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HTTP/2 200 server: nginx date: Tue, 30 Dec 2025 17:34:54 GMT content-type: text/html; charset=UTF-8 vary: Accept-Encoding x-hacker: Want root? Visit join.a8c.com/hacker and mention this header. host-header: WordPress.com vary: accept, content-type, cookie link: ; rel=shortlink content-encoding: gzip x-ac: 3.bom _dca STALE alt-svc: h3=":443"; ma=86400 strict-transport-security: max-age=31536000 server-timing: a8c-cdn, dc;desc=bom, cache;desc=STALE;dur=2.0 Explaining mathematics – How to explain mathematical concepts in a way that students can understand Skip to content

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Explaining mathematics

How to explain mathematical concepts in a way that students can understand

Latest posts on my Blogger blog 29/4/25

April 29, 2025September 21, 2025JoelLeave a comment

For more of these summaries, you can search this blog for “Latest posts on my Blogger blog”, or click this search link

Here are some links to my most recent posts on my Blogger blog, where I am able to use MathJax.

Currently I mostly write posts on my Blogger blog, and then link to them from this WordPress blog.

I’ll periodically post updates on this blog, with links and brief descriptions.

The latest series of posts is about the Cauchy-Schwarz inequality. Before that there was a post about measures, outer measures and measurable sets which I hope to follow up on once I have caught up on some urgent tasks.

Continue reading →

TeachingLevel 1, Level 2, Level 3, Level 4, Mathematics

Latest posts on my Blogger blog 5/2/25

February 5, 2025JoelLeave a comment

For more of these summaries, you can search this blog for “Latest posts on my Blogger blog”, or click this search link

Here are some links to my most recent posts on my Blogger blog, where I am able to use MathJax.

I’ll periodically post updates on this blog, with links and brief descriptions.

Continue reading →

TeachingLevel 1, Level 2, math, Mathematics

A rule of thumb for when to use the Ratio Test

December 12, 2024JoelLeave a comment

My latest post on my Blogger blog is about when you should use the Ratio Test (for sequences and/or series) and, perhaps more importantly, when you should definitely NOT use the Ratio Test. I describe this briefly as a rule of thumb for when to use the Ratio Test

TeachingLevel 1, Level 2

Latest posts on my Blogger blog 6/4/24

April 7, 2024November 14, 2024Joel1 Comment

For more of these summaries, you can search this blog for “Latest posts on my Blogger blog”, or directly see https://explainingmaths.wordpress.com/?s=%22Latest+posts+on+my+Blogger+blog%22

Here are some links to my most recent posts on my Blogger blog, where I am able to use MathJax.

I’ll periodically post updates on this blog, with links and brief descriptions.

My latest set of posts is a series of 6 posts (intended for mathematics undergraduates in their 3rd or 4th year) on the Hahn-Banach extension theorem for continuous linear functionals on normed spaces.

Continue reading →

TeachingLevel 3, Level 4

Latest posts on my Blogger blog 16/3/24

March 16, 2024November 14, 2024JoelLeave a comment

For more of these summaries, you can search this blog for “Latest posts on my Blogger blog”, or directly see https://explainingmaths.wordpress.com/?s=%22Latest+posts+on+my+Blogger+blog%22

Here are some links to my most recent posts on my Blogger blog, where I am able to use MathJax.

I’ll periodically post updates on this blog, with links and brief descriptions.

Continue reading →

TeachingLevel 1, Level 2

Optional additional resources for first-year analysis

March 14, 2024November 14, 2024JoelLeave a comment

I’ve recently finished teaching 5 weeks of Analysis as part of our new first-year Core Mathematics module. I’ve made some additional resources available to them on a Moodle page. I expect to add further resources in due course! But here is the current version of that page.

Continue reading →

TeachingLevel 1, Level 2

Further Topics in Analysis optional additional resources

March 14, 2024November 14, 2024JoelLeave a comment

My Level 4 analysis module has been through various codes and titles. Currently it is MATH4047 Further Topics in Analysis (FTA for short).

I have made some optional additional resources available to them via a Moodle page.

Here is a copy of that page!

Continue reading →

TeachingLevel 3, Level 4

Latest posts on my Blogger blog 11/3/24

March 11, 2024November 14, 2024JoelLeave a comment

For more of these summaries, you can search this blog for “Latest posts on my Blogger blog”, or directly see https://explainingmaths.wordpress.com/?s=%22Latest+posts+on+my+Blogger+blog%22

Here are some links to my most recent posts on my Blogger blog, where I am able to use MathJax.

I’ll periodically post updates on this blog, with links and brief descriptions.

Continue reading →

TeachingLevel 1, Level 2

8×8 Othello “solved”

February 21, 2024November 14, 2024JoelLeave a comment

This is amazing! I never thought I’d see this without some major breakthrough in computational power. It looks like, as long suspected, 8×8 Othello is a draw with perfect play.

https://arxiv.org/abs/2310.19387#:~:text=The%20challenge%20of%20solving%20Othello,players%20lead%20to%20a%20draw

Continue reading →

Teaching

Latest posts on my Blogger blog 18/2/24

February 18, 2024November 14, 2024JoelLeave a comment

For more of these summaries, you can search this blog for “Latest posts on my Blogger blog”, or directly see https://explainingmaths.wordpress.com/?s=%22Latest+posts+on+my+Blogger+blog%22

Here are some links to my most recent posts on my Blogger blog, where I am able to use MathJax.

