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Main Argument: Teaching that impacts students' lives beyond the classroom has to prioritize both student and local teacher autonomy.

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Main Argument: Teaching that impacts students' lives beyond the classroom has to prioritize both student and local teacher autonomy.

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Subtopic 1: Teaching Through Problem Solving vs. Teaching Problem Solving:

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Subtopic 1: Teaching Through Problem Solving vs. Teaching Problem Solving:

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Subtopic 2: Student-directed vs. Teacher-directed Classrooms

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Subtopic 2: Student-directed vs. Teacher-directed Classrooms

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Subtopic 3: Teacher-Created vs Centrally Planned Curriculums

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Subtopic 3: Teacher-Created vs Centrally Planned Curriculums

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Many students don't have the ability to persevere and try new things when either the first thing they can think of to try doesn't work, or when they don't know at first what to do.

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In all these cases, the lack of faith in the person's agency and power to solve non-trivial problems becomes a self-fulfilling prophecy, as structures form to accommodate an "average" that cannot solve meaningful problems on their own, forcing those who can or who are learning to do so to find resources outside of schools/workplaces.

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When kids don't have the ability to solve rewarding problems on their own, they are more likely to turn towards "easier", more instantly gratifying forms of digital stimulation that leave them bored and unfulfilled in the long run.

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When workers don't have the ability to solve rewarding problems on their own, it results in middle and upper managers micro-managing workers or giving them watered-down tasks. It leads to lost productivity in companies, and it creates reward structures (say, promotions) where success is based off of demonstrations of compliance and social jockeying.

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When students don't have the ability to solve rewarding problems on their own, it results in teachers watering down curriculums and micromanaging the steps of assignments. This happens most often in high schools but is also found in some university courses as well.

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Any problems that are either fulfilling to solve, or that will be actually worthwhile to solve in industry, will not be exercises of the forms taught in school, where the solutions are known in advance. As shown by international test rankings, Western countries today are very much failing to give kids the perseverance of the craftiness to solve non-trivial problems.

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Many high schools today will have language about fostering "problem-solving skills" in their curriculums, and that the teachers use in their lessons. These schools use the same core teaching methods (Didactic instruction), but add in disjointed "extra" lessons that have students draw diagrams for exercises they could have solved without the diagrams, use guess-and-check for, say, an equation with a known closed-form solution like quadratics, or etc. Well-meaning teachers who try this accept the top-down structure while differing on the details, naively expecting different results.

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If a classroom is like a solar system, the Didactic model makes the teacher the sun.

Proponents of the Didactic model claim that class time is used more "efficiently" when teachers give formulas or concepts directly to students through instruction or through examples, rather than restructuring classes to be about student-centered collaborative derivations.

Immediately, this fails because it prioritizes the metric of class time "efficiency" instead of the more valuable goal of teaching information to students in a more memorable (deeper encoding) way while also developing the skill of how to figure out things for themselves.

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The US's spoonfeeding teacher-centered methods shortsightedly try to cover more material by having teachers directly give formulas, sacrificing long-term mathematical maturity. Didactic instruction fails even on the class-time metric. Japanese students are on average one to two grade levels ahead of American students in mathematics. A big cause of this is that figuring out derivations for yourself (or even as part of a group) as Japanese students do makes concepts more intuitive. A side effect of this is that they require fewer in-class practice days than American students, and can absorb newer topics more quickly.

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A pre-requisite for teachers improve at teaching instead of stagnating is to allow them the power to make (significant) variations in the curriculums they teach based on their own judgement.

Teachers who use Lesson Study get direct, real-world, experience and can see practically the results of variations in what they present, how they present it, how they address students, when they move on, etc. They get feedback from their peers.

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What the lesson study approach does is account for something obvious, but not said enough: Teaching is a skill, so it should be learned by practicing it AND getting immediate feedback just people do to master any other skill. In particular, this means that most improvement in teaching will not come from going to large teaching conventions, reading books about educational strategies, lesson planning, online modules, or other itemized "professional development" requirements.

Unfortunately, these efforts make up most of the (busy) work of American high school teachers today.

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"Few curricula are designed to support such lessons; most are designed to support fairly direct instruction by the teacher."

Any American teacher who wants to try deviating from a centrally planned curriculum way will have to swim against the current of established materials and the bureaucracy that reinforces them. Systems which bog down teachers with busy work strap teachers of their agency the same way depersonalized curriculums strap students of theirs.

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"During most Japanese lessons, the textbook is closed, but the textbook shows how the authors think the lesson might play out."

I thought this was interesting since this is really different from what textbooks in the US focus on. Japan's textbooks focus on the *act\* of teaching, and detail how the act might play out so that the teachers have a base to go off of. US textbooks don't include hypothetical lesson scenarios - they focus on the *content* of what the teacher teaches, based off of the Didactic model. Students work exercises from the textbook.

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"The teacher then displays the pictures in Figure 3. “What do you think?” the teacher asks, as he puts them up one at a time for dramatic effect."

The first thing the teacher does in the lesson is set up the problem and pose it to the students. He allows the students to figure out how to formalize the problem in the precise mathematical way of per-unit division. When students learn how to take a "natural" problem and derive a formalization for themselves, the formalization feels natural as well. Contrast this with giving students the formalization first, and then showing them how it applies to the problem: the problem then feels artificial; constructed merely to display the day's formula.

