If you are interested in actioning what you read below, you can explore your options at [[Book Tutoring Sessions]]. You can also read about [[Daruma Dev Lab]], where students learn Coding, Maths and Physics. --- # Pedagogy Overview (Our Tutoring Methods) Below, I'll provide a detailed outline of how tutoring sessions historically evolve, what I believe is most important when teaching, and what my unique expertise can offer. ![[Website Pictures - Overview.webp]] --- ## Stage 1: First Sessions I believe in a conversational style where the focus is getting students used to reflecting on their knowledge and talking about Maths. In a class, they are one of many, and in that environment, it is difficult to develop Mathematical oracy (verbally expressing your understanding). While this skill is a Department for Education priority, it's something we don't have enough time for in the average classroom. This is why it's one of the most underserved skills in mainstream education and a great early focus to assess where a student currently is. This begins with foundational questions that aim to identify any gaps or inconsistencies. I'll pose small problems that probe their core concepts, like: "How does multiplication work in these different contexts? What's the same and what's different? Now try to explain why with your own examples.". We'll follow the path their answers point to and identify the most valuable areas of discussion. The way we learn at school leans towards having a toolbox of methods (when you see this kind of question, try this method). However, this is our opportunity to find the gaps (which all students have, as they are not yet experts) and deconstruct their knowledge by discussing it with an expert. It's always interesting to find the quirks that even top students haven't yet had the opportunity to examine. #### Example of difference between in-class and tutoring ##### In-Class Q & A _Question:_ $5y^3 \times 4y^2$ _Answer:_ $20y^5$ _Feedback:_ tick = right answer. ##### Tutoring Discussion _Question:_ Can you break down why both sides of this equation are equal? $5y^3 \times 4y^2 = 20y^5$ _Follow-Up Question:_ Say someone wasn't sure, how could we prove to them that they are equal? $5 \times y \times y \times y \times 4 \times y \times y = 5 \times 4 \times y \times y \times y \times y \times y$ _Follow-Up Question:_ So let's check what would happen if we set `y = 2` or `y = 5`? $5(2)^3 \times 4(2)^2 = 20(2)^5$ _Extension Question:_ How might we apply similar logic to these other examples? $y^5 \times y^{-3}$ _Extension Question:_ You're going to think outside the box a bit more from here, so just start talking me through your thoughts. $y^{1/2} \times y^{1/2}$ --- ## Stage 2: Focusing on Key Areas After we've reflected and assessed together, we'll drill into foundational concepts for a few sessions to solidify our understanding. The best way to practice Maths is ultimately to do Maths, so here we try to build on what they are already doing in class. However, in a classroom, a teacher has to find activities appropriate for many students of different ability levels; here, we aim to tailor the focus and difficulty of problems to a student's unique knowledge. This usually starts with me explaining and introducing a topic from base principles, creating small questions as we go to assess how fast we should go and where we should head. Here, we're deciding what scaffolding the student needs to get started and then working together to remove those supports. But we're always ready to improvise. If a student comes with a topic that recently confused them, we'll investigate together and adapt our discussion to their personal obstacles. These are often the liveliest and most fun sessions for us both, as students have the strongest opinions and understanding of what they are missing in the troublesome topics. --- _E.g., "What exactly are Sin & Cos? We just started using them in class but I don't get what they actually are."_ ![[Website Pictures - Cos & Sin Example.webp]] --- When the time comes, we'll reinforce by practicing questions together, discussing ideas and methods openly. This will sometimes mean working through my premade resources, other external questions, or eventually, real exam questions. The goal will always be to get students independently problem-solving, and I will challenge them by continually adding one more bit of complexity or approaching from another angle. These chains of problems allow us to move at the right pace for any student, only moving forwards when we're both confident. The final pieces to robust individual confidence are: self-marking by examination standards, working under time constraints, identifying errors in an answer, and critiquing their ability in individual topics. All of this develops into what we call metacognition strategies, which set up students to interrogate their own knowledge and identify how to develop and revise independently. --- ## Stage 3: Long-Term Journey The longer we work together, the more topics we can cover and understand, but the best outcome is the building of a student's independence and confidence. This sets them up to keep developing as resilient, self-motivated, and ambitious learners in whatever they choose to pursue. ### How does tutoring help get us there? In schools, we generally have to teach broadly and cover as much material as possible. This is because we have lots of topics to cover and are always wishing for more time. School curriculums are often organised into a few weeks on each topic, revisiting it in the next school year with more depth. Even newer structures like spiral curriculums rely on breadth, as there is so much to cover. When tutoring, we have a lot more freedom to adapt to a student's needs. We've already discussed the benefits of tailoring pace and questions to explore their understanding, but we can also design the structure of our external learning to complement what they already do. We can prioritise depth instead of width. This means we can revisit and break apart their assumptions from prior learning, or sometimes introduce new concepts ahead of time. Rather than revisiting topics after a gap of a year in between steps, we have the time and resources to lay out the whole thread that goes through their schooling. _Example 1:_ We learn about adding fractions in primary school, but what if they have algebra in them, or we have to rearrange and solve a group of them? _Example 2:_ We learn how to use sequences from learning to count or our multiplication tables, but what if we need to write and manipulate them as algebra, or investigate where a sequence goes if we keep counting infinitely? --- ![[Website Pictures - Different Directions.webp]] --- The more connections we can build, and the more perspectives from which we understand a concept, the stronger the knowledge is. This is true of how our memory works, and is what transitions a skill from something we consciously work to remember into a subconscious intuition. But it's never more true than in Maths, the subject often affectionately referred to as the language of the universe, where we can make connections to almost every other field—a pursuit I've been enjoying for decades. --- ![[xkcd - 435.webp]] I remember seeing this years ago and still think about it. Credit: [https://xkcd.com/435/](https://xkcd.com/435/?utm_source=gemini) --- ### How do we go deeper? Here is a simple question that we can keep revisiting as we grow in experience: _Can you present me a mathematical argument for which is bigger?