Designing Opportunities for Mistakes

An abundance of research and blog posts exist that point to the learning that happens when our students make mistakes. This week, I wanted to explore some ways to use this knowledge in the classroom/as a class. There are certain ways that educators can set the whole classroom environment to embrace mistakes, but going further I want to find ways that teachers can create math questions/scenarios that leave room for mistakes to promote learning.


As I read about mistakes, I kept thinking, “OK great we need mistakes to happen, but I can’t force my students to get it wrong…so what do I do?”. I could not find many articles specifically on how to create openings for mistakes in the classroom. However, many of the ways we can do this, we already are familiar with. The following is a list of key ideas I compiled that I believe are essential to creating opportunities for mistakes in the math classroom (or all subjects!)

1. Encourage students to seek challenges

Carol Dweck – the pioneer of the growth-mindset – says, “I want challenges to become the new comfort zone” (PERTS, 2016). If we can encourage students to seek challenges, then the mindset of the educator and the student are aligned. As we discussed last week, there are many ways to encourage this growth mindset beginning with something as simple as praising based on ability (“You’re smart!”) but on effort (“You worked hard to achieve this”).

2. Make mistakes part of their work

This idea from Kelly O’Shea asks students to make an intentional mistake in their solution to a problem. They are able to choose a mistake that one of their group’s members made along the way to finding the solution, or create a mistake they feel other students might make. Unintentional mistakes are also welcome (as always!). Explicit mistake activities are a great way to get students to ask questions about mistakes rather than just pointing them out. This also asks students to reflect on the experience and review how struggle can be part of the learning process. My professor, Dr. Shelly Vohra, also highlighted an exit strategy that asks students to choose a favourite mistake they made during their math activity – a similar strategy to the one above.

3. Give work that encourages mistakes

This video highlights work with younger learners, but is very applicable for all ages. Let’s look at steps we can take to give work to encourage mistakes appropriately…

  • Find your student’s Zone of Proximal Development (ZPD). Vygotsky, in this paper, says that the ZPD is the difference between what the learner can do with help and what the learner can do without help. Instruction and questions should be catered to the ZPD so that our students can struggle, but struggle appropriately. The question is, how can we differentiate this for all students? Answer – question types!
  • In her blog post, “Open Ended Questions in Math”, Dr. Vohra explores the benefits of using multiple question types in the classroom – specifically open-ended and open-routed questions. Here is a chart she developed based on the work by Marian Small.
  • Within open-ended and open-routed questions we can also assign parallel tasks. Parallel tasks are sets of related tasks that explore the same key idea but are suited to different levels of student readiness. The Ontario’s Capacity Building Series Differentiating Mathematics Instruction suggests that using open-ended, open-routed and parallel tasks are imperative to differentiated math instruction.

If we use questions that have either multiple solutions or multiple ways to get the answers, we leave MORE freedom for students to mistakes. In my opinion, this is a beautiful way (theoretically) to encourage mistakes through the work educators give students!


In our professional course setting, we worked in groups to take closed-type questions from popular math textbooks and change them into either an open-ended or open-routed questions with a parallel task. This seems like a solid step towards creating improved math instruction techniques. Creating open-type questions takes a lot of time and perhaps workshops like these would be useful to in-service teachers as well. There are some resources with open-type questions, but what do you think about in-service workshops focusing on building an open-ended question database? Also, please share any more resources you may have OR any ways you leave room for mistakes for learning in your classrooms!

– A


O’Shea, K. (2012, July 5). Whiteboarding Mistake Game: A Guide. Physics! Blog!. Retrieved from

Ontario Ministry of Education. (2008). Monograph:  Differentiating Mathematics Instruction. Toronto, ON: Queen’s Printer for Ontario.

Ontario Ministry of Education. (2011). Monograph:  Asking Effective Questions. Toronto, ON: Queen’s Printer for Ontario.

PERTS. (2016). Make Challenge the New Comfort Zone. YouTube Video. Retrieved from

Vohra, S. (2015, April 30). Open Ended Questions in Math. [Image]. Retrieved from

Vygotsky, L. S. (1978). Interaction between learning and development (M. Lopez-Morillas, Trans.). In M. Cole, V. John-Steiner, S. Scribner, & E. Souberman (Eds.), Mind in society: The development of higher psychological processes (pp. 79-91). Cambridge, MA: Harvard University Press.


Don’t Call Me “Smart”!

Last week we looked at rebranding math – shattering the stereotypes and beliefs that have clouded the minds of many students. This week, we naturally transition our focus onto the mindsets of our students and the messages educators and parents/guardians convey. Quite simply, we’re looking at “fixed” vs “growth mind-sets”. A “growth mind-set” is one that understands the correlation between effort and achievement and has a strong will to work hard to improve and grow. A “fixed mind-set” is a belief that talents or intelligence are pre-determined fixed traits that can only improve so much.


