Exploring Mathematical Concepts

If I had to restart my GATE journey, I would begin Linear Algebra differently. Back then, like most aspirants, I focused only on what was “needed for the exam.” I memorized the formulas for determinants, ranks, eigenvalues, and matrix tricks. And yes, that worked. You don’t really need deep intuition to solve GATE CS linear algebra questions. You can get full marks by applying procedures mechanically. But looking back, I realize I missed something important. If I had started with Gilbert Strang’s Linear Algebra lectures on YouTube (MIT OCW), I would’ve built a foundation that goes beyond exam shortcuts. Strang doesn’t just teach you how to compute—he makes you see what’s happening. Why multiplying matrices is more than a symbolic exercise, why the determinant collapses to zero when rows are linearly dependent, what span and basis actually mean geometrically. This kind of understanding may not give you a direct edge in GATE marks. But here’s the catch: once you step into an IIT classroom, especially in courses like Machine Learning, this intuition becomes gold. Suddenly, terms like “vector spaces,” “projections,” “orthogonality,” and “basis transformations” are everywhere. And if you’ve only learned linear algebra as a bag of tricks, you’ll feel lost. That’s why, if I could start again, I’d tell myself: For GATE → yes, practice shortcuts and tricks, because time matters. But also → invest parallel time in Strang’s lectures. Don’t ignore intuition just because the exam doesn’t ask for it. Because exams end in a few hours. But the way you understand math will follow you for years. Here Is The Link 🖇️ - https://lnkd.in/d659c4Wu --- #GATEPreparation #GATE2026 #GATECS #GATEJourney #GATEExam #GATEAspirants #ExamPrep #LinearAlgebra #GilbertStrang #GilStrang #Mathematics #MathIntuition #MatrixAlgebra #VectorSpaces #Determinants #BasisAndSpan #MachineLearning #MLMathematics #MathForML #DataScience #AI #MLJourney #IITJourney #EngineeringLife #StudyTips #StudentLife #ConceptOverFormula #LearningNeverStops #LifelongLearning #AcademicGrowth #FutureReady

🤯 𝗦𝘁𝘂𝗰𝗸 𝗶𝗻 𝘁𝗵𝗲 ‘𝗪𝗵𝘆’ 𝗼𝗳 𝗠𝗮𝘁𝗵𝘀? 𝗟𝗲𝘁’𝘀 𝗖𝗵𝗮𝗻𝗴𝗲 𝗧𝗵𝗮𝘁! 𝗪𝗵𝗲𝗿𝗲 𝗺𝗮𝗻𝘆 𝘀𝘁𝘂𝗱𝗲𝗻𝘁𝘀 𝗮𝗿𝗲: Sitting with a complex maths problem, staring at the paper, feeling frustrated. They’ve memorized formulas, but when it comes to breaking down a tricky question – they freeze. ❄️ 𝗧𝗵𝗲𝗶𝗿 𝗱𝗿𝗲𝗮𝗺 𝗼𝘂𝘁𝗰𝗼𝗺𝗲? They want to approach any problem with confidence. To see complexity and think, “I’ve got this.” To understand every step—not just apply it. 𝗪𝗵𝘆 𝗗𝗼 𝗠𝗼𝘀𝘁 𝗦𝘁𝘂𝗱𝗲𝗻𝘁𝘀 𝗦𝘁𝗿𝘂𝗴𝗴𝗹𝗲? Most students fail because they focus on what to do, not why they’re doing it. They memorize steps but miss the logic. So, when they face an unfamiliar problem, they get lost.  • They don’t question the question.  • They skip the ‘why.’ 𝗧𝗵𝗲 𝗦𝗼𝗹𝘂𝘁𝗶𝗼𝗻: 𝗕𝗿𝗲𝗮𝗸 𝗗𝗼𝘄𝗻 𝗣𝗿𝗼𝗯𝗹𝗲𝗺𝘀 𝘄𝗶𝘁𝗵 ‘𝗪𝗵𝘆’ 𝗧𝗵𝗶𝗻𝗸𝗶𝗻𝗴 🧐 Teaching students to ask “why” at each step transforms their understanding. Here’s how it works: 1️⃣ 𝗦𝘁𝗮𝗿𝘁 𝘄𝗶𝘁𝗵 𝘁𝗵𝗲 𝗣𝗿𝗼𝗯𝗹𝗲𝗺, 𝗡𝗼𝘁 𝘁𝗵𝗲 𝗦𝗼𝗹𝘂𝘁𝗶𝗼𝗻: Instead of rushing to plug in formulas, encourage them to ask: • “What is this problem actually asking?” • “What information do I have?” • “What do I need to find?” 🧠 𝗧𝗶𝗽: Break the question into smaller parts. Each part should be a mini-problem to solve. 2️⃣ 𝗤𝘂𝗲𝘀𝘁𝗶𝗼𝗻 𝗘𝗮𝗰𝗵 𝗦𝘁𝗲𝗽: When they apply a formula or make a calculation, they should ask: • “Why am I doing this step?” • “How does this help me get closer to the solution?” 🔍 𝗘𝘅𝗮𝗺𝗽𝗹𝗲: Solving an equation? • Why do we move variables to one side? • Why do we simplify terms first? 3️⃣ 𝗥𝗲𝗳𝗹𝗲𝗰𝘁 𝗔𝗳𝘁𝗲𝗿 𝗦𝗼𝗹𝘃𝗶𝗻𝗴: Once they reach an answer, teach them to look back and ask: • “Did every step make sense?” • “Why did this method work?” • “Could I explain this to someone else?” This reflection cements learning 𝗧𝗵𝗲 𝗧𝗿𝗮𝗻𝘀𝗳𝗼𝗿𝗺𝗮𝘁𝗶𝗼𝗻 🌟 When students adopt ‘why thinking,’ they gain confidence, reduce mistakes, and develop deeper mastery of concepts. This approach shifts them from memorizing to truly understanding, paving the way to top grades. Let’s help them question the question. The answers will follow! 👍 Like | 💬 Comment | 🔁 Repost | 👤 Follow me, Faisal Naqvi #MathsMastery #GrowthMindset #QEDTuitions #CriticalThinking #ProblemSolving #WhyThinking #ConfidentLearners

