Mastering Maths: Teaching For Understanding In A Tutoring Session

In this blog post, we explore my 9 essential principles for effective maths teaching and learning. Aimed at giving prospective parents an insight into what happens during my tutoring sessions, these principles have been forged over years of experience  building confidence, reinforcing understanding, and addressing misconceptions that students bring to the classroom. From starting each session with a recap to tailoring customised homework, we discuss the importance of mastering foundational skills, personalising explanations, and familiarising students with exam-style questions. By wrapping up with reflections and insights, we help students recognise their progress and deepen their learning. Lets look at these key principles so you can better understand how I can help you.

 

My 9 Key Principles for Effective Maths Tutoring

1.    Recap and Warm up

The most important part of any lesson is the opportunity to recap previously learned material. This not only helps students revisit key concepts but also addresses any misconceptions or gaps in their memory, which naturally occur over time. Forgetting and then re-remembering is a crucial aspect of the learning process. I start each session with five quick questions that act as a warm-up, priming students’ minds for the lesson ahead. This also gives me a chance to quickly check and mark the homework they bring in, ensuring we stay on track with their progress.

2.    Individual Feedback On Purposeful Homework

Homework is a valuable opportunity for students to work on their own, building confidence as they practice and reinforce what we’ve covered together. I encourage students to complete their homework outside of our sessions so we can make the most of our time for teaching and deeper learning. After all, I want them to get the best value from our time together. During our homework feedback, we go over any mistakes or tricky areas, reteaching any misconceptions before moving forward. I design homework to be accessible and achievable, tailored to each student's needs and understanding in the previous session, ensuring they feel confident in their progress every step of the way.

3.    Mastering The Foundational Basics

Most new mathematical concepts build on foundational skills, so it's essential that students master these basics first. Take Pythagoras' theorem, for example—before diving in, students need to understand how to square and square root numbers, and how this process is represented geometrically. To ensure this, I briefly assess these key skills before starting the lesson. This helps me make sure that all the language and mathematical ideas I'll be using in my explanation are clear and accessible, setting my students up for success as they tackle the new concept.

4.    A Tailored Explanation

All the preparation with previously learned material leads up to the moment when we introduce a new concept. This is where everything comes together. There are many ways to teach new topics, and I always tailor my explanation to match each student's current level of understanding. My explanations are carefully considered and planned to give students the best chance to succeed. I strongly believe this is where my expertise lies and where you get the best value from a session. I also use a variety of visual and physical aids to support their learning, as there are multiple ways to grasp a concept. My focus is on fostering deep conceptual understanding and mastery, rather than relying on rote memorisation.

5.    Check For Understanding

After explaining a new concept, I ask students a few quick, low-stakes questions to check their understanding. Simply asking if they "get it" isn't enough—I need to see evidence through their mathematical work that they’ve truly grasped and retained what I’ve taught. This approach ensures that they're ready to move forward with confidence, and it helps me identify any areas that might need a bit more attention before progressing.

6.    Independent Practice And Feedback

Next, students have the chance to put their understanding into practice, reinforcing what they've learned. The difficulty of the questions is carefully tailored based on the feedback I gather from the previous low-stakes checks. This ensures that the questions are at the right level to build both fluency and confidence. There's nothing worse than giving students questions that are too difficult, which can make them feel discouraged and erode their trust in the learning process. I believe it's essential for my students to feel like they're improving during each session, and that only happens through independent and deliberate practice that’s both challenging and achievable.

7.    Exam Preparation

Now students are more confident with the new concept, we can start applying it in different contexts. Past paper exam questions, especially those near the back of the paper, are an excellent way to do this. Exposure to these types of questions, with guided support from someone who knows the course and exam structure inside out, is incredibly valuable. I help students understand how the questions are framed, what the examiner is looking for, and how to approach them to maximise their potential, given their current level of understanding. Familiarity with these exam-style questions boosts their confidence and significantly increases their chances of achieving the best possible grade. Teaching and learning is something I feel very passionately about, however I also see preparing them for the exam as incredibly important.

8.    Reflection On Learning

At the end of the session, I do a quick informal check-in to highlight what we've accomplished together. This not only boosts their confidence but also reinforces their hard work and progress over the hour. It’s also a great opportunity for students to share any reflections they have about the lesson, allowing them to express their thoughts and further solidify their understanding.

9.   Customised Homework for Practice and Retention

I assign tailored homework that’s created during the independent practice section of the lesson. I make this homework as I’m assessing their level of understanding. This individualised homework is designed to reinforce everything we’ve learned during our session and must be completed before the next meeting. It's essential for students to finish it, as it helps maintain the flow of the lessons and allows us to maximise our time together in the following session. By doing so, we ensure that each hour builds effectively on the last.