Perhaps the most famous researcher and author on this subject is Carol Dweck, a Stanford University professor who has spent many (40) years conducting research on the ideas behind the mindsets. She advocates that the Secret to Raising Smart Kids is that we shouldn’t “tell kids that they are [smart]”, rather help them focus on the process (Dweck, 2015). This highlights the dangers of praising students on fixed qualities, such as being talented or smart, because it stifles their efforts to grow and paints mistakes as failures instead of opportunities to learn and improve.

The test results, especially in visual form, really highlight the importance of the messages we give our students… but is it enough to just praise efforts? As we discussed this in our professional learning environment, I couldn’t help but think – if we only praise children for their efforts than we miss the overall objective of actually growing.


Turns out, Carol also realized that many people seem to have misinterpreted the results of her studies. Educators cannot simply claim they have a growth mindset without conveying positive messages that promote learning. Messages and message-framing are important in and outside the classroom. You can find thousands of growth mindset messages online with a simple search, but I believe an equally important aspect of these messages is the consistency with which they are delivered. The educator AND the parents/guardians all need to convey similar messages and beliefs.


(Espinosa, 2015)


In my grade 7 placement last year, my mentor teacher did not allow the students to say, “that was easy”. I loved this rule. “Easy” is a perspective, not a fact and stating something seems easy does not foster a safe environment for learning. If we want students to value effort and view mistakes as learning, the classroom needs to be set up for this. If a student is finding things “easy”, then the educator needs to find a way to help that student grow – a next step for learning or a problem situated in their personal zone of proximal development. Perhaps there is even a value in teaching our students to complement each other on the growth they see in their peers and friends.


“Growth mind-set” is clearly a buzz word in education these days, but for good reason. I hope as an educator I can apply strategies in the classroom and deliver the messages to create this growth environment, especially in the math classroom. There is no “math person”, rather every student can learn math because our brains have the capacity to grow! Please let me know some strategies and messages you have found to be successful in your math classroom (or classroom in general)!

– A


Anderson, J. (2016, January 12). The Stanford professor who pioneered praising kids for effort says we’ve totally missed the point. Retrieved from

Dweck, C. S. (2015, January 1). The Secret to Raising Smart Kids. Retrieved from

Espinosa, O. (2015, June 18). Try one more time. [Image]. Retrieved from

Ragan, T. (2014, January 30). Carol Dweck a Study on Praise and Mindsets. [Video]. Retrieved from

Rebranding Math

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Andrew Morris, 2016.

New Year, New Problems

As the new school year begins, the coverage in the media has quickly switched from back-to-school excitement to current issues in education. The most popular news comes on the back of the latest results from the EQAO testing which revealed a decrease in mathematics scores. The Canadian Press (2016) reports that this year, in Ontario, only 50% of students achieved a score at the math standard – a drop from the 58% in 2012. As a result, the government has put forward a 60-million-dollar plan to improve students’ test scores in mathematics. There is much to say about this (including my opinion on standardized testing), but check it out the link to explore some more about this plan. As I head back to the first ever* second year of teacher’s college, this news is extremely relevant and important.

Tell Me How You Really Feel

It is not unusual to hear how much students hate math class. In the above video, we can hear the negative opinions and beliefs about math and even gain some insight to how they are formed. Quite simply, students find math useless and difficult. This can lead to a cyclical process where students receive poor grades in math leading to decreased self-efficacy and increased perceptions of difficulty which again lead back to poor results. Another aspect are the stereotypes and perceptions associated with mathematics. North American media often portrays math as a negative subject and math is often stereotyped as a subject for boys, not for girls. Many people also believe that they are not a “math person” (which is just not true). My point is, there are multiple factors that contribute to this perpetual distaste and hate for math. However, the outlook for educators shouldn’t be bleak. In the Discovery Education video above, we hear that students who enjoy math enjoyed it because of what their teachers did! Change in attitude and standardize testing results will always require support from multiple environments in a student’s life, but there is at least one intervention point that we have control over – our classrooms.

Next Steps / Reflection

Where do we often go wrong? One of my classmates, Aaron Strong, suggested that math is often taught in a “vacuum” and is isolated from other subjects. This vacuum implicitly creates irrelevancy for students – math is meaningless. I also believe that this is a huge issue in our math education today. Change in the way teachers approach math will not be easy. There are many in-service teachers that are very set in their ways of teaching. As pre-service teachers I think we have an increased responsibility to bring new and fresh ideas to our schools.

This week has been useful in reopening my mind to the strategies I need to use to increase my resource pool and start thinking about math from an integrative viewpoint. As we continue to explore math this year, I will blog about the ways in which I hope to rebrand math because: (1) anyone can do it, (2) it is applicable in our lives and (3) yes, it can be fun!

– A

*My cohort is the first to enter a new two-year Bachelor of Education program in Ontario. Previously, the B.Ed. was achievable in one year of study.


Discovery Education (2015). How do you really feel about math? YouTube Video.

The Canadian Press (2016). Only 50% of Grade 6 students met the province’s math standard.

Queen’s Printer of Ontario (2016). A renewed strategy for math.