Juniors and seniors choose among a fascinating array of electives. Juniors explore texts centered on broad themes, such as identity, war and conflict. Reading helps Lower School students build an academic foundation and learn about the world around them. They have the confidence to investigate, form conjectures, make predictions, develop strategies, and analyze and verify what works and why. Students up through grade eight progress through a program based on the Singapore mathematics problem-solving approach. Students connect numbers and number concepts.

By fourth grade, they synthesize concepts of area, estimation, mean, median and mode to complete the Dot Project, determining the total number of dots on all carpeted areas of the school. Students in eleventh and twelfth grades create intricate art projects representing concepts like fractals. In each division, mathematics courses emphasize a spirit of inquiry and collaboration, as students learn to solve novel problems while presenting and defending their ideas.

Unlike students in traditional math programs who cycle rapidly through many topics in a school year, Blake students take time to study each topic deeply, mastering it before moving on. They may also retest themselves on specific skills, allowing for ongoing improvement. The math faculty includes teachers who are doubly qualified in mathematics and such disciplines as sociology and astrophysics. Fun with numbers need not be confined to the classroom. A majority of students take calculus or explore subjects like statistics before graduating. College-level electives include number theory, graph theory, logic, game theory, discrete systems, advanced geometry, math research, software design and more.

Math integrates tradition with technology. Students tackle calculus using interactive technologies. Why does it matter? These questions lie at the heart of Blake science. Younger students explore their world through creative play that integrates scientific thinking. Pre-kindergarteners observe principles of motion by experimenting with marbles and ramps. Older students apply the scientific method explicitly. Again and again, they ask questions, form hypotheses, conduct research and analyze data. Upper School students can challenge themselves with Advanced Placement courses, independent studies and post-AP courses.

Blake students know science is intricately connected with ethics, public policy, economics and culture. Fourth graders study how water evaporates inside a plastic bag and connect this knowledge to the effects of global warming on Minnesota lakes. Middle School students. Children integrate science with other subjects. Using their understanding of light and mirrors, third graders engineer a way to illuminate the interiors of the pyramids.

Fourth graders write about the water cycle from the perspective of a water droplet. Science also informs instruction in the arts, with students completing projects such as building functioning marimbas. Upper School students continue foundational work in biology, chemistry and physics, focusing on learning through inquiry. Electives include astronomy, anatomy, physiology, environmental science and forensics.

Experimentation is at the heart of Upper School science. In state-of-the-art laboratory classrooms, students learn to design experiments, analyze results, work with others and share results. About half of all Upper School students take more than four years of science courses, with many pursuing AP-level courses, independent research and semester-long projects. Empathy and ethics inform scientific learning. Seventh-grade students explore the physics behind force and motion on the playground and use their understanding to design all-inclusive playground equipment.

Students connect science to their everyday life. Physics students use their understanding of electrical circuits and magnetic forces to build devices, while chemistry students manufacture hand lotions to study colloids, suspensions and emulsions. Students learn explore U. Eighth-graders design and creto new experiences. Social studies also reinate memorials and think critically about the forces the research, critical-thinking and purposes of historical memory.

Ninth-gradcommunication skills students need to be ers can take a history course that connects great problem-solvers and creators. With each year at Blake, stuIn grades eleven and twelve, dents build upon their growing students might elect to do understanding of the world. They also have empathy, modern global issues — including peace compassion and a deep understanding of and security, human rights, development, how the world works.

Lower School students learn early on how social studies is connected with their own lives. For example, third graders learn about maps and cartography by mapping a place with personal meaning for them. Middle Schoolers engage in challengebased learning around topics of peace and security, contemporary immigration, economics and social responsibility. Social and political questions are tied to ethical ones. Moral Issues, an upper-level elective, explores this connection.

Students consider modern social controversies in conjunction with classic works of philosophy. Falling in love with one is something else. Blake provides intensive, individualized language instruction, giving students every opportunity to fall in love with another language. Research has demonstrated that students achieve fluency more easily if language instruction begins before age seven.

