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Math, Better Explained

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Slope-intercept form is written in a very mathy way: y = mx + b. Let's turn this into plain English.
July 13th, 2018 |

June 14th, 2018 |

In English, we often drop the subject of a phrase, such as in these signs: Who are these signs written for? It's really *You, stop* or *You, yield* or *You, be alert for bears* (I'm not doing it). After internalizing a language, we can take hints without explicit instructions. But, to put it politely, math isn't usually well-internalized. Let's get clear about who the "math signs" are referring to.

May 9th, 2018 |

In class there's the lesson about a specific formula, sure, but the meta-lesson is how well the experience went. What worked? What didn't? How can we get more of the first and less of the second?

May 3rd, 2018 |

Here's a talk (video and transcript) on "Math And Analogies" I did for the Center on Contemporary Art in Seattle. The gist: good analogies make math a joy.

March 30th, 2018 |

I had another great chat with Nasos Papadopoulos on the wonderful MetaLearn podcast. Einstein’s Theory of Special Relativity has many consequences, including the famous equation E = MC² (Energy = Mass x Speed of Light²) and the notion that space and time are relative (observers can have different experiences of how large something is, or how much time elapsed).

Most lessons offer low-level details in a linear, seemingly logical sequence. Better Explained focuses on the big picture — the Aha! moment — and then the specifics. Here’s the difference:

I know which approach keeps my curiosity and enthusiasm. The general strategy is the ADEPT Method:

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Learning isn’t about memorizingfacts to pass a test. It’s about unlocking the joy of discovery when an idea finally makes sense. If this approach resonates with you, welcome aboard.

I enjoyed math until a poorly-taught class nearly destroyed that passion. A last-minute Aha! moment showed me math could make sense, even be enjoyable, when presented with:

I share explanations that helped, hoping they help you too. I’m thrilled that BetterExplained now reaches millions every year, and has appeared in blogs for the New York Times and Scientific American. Read more…

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The rich phenomenology of the progressive aphasias presents a substantial clinical challenge and a unique window on the neurobiology of language. Accurate clinical diagnosis of the patient who presents with word-finding difficulty requires an appreciation of the taxonomy of the progressive speech and language syndromes and a systematic approach based on the principles of structured history-taking and examination, analogous to those that guide other areas of neurological practice. Here we have presented an approach to the clinical analysis of word-finding difficulty, both to assist diagnosis and to set clinical symptoms in the context of experimental evidence concerning the organization of the language system. However, any such analysis exposes problems that will only be resolved by a more detailed understanding of the pathophysiology of the progressive aphasias.

The progressive aphasias are more than the sum of their neurolinguistic parts: these are diseases of neural networks, distributed both in space (functionally connected brain regions) and time (evolution of deficits). Although broad correlations can be established to guide clinical localisation ( Fig. 3 ), few if any clinical deficits are specific to dysfunction in a single brain region, while a particular brain region often participates in the development of several different syndromes. The search for correspondences between clinical syndromes and regional brain atrophy in the progressive aphasias is analogous to classical attempts to correlate acute aphasic syndromes with focal lesions. The language models of classical neurology that emphasized discrete cortical centres in the mediation of specific language functions (Lichtheim, 1885 ) have given way to neurolinguistic accounts that emphasise distributed functional networks (Levelt, 1989 , 2001 ; Hillis, 2007 ). Despite longstanding interest in the so-called disconnection syndromes, the science of distributed neural networks has yet to be widely translated to clinical practice, yet this may hold the key to understanding the phenomenology of the progressive aphasias and the ways in which they depart from the acute syndromes of vascular disease. At present, the functional consequences of neural network disintegration remain difficult to predict a priori , and the mechanisms by which they give rise to clinical syndromes have not been elucidated in any detail. The progressive aphasias have thrown up fundamental issues that are often difficult to reconcile with classical models of language localization: the SD syndrome, for example, clearly illustrates the fundamental importance of the anterior temporal lobe in language, yet the relations of this region to the ‘classical’ language cortex in Broca's and Wernicke's areas within the wider language network remain problematic. In line with network accounts of the acute aphasias (Hillis, 2007 ), the overarching challenge of future work in the degenerative language syndromes will be to characterize particular syndromes as ‘pathway-opathies’ or dynamic profiles of correlated atrophy across brain regions. This perspective will be constrained by models of healthy brain function (see for example, Binder et al ., 2005 ), and may help to resolve the many apparent discrepancies of structure: function correlation in degenerative disease (see for example, Nestor et al ., 2006 ).

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