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Reading @LongFormMath's book "Proofs" has been a delightful experience. Not only did I learn a lot about proofs, I also noticed some great pedagogical features that I wish were a part of every math textbook. Here's my list: #mathreads (1/n) 
1. Throughout the book, Cummings shows several ways of thinking about the same thing. This is tremendously helpful.
2. Cummings acknowledges when something may be confusing or difficult to understand. This is not only very comforting, it also makes me read more carefully.
2. Cummings acknowledges when something may be confusing or difficult to understand. This is not only very comforting, it also makes me read more carefully.
3. Every chapter ends with an "open question" - a mathematical conjecture not yet proven. This is such a wonderful way of spreading the joy and wonder of mathematics!
4. Before giving a formal proof, Cummings shows the intuition behind the argument.
4. Before giving a formal proof, Cummings shows the intuition behind the argument.
I’ve recently read @mpershan's book “Teaching math with examples”. Here’s what I learned, noticed and wondered. Thread. #mathreads
1. Worked examples are completed solutions that we ask students to learn from. They are preferably used in the beginning of the learning cycle.
2. Having read Liljedahl’s Thinking Classrooms, I found a lot of differences between the two approaches. But a common factor is the emphasis on student thinking: “Students don’t learn from a worked example; ss learn when they think actively and deeply about a worked example.”
The book “The unfinished game” by @profkeithdevlin, is a fascinating account of the history of probability theory. Here are a 13 things that I learned: #mathreads #iteachmath
1. Probability theory is often considered to have started with an exchange of letters between Pascal
1. Probability theory is often considered to have started with an exchange of letters between Pascal
and Fermat in 1654. In the letters they try to solve a tricky gambling problem called “The problem of points”.
2. Before their exchange, many learned people (including leading mathematicians) believed that predicting the likelihood of a future event was simply not possible.
2. Before their exchange, many learned people (including leading mathematicians) believed that predicting the likelihood of a future event was simply not possible.
The future was a matter determined by God. In that sense, Fermat’s and Pascal’s ideas were a leap of thought.
3. The first known attempt to discern patterns in games of chance, dates to around 960, when Bishop Wibold of Cambrai, enumerated the 56 outcomes when 3 dice are thrown
3. The first known attempt to discern patterns in games of chance, dates to around 960, when Bishop Wibold of Cambrai, enumerated the 56 outcomes when 3 dice are thrown
I just finished the book Heavenly mathematics. Here are nine things I learned, and one thing I still wonder: #mathreads
1. The word trigonometry comes from the book “Trigonometria” by Bartholomew Pitiscus in 1600.
1. The word trigonometry comes from the book “Trigonometria” by Bartholomew Pitiscus in 1600.
2. To describe the position of planets and stars on the celestial sphere, one needs a coordinate system. There are (at least) three different ones, using in turn the celestial equator, the suns trajectory (the ecliptic) or the horizon, as its base. -->
Determining the coordinates requires solving spherical triangles – hence the need for spherical trigonometry. Translating between the coordinate systems was one of the primary tasks of ancient astronomers.
Here are ten things I’ve learned from reading Joseph Mazur’s “Enlightening symbols”: #mathreads #historyofmath
1. Mathematical symbols are relatively recent creations. Many of the symbols we use today took form in the 1400s-1600s.
1. Mathematical symbols are relatively recent creations. Many of the symbols we use today took form in the 1400s-1600s.
2. From the beginning mathematics was rhetorical. Even the numbers themselves were often written as words. With time, common mathematical words were abbreviated, by omitting letters, thereby becoming a sort of symbols. For instance, p instead of plus and m instead of minus.
3. Exactly how the numerals we used today evolved, is very uncertain. There is little archeological evidence. What we do know is that the idea of our decimal system was transferred from India to the arabs and on to Europe.