I’ll periodically post updates on this blog, with links and brief descriptions.

Continue reading →

TeachingLevel 1, Level 2, Level 3, Level 4

Measure and Integration 2006-7-8 update

February 14, 2024November 14, 2024JoelLeave a comment

Thanks to the local team at the University of Nottingham for helping me to track down my missing Measure and Integration mp3 files. I’m pleased to say that they were all safely backed up. I am now hosting the mp3 files on this blog.

Yesterday and today I’ve been working on the old Measure and Integration module page (from 2006-7-8) at https://tinyurl.com/UoNG13MIN, and I think that I have updated most of the links. But I might have missed a few, so please let me know if you spot any broken links!

TeachingLevel 3, Level 4

Screencast and audio links from this blog.

February 4, 2024November 14, 2024JoelLeave a comment

Hi everyone,

An old server was recently decommissioned here at Nottingham (I was warned!)
I thought that I had safely transferred all of my screencasts and podcasts to somewhere sensible, and to some extent that is true. What I had forgotten was that this blog still has lots of links to the old server.

Continue reading →

TeachingLevel 3, Level 4

Nottingham Symphony Orchestra Concert 9th March 2024 (OT)

January 20, 2024JoelLeave a comment

My wife Uta and I both play violin in the Nottingham Symphony Orchestra.

Our next concert, An evening of European Classics, will be on 9th March 2024, in the (Nottingham) Albert Hall.

Here is our flyer (as screenshots and, further down, as a PDF).

1739-nso-flyer-euro-classic-march-2024 Download

MusicMusic, Nottingham

Piazza “less than” bug?

January 14, 2024February 3, 2024Joel1 Comment

There appears to be a bug in (or a feature of) the Piazza mobile app (on android at least) that causes posts to be truncated at any strict less than sign inside maths.

It looks fine on a desktop web browser. (It is probably html-related.) But don’t try switching to the mark-up editor (even on the desktop web browser) as it literally deletes everything from the less than sign onwards!

Typing < inside maths in the Rich text editor on the desktop browser doesn’t help, because Piazza just converts it to the strict less than sign, which then breaks the app again.

Typing < in the Plain text editor on the desktop appears to be safe both for the desktop and for the mobile app.

This might be just a temporary bug/feature, but it may be just as well to be aware!

Note added 3 Feb 2024

I think a more efficient solution is to use the LaTeX code \lt instead of the less than sign.

Interestingly the greater than sign doesn’t appear to cause trouble. But < is probably interpreted as the start if of an html tag

I have notified Piazza of the issue, so hopefully they will fix this, or issue guidance.

Teaching

Latest posts on my Blogger blog 26/12/23

December 26, 2023November 14, 2024JoelLeave a comment

For more of these summaries, you can search this blog for “Latest posts on my Blogger blog”, or directly see https://explainingmaths.wordpress.com/?s=%22Latest+posts+on+my+Blogger+blog%22

Here are some links to my most recent posts on my Blogger blog, where I am able to use MathJax.

I’ll periodically post updates on this blog, with links and brief descriptions.

Date	Title and link	Description
24/10/23	Converses and negations	Discussion of some specific examples to help understand the difference between converse and negation
24/10/23	Discussion of the proof that the uniform norm really is a norm	A detailed proof that the uniform really is a norm, including some comments and warnings on possible pitfalls along the way
23/12/23	Connectedness of unions	Discussion of a standard result concerning unions of connected sets, relevant to the theory of connected components of topological spaces
24/12/23	Totally bounded metric spaces	Discussion of the fact that a metric space X is totally bounded if and only if every sequence in X has a Cauchy subsequence. Also some initial discussion of uniformly separated sequences.
25/12/23	Sequences that have no Cauchy subsequences	Discussion of the fact that a sequence in a metric space has a Cauchy subsequence if and only if it has a subsequence which has no uniformly separated subsequence. Thus a sequence in a metric space has no Cauchy subsequence if and only if every subsequence has a uniformly separated subsequence.
25/12/23	The Christmas Equation	Taking a short break from total boundedness and Cauchy sequences, this is a repost of some funny maths I saw on QI a few years ago
25/12/23	Total boundedness for countable metric spaces	Some musings on the special case of countably infinite metric spaces. Not particularly conclusive! But there are various reformulations of the results above, and a note that imposing a well-order enables us to do without the usual sequence of choices. Also some discussion of uniformly separated subsets.
26/12/23	Conclusions on Totally Bounded metric spaces and Cauchy sequences	A summary of all the facts from the posts above, relating total boundness, uniformly separated sequences and subsets, and Cauchy subsequences

TeachingLevel 1, Level 2, Level 3, Level 4, Mathematics

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My blogger blog
Selected posts from this Wordpress blog, but using MathJax to display the maths

Uta Feinstein's art
Selected artwork by my wife, Uta

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