_ $\text{Firstly}\quad 3/2\quad or\quad 4/3$ $\text{Secondly}\quad 7/8\quad or\quad 8/9$ _How many different arguments can you come up with?_ I've seen many different ways of making mathematical arguments for this topic that vary from primary school to university level. The point isn't for every student to memorise all the arguments, but for them to explore by thinking out loud and having a space to think deeper while being guided by someone with experience. These are the kinds of threads we can keep revisiting, making connections in different directions. I have a large bank of brain teasers and open questions like this which promote deeper thinking—something we don't get enough time for in a classroom. ### How do we go wider? Maths has always been the core of my education, even when I wasn't sure if I wanted it to be. In the end, it has taken me to many wonderful places, and I've often been in fields where the people around me say they wish they had done more of it. I'm still amazed how many students say something along the lines of, "I won't need Maths when I am older, what's the point?". I can't quickly summarise all the feelings I have about this, but here are a few headlines: - By practicing Maths, you are practicing pure abstract problem-solving, learning about how rules, systems, and logic all interact. This develops your mind to be better at functionally anything else. - The respect the subject has from academics and employers is potentially unmatched. This means it opens a huge number of doors if you work with it—not against it. - The number of times I have encountered GCSE/A-Level Maths in adult life and other fields is countless. My personal favourite is games programming and game design, where it is as important as anything else. This is often a favourite for young people who don't realise the connection and makes for fun examples. --- ![[Website Pictures - Video Game Example.webp]] --- By bringing Maths into other fields, we not only deepen our understanding of the topic, but we also make it more interesting and create intrinsic motivation where we naturally enjoy it. This is a fantastic field of subjects with something to interest anyone. If you need more motivation here, my other favourite examples are in Physics. For instance, Albert Einstein developed his theory about how gravity affected space and time, but he didn't have the Maths to represent it—until his mathematician friend Grossman did some research and found an existing branch of Mathematics which could use tensors to represent movements in four dimensions instead of three! ### Coaching, Revising and Thriving under pressure Last but not least, one of the most effective things we can do to boost a student's performance is to make them feel confident. I've found that students regularly achieve below what they can because they get anxious or stressed, causing them to forget what they know or do things they wouldn't normally. I know what it is like to be an uncertain student, but through good teaching when I was a student, I learned great revision techniques that built my confidence until I got A*s in Maths from GCSE to A-Level. I believe coaching is a really important skill for me to always be developing when working with young people. Fundamentally, exams pose a stressful time in a young person's life. It's the beginning of independence and the need to start succeeding on their own, isolated in exam conditions for the first time. Teaching them effective exam technique and revision strategies is absolutely essential to achieving their potential. They need to build problem-solving skills and strategies so that when exams come around: 1) They can manage emotions and self-regulate stress, staying calm under pressure. 2) They can practice improvising and making a plan to overcome uncertainty. 3) They can direct themselves down the most successful path (e.g., "I'm good at X, so I'll focus on that first for the most marks; we'll have a go at Y last and see what marks we can pick up"). 4) They know the difference between how short-term and long-term memory works (e.g., "There's no point in covering something new the night before, so I'll just skim this list of core topics and use that as a way to recall all the different methods I know"). 5) They understand the exam landscape (e.g., "This is a list of all the topics that can be asked, this is what came up on the most recent exams, and this is what the experts think is most important to be secure on this year"). 6) They have faith in their resilience, having tried lots of problems they didn't know how to do at first. All these skills develop them not just into capable students ready for the challenges ahead, but also well-rounded, positive people ready for what comes after. --- # Final Thoughts Thanks for reading if you got this far. As you can probably tell, I'm passionate about these subjects and really enjoy sharing that enthusiasm with others. I love what I do, and one of the great things about teaching is you are always learning yourself, too. I plan to keep growing these projects and this community as long as I can. This website's primary purpose is to serve as a homepage for booking online education sessions. Over time, it will hopefully also grow into a significant bank of resources for explaining and practicing Mathematics, Physics, and Coding, as I collate and develop my personal resources into activities, articles, write-ups, and e-books. Please share it around, as we rely on word of mouth to keep growing. Thanks for everything! --- # Any Questions? 📩 **Write to us at:** `[email protected]` 🚀 **Book Sessions & View Options:** [[Book Tutoring Sessions | Book Tutoring]] 👨‍💻 **Game-Making Club**: [[Daruma Dev Lab]] 💃🕺 **Who are we**: [[Services/AI/About Us]] 👺**Problems setting up?** [[Troubleshoot Tech Issues]] 🏄‍♂️**Cancelling or rescheduling?** [[Services/Terms and Conditions|Terms and Conditions]]