Students entering the Middle School have a solid foundation in Spanish and the skills to master other languages. Effective instruction draws connections between language study and other subjects. Spanish language instructors work with teachers in other subjects to identify these cross-curricular connections. Blake recognizes that students enter school with varying levels of language exposure and that each student learns at his or her own pace.

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## Exploring Maths

Therefore students in the Middle School and Upper School language programs are placed not by grade but by level of proficiency. Some students arrive already familiar with one or more of the languages Blake offers and are placed in more advanced language classes. Through rigorous study of world languages, Blake students develop communication and critical-thinking skills. They also acquire an essential skill for becoming global citizens. Most students continue with language study into their junior and senior years, even after meeting graduation requirements. Middle School language studies are integrated with cultural events.

The Culture Club gives students a chance to celebrate international holidays, listen to guests discuss life in other countries and enjoy the cuisine of these cultures, too. Students learn about the nuances of translation through Latin texts such as Cicero and Catullus. They learn about the religious and philosophical beliefs of ancient Greeks by delving into the Athenaze book series. Students receive one-on-one attention. A journalist covering conflicts in South Sudan?

A medical doctor addressing global health concerns? Through global programs Blake students develop the ability to think critically and to collaborate effectively. Blake teaches students to communicate across cultural boundaries and to understand multiple perspectives. This focus on global education includes several initiatives: semester and yearlong study-away programs; short-term global immersion experiences; collaborations with local schools and nonprofits that serve diverse communities; partnerships with schools in South Africa and India; and opportunities to host international.

For example, students in kindergarten and second grade have formed friendships with the predominantly Somali-American students at the Cedar Riverside School. These initiatives share a goal: to help students broaden their worldview and value our interconnectedness. Global programs complement social studies and languages curriculum and provide opportunities for service. Middle School students take part in the World Savvy program, developing plans to address global issues, such as food distribution, sustainability and healthcare access.

Upper School students might participate in Model United Nations, take a service trip to Ecuador or study in Switzerland. Those are just a few of the many ways Blake students expand their world and become global citizens. World Savvy helps students develop — and possibly implement — solutions to global problems. Recently, sixth graders studied healthcare access in Southwest Asia and proposed a school drive for medical supplies.

Their plan was selected to receive funding. All students should be able to study abroad if they desire. To this end, Blake waives most of the tuition for students who spend a semester or year studying at another school. Lower School students study curriculum. Participation basic concepts such as line, color and unity, in any community of applying these not only in art classes but artists involves observing and analyzing also in academic projects.

Middle ations to impact an audience. These skills School students receive broad instruction are fundamentals, not niceties. Upper School stugarten through grade five, dents take electives as diverse Blake students have regular as 20th century art history, elective courses in classes in visual arts and animation, ceramics and game the visual arts offered music.

Arts exposure through development. All visual arts to students in grades six to twelve daily activities begins in disciplines have dedicated stupre-kindergarten. Later, sixth dio space. Furthermore, Blake graders take visual arts and wood studio as hires only active working artists for its arts part of their course rotation; many choose faculty, providing an invaluable resource to continue with visual arts throughout for emerging student artists. Middle School. Upper School students Asking questions, collaborating, solvtake at least four semesters of arts classes.

Arts edu Students with a passion for visual arts cation is a vital part of the process. Students learn artistic techniques not only from their teachers but from older students as well. For example, a third grade class might visit the Upper School ceramics studio to get hands-on training in pottery from teenage mentors. Middle School students create, experiment and explore art across several media, cultures and eras.

They also receive broad instruction emphasizing core skills in drawing, painting, carving and 3D constructions. Students are regularly awarded in prestigious programs such as the Minnesota Scholastic Art Awards. Upper School art students work on complex projects and refine their work through group and one-onone critiques. Lower School students are serious about their artistic endeavors and take pride in the act of creating.

All students have general music and theatre classes from kindergarten through fifth grade. They learn to take risks and express themselves, with the goal of connecting with others. Most Middle School students participate in a performance-based class — band, orchestra or choir. Learning the art of performing begins in pre-kindergarten with role-playing. Fourth graders produce short scenes and plays, building to a musical production in fifth grade.

