A Year of Mathematical Freedom

As a student, mathematics was a confusion of disjointed concepts to be memorised from page upon page of joyless, sterile and seemingly endless textbook problems. It was about speed and rules and conformity.

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A Year of Mathematical Freedom

I hated maths at school.

As a student, mathematics was a confusion of disjointed concepts to be memorised from page upon page of joyless, sterile and seemingly endless textbook problems. It was about speed and rules and conformity. Correct answers were currency. Every test score, grade and report comment only served to reinforce that I wasn’t a math person. My experience never deviated from the traditional.

Fast forward to today, I love teaching maths. Love it. My approach is pretty simple; I try to be the teacher I needed in school. If I could go back in time and give advice to my younger self, I’d let that little kid know he is a maths person. He needn’t feel dumb because eventually, he would begin to connect the dots rather than merely collect them. I’d tell him, absurd as it might seem at the time, that one day he’d be teaching the class.

In an earlier blog post called Self-Reporting Grades In Maths, I made this claim:

“Taking grades, scores, percentages, and competition out of our maths lessons has had a significant impact. And I’m not just talking about growth and achievement; I’m talking enjoyment. Students like learning maths in this environment. They don’t avoid “the struggle”, there is less fear of failure, in fact, we rely on it. Failing forward is the common factor in our room; if we aren’t struggling and making mistakes, then we probably aren’t learning.”

A couple of years later, I stand by this statement and I put it down to one thing; mathematical freedom. I’ve weaved some student responses (with permission) into this post because, to often, their voice and opinions are missing when educators discuss the impact of teaching and learning in their classrooms.

Mathematical Freedom

“I’ve definitely found Maths different to previous years because before it was sitting down and simply filling in a worksheet and waiting for a score, but this year we’ve been given more choice over what we do and how we choose to do it. I’ve enjoyed doing more hands-on activities and collaborations.” — C.H.
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Freedom Scales c/- Youcubed — Are all of the situations on the left of this continuum wrong or bad? I don’t believe so. But for me, the best learning happens in the situations on the right-hand side.

According to Dr Jo Boaler — Professor of Mathematics Education at Stanford University- mathematical freedom involves creating environments where learning is more highly valued than performing. Boaler expands this concept in the 2017 article “Math Class Doesn’t Work. Here’s the Solution.”

In my classroom, mathematical freedom looks like:

  • No grading day-to-day learning
  • Formative feedback
  • Regular self-assessment
  • Low floor/high ceiling tasks
  • Authentic assessments
  • Student controlled evidence of learning portfolios

Unlearning the performance culture that has stripped creativity, joy and wonder from an essential life subject might be the biggest challenge for educators who, like me, only experienced maths as highlighted in the left-hand column of the Freedom Scales featured above.

Believe You Can And You’re Half Way There…

“In previous years I doubted myself in maths thinking I wasn’t good enough. But when I came into this classroom my teacher understood me and helped me realise that I was pretty good at maths.” — P.J.

Helping students find their mathematical strengths is a big part of the job. How important is a little belief? Priceless. All kids bring something to the table, taking the time to help them identify their math qualities and strengths, rather than focusing on deficits, is time well spent. And it does take time, there is no quick fix. As Carol Dweck points out, one of the biggest misconceptions about growth mindset is,

“That it’s easy to implement. It isn’t. It’s really hard to pass a growth mindset on to others and create a growth mindset culture. It’s not about educators giving a mindset lecture or putting up a poster — it’s about embodying it in all their practices.”

We should be purposely helping students to understand themselves as mathematical thinkers. The speed + fluency = excellence stereotype is difficult to overcome, particularly when many parents with fragile math identities unknowingly perpetuate the same messages at home as a result of their own maths trauma experiences in school. The diversity of skills and thinking that students bring to the classroom is something to be celebrated, and that starts with us.

If you have quick mental recall in maths that’s great, but it doesn’t make you superior to others. Some very famous mathematicians are slow in figuring out answers” — D.S.

Open = Trust

“I think the Open Middle and DESMOS stuff is great because it lets people know that there are many ways to answer questions.” — C.H.

Open — or low floor/high ceiling- problems are similar to essential or non-googleable questions because they encourage the use of diverse strategies and pathways to arrive at a variety of answers depending on the learner’s understanding. We frequently use open problems from DESMOS, Open Middle, Youcubed and 3-Act problems. Why? I think it boils down to trust.

When I look at the right-hand column of the Freedom Scales, I see trust. Working on problems as long as it is useful; collaborating; sharing thinking; valuing different interpretations and methods and answers. Open problems and organisational flexibility provide students with the opportunity to have a sense of control over learning, seek to learn and improve for intrinsic reward, and the ability to connect learning with their own goals or interests, what Dan Pink describes as autonomy, mastery and purpose.

The Depth Of Knowledge (DOK) required to solve open problems also plays an important role.

“Depth Of Knowledge is like the depths of the sea. You have shallow stuff like DOK1 problems which can be worksheets or simple math problems that have a right or wrong answer. You don’t really have to think about them too much. DOK2 & 3 is the deeper stuff when you need to dive deep and put a lot of thought into your answer. There are lots of different ways of doing it and somebody else might have a different point of view.” — E.K.
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Depth Of Knowledge matrix from Robert Kaplinsky. Problems with a high DOK have answers that aren’t immediately obvious to the learner. They often require reflection on trial and error mistakes to inform future answers.

Problems with a high DOK ask students to demonstrate conceptual understanding, low DOK problems often require procedural skills and fluency. Again, this can be demonstrated by the differences between the two columns of the Freedom Scales. Low DOK tasks can be done alone, sitting at a desk, following a procedure. High DOK tasks encourage collaboration, discussion, and creative solutions. Both have value, but which one do you think learners enjoy doing most?

Learning > Performing

It doesn’t explicitly rate a mention on the Freedom Scales, but arguably, de-emphasising grades has had the biggest impact on learning in our room. Freedom from the constant judgement of scores and grades is something that I can offer in my room.

“Giving people grades isn’t always a good thing because if they get a bad score it can lower their self-confidence. Removing tests has definitely been positive and lifted the stress off my shoulders.” — C.K.

In the past, I’ve been guilty of pushing through content at breakneck speed to ensure we covered as much of the curriculum map as possible. Didn’t quite grasp that concept? Sorry, maybe next year.

When I gave myself permission to slow down, I also gave my students the time and space to struggle, wonder, explore and create rather than “learn and burn” to pass a test. In fact, we pretty much stopped testing at all. When test scores are off the table, standardised norms, competition, and cheating become redundant. Opportunity replaces compliance. Alfie Kohn has this to say about testing,

“There’s a difference between using them to figure out who needs help — or, for more thoughtful teachers, what aspects of their own instruction may have been ineffective — and using them to compel students to pay attention and complete their assignments.” — Alfie Kohn, Why the Best Teachers Don’t Give Tests

Students are willing to persevere with challenging tasks at the edge of their understanding because there is no point scoring to be done, no punishment for failed attempts, only progress to be made.

“Removing tests has impacted me in a good way because knowing I couldn’t answer all the questions used to make me feel like I suck at maths.” — A.T.

So was our year of mathematical freedom a success? I observed students become more confident, persistent, collaborative, and engaged in the learning. Interestingly, every student passed. Even the ones who school normally says shouldn’t; the learner with attendance issues, the students with disabilities and difficulties, the kids with test anxiety, all of them. Not because I lowered the bar, because they deserved to.

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