Last week, I sat down to mark a quiz on radicals and exponents that I had given my grade 10 class. I knew that students had found the quiz difficult (there were gasps and shrugs from each student at some point during the quiz), but I was still surprised by the results. Problems that required skills that I had demonstrated accurately in class were answered incorrectly, and students made errors that suggested misconceptions. What was most discouraging was that both my students and I knew that their performance on the quiz did not reflect the significant amount of work they had done to prepare.
As I mulled over the results, and contemplated what to do next, two thoughts emerged:
- Students saw questions that looked a bit different from what they had seen in their homework and they panicked. As soon as the sinking feeling of I don’t know how to do this crept in, they weren’t able to employ the skills that they had used successfully several times before.
- Students had learned how to use specific skills to solve certain types of problems, but didn’t know how those skills could be applied together. For example, they knew how to convert a fractional exponent to a radical, and they knew how to rewrite a number with a negative exponent as a fraction with a positive exponent, but they didn’t know how to deal with an exponent that was a negative fraction.
With these two observations, I was reminded of my greatest weakness as a math teacher: as someone who is predisposed to think quantitatively, logically and to make connections between math skills, I often forget that many (possibly even most) students have a different cognitive disposition and need to learn how to think mathematically. In other words, the strategies that have largely contributed to my success in math are strategies that are intuitive to me. Being aware of my (subconscious) intuition, and making my strategies explicit/visible, is essential in helping students to develop those same strategies.
So, rather than asking my students to do corrections on their quiz to see if they can improve their score (which is essentially asking them to do the same thing but expecting a different result), I have attempted to map out what the skills and content look like in my brain (radicals-and-exponents-concept-map). My hope is that by making my thinking more visible, students will be able to articulate and deepen their own thinking in order to enhance their understanding.
Reflection is woven throughout the Middle Years Program. At the very core of the program, as in all IB programs, being reflective is one of the ten Learner Profile attributes that we cultivate in our students. Similarly, developing specific reflections skills is one of the Approaches to Learning skills that is developed across the IB continuum. The ability to reflect in discipline-specific ways is also embedded many of the MYP subject areas. For example, in English, students are taught to “produce texts that demonstrate insight, imagination and sensitivity while exploring and reflecting critically on new perspectives and ideas arising from personal engagement with the creative process” (criterion C). In their arts courses, they “create an artistic response that intends to reflect the world around them” (criterion D) and in science, students regularly reflect on the implications of science (criterion D).
The design and physical and health education (PHE) courses address reflection in even more detail. In PHE, students develop the ability to “explain and demonstrate strategies that enhance interpersonal skills; develop goals; apply strategies to enhance performance; and analyse and evaluate performance” (criterion D). In design (including courses as varied as theatre tech, coding, film and creative writing), students learn to “design detailed and relevant testing methods, which generate data, to measure the success of the solution; critically evaluate the success of the solution against the design specification; explain how the solution could be improved; and explain the impact of the solution on the client/target audience.” (criterion D).
With this variety of reflective skills permeating the whole program, and embedded in subject-specific objectives and assessment criteria, students get regular practice and feedback about the development of this essential skill set. Moreover, because the ability to reflect is explicitly stated among the learning objectives – and therefore included in the assessment framework – it is something that is intentionally taught, rather than something that is implicitly expected.
As members of the IB community, teachers at my school spend a lot of time planning and implementing strategies for cultivating the Learner Profile attributes in our students. This is exactly how it should be. The Learner Profile is central to the mission of IB and, as such, central to the mission of any IB school. Moreover, investing time and energy into proactively developing these attributes is a positive, strengths-oriented approach that is much more inspiring than a deficits-oriented approach. Thus, when it comes to the Learner Profile, our guiding question is “What experiences can we incorporate into our instruction that will help students to develop the Learner Profile attributes?”
From time to time, however, I think there is value in asking the more critical question, “What elements of of our instruction prevent students from developing the Learner Profile attributes?” This sheds light on some of the practices that might (unintentionally) be undermining our efforts in other areas.
In my own practice as a math teacher, I can think of a few habits that, unchecked, get in the way of my primary goal:
- Teaching students tricks or shortcuts undermines their thinking skills if they are not required to first make sense of the algorithm.
- Organizing assignments by topic or skill limits students’ thinking, as they follow the same procedure without having to identify the best skill to use in a given situation.
- Providing all the necessary information for students to solve a word problem undermines their inquiry skills, and passes up a valuable opportunity for students to generate questions rather than answering questions supplied by the teacher.
- Having students consult the answer key (or the teacher) immediately after solving a problem detracts from the value of reflecting on the cues available to determine if the solution is reasonable.
None of these habits is inherently bad, and they are all appropriate in some circumstances; however, if they become the standard operating procedure, students lose valuable opportunities to develop important traits. Imagine the unintended message we would send to students if the habits described above were entrenched in the routines of the class: repeat the process your have been shown (don’t think about what to do, how to do it or why it works), let the teacher ask the questions (don’t generate questions of your own), only the teacher has the answer. Together, these messages reduce students’ independence and self-efficacy.
