A deep dive into Rob Vingerhoets' approach to mathematical proficiencies and how they can transform your classroom
Beyond the Basics: Understanding Mathematical Proficiencies
When mathematics consultant Rob Vingerhoets describes mathematical proficiencies as "the thing" in mathematics education, he's highlighting a fundamental shift in how we should approach teaching maths. In a recent podcast conversation with host Phil, Rob unpacked the four proficiencies embedded in the Australian Curriculum that many teachers either misunderstand or overlook entirely.
These four proficiencies—fluency, reasoning, understanding, and problem solving—aren't mere add-ons to our teaching practice. As Rob emphasises, they are the very foundation of effective mathematics education.
What Are the Four Proficiencies?
Fluency extends far beyond simply knowing multiplication tables. It encompasses a broader number sense and the ability to articulate mathematical thinking. It's about mental computation, effective and accurate use of mathematics, and the capacity to explain your mathematical process.
Reasoning involves justifying and explaining your mathematical approach. It's about answering the "why" and "how" questions: Why did you do it that way? How did you arrive at that answer? It's being able to provide evidence for your mathematical decisions.
Understanding means truly grasping mathematical concepts rather than memorising procedures by rote. As Rob recalls from his own education with Sister Colette in Bendigo, he could recite "six sevens are 42" without understanding why. He describes himself as having been "as deep as a puddle" in his mathematical understanding despite his strong memory for facts.
Problem Solving is the application of mathematical knowledge in varied situations. It's the ability to use reasoning, understanding, and fluency to tackle mathematical challenges and find solutions—ideally multiple solutions.
Rob also mentions a potential fifth proficiency that some educators discuss: productive disposition. This relates to student motivation and engagement with mathematics—having a positive mindset that encourages persistence and enjoyment in mathematical tasks.
From Theory to Practice: Implementing Proficiencies in the Classroom
The most powerful insight from Rob's discussion is how these proficiencies are intertwined. A well-designed mathematical task naturally incorporates all four proficiencies, engaging students at multiple levels of thinking.
Transforming Closed Tasks into Open-Ended Learning Opportunities
Rob provides a brilliant example of transforming a closed task into an open-ended one that engages multiple proficiencies:
Instead of asking students to solve "12 + 13 = ?" (a closed task), flip it around to "The answer is 25. What two numbers can you add to get 25?"
This simple shift creates:
- Multiple correct answers (25+0, 24+1, 23+2, etc.)
- Opportunities for pattern recognition
- Different entry points for students of varying abilities
- Natural differentiation
- Engagement with all four proficiencies
Rob explains how this open-ended approach reveals student understanding: "A kid who responds with 25+0, 24+1, 23+2 demonstrates total understanding, and their problem-solving skills of patterning are to the fore." The task can be extended further by asking for three numbers that add to 25, or introducing algebraic thinking.
The Three Cookie Challenge: Engaging Young Mathematicians
Perhaps most eye-opening is Rob's "three cookie challenge," which he uses with foundation (prep) students. He simply draws three cookies and a stick figure on the board, then waits for students to respond.
The progression goes:
- One person gets all three cookies (3 ÷ 1 = 3)
- Two people share three cookies (3 ÷ 2 = 1½)
- Three people share three cookies (3 ÷ 3 = 1)
- Four people share three cookies (3 ÷ 4 = ¾)
Rob notes that foundation students consistently solve this problem, recognising that with four people, "nobody gets a whole cookie anymore. You get three quarters of a cookie." This challenges our often low expectations of young learners and demonstrates that even 4-5 year olds can engage with fraction concepts when presented appropriately.
Auditing Your Teaching Against the Proficiencies
One practical strategy Rob recommends is "auditing back" your planned lessons against the proficiencies. Before teaching a lesson, ask yourself:
- How much reasoning is in this task?
- Will students be explaining their thinking?
- Are they developing understanding of the concept?
- Are they solving problems or just computing answers?
This audit can help you evaluate the quality of tasks and resources. Rob is particularly critical of worksheet-based approaches: "If you look at worksheets and line them up against the proficiencies and do your audit against the proficiencies, they're nowhere near it."
