Mathematics

Additional Topics

There were many topics that used to be taught in the US in Mathematics which have been discarded from the curriculum. It would be lovely to see those topics available. 

Spherical Trigonometry .   This used to be taught as part of a more extensive trigonometry curriculum. It would be a great and accessible area of mathematical study for students who have studied algebra and geometry. 

Analytical Geometry This used to be a full year course of study. Those topics could be put back into the Khan curriculum to challenge those students who love math but have not yet taken Calculus, or as recreation for those currently in Calculus. 

Finite Mathematics Topics not taught extensively elsewhere in the Khan curriculum and found in Finite should be available to students. 

Discrete Mathematics An introduction to this topic should be available to students in mathematics. 

Introduction to Mathematical Proofs The difficult questions in math are NOT ....the questions which involve plugging in. Students should be trained in how to prove things in geometry and algebra and with available modules continuing on up through training in multi-variable calculus, differential equations, and linear algebra. If not, then students trained in plug in algorithms hit a brick wall of understanding on trying to reach beyond this level of mathematics. Think about a proof course where modules are suggested during or on completion of other courses.

Forgotten Techniques How can one manually take a square or cube root? How can a student manually figure out the sine of an angle? There are techniques such as these which should be available....especially to international students. To truly be international, use of trig tables, logs and the slide rule should be available. 

History of Mathematics Modules should be added to this course and recommended to students as they learn about new math techniques. Should human history be just about the names of political leaders, of wars, and of battles? What was the history of what humans thought they knew? Regarding mathematics....what were the questions that developed. Why did those questions develop in the time and place that they did? What were the early answers? Why were new techniques developed? What was wrong with the prior methods? Were the techniques altered later? Why? Even elementary age children could be exposed to modules concerning the early history of counting. When did humans begin to mark down records that kept track of counted items? Was zero always thought of as a number? What was the history of negative numbers, fractions, decimals etc. For middle schoolers, ideas such as the history of square and cube roots could be added as modules. In other words....if there are new techniques added, the history of how that idea was developed and why it was initially developed should be available Ex why and how was the concept of logarithm developed? What was the academic question under consideration that lead to the development of logs. What caused the introduction of the natural log function? What was the question where this technique was needed? Why was the sine function (and other trig functions) developed? How did workers before trig tables work out the numerical value of a trig function for a given angle? 

Additional Math Topics Here I am thinking about topics that tend to show up in mathematical competitions for students. There have been topics printed in series such as the "New Mathematical Library" ex continued fractions by Olds. That series was purposefully pitched at high school students to supply new areas for exploration. Sadly the series is no longer in print. 

Great Books in Mathematics There are high schools and colleges which base their curriculum on the "Great Books". This implies that courses working thorough ALL of Euclids elements, Archimedes, Appollonius, and Nichomachus....should be available. These are important for understanding later work in "Great Books" series. There should be modules teaching mathematics techniques as used in science books in the Great Books series. Could students be prepared to read the Principia? I would hope so. Should anyone willing to learn the prerequisites...be able to learn the mathematics used in Gallileo's work or that from Special Relativity? 

Mathematics in Art There are many topics here regarding proportion, dilation, inversion, and prospective that should be available to students. Repeated mathematical patterns, optical illusions and other similar topics could be of interest. 

Mathematics of Daily Living  There could be a high school to adult course regarding basic use of mathematics in the world....from balancing a check book, to figuring out gasoline costs of a trip, measuring in carpentry, to multiplying or dividing portions in a recipe. If this is kept modular, students could review a specific topic. Which bank account is better? When could that loan be repaid? Prerequisite topics to understand a module should be linked ex if some beginning algebra and or geometry modules should be viewed first. Such links would help answer the question from those other courses "where will I EVER use this stuff?"

Word Problems  For many students there is a marked difficulty in translating paragraphs into math equations. Modules drilling on words that mean add such as " and 3 more", "bought 3 more", "found 3 more" etc should be discussed. The same goes for words that imply subtraction, multiplication and division. Another KEY problem for students is to figure out what in the paragraph is not relevant information. If the question boils down to 5 plus 3, does it matter if ...it is sheep, apples, boxes, students etc? Does it matter if it is raining, Tuesday, the farmer is wearing a brown jacket, he is your uncle Bob etc? Text books are NOTORIOUS for NOT drilling this key skill. What data is irrelevant to solving the problem? Students need to make the transition that 5 is 5 no matter if it is apples, students or cupcakes. Show 5 objects. Show pictures of different objects. Does the number word change? You may think this is an obvious question but realize that different cultures through history have not considered it so obvious. Think of our different words for 2 people (couple), 2 shoes (pair) etc. Even words for groups of things and animals differ depending on which type. Another concept that cant be assumed is that 3 plus 5 stays the same no matter which type of objects are added. Also do students really know what you mean by addition? Can they take objects and show you....3 plus 5? Can they show you 3 times 5? Gather rocks, put them on the table and tell the students to show you 3x5. Try again with donuts and paper bags, or drawing animals in animal pens. Students arrive at college and junior college having memorized "math facts" with no real idea what these mean. Can they use pebbles and show 12 divided by 4? Could they show this on a piece of paper using areas? Ask most adults who think they understand multiplication and fractions to show you by drawing areas, slicing bread or some other manual method....what is meant by 1/3 times 2/5. Typically no algorithm pops up, and students taught fact memorization then plug and chug...really struggle with a basic question such as this one. Dont accept an algorithm based answer such as writing down the fractions and doing a fraction multiplication algorithm. What does it really mean to multiply?  Most schools teach math by putting equations on the board and having students work from there for years. There is way to little practice translating stories into math equations and eliminating unneeded information. Then students who...when faced with such a translation exercise perform as expected given little to no practice in that type of question do poorly, the student is simply told they are bad at mathematics. Nonsense! Translation of English to math is a skill and improves with practice and understanding. In this math section, once the English is translated to math equations, move on. These students need rapid drill in translation, not plugging in. Granted there should be an option to click to see the worked solution IF the student sees a question that is especially interesting. 

Courses After Calculus For the student who has completed multivariable, linear algebra, and intro to differential equations, what are the mathematics courses typically offered to undergraduate students. What are the overall ideas? The point is not to get bogged down here but to develop interest and to gain an understanding of how broad mathematics is. What would a student encounter when seeing number theory vs topology, complex variable analysis vs nonEuclidian geometry, differential geometry vs advanced analysis? What type of questions are considered by this branch of mathematics? Are there other areas of math for this this material is a foundation? 

In other words, re math....MORE MORE MORE MORE MORE!

There is plenty of room for additional material to help understanding, challenge bored students, or to have adults continue learning.