Parents and teachers alike are constantly making decisions about how to help students develop character and intellect. There are too many decisions to make a pro-con list for every one; however, occasional reflection about the about the extent to which our habits support our goals will help to ensure our efforts yield the best possible result. When teachers and parents establish routines that are consistent with a shared goal, we support each other in developing students’ full potential.
Over the weekend, I had the privilege of facilitating a session with IB Primary Years Program (PYP) teachers about inquiry in math class. Because we all work in British Columbia, we share the opportunities (and challenges) of delivering the new BC curriculum within the IB framework. As I prepared the session, the image of a double-headed arrow kept coming to mind.
I have just finished a series of patterns investigations with my math 10 class. In this specific series, students explored the connections between quadratic equations (in both vertex and factored form) in order to connections between the equation and the features of the graph (like the position of the vertex, and intercepts).
These kinds of activities are a regular feature of math in the IB Middle Years Program (MYP) in which students are required to consider a series of specific situations to identify a pattern, describe it as a general rule, verify that their rule works and then justify why it works.
I’ve had a love-hate relationship with this kind of task since I started teaching MYP math almost 10 years ago.
I hate investigations because…
- Students find it difficult to approach novel situations.
- I find it difficult to prepare students for novel situations.
However, I love investigations because…
- Investigating is what mathematicians do. While math students typically spend more time doing tests (i.e. using specific examples to demonstrate that they understand a general rule that someone else discovered), mathematicians are on the frontier of searching for new pattern, generalization and rules. By engaging students in this process (even if they are discovering patterns that are new to them, but well-known to others), they are engaging in the authentic work of mathematics.
- Investigations support inquiry skills. In looking at a variety of specific situations in order to find trends, patterns and generalization, students develop strategies of problem-solving, visualization, hypothesizing and generalizing. These skills support similar processes in other disciplines, like finding trends in a data set, identifying themes in a work of literature or cause-and-effect relationships in history.
- Investigations foster independence. By looking for patterns and general rules, students develop the confidence to use what they know as the basis to discover new things, making them less reliant on the teacher as the source of knowledge.
- Investigations cultivate persistence. Not all problem-solving strategies will work in every situation. The more experience students have investigating patterns, the more comfortable they will be with trying an approach and switching to a different strategy when necessary. Rather than seeing this as a set-back, they will accept it as a normal part of the process.
As you can see, the benefits of investigations outweigh the challenges. Promoting a classroom culture in which students are willing to take a risk on an unfamiliar problem and persist with challenging work requires on-going effort, but I believe the benefits are on-going as well.
Last spring, I had a conversation with a parent that has stuck with me ever since. She initiated a meeting with me because she wanted a textbook (or similar resource) so that she could study the same math content as her child in order to be able to help with homework. At first, the request sounded totally reasonable, and even admirable. I love the idea of a parent modeling life-long learning by learning along with her child; however, I felt uneasy about the request and couldn’t put my finger on why.
Stalling for a bit of time to figure out my own confusing reaction to the request, I asked the parent why she felt responsible for helping with homework (again, a weird question since I’m totally in favour of families supporting learning at home). She explained that both her parents were teachers and were very helpful to her as a student, especially when a teacher had covered content too quickly, or when she didn’t understand the way a teacher had explained something.
That’s when the penny dropped. As a teacher, one of the reasons I assign homework is to gauge whether students understand the content and whether they’re ready to move on. When parents help with homework, the product that I see gives the impression that everything is going well… even when it isn’t. As much as this parent appreciated the help of her parents, her teachers never found out that they weren’t meeting her needs: the homework was complete and correct, indicating (perhaps incorrectly) that the pace and approach were fully effective.
In contexts where there are marks to be earned from homework, there is an incentive in helping children earn as many of those marks as they can; however, if homework is really going to be about learning – the student learning the content and the teacher learning about students’ progress – then that kind of incentive needs to be removed (as I have argued before). Students must be able to present their skills honestly so that teachers can give helpful feedback and adjust their instruction.
So, how can parents help more helpfully? Here are some ideas that come to mind…
- instead of correcting work, prompt your child to check their own work using appropriate resources and strategies (like an answer key, spell-check)
- if your child is unsure whether something is correct, or why something is correct, prompt them to follow-up with their teacher before school or during class (ideally before the assignment is due)
- remind children to use the feedback that they have received… this could mean doing corrections (just for the sake of learning, not for extra credit), or perhaps they can use the feedback to improve subsequent work
If you have additional suggestions, please post them in the comments.
This post is adapted from a piece I wrote for my school newsletter in the Fall of 2016.
I have borrowed the title of this article from an excellent book by Jo Boaler. A researcher in from Stanford University, Dr. Boaler draws compelling connections between her work in mathematics education and the work of Carol Dweck, another Stanford researcher, who studies the growth mindset. Dweck draws an important distinction between a growth mindset and a fixed mindset. Those with a fixed mindset believe that their intelligence is innate and cannot be expanded, whereas those with a growth mindset believe that they can increase their intelligence through effort.
A growth mindset is essential for learning in mathematics. Research has shown that students with a fixed mindset tend to avoid challenging work. Interestingly, this tendency is particularly pronounced in high-achieving students, likely because an inability solve a challenging problem is a threat to their self-concept as a smart person. Conversely, students with a growth mindset see challenging problems as an opportunity to extend their skills and develop their intelligence. Continue reading