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25/09/2026

Our social focus this week is on ‘maintaining appropriate challenge’ and it’s easy to hear that phrase and picture one particular kind of lesson - an open problem, students working in small groups or perhaps a rich task on display on a screen or whiteboard. But challenge isn't a genre of lesson, rather it's a decision that you make however you are teaching. It's also a decision worth revisiting mid-lesson when you ask yourself whether the level of challenge is right for everyone in the room.

Where a lesson takes a more explicit approach, you might be asking yourself:
▪ how demanding is this worked example for the students in this class?
▪ what tells me a student is ready for scaffolding to be removed and how do I remove it?
▪ when I check for understanding, is the question pitched so the answer actually tells me something valuable?

Where students are working on a more open or investigative task, the questions are different:
▪ can every student find a way in, and what happens to the one who can't?
▪ what prompt opens a different door into the same mathematics for a student who is stuck?
▪ what deepens the mathematics for a student who finds the task straightforward?

These questions look different but lean into the same process. The support a student needs when an idea is new will hold that same student back once they have grasped it. Left in place, a scaffold stops being support and becomes a ceiling. So in both these lessons the work is identical: pitch the support to what this student already knows, then take it away as they learn. That is what 'maintaining' means when we talk about maintaining appropriate challenge.

If you want to take this further, research with Australian early years teachers looked at how they actually use enabling and extending prompts, including the finding that most waited at least five minutes before making an enabling prompt available.
To better understand what maintaining challenge might look like in a more explicitly taught lesson, Mathematics Hub's guidance covers worked examples, while this Evidence Snack from Peps Mccrea explores the idea of fading. Links are below:

Using enabling and extending prompts in the early primary years, James Russo and colleagues, MERGA:
🔗https://files.eric.ed.gov/fulltext/ED623752.pdf

Worked examples, Mathematics Hub:
🔗 https://www.mathematicshub.edu.au/plan-teach-and-assess/teaching/teaching-strategies/worked-examples/

Concreteness fading, Peps Mccrea
🔗 https://snacks.pepsmccrea.com/p/concreteness-fading

25/09/2026

Many teachers feel the pressure of a mathematics curriculum packed with content and the need to cover a wide range of separate procedures, each requiring enough practice for learners to remember and perform them on demand.

But is there another, more conceptual, authentic and satisfying way? Could we identify a small number of BIG Ideas, and make those the main focus?

That's the proposition Professor Colin Foster brings to The Keynotes Conference: Perspectives on Pedagogy on Saturday 17 October. His session, Identifying and Teaching the BIG Ideas in School Mathematics, names five ideas - Thinking Multiplicatively, Thinking Algebraically, Thinking Geometrically, Understanding Functions and Graphs, and Modelling - and asks what teaching could look like if we built everything around them. Explore the framework in detail at:
🔗 www.bigmathematicalideas.org

Supported by the Australian Council for Educational Research, the conference begins the evening before, on Friday 16 October, with the free Hanna Neumann Lecture from Professor Catherine Attard.

Join the Facebook event:
🔗 https://www.facebook.com/share/1NdzB45N5v/ or
Learn more and register:
🔗 aamt.edu.au/keynotesconference2026/

24/09/2026

On September 16, 2011, an anonymous user posted a math question to the "/sci/" board of 4chan about The Melancholy of Haruhi Suzumiya, an anime series originally broadcast in nonlinear order, which had led fans to develop a habit of rewatching its 14 episodes in every possible sequence.

The question was simple to state and brutally hard to solve: what's the shortest possible string of episodes you'd need to watch to see the first season in every conceivable order? In mathematical terms, this is called a superpermutation, and finding a general formula for the shortest possible superpermutation of any size had stumped mathematicians for at least 25 years.

In under an hour, another anonymous poster replied with an answer, not a full solution, but a rigorous proof establishing a new lower bound. The proof showed that for a set of any size, and specifically for Haruhi's 14 episodes, a viewer would need to watch at least 93,884,313,611 episodes to see every possible ordering, roughly 4.3 million years of continuous viewing.

The post sat largely unnoticed outside anime fan circles for seven years. That changed in October 2018, when mathematician Robin Houston stumbled across the proof on a fan wiki and realized its significance, tweeting that the best known lower bound for this decades-old problem had been established by an anonymous poster on a board "mainly devoted to anime."

Houston, along with mathematicians Jay Pantone of Marquette University and Vince Vatter of the University of Florida, independently verified the anonymous proof and confirmed it held up to rigorous scrutiny. The timing lined up with a separate breakthrough from science fiction writer Greg Egan, who had just published a new upper bound for the same problem, drawing renewed attention to the puzzle as a whole.

When Houston, Pantone, and Vatter formally published their refined version of the proof, first in the On-Line Encyclopedia of Integer Sequences and later in an academic paper, they credited the original source exactly as it had appeared: listed as first author, "Anonymous 4chan Poster."

22/09/2026

【問題解決了,你幸福嗎?】

#封面
全文🔗見留言

Anthropic工程師預警AI會殺光人類後,AI兩大巨頭 和 宣布會放慢步伐,他們葫蘆裏賣什麼藥過去一星期已引起議論紛紛,也不多談了。這時候,或許值得回顧一下,兩家公司突然爭相承認AI很危險前,這幾年來是如何渲染AI可以造福人類的。AI企業投入巨大資源解決科研問題,其目的為何? #解決問題是否就等同人類幸福大幅提升(當然要先假設AI不會大開殺戒)?

