We know that our emphasis on the rote learning of mathematical processes is not facilitating the sort of deep-understanding of mathematics that our students need for success. Research from the Office of Australia’s Chief Scientist examined the approach taken to mathematics in 619 Australian schools achieving outstanding improvement in NAPLAN (a national standardised numeracy and literacy assessment) numeracy scores over a two-year period. A significant finding from this study was that “87% of case study schools had a classroom focus on mastery (i.e. developing conceptual understanding) rather than just procedural fluency.”
Our approach in mathematics, more so than in other disciplines is to remove any struggle that our students might experience. I have compared previously the typical approach taken to teaching mathematics with how we programme a computer. When this approach is the norm we shouldn’t be overly surprised then when our students consider mathematics to be all about learning a set of rules that they need to apply in the right order so as to output the correct response.
The product of most computational thinking, an algorithm is in essence just a step by step list of instructions that can be followed by a human or machine. An algorithm is defined by Google as "a process or set of rules to be followed in calculations or other problem-solving operations, especially by a computer". This approach is adopted in mathematics lessons when we teach students the steps they will then take to solve problems that we have assigned them
The “I do, we do, you do” sequence often empowers the coding of our students. Typically, the lesson begins with the teacher presenting the required method to the students. The teacher begins with step one being demonstrated on the board. Once step one is complete, the teacher demonstrates step two, and then step three and sometimes steps four and five. With triumphant zeal the teacher indicates the correct answer with a flourish of whiteboard marker and perhaps a double underline for effect. In phase two the students copy the process they have been shown with the teacher looking on to ensure the steps have been followed accurately. Naturally there are some bugs and errors that require correction. By the end of the lesson most students are able to accurately follow the steps and arrive at a desirable answer even if some of the numbers are changed.
This tightly controlled method reduces mathematics to a process and the learning of mathematics requires little from the learner other than memorisation. The impact of teaching mathematics as a set of rules to be memorised is reflected in the thinking of mathematical educators such as Ed Southall who writes:
Mathematics gradually became a mysterious entity, whose rules and steps I was expected to unquestioningly memorise – which I dutifully did. However, the process of storing numerous algorithms and their quirky properties became increasingly tedious, and I fell out of love with the subject that once intrigued and excited me. (Southall, 2017 p1)
If we deployed similar methods in the language arts we would not be surprised if we produced students who could decode text but struggle to infer meaning and have no love of reading. A librarian colleague who ignites a passion for reading in the lives of the young people who spend time in her Library refers to readers who “bark at text”. They are able to make the correct noises, but their understanding of the literature they engage with has been stifled by an overemphasis on the rapid reading of increasingly lengthy texts. When we emphasise mathematics as a discipline of procedures, we produce learners who “bark at numbers” with little understanding of their beauty, complexity and mystery. We produce mathematicians with limited knowledge and with no capacity to delve into mathematics for which they have not yet been programmed.
How might we move beyond this scenario? How might we teach in ways that foster a love of mathematical exploration?
We begin by shifting our thinking away from an emphasis on the correct solving of mathematical questions or problems and focus instead on mathematical reasoning. We invite our students to be mathematicians and we explain that the work of the mathematical mind is to reason with numbers. We create opportunities for students to play with mathematical concepts, to look closely, think slowly, test ideas and see what happens. We flip the instructional model on its head so that the pattern is “you do, we do, I do”. We spend more time on the first two phases of this cycle and we wheel out the “I do” only at the point of need, when only with access to some new knowledge will the learner find a way forward.
Above all else we value mathematical reasoning. We create opportunities for our learners to share their thinking and explain what they are doing. We often ask questions like “What makes you say that?” or “What do you think is going on here?” or “Explain the approach you are taking here?”. Whether the answer is right or wrong, we are more interested in the thinking that our students are engaging in. We create a classroom culture that values mathematical reasoning more than it does it answers. Indeed, if a question is answered and it solution can not be adequately explained our exploration continues until it can be.
Mathematical reasoning is defined by the writers of the Australian Curriculum as follows:
Students develop an increasingly sophisticated capacity for logical thought and actions, such as analysing, proving, evaluating, explaining, inferring, justifying and generalising. They are reasoning mathematically when they explain their thinking, deduce and justify strategies used and conclusions reached, adapt the known to the unknown, transfer learning from one context to another, prove that something is true or false, and compare and contrast related ideas and explain their choices.
While the prospect of teaching mathematical reasoning may seem daunting, it does not need to be. Including time for a Number Talk within each mathematics lesson is a simple step in this direction and doing so sends a powerful message to students about the nature of mathematics. Many teachers are already planning a sequence of lessons that would be facilitative of mathematical reasoning if they taught the planned programme in reverse order (beginning with the problem solving task that is traditionally used as the extension activity). When we make these changes and in doing so create opportunities to become observers of our students in the act of mathematical thinking we change the culture of our classrooms.
Careful questioning within a classroom culture where thinking is the norm, combined with an expectation that methods will be debated and that answers alone are insufficient provides the right environment for mathematical reasoning to thrive.
By Nigel Coutts
Questions that focus on mathematical reasoning:
What's going on here?
What makes you say that?
What are you noticing?
What do you wonder?
Tell me something about the problem.
Forget about the question for a second. What's going on in this situation?
What do you estimate the answer might be?
What do you predict the solution might look like?
Questions that direct students towards mathematical reasoning while problem solving:
Can you read the problem aloud again?
Let's go back to the question for a second. Is everything still making sense?
Let's refresh our memories about what each of these numbers represents. What does "this" mean?
Let's put numbers aside for a second and think about the units. Do they check out?
Let's try to visualise what's going on in this problem. Does that seem possible?
Can we visualise this in another way? What do you notice now?
What do we know? What don't we know?
What is not the answer? Why?
What makes you say that?
Is there a pattern here? What is it? Can you describe it, draw it or make it?
Southall, Ed. Yes, but why? Teaching for understanding in mathematics (p. 1). SAGE Publications.