student approaches/solutions to make connections|
|Focus mostly on underachievers.||The entire class is engaged in the math.|
|Cognitive level is usually lower.||Cognitive level is usually higher.|
(Foster & Poppers, 2009)
I offer the two classroom activities as examples to help us start talking about changing the mathematics culture in our classrooms and schools so that all students, including future teachers, have experiences that support them in forming productive mathematical habits of mind.
To educate our students to become mathematicians and teachers we have to do more than role-model mathematical practices, we have to create the environment where students engage in them, and we have to talk more about what we are doing and why. We have 4 years to help our students replace bad mathematical habits (speed, answer-getting, anxiety) with productive ones (sense-making, perseverance, use of tools and structure). This is our responsibility, but maybe even more importantly, this is our opportunity.
Boaler, J., & Dweck, C. S. (2016). Mathematical mindsets: Unleashing students’ potential through creative math, inspiring messages and innovative teaching.
Connors, R., & Smith, T. (2012). Change the culture, change the game: The breakthrough strategy for energizing your organization and creating accountability for results. [Also https://www.partnersinleadership.com/insights-publications/changing-your-culture/]
Common Core State Standards: http://www.corestandards.org/Math/Practice/
Floden, R., and Meniketti, M. (2005). Research on the effects of coursework in the arts and sciences and in the foundations of education. In M. Cochran-Smith and K. Zeichner (Eds.), Studying teacher education: The report of the AERA panel on research and teacher education. Mahwah, NJ: Lawrence Erlbaum Associates
Foster, D. and Poppers, A. (2009). Using Formative Assessment to Drive Learning: http://www.svmimac.org/images/Using_Formative_Assessment_to_Drive_Learning_Reduced.pdf