Sixth graders take theatre as part of their arts rotation. Seventh and eighth graders may enroll in acting and playwriting electives. Upper School thespians take courses in acting, playwriting, theatre production and musical theatre. A capstone program is the fifth grade class play. Composition and improvisation abound. Students may study violin or cello as part of the Lower School strings program. Middle School students can choose among offerings such as chorus, band or strings.

Upper School students may continue choral and instrumental studies. Blake offers student-led a cappella groups and a jazz combo. Middle School students may join the debate team. A variety of debate courses exist in Upper School. In the regular apart from a few specialized design people in research, no one uses math even in engineering.

A liking for math may infact be a curse. Very nice articles which clearly unleash the learning stages. Transition from is done by most of the people.

But from requires a lot, which also decides your field of interest. Considering a simple example finding an area. At rigorous stage is just applying the regular method to get the result. Heuristic is just an educated guess, to say whether its gonna increase or decrease based on the shape change….

Then, I saw mountains were not mountain… […]. Hi, Terence Tao. I always loved physics and mathematics, and I am an aspiring primary and secondary education mathematics teacher in Brazil. I can follow the given proofs I read, and sometimes even find alternative proofs with a similar strategy of the one I read.

Sorry about my English. Well, I think I solved my problem. Maybe kind off-topic, sorry. If you get good at the mechanical aspects of proving things, you can fill in the details of the proof as long as you remember the idea. So when I read a proof, I first try to skim it, ignoring everything that looks boring, and try to spot the key ideas.

Once I know these I can often fill in the details myself.

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At least this is true for fairly easy theorems. Toby Bartels. You were focusing too much on the rigour. Stand back and let mountains be mountains again let lines and planes be lines and planes for a while, instead of tables and beer mugs , before you decompose them into logical fundamentals for the proof. Like a lot of students in my class, I find this course very hard compared to other courses such as abstract algebra. Is the main goal of real analysis course teaching us how to prove theorems rigorously? Can anyone give me some advices about studying analysis?

Thanks a lot! Does 0. Girls' Angle. Understanding this can save a lot of time and not just in math. Great article Dr. The real excitement in mathematics comes when all the mysteries of stage 1 are finally explained in stage 2. Frankly speaking I have been unable to understand why mathematics educators try very hard to separate the computational aspects from the theoretical foundations.

As an example in case of calculus all the mystery i. But you are not very impressed by what can look like magic, because you know the trick. The trick is that your brain can quickly decide if question is answerable by one of a small number…. Mathematics by samadhi - Pearltrees.

For instance, calculus is usually first introduced in terms of slopes, areas, rates of change, and so forth. The emphasis is more on computation than on theory. This stage generally lasts until the early undergraduate years. Ronnie Brown. Dan Schmidt. This is very analogous to the progression of stages that chess players go through.

Roughly, beginners are in stage 1, strong amateurs are in stage 2, and professional players are in stage 3. Beliau bilang, kalau pembelajaran matematika itu bisa dibagi menjadi 3 fase: […]. Postulates, Proofs, and Obviousness Girls' Angle. Your question also implicitly asks about the relationship between […]. How does one develop intuitive learning? And what are the key differences between intuition based learning and learning by proof?

And which one do you prefer? You need to have both intuition and rigor to really do well in mathematics. So you can r…. A mathematics education should not just be about learning to write neat answers to exam questions! I see most of the answers here have been recommending AOPs and problem solving books. In the name of creativity, do not let the rules of mathematics impede your intuition. It is only with a combination of both rigorous formalism and good intuition that one can tackle complex mathematical problems; one needs the former to correctly deal with the […].

There are concepts and theorems named after people throughout mathematics. You might hear a few stories, e. But anyone who gets into upper-level mathematics inevitably begins meeting other people in their field, seeing the human side of mathematics, and learning the history of mathematics, or at least their corner of it. In the end, one learns that mathematics is a deeply human endeavor, full of interesting people, many of whom are still alive! Some mathematicians even have blogs!