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The feeling of agency, (i.e. that the day's formula is not some magical rune handed down from on high but instead a natural thing they could have figured out for themselves through their own action/deliberation), is best given when you let students figure things out for themselves. The lesson study approach tries to give all students this feeling: Students with partial but not full solutions feel agentic when the teacher writes their ideas on the board to motivate further discussion. Other students talk with peers about the problem and see their peers figuring it out collaboratively, so it does not feel out of reach.

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As the article says, "The teacher's role in the lesson can be compared to the role of a film director, who carefully stages each scene and makes cuts between cameras to create the desired effect."

This reminded me of one thing Einstein said: "Education is not the learning of facts, but the training of the mind to think". The teacher's role becomes about drawing out the capacities of the students. When students have differing ideas, the teacher has the students exercise their executive function by having THEM conclude what type of formalization is more practical to use.

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"Teaching Mathematics Through Problem Solving" by Tom McDougal and Akihiko Takahashi demonstrates both the power of problem-solving pedagogy in mathematics education and the problems highlighted by western mathematics education that arise in contrast. In Western educational discourse, the idea of problem-solving is nominally valued. As the article states, the USA's Common Core's "very first Standard for Mathematical Practice" emphasizes the importance of students' ability to "understand problems and persevere in solving them,"

Traditional instructional models, in particular the "**I do, we do, you do**" approach, create excessive student dependence on teachers and a reluctance to persevere in problem-solving independently. The article advocates for a student-centered approach where learners actively engage in problem-solving tasks crafted by the teacher, who takes a more deliberate, behind-the-scenes approach. This approach aligns with the Japanese model of lesson study, which is built on the foundation of teacher peer collaboration, deliberate practice, and reflection to improve teaching effectiveness based on local real-world learned experience. (contrast with centrally planned top-down implemented teaching reforms in the West)
Through lesson study, teachers craft lessons centered on problem-solving activities, guiding students through the process of tackling mathematical problems and fostering a deeper understanding of mathematical concepts. This bottom-up reform has made it so that "today, most elementary mathematics lessons in Japan are organized around the solving of one or a very few problems".

"**Teaching through problem solving**" is a key idea in the article, contrasting with the conventional method of "teaching problem solving" found in American classrooms. While the latter focuses on teaching specific strategies divorced from the main material of the curriculum, the former aims to completely redesign curriculums so that they allow students to discover mathematical ideas through problem-solving activities. This approach gets students to think critically (and most importantly, think for themselves) so they can apply their mathematical knowledge in novel contexts, giving them perseverance, resilience, and a sense of agency in problem-solving.

The narrative of the article occurs through a hypothetical fifth-grade math lesson, providing a picture of what a classroom looks like when it is redesigned in this way. By presenting students with an open-ended but carefully crafted problem involving rabbit cages, the teacher guides students through the process of investigating the problem and developing strategies for solution through both whole-class discussions and independent exploration. All in all, the article makes a case for the fact that teaching approaches founded on student autonomy improve student performance.

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One issue (beyond just this project) I've had is that I come up with ideas faster than I formalize them (i.e. have them typed out into direct writing and have all the connections between my ideas made explicit and not just as implicit intuitive ideas). I was pleasantly surprised at how useful the Padlet ended up being for organizing and polishing up my ideas. It's a nice complement to Obsidian, which I've been using as my drafting space. The formalization process was useful for two reasons. First, it told me when ideas I had weren't ironed out. I had to scrap a few intuitive ideas that I realized didn't have much backing when I was polishing them. Second, it made the ideas I did have clearer.

One thing I did before I typed out the map was that I talked about my thoughts on this with family. I talked with my parents, (both Spanish high school teachers) and my sisters. This was helpful to do. First off, I think it's useful to write like you talk, so actually talking about what you have to say is a good first start. My parents gave me some context and talking to my sisters was a good litmus test to see if I could talk about these ideas clearly enough (meaning without jargon) so that they, as laymen, could follow along and give their own thoughts.

I think the ideas I had about this article fit better into a Padlet than they would have if I had written an essay about it. I liked the fact that we could choose our own sources/topic, since I was able to choose something I would have been just as interested to think about on my own time. When I looked back on the final Padlet with all my ideas typed out and connected to each other, it felt pretty satisfying to see them organized into a map.

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1) On a local scale, lesson study groups give a clear model for how to decentralize education. But in a country like the US, is there any way to promote decentralization on a mass scale without falling back into creating another centralized education system?

2) Besides educational system bureaucracy, what other entrenched aspects of the "system" keep education centralized? (relation to article on textbook writer's inside story + anything else?)

3) I touched on loss of agency in the workplace - is there anywhere else in society where both the people/systems are losing their ability to act with / allow for agency? What would the implications be?

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1) Lockhart, Paul. "A Mathematician's Lament." Bellevue Literary Press, 2009.

2) Ansary, Tamim. "A Textbook Example of What's Wrong with Education." Edutopia.org, 2004.

3) Gatto, J. T. Dumbing us down: the hidden curriculum of compulsory schooling. Philadelphia, New Society Publishers, 1992.