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.
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.
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
Inspired by this segment of the BBC’s series, The Code, I recently tried replicating the jelly bean experiment during my school’s science fair.
I filled an old pickle jar with jelly beans and put it on display as students, teachers and parents explored the science fair. All science fair visitors were invited to estimate the number of jelly beans in the jar for a chance to win all the jelly beans. Meanwhile, I collected the estimates to determine whether the average estimate approached the actual number as more and more people participated.
In order to streamline the data collection, I had students enter their estimate in a google form. The estimates were automatically collected in a spreadsheet that I could analyze. It only took a few minutes to graph the results:
The display was a popular exhibit at the science fair and appealed to both elementary and secondary students, as well as parent visitors. As it turned out, the three best estimates all came from elementary students. I just presented the results at the elementary school assembly and it was a big hit. It was great to be able to present scientific evidence that two heads (or many heads) are better than one!
Would you like to give this a try at your school? Here’s a sample Google Form that you can use to get started.
This fall, I have been using the Problem of the Week resource from the Centre for Education in Math and Computing at the University of Waterloo. Every Wednesday, my grade 8 students get 40 minutes to tackle one of the problems and it’s great!
I love that students are required to draw on math skills that they have learned in previous units. Too often, students think they can forget skills just because they have moved on to another unit. I intentionally choose problems are are not necessarily related to the current unit. Students know this and expect to have to select relevant math skills and strategies.
I love that students can’t check the answer in the back of the book. To be honest, the first time I tried Problem of the Week with my students, I didn’t have time to work out the solution in advance. That turned out to be serendipitous: when students asked if they had the right answer, I had to respond with, “I don’t know. Is it right? How would you know?”, which encouraged some really great discussions. Now, students are in the habit of talking to each other to justify their solution and their approach.
By trying to explain their reasoning to a few classmates, students are better able to present their solution logically, which bring me to my next point…
I love that this is both accessible and challenging for all students. It’s a gross over-simplification, but I do seem to have two groups of students in my class: those who solve problems quickly but struggle to explain how they did it, and those who reach their solution by working it out carefully on paper. The problem itself is a challenge for the latter, and requiring students to write out their solution extends the abilities of the former.
If you’re interested in using Problem of the Week in your classroom, click here to subscribe to receive weekly problems from the CEMC.
Many parents believe that strong math skills, like strong literacy skills, are essential to their child’s success. While many of the activities that support literacy development- like bedtime stories, conversations at the dinner table and journal writing – are part of family routines, additional support for math is often outsourced to tutors or special programs. There are many resources and programs geared towards families who are looking for extra support, or extra challenges for their child. Here are some features that I think are essential to a good math program.
- The program structure should value improvement more than achievement. Many students believe that some students are good at math and others aren’t – a mindset that is damaging for students regardless of the group in which they think they belong. Rather than celebrating the “smart kids” who grasp concepts quickly, it is better for all students if the goal is improvement, regardless of their individual starting point. This encourages all students to reach their full potential.
- Students should understand the reason(s) behind the algorithms. Algorithms (or processes) are a great way to speed up computation once students understand why they work. For example, once students understand that 3 x 7 is the same as 3+3+3+3+3+3+3 which is also the same as 7+7+7, then they can memorize the 3 x 7 = 21 fact. However, if they only learn that 3 x 7 = 21, then it will be harder for them to apply that fact to solve problems.
- Students should be encouraged (and equipped) to explain their reasoning. This could be a simple as showing the steps they took to solve a problem or a complex verbal justification of their approach. Regardless of the format, students need to by able to articulate how they know that their solution is valid.
- Students should practice representing quantities in different forms. Whether it’s moving from standard form to expanded form, using a diagram to represent a word problem, or expressing a function algebraically and graphically, students need to learn how to move between forms of representation. This gives them a range of strategies to employ when it’s time to model and solve problems.
- Students should have fun. Math is both powerful and beautiful, and students should get glimpses of this. Board games, puzzles and rich problems are great ways for students to explore and enjoy the beauty and power of math.