I’m an avid fisherman. There are times when I like to navigate a new spot and discover on my own the in’s and out’s of a place: how the tides work, where there’s structure that holds fish, depth changes, bait preferences, etc. I’m a better fisherman due to these experiences. However, there are other instances where I need to be told something directly like how to tie a uni-to-uni knot to attach my leader. This year I had the opportunity to present at NCTM’s annual conference on this same line of thinking in mathematics. In the session we discussed supporting students in productive struggle, specifically navigating the decision of when to tell. Here’s a recap of the session:
We know that productive struggle is an important instructional practice in teaching mathematics, but what is productive struggle?
“Effective teaching of mathematics consistently provides students, individually and collectively, with opportunities and supports to engage in productive struggle as they grapple with mathematical ideas and relationships.” (NCTM, 2014, p. 48)
Just as important, what is not productive struggle?
“…when students ‘make no progress towards sense-making, explaining, or proceeding with a problem or task at hand.'” (NCTM, 2014, p. 48)
So, students should be grappling with the mathematical ideas and relationships and if they’re making no progress, then the struggle is no longer productive.
Below are five tips for engaging students in productive struggle, two of which are specific about when to tell.
1. Anticipate the source of the struggle. Beware the “Expert Blind Spot.” (Wiggins & McTighe, 2005, p. 51)
Wiggins & McTighe (2005) describes the “Expert Blind Spot” as when an expert is so good at something that they fail to see a situation from the perspective of a novice. One of the best ways to avoid this is to do the math ourselves, trying to approach the task as your students would approach it. This gives us insight about what specific aspects of a task students might struggle with and what could be problematic for a novice. Also, this allows us to think about representations the student may use or misconceptions they may bring to the table.
2. Listen closely. Then ask questions based on student thinking.
The act of listening is a natural way to support someone when they’re struggling, just being present and empathetic. Also, John van de Walle (2006) suggests that we listen closely before doing anything else. We need to be sure that we really understand a student’s thinking before asking questions to move them forward in their thinking. This ensures students are moving forward in a way that makes sense to them.
3. Be specific about what you want students to struggle with.
The standards for mathematical practice are an excellent source of specific struggle for students. Consider MP 1 (make sense of problems and persevere in solving them) and MP 7 (look for and make use of structure). There are some wonderful “struggle words” in just these two mathematical practice standards like “make sense,” “persevere,” and “look for.”
4. Tell when it’s symbolic.
If it’s a mathematical symbol which carries an agreed-upon meaning, then this is a situation in which we need to tell. Also, we want to be careful about what we tell. For instance, = does not indicate that an answer comes next; it means that both sides of the symbol have the same value. Here it is important that we give students well-crafted, universal definitions that will serve them well into future grades. The above definition of the equal sign is functional through elementary school and into middle and high school, as well.
5. Tell if no progress is being made on the task,… but just enough.
I know in the past I’ve made the decision to tell and proceeded to give away too much. When we decide that telling is in order, we have to exercise restraint and give students just enough for them to begin making sense of the problem and moving forward on the task.
During the session there was authentic dialogue around navigating the decision to tell. Professionalism is another principle that NCTM proposes we put into action, not just arriving on time and being polite, but sharing the details of our practice in order to hone our craft. Look for opportunities to discuss productive struggle and the decision to tell with colleagues. Some great questions to consider are:
Is this an instance in which I need to tell? If so, how much support does the student need to make sense of the problem and begin making progress on the task?
What question(s) could I ask to clarify how the student is thinking about the task?
What question(s) could I ask to the student to move them forward in a way that builds on their thinking?
References
Leinwand et al. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA. The National Council of Teachers of Mathematics.
Smith, M.S. & Stein, M.K. (2011). 5 practices for orchestrating productive mathematical discussions. Reston, VA. The National Council of Teachers of Mathematics
Van de Walle, J. A. & Lovin, L. H. (2006). Teaching student-centered mathematics: Grades 3-5. United States of America: Pearson Education, Inc.
Wiggins, G. & McTighe J. (2005). Understanding by design expanded (2nd Ed.). Alexandria, VA: Association for Supervision and Curriculum Development.