The sad thing, to me, is that the human element of mathematics, the historical anecdotes, the fun of talking about mathematical ideas with other people… are all but stripped out in the early stages of modern math education. Maths and Writing Creative Culture Kenya. Was it by empirically measuring the sides of many right triangles? Or use other axioms? Or, are transformational proofs in geometry also considered synthetic geometry?

My philosophical! I won several math and physics competitions on primary and high school. The problem I see is that many things in papers and lecture notes are described in very terse, plain rigorous way. Even the maths lecture at the college was presented in this way. In constrant with this, when something is described by examples, problems, analogies, intuition and induction or by telling the patterns, even understanding the most sophisticated concept becomes easy.

After the intuitive foundations are built then we can generalize to N, other fields, etc, and discover the corner cases and gotchas, and backtrack towards the axioms or towards the desired level. But in real life I see that things are explained the other way around: axioms, definition, definition, definition, definition, definition, theorem, proof, theorem, proof, theorem, proof, theorem, proof, definition, definition, definition, definition, definition, theorem, proof, theorem, proof, theorem, proof, theorem, proof.

Because all look like meaningless symbol folding. What can I do with this? While you can do the same by reading the nicely commented source code, without even knowing anything about the gory details of the hardware. Personally I cannot think of differentiation without visualizing the slope or thinking of a rate of change; also I cannot think of integration without visualizing the area under the graph, or thinking of summation.

## Blake's Go Guides

I like maths, I like puzzles, I like solving problems, I like understanding world. I can prove many theorems. I can use mathematical tools when I need to. But excess rigor is not for me. But heck I discovered, reinvented many things way before I learned about it. Because the fundamental definitions, theorems, etc are just one step away from the topics discussed, whereas hardware and software need some bridging. I think one reason why the rigor is necessary is to ensure that one knows what it might be used for and, secondly, to demonstrate the manner in which one establishes a concept.

This, I think, would allow the student perhaps replicate this rigorous path with a different concept perhaps a new one. Rigor remains just as impenetrable for us as for you, though after a while one gets used to it. It does, admittedly, have that magic trick effect when you produce some useful result. Actually most mathematicians make very poor programmers and engineers.

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Three quarters subOptimal. There is also a speech given by Atiyah about the style of Mathematics, in witch he talked about the intuitive mathematician. One analogy I like is that between describing a route to the station in terms of the landscape e. Of course, Grothendieck was a master at this. And of course the modern language for discussing mathematical structures is category theory. I also feel there is not enough discussion about methodology. Chapter 1: Dimension Complex Analytic. I used this idea in the middle of the semester last year and the students were fascinated with the idea.

End-of-semester surveys generally had it as one of their favorite topics and one they wanted to learn more about. Furthermore many of the students bought into the idea that in order to explore an arena like higher dimensions, where we lacked a great deal of intuition, it would be necessary to think more carefully and try to formalize some of our intuition from lower dimensions. This incidentally is what I think of when I think about mathematical rigor.

Nice article. The Value of Science is a very influential reading to me. I always found myself a bit loser for being somewhat stuck at the stage 1 of mathematical reasoning. Hi Terry, Should formalism be introduced strongly at the middle school and high school level?

## Five Senses Education

Or only people in graduate or advanced undergraduate should be introduced to formalism. I am asking as a person wanting to teach mathematics to middle school and high school students. Toward an Understanding of Mathematics Empathic Dynamics. Here, I am more interested in advanced undergraduate and graduate learning rather than earlier concepts that a lot of the education literature is focused on. How to study math to really understand it and have a healthy lifestyle with free time? Crescent Yemen. It sounds relevant to you, too: terrytao.

Indeed, we might think of intuitive, unconscious thought as a sort of tennis partner with slower, conscious reasoning — a back and forth. The intuition provides material to the conscious mind and the conscious mind processes that information, which sculpts and corrects the intuition. David MW Powers.