Benefits of Proficiency-Focused Teaching
Natural Differentiation
Open-ended tasks naturally accommodate different proficiency levels and allow for differentiation. As Rob explains, "Because an open task, by its nature, you can move up and you can move down."
This is particularly valuable in mixed-age classrooms. Rob shares his experience in a regional school with combined year levels: "In that grade 3-6 class, it'll be grade three who are smarter than some of the grade fives... because age is such a false thing to intelligence."
A single well-designed task can engage an entire class, regardless of age or ability level. Rob's car park problem (involving 30 wheels and different vehicles) successfully engaged all students in a combined prep-1-2 class.
Reduced Planning Load
Contrary to what you might expect, teaching with a proficiency focus can reduce your planning burden. As Rob notes, "I love open-ended tasks because it minimises my planning. I've got the one task to cater for all my kids."
This approach also saves on photocopying: "The poor photocopier if you've got teachers that don't have their head around differentiation. That photocopier is working overtime."
Improved Assessment
Rich tasks naturally generate meaningful evidence of student learning. Through observing students working on these tasks, teachers gain insights into their proficiency development that standardised tests can't capture.
Rob emphasises the importance of listening to students explain their thinking: "So much of how I assess a kid around the proficiencies is just listening... how they respond to questions like 'Why did you do it like that?' or 'How did you go about getting that answer?'"
Preparation for Future Success
Perhaps most importantly, mathematical proficiencies prepare students for life beyond school. Rob points out that employers aren't looking at test scores; they want employees who are:
- Good communicators
- Problem solvers
- Logical thinkers
- Collaborative workers
- Creative thinkers
These are precisely the skills developed through proficiency-focused mathematics teaching.
Overcoming Barriers to Implementation
Rob identifies several barriers to implementing proficiency-focused teaching:
Rigid Teaching Approaches
Rob is critical of heavily scripted, lecture-based approaches to teaching mathematics: "Lecturing, there is just so much research to say it's ineffective. It's actually a highly ineffective way to teach a human anything."
He's particularly concerned about approaches that begin with teacher lectures followed by worksheets: "When you start with a lecture, you're actually saying, 'Well, you won't know anything until I tell you.' Well, that's not right."
Over-Reliance on Poor Resources
Rob identifies several resource types that can hinder proficiency development:
- PowerPoint-heavy teaching ("Teaching can be death by PowerPoint")
- YouTube videos used uncritically
- Generic worksheet packages
- Topic-by-topic approaches that fail to show connections in mathematics
Low Expectations of Students
Rob expresses frustration at the consistently low expectations we have of young learners: "I can't get over how poor, how low the expectation sometimes is of our four, five and six year olds. They are so capable and we often give them dribble."
Moving Forward: Practical Steps
For teachers wanting to embrace a proficiency-focused approach, Rob offers several practical suggestions:
- Audit your current teaching against the proficiencies and be honest about what's working and what isn't.
- Start with engaging tasks - they don't all need to be open-ended, but they should be rich and engaging.
- Ask powerful questions like "Why did you do it that way?" and "Where's your evidence?" Don't accept "I just did it in my head" without further explanation.
- Start young - introduce proficiency-focused teaching from foundation year onwards.
- Access quality resources - Rob mentions his own website as well as resources from Michael Minas or New Zealand Maths, and other sites.
- Join professional learning opportunities - Rob is running webinars on each of the four proficiencies this year.
Conclusion: Making the Proficiencies Central
Throughout the conversation, Rob returns to his central message: the proficiencies are "the thing" in mathematics education. They aren't just supplementary elements but the very purpose of our teaching.
As he colourfully puts it, "They're not just the bit of mayo that you put on your maths salad. They are the salad."
By embracing the proficiencies and designing our teaching around them, we can transform mathematics education from a rote-based, worksheet-heavy experience into something genuinely engaging and powerful for all students.
For teachers looking to improve their mathematics teaching, focusing on the proficiencies offers a clear path forward. Start by auditing your current practice, seek out quality resources, and remember Rob's advice: "Rich, engaging tasks are the gift. They are the thing."
For more information about Rob Vingerhoets' approach to mathematics teaching and to access resources, visit his website or email him at rvec@bigpond.com. He is also running webinars throughout the year focusing on each proficiency.