The goal of this four part blog series is to consider the following questions:
- What is it about the way we talk about mathematics that promotes the acceptance of "I can't do that"?
- What might we do to shift the conversation?
Imagine being a student who is still learning sitting in a mathematics classroom and trying to navigate the lesson.
You might wonder: What is the teacher saying I should do? Did my classmates solve it the way I did? Will the other students laugh at me when I try to explain how I solved the problem?
Now imagine being the teacher of the math class.
You might be asking yourself: What can I say so that all my students will understand what to work on? How do I know that all my students understand how to solve this problem? How can I help students share their mathematical thinking in productive ways?
While we consider the mathematics classroom, it must be understood that numeracy goes beyond the mathematics teacher’s responsibility. By definition, “Numeracy is the ability, confidence, and willingness to engage with quantitative or spatial information to make informed decisions in all aspects of daily living” (Alberta Government, 2015).
So why the focus on mathematics? “Mathematics is the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations” (Merriam-Webster, 2020). It is a foundation for numeracy.
In Alberta, the Program of Studies makes this connection:
“The main goals of mathematics education are to prepare students to:
- use mathematics confidently to solve problems
- communicate and reason mathematically
- appreciate and value mathematics
- make connections between mathematics and its applications
- commit themselves to lifelong learning
- become mathematically literate adults, using mathematics to contribute to society.
Students who have met these goals will:
- gain understanding and appreciation of the contributions of mathematics as a science, philosophy and art
- exhibit a positive attitude toward mathematics
- engage and persevere in mathematical tasks and projects
- contribute to mathematical discussions
- take risks in performing mathematical tasks
- exhibit curiosity” (Alberta Education, 2016).
So how might we help students overcome the belief that “I can’t do math” in order to achieve confidence and competence with numeracy?
Reflecting on Our Beliefs
First, we may wish to reflect upon and examine our own beliefs as educators regarding mathematics learning and teaching. One such example of a tool for this is a self-assessment adapted from a self-assessment adapted from First Steps in Mathematics.
Facilitating Growth Mindsets
Next, we may wish to consider how we facilitate a growth mindset for students within our mathematics classrooms. Jo Boaler and YouCubed out of Stanford, among others, have done a great deal of work describing a growth mindset in mathematics. However, one must consider the practice of the theory.
How do we build and foster a classroom that values growth mindset? What steps are taken to build a culture of mathematical thinking? What feedback do we provide to our students when they say they cannot do well in math? What comments are made when our students make a mistake? How do we respond when a student approaches a problem in a different way than we might anticipate? How might we react when a comment is made to the effect of “That’s not how so-and-so showed us to do it!”? It is in our actions as educators, the modeling of growth mindset, that we will build this capacity in our students.
Empowering Our Learners
If I could do away with one phrase in mathematics classrooms, it would be “Show your work”. Of all the things I was ever asked “why” about, this was the most frequently queried.
To some students it may imply a lack of trust. To other students it conveys a desire to “work” or to “punish”. For others, it initiates anxiety as they do not know how to communicate their thinking. Regardless, the phrase does not explain to the student what is desired: the evidence of the thinking that went into finding the solution.
Math teachers are trying to understand the misconceptions that students are operating with or the mistakes they are making. And if the answer is the only thing that matters, we may overlook the fact that “Students may get the right answer for the wrong reasons”. (Western Australia Minister of Education and Training, 2007).
So how do we help our students understand that we are trying to do our best as their teachers to help them be their best as learners?
It may empower them if we change the wording of “Show your work” to better articulate the why, with phrases such as:
How do you know?
Demonstrate your thought process
Explain your thinking
It may also be worth a conversation to explain that process is as important as product. There is a time and place when yes, the answer must be absolutely correct, but in learning about how to get a correct answer, the emphasis should be on the process.
We may also wish to explore the power of the word “might”. By inviting students to respond to a might question, it allows them to enter a problem wherever they are at, without fear of there being explicitly only one way. For example:
How might you approach this problem?
What might be some possible solutions to this problem?
How might you ensure your answer is correct?
How might you know this process will always work?
I am a believer in the word might because it encourages thinking, opens up to many answers, and empowers students to take risks as they learn.
Inviting Students to Engage with Their Thinking
There are many other ways aside from the word “might” that invite students to engage with their thinking. The following are only a few examples with links to further resources should you wish to explore.