This is an excellent way of looking at the way we treat mathematics, but different individuals will have different areas of expertise and be in different corners for different areas of mathematics or different application areas. Weaning ourselves away from grounded applications, to developing more general understandings, models and formalisms, is key to the power of mathematics, and often means navigating uncharted waters without the benefit of intuition, but ideally will lead to develop better deeper intuitions that in pure mathematics can go beyond real world applicability.

At conferences, authors who are challenged on the appropriateness and applicability of a model often show that they are unaware or unmindful of the assumptions and only understand the model in the formal mechanical sense of being able to manipulate equations, verify derivations and produce proofs within the bounds of the model.

This perhaps characterizes the dangers of an applied hybrid of heuristic and rigorous approaches and pinches the square into a figure eight as these are brought together in the same individual, or even the ubiquitous paradigm of an entire field. I mean that an individual has sufficient rigour to go through the motions at a the model level, but is applying canned heuristics at the application level, without either pre- or post-rigour intuition being in evidence, without addressing satisfaction of assumptions or performing sanity checks on conclusions.

In some cases, whole fields are operating in a kind of unsound limbo, because a formal but inappropriate model takes precedence over common sense intuition and understanding of the boundary cases and the impact of assumptions. Good point, David. I can do abstract reasoning when interpreting and using linguistic tools like metaphors and metonymy. I think language is underrated as a math tool. Every time I see greek letters in an explanation I translate it to language and then to some intuition, if needed, but I always do the translation step. If some concept from pure math is explained by a textual description, I tend to capture it easier and I can instantly map to a bunch of real-world applications.

I know, someone will say that natural language is ambiguous, but so is the actual academic math. In short, I think pure math i. You can do similar reasoning with plain natural language, so you could reach a broader audience and allow more intuitions to emerge — even pure math intuitions. Proving my hypothesis is the harder part, since I should use pure math, but I think I could contribute with some empirical results for motivating a mathematician to try something on this path.

There is room for all kind of abstract thinking mathematicians. Age - Page 6. I would encourage you to read this short little article by a grown up prodigy on mathematical […]. There is no connection between maths and rigour. Opera singers or ballet dancers, for instance, also require insane amounts of rigour and precision in the profession of their skill.

However, no one actually thinks of classical music or ballet dancing in such terms. At the same time, rather reductively and discriminately, math is perceived in this way, almost to the point of confusion. I prefer intuitive insights or visual explanations over rigorous proofs any day. One is the distinction between art and craft. So the final rigour in a proof is the craftmanship of a mathematician, making sure that everything works as claimed. No patches needed though they sometimes are. Another is that proofs are like describing a route, and this is in a certain landscape.

How much detail should you give when describing a walk to the station? You do not want to describe all the cracks in the pavement, but you do want to warn of dangerous manholes. One of the jobs of mathematicians is t build a landscape in which proofs, routes, can be found. A good test of a future mathematician is not necessarily current level of performance, but do they actually want to know why something is true.

Thanks to David Roberts for this reference. Discourse of Mathematics im too lazy for prose. Rigor vwkl. And Peter Woit is always arguing that the way to make progress in physics is by a better […]. I am of the belief that strong intuition, and importantly, in any IQ range , allows for, enhancements high level problem solving ability.

Intuition can be just as important as analysis. It is what sets analysis up. John Gabriel. Intuition is very dangerous, no matter how intelligent one is. Rigour is very important in mathematics. Of course there is little or no rigour in mainstream mythmatics. Analysis is not rigorous, never was rigorous, will never be rigorous.

Fracking and the climate debate, universal daemonization, mathematical rigour, the cult of genius and the ban on trolling scotchverdict. Read next Education and outreach Blog What teaching physics in Kenya taught me about inspiring students to study maths and science. Discover more from Physics World. Everyday science Blog Autonomous particle accelerators, how many superheroes are needed to make a film successful? Culture, history and society Blog Lindau Fighting the threats nationalism poses to science. Everyday science Audio Battle of the Elements round three, ultracold atoms in space, life on other planets.

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