𝗠𝗼𝘀𝘁 𝗺𝗮𝘁𝗵 𝗺𝗶𝘀𝗰𝗼𝗻𝗰𝗲𝗽𝘁𝗶𝗼𝗻𝘀 𝗮𝗿𝗲 𝘁𝗿𝗲𝗮𝘁𝗲𝗱 𝘁𝗵𝗲 𝘀𝗮𝗺𝗲. 𝗧𝗵𝗮𝘁’𝘀 𝗮 𝗽𝗿𝗼𝗯𝗹𝗲𝗺, 𝗮𝗻𝗱 𝗵𝗲𝗿𝗲 𝗶𝘀 𝘄𝗵𝗮𝘁 𝘁𝗼 𝗱𝗼: Not all misconceptions come from the same place and if we don’t diagnose them correctly, we address the wrong thing. After coaching across multiple classrooms, here are 3 common types of misconceptions, and what to do to address it: 𝟭. 𝗖𝗼𝗻𝗰𝗲𝗽𝘁𝘂𝗮𝗹 𝗠𝗶𝘀𝗰𝗼𝗻𝗰𝗲𝗽𝘁𝗶𝗼𝗻𝘀 Students don’t fully understand the why behind the math. ➡️ Example: Believing a larger denominator means a larger fraction. What to do: 🧱 Rebuild understanding using visual models (number lines, area models) 📈Use the CRA progression (concrete → representational → abstract) ❓Ask: “Why does this make sense?” 𝟮. 𝗣𝗿𝗼𝗰𝗲𝗱𝘂𝗿𝗮𝗹 𝗠𝗶𝘀𝗰𝗼𝗻𝗰𝗲𝗽𝘁𝗶𝗼𝗻𝘀 Students misapply or incorrectly execute a process. ➡️ Example: Adding denominators when adding fractions. What to do: 🎯Identify the exact step where the error occurs 🗣️Model the correct process with think-alouds 📍Give targeted practice at the point of breakdown (not the whole problem) 𝟯. 𝗟𝗮𝗻𝗴𝘂𝗮𝗴𝗲-𝗕𝗮𝘀𝗲𝗱 𝗠𝗶𝘀𝗰𝗼𝗻𝗰𝗲𝗽𝘁𝗶𝗼𝗻𝘀 Students misunderstand the math because of vocabulary or phrasing. ➡️ Example: Misinterpreting words like “of,” “per,” or “difference.” What to do: 🔃 Pre-teach and revisit key vocabulary in context ✍🏿 Incorporate math writing and discussion routines 💡 Ask students to explain their thinking in their own words Strong instruction doesn’t just fix mistakes. It identifies the type of misconception and responds with precision. Because when we get clearer about the problem our instruction gets sharper. And student learning accelerates. ______ ♻️ Repost if this is a conversation your math team needs to have ➕ Follow for more on math instruction, coaching, and leadership systems 📬 Join my newsletter The 3-1-4 3 insights. 1 strategy. In 4 minutes. For leaders committed to strengthening math instruction and student outcomes. Link in comments ——— Hi, I’m Dwight Williams. I help schools and districts strengthen math instruction through coaching, curriculum support, and data-informed systems that drive student confidence and achievement.