My Favourite No
An warm-up activity where students solve a problem and the teacher reviews an incorrect solution to reaffirm learning and overcome misconceptions. It promotes the acceptance of mistakes and demonstrates how errors are crucial to making progress.
Opportunities to engage in structured conversation with peers to review the accuracy and efficiency of solutions.
Regular journaling prompts that explore attitudes toward, learning of, and processes for mathematics.
Questions that promote mathematical discourse.
Rich activities that are authentic and accessible.
The Use of Manipulatives
This is a topic I have defended most of my career. I do not believe there is a time when manipulatives should be taken away from students. The outcomes in the Program of Studies speak to solving “concretely, pictorially, and symbolically”, not in one singular way. There is an efficiency that grows as students move from the concrete manipulative to the drawing to the symbols; along with this comes a confidence in their mathematical ability. However, manipulatives are a tool that empower students: it may be how they enter a problem or how they review their symbolic reasoning. They make learning accessible for some: they provide an opportunity for visual and tactile learners to truly engage in mathematics. They extend learning for others: some manipulatives require the implementation of spatial reasoning in order to make sense. Please make manipulatives available to all learners and normalize their regular and ongoing use in mathematics classrooms.
“Learning is built on existing knowledge” (Western Australia Minister of Education and Training, 2007).
It is important to find out what our students know: then we can help them scaffold a path to demonstrate what the outcome identifies. How are we describing assessment to students and providing feedback on those assessments?
“Learning requires that existing ideas be challenged” (Western Australia Minister of Education and Training, 2007).
In order for there to be new learning, sometimes the old learning needs to be deconstructed and reconstructed to make sense. How are we providing opportunities for students to share what they already know and talk through what may be confusing or conflicting?
“Learning occurs when the learner makes sense of the new ideas” (Western Australia Minister of Education and Training, 2007).
2 + 2 may not equal 4 until a student is able to make sense of what 2, what addition is, what a sum is, etc. How are we inviting students to make sense of the world around them? What authentic learning opportunities are we engaging students with?
“Learning involves taking risks and making errors” (Western Australia Minister of Education and Training, 2007).
A growth mindset is a key characteristic of a resilient learner. How are we inviting students to take risks? How are we addressing errors?
“Learners get better with practice” (Western Australia Minister of Education and Training, 2007).
Practice makes permanent. How are we helping students to understand the importance of practicing mathematics?
“Teachers cannot control what and when students will learn” (Western Australia Minister of Education and Training, 2007).
Patience is a virtue. How are we planning for teaching and reteaching and revisiting ideas? How are we addressing the same question that may come up repeatedly?
Some Final Thoughts
These thoughts come from a place in my mind where years of teaching experience, reading, watching videos, attending conferences, searching resources, taking courses, and so on have come together. I have done my best to attribute my learning to those who I know deserve the credit.
I tend to ask questions as I know I do not have all the answers. It is these questions that I hope spark reflection, contemplation, and conversation.
Consider the following when planning for instruction:
- Integration of the mathematical processes within each strand is expected.
- Learning mathematics includes a balance between understanding, recalling and applying mathematical concepts.
- Problem solving, reasoning and connections are vital to increasing mathematical fluency and must be integrated throughout the program.
- There is to be a balance among mental mathematics and estimation, paper and pencil exercises, and the use of technology, including calculators and computers. Concepts should be introduced using manipulatives and be developed concretely, pictorially and symbolically.
- Students bring a diversity of learning styles and cultural backgrounds to the classroom. They will be at varying developmental stages”
(Alberta Education, 2016).
Overcoming innumeracy and the social acceptability of “not being good at math” will take time. It starts with the students that we have now, and it will continue through conversations with parents, colleagues and leaders.
Alberta Government. (2015). Numeracy definition poster - Black and white. Retrieved from https://education.alberta.ca/media/159476/numeracy-definition-poster-blackwhite.pdf
Alberta Education. (2016). Mathematics kindergarten to grade 9. Retrieved from https://education.alberta.ca/media/3115252/2016_k_to_9_math_pos.pdf
Merriam-Webster. (2020). Mathematics. Retrieved from https://www.merriam-webster.com/dictionary/mathematics
Western Australia Minister of Education and Training. 2007. First steps in mathematics: Number course book, p. vii. Pearson Education Canada Inc: Toronto, ON.