AI其中一個將會為人類帶來的福祉便是 #讓大家可以躺平。OpenAI總裁Sam Altman 2021年寫了一篇名為Moore's Law for Everything的網誌,稱AI發展一日千里,可以比人類做更多工作,估計不出10年,AI可以生產足夠的財富,每年支付美國每名成年人1.35萬美元,而這筆錢將會不斷增加,人類可以不工作。至於Anthropic,總裁Dario Amodei也經常勾勒願景。例如今年8月,他聲稱未來5至10年,AI可以醫治大部分人類疾病。

究竟AI先會殺死我們還是治療我們,大概要待AI巨擘們先統一口徑。這裏只想談談AI破解科研難題。兩家AI企業為了顯示自家的模型更聰明,一直投入不少資源於科研。「納維-斯托克斯存在性與光滑性」問題是千禧年大獎7個難題之一,破解者可獲100萬美元獎金。OpenAI 9月8日宣布, —斯托克斯一個證明。OpenAI為此投入至少650萬美元的運算資源。OpenAI為什麼突然熱中解決這條難題?當然不是為了獎金,也不是出於好奇心,而是為了競爭。

OpenAI承認, #他們是聽聞有人快將破解難題才開始集中火力解題。紐約大學數學家Tristan Buckmaster跟Anthropic的 Levent Alpöge一直合作研究破解該難題。兩人只是出於興趣合作,與所屬的機構無關,他們亦有使用OpenAI和Anthropic的工具。據兩人聲明所述,9月3日,社交網流傳納維—斯托克斯題快將破解的消息,又知道 ,Buckmaster便主動電郵OpenAI領導解題任務的Sébastien Bubeck,強調自己的解題無關Anthropic,完全自己掏荷包支付開支,包括使用OpenAI工具的費用。不過雙方最終不歡而散。據Buckmaster,OpenAI提出方案,同意由他先發表成果,並在日後發表的論文中擔任第一作者,但前提是成果歸屬必須移除Alpöge的名字,並須註明該問題用了OpenAI模型解決。Buckmaster拒絕,覺得OpenAI純粹想收買他,並在9月7日匆匆將未完成的算式, 。OpenAI則在9月8日公布成功解題。
.....

OpenAI解題後,陶哲軒等25名菲爾茲獎得主發表聲明,題為「 #數學領域中人工智能的嚴重錯置」(A Severe Misalignment of AI in Mathematics),指AI企業將解題當作AI能力指標,除了帶來成果歸屬及抄襲等問題外,把解題當作終極目標,反而可能傷害數學這門學科的根本目標。 #數學的根本目標是什麼?那就是概念理解和洞見。聲明稱,那些著名數學問題就像地標與燈塔,供人們作參照,衡量人們是否加深了對數學世界的理解。解決一個問題,通常意味出現了新的洞見與有趣的方法;數學社群隨後會展開漫長而艱辛的討論、通過教學等方式鑽研這些成果。在這個過程中, #人與人之間的互動與學問傳承相當重要,而人類學問正是這樣日積月累。聲明說:「如果以愈來愈快的速度大量生產數學陳述,這種做法可能不是為新思想注入生命,而是在摧毁原本能孕育新思想的肥沃土壤。」

犬儒者也許會覺得,數學家只是「妒忌」AI比他們聰明,擔心被AI搶飯碗而已。當作家和畫家抱怨AI大規模生產所謂「藝術作品」時,大家也會以「搶飯碗」形容其憂慮。但箇中涉及的並不僅僅是飯碗問題。數學家聲明更關注的是 #人類智識工作(intellectual work)所面臨的普遍威脅。人類動腦筋不僅旨在產生最終答案或成果,更在於培養理解力,以及構思新問題與想法的能力。
.....

陶哲軒以小孩的遊戲與成年人世界的對比作引子。小孩在好奇心驅使下經常透過玩遊戲探索世界、享受樂趣,在成人看來,這些遊戲都費時失事。陶哲軒說,成人世界講求效率,不能再玩遊戲,但科學——尤其是純數學這樣基礎的科學——卻是例外,孩子氣的好奇心可以有「嚴肅」的用途。數學家可以提出一些看似瑣碎及沒有直接實際用途的問題,如果這些問題選得好,科學家或數學家可以憑着成人的紀律與專業,再加上孩子氣的天馬行空與無懼,展開極有成果的探索。
.....

陶哲軒的話不難理解。正如我們去旅行,當然會安排目的地,但旅行並不旨在達到目的地而已,如何達到目的地所經歷的往往是旅行最有收穫的部分。當AI以極有效率的方法幫你解題、幫你達到目標、奪走你探索的樂趣,你還可以做什麼?Sam Altman的答案是,你可以躺平了。但這真是你想要的幸福嗎?你寧願在草地上自由跑來跑去探索,還是有個沒趣的成年人把一切答案告訴你、叫你不用玩了?陶哲軒一句話很有意思,他說:「在這個廣泛運用AI的現代,我們尤其要記得, #當孩子是怎麼樣的感覺。」這句話不止適用於數學家,也適用於你和我。

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文//林康琪
圖//吳浚匡
編輯//王翠麗

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每期「 #星期日生活」隨《 #明報》出版,買份報紙就可以睇到全部內容~

19/09/2026

剛剛講到了數學,之前有用全家炸雞形容過0跟null,現在看到這樣的比喻,覺得真的很合適。

15/09/2026

#數學課程研究
〈幾何在十二年國教中的重新定位〉
長久以來,中學幾何教育在許多人的印象中,往往等同於三角形、圓、角度、輔助線、證明與追角技巧。然而,這種理解其實過於狹窄,也逐漸脫離了幾何在人類文明中的真正角色。幾何從來不只是圖形技巧的集合,而是人類理解空間、位置、方向、比例與世界結構的重要方法。若從現代科學與教育的角度重新檢視幾何課綱,我們會發現......

https://shann.idv.tw/edu/260831.html

〔圖文不符:光頭山,郭英豪攝,2026年6月發布於FB〕

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