Some personal advice for those learning Mathematics: (1) Work through topics in parallel, and do not do too much new theory on the same topic in a day. Allow your mind time to subliminally process ideas and techniques, and move on to a different topic after a chapter. Reading 100+ pages of new material on the same topic in a day is a deceptive way of feeling productive. (2) Learning theorems without doing exercises is a near-pointless endeavour. One rarely gets to directly apply a theorem in its exact form in mathematics—it has either already been done or is quite a trivial exercise. The methods and the techniques used to construct those theorems teach you far more; it is the proof that will enrich your mathematical knowledge, not the recollection of the theorem itself. (3) In pure mathematics, less is (often) more. There is non-homogeneity in learning different areas of mathematics. For example, in more calculation and derivation heavy subjects, I would argue that volume and breadth of examples, even without concrete understanding after solving a question, is okay, given you have many more questions to solve of similar type. In more abstract modules, doing 10 proofs in immense depth, where you understand the nuances of the problem, can visualise it, can understand the logic of each step, and develop a generalisable proof “toolkit”, is far more useful than blitzing through 100 proof solutions with subpar understanding after each one. The latter approach was a mistake I made many times. (4) Visualise each theorem so you can reconstruct it at any time. This builds on points 2 and 3, but many textbooks and lecture notes will describe a theorem in an alternative style, use unconventional notation, etc, and a useful way to (quickly) deal with those differences is by recalling the visual map you built when learning the theorem. This takes a lot of time, it is hard, and sometimes slightly overkill, but it will improve your mathematical clarity in the long run. Think of this as stripping away any dependence on particular notation / style, so you’re forced to understand the essence of the theorem. There are several more pieces of advice I have, and I will share them in the near future. The photo is of the common room in the Mathematics department (Andrew Wiles building) at Oxford.

We're asking the wrong question when we help children with math. And it's limiting their mathematical growth in ways we don't realize. Most of us default to "What answer did you get?" when we should be asking "How did you get it?" This simple shift in questioning has the power to transform mathematical learning both at home and in our classrooms. When we focus solely on correct answers, we miss the opportunity to understand and strengthen the thinking processes that lead to mathematical understanding. But when we ask children to explain their reasoning, something remarkable happens. Here's what changes when we prioritize process over product: → Children become more aware of their own mathematical strategies → They develop confidence in their problem-solving abilities → They learn that there are often multiple valid approaches to solving problems → They begin to see themselves as mathematical thinkers, not just answer-producers Students who regularly explain their mathematical thinking show deeper conceptual understanding, improved retention, and stronger problem-solving skills across all areas of mathematics. In practice, this means asking questions like: • "Can you walk me through your thinking?" • "What strategy did you use to solve this?" • "How did you know to approach it that way?" • "Could you solve this problem a different way?" For parents: This transforms homework time from answer-checking to thinking-building. Your kitchen table becomes a space for mathematical discourse. For educators: This shifts classroom culture from performance-focused to learning-focused, where students feel safe to share their reasoning and learn from each other's approaches. The impact extends beyond mathematics: When children regularly articulate their thinking, they develop metacognitive skills that benefit them across all academic areas and in life. To my fellow educators and parents: How are you encouraging mathematical conversations in your environments? Because when we value the journey of mathematical thinking as much as the destination of correct answers, we raise children who see themselves as capable, confident problem-solvers.

*** Books Review: Learning How to Learn & A Mind for Numbers **** * Barbara Oakley, PhD (Professor of Engineering) published her famous book: A Mind for Learning in 2014 which was aimed at providing students with effective strategies for mastering math and science concepts * Later, a MOOC on Coursera was lauchned titled 'Learning How to Learn' and subsequently a book with same title was published in 2018 * All these resources provide very similar recommendations, although the first book: A Mind for Numbers is more detailed and which 'Learning How to Learn' provides more generic guidance for broader audience * I read these books a couple of years earlier and have been trying to implement their strategies in my continuous learning journey ************ SUMMARY / RECOMMENDAIONS ********** Below are my keys takeways from these books (although not in the same order as wrote by the author) * Mindset: Any person (not just gifted individuals) can learn math & science by adopting right learning strategies * Habits / Routine: follow a balance routine which includes study, physical exercise and adequate sleep. Good sleep and physical activities have postiive impact on our learning abilities (and on our happiness / wellbeing) * Avoid multi-tasking: focus on one task at time, breakdown larger tasks in small steps (use pomodoro method: 25 min of activity followed by 5 min break) * Simplify First: use analogies / visulatization / intution / common sense to grasp complex topics (lenghtly, complicated equations are not helpful as first step in learning a concept) * Learn in chuncks - by interconnecting pieces of information together * practice and repetition are very effective in learning * use nmenomics to retain important information & facts * view mistakes as integral part of learning * actively solve problems by applying concepts rather than passively re-reading and highlighting text * benefit from both focused and diffuse modes of learning *********** These books have profoundly shaped my approach to continuous learning, and I highly recommend them to anyone looking to sharpen their skills and achieve personal growth. Have you read either of these books or taken the Coursera course? I’d love to hear about your experiences or favorite strategies! ************ #LearningStrategies #ContinuousLearning #BookReview #ActiveLearning #PersonalDevelopment

Try the simple shift to retain more of what you study. If you’re reading your notes and saying, “Yeah, I get this,” you’re not actually studying. Instead, close your notes and teach the concept out loud. Can’t explain it? You don’t truly understand it. This is called the Feynman Technique, and it forces you to make concepts simple and clear in your own words. When you hit a gap, go back and review. Your retention will skyrocket.