Project Mathematics

Change of plan to follow shortly...

Outline of the eventual website

The idea for the website, which will eventually have sibling mobile supporting apps - is that of a maths education website that wholly encompass all aspects of a student's need delivered in a consistent way.

The MVP will contain

• A course that covers all topics for 11 plus maths exams.
• High quality video tutorials on each topic.
• Unlimited supply of online questions and detailed solutions (solutions given in stages following hints).
• Track student progress through the course track. Each topic has a base level of difficulty a student must achieve in order to move onto topics for which the current topic is a prerequisite.
• An achievement system - yet to be decided (points? badges? levels?)
• For each student, on a given topic, the website generates questions on the same topic of higher level of difficulty based on what the student has completed already.
• Student profile and parent dashboard.
• Unlimited supply of computer generated worksheets based on the student's current level on any topic, together with detailed solution sheet, for if the student/parent wish to do handwritten practice.
• Unlimited supply of computer generated 11 plus practice papers based on the student's current level across all 11 plus topics, together with detailed solution sheet.

Post MVP course list (these will definitely happen)

All additional course tracks will follow the same structure of the 11 plus course - videos, online questions, offline worksheets and practice paper.

• GCSE Maths course (mostly computer generated content)
• Maths problem solving course based on GCSE level of knowledge (non-generated problems)
• A-level Maths course (mostly computer generated content)
• A-level Further Maths course (mostly computer generated content)

Post MVP feature ideas / wish-list (some of these may happen)

• ebook with printable worksheets and practice papers on any topic. This can be a separate product line. The idea is that the contents of the website will more than serve function of current school text books and much more.
• teacher portal - allow teachers to set up teaching plans, so that the website generate content based on the teacher's teaching plans. This is also a separate product line. This can also be extended for school usage.
• one-to-one sessions have tutors/teachers on the site giving either written comments on student's practice paper or online lesson to discuss the student's work.
• weekly fun maths-jam lead by a teacher of the site, live-streamed session (akin to twitch), discuss an interesting problem with students.

Ok before we get too carried away...here and now...

Outline of this repository

This repository contains the ruby-written maths content generation engine that will eventually power the website described above for maths learning for students between ages 8 - 18. Currently making maths 11 plus (key-stage 2) topics.

Structure

Each topic is a class which implements the following public API (public methods):

self.generate_question(parameters)  #parameter is a hash
# return hash => {question: question,solution: solution}

self.latex(question)  #the argument is the returned hash from previous method
# return hash => {question_latex:question_latex,solution_latex:solution_latex}

The returned hash contains the latex string for the question and the latex string for the solutions.

QuestionGenerator module

This module is responsible for two methods which invokes topic question generators from all the topic classes to produce arrays of questions for worksheet and practice paper.

LatexPrinter class

This class is responsible for rendering the generated questions array from the QuestionGenerator module together with some other information about the practice paper or worksheet (e.g. student name, paper serial) into latex string. One for questions, one for solutions.

Equation class

This does not do too much at the present, apart from holding a left-hand-side expression and a right-hand-side expression. It may have more responsibilities to solve certain generic equation forms later.

Expression and Step class

These two classes are central and especially important, as almost every topic class will utilize these two classes to do equation solving.

• A mathematical Step consists of an operation (+,- etc), value, and direction.
• A mathematical Expression consists of an array of Steps.
• For example x + 2 is an Expression consisting of two steps: (no-operation,'x',right) and (add,2,right).
• x + 2 can also be expressed as these two steps: (no-operation,2,right) and (add,'x',left).
• The value of a Step can also be an Expression, thus 11 - (x+2) can be expressed as (no-operation,Expression(x+2),right) and (subtract,11,left).

This system allows most maths expressions to be expressed flexibly in many different ways depending on need of the context.

Some of the critical methods that Expression need to implement are:

• expand to expand something like (x+2)(y-3) into a sum xy+2y-3x-6, and any such expressions such as ((x+2)(y-3)+4)(a-b+5).
• latex return latex string representation of any expression.
• expand_to_rsum short for expand to a sum of rationals. This is a more general way to expand, it will expand an expression involving fractions/division into a sum of fractions. Note it will need to deal with fractions inside fraction situation - however many layers.
• simplify cancelling terms, collecting like terms.

The above methods are also the most challenging to write, I have spent too long writing and rewriting them. And an effort to refactor these methods is the main reason for all the rewrites. In my latest attempt (5th? 6th? lost count...), I feel I may have it.

Help sought!: I am still looking for improvements to make these methods more readable and extensible for the future. I am still open to rewrite (completely!?!?!) / reorganization if it will make significant improvements.

Latest Progress and Ordered To Do List

To do:

• Write a more generic linear_equation solver for those with x on both sides.
• Make generic linear equation solvers.
• Tidy-up methods in expression class to have more single responsibility.
• Create Ratio & Percentage topic class.

Topics to create

The stars denote difficulty

• Ratio word problems ***
• Percentage word problems ***
• Addition and Subtraction of Integers **
• Long multiplication and division ****
• Decimal arithmetic ****
• Basic Sequences **
• Area & Perimeter *****
• Transformations *****
• Integer properties *
• Negative numbers **
• Temperature negative number problems ***
• Simultaneous Equations ***
• Shopping word problems ***
• Simultaneous equation word problems ****
• Pattern recognition with basic shape patterns *****

Latest Macro Ideas

• A class such as LinearEquation, AgeProblem, Ratio which produce questions for those topic, should only have class methods, and should never need to instantiate any instance of itself, as such they are better served as ruby modules. Whose end goal is to self.generate(options={}) with return value of a hash of question latex and solution latex.
• A module for a topic such as AgeProblem, Ratio will make use of classes instances from things like Expression, Equation.
• Therefore Fraction class should be split responsibility in two classes in two parts, one part deals with the basics of Fraction whose instances are used as values of Expressions and can be used in any other topic module. Whilst the question generators into its own FractionQuestion module.

Example of generated linear equation worksheet solutions

Example of generated age problem worksheet solutions

Example of generated latex string script

Ideas for future improvements (These are NOT necessary for either than Prototype or final versions.)

• This one is not needed at least for all the maths upto university, so to remind me of this idea once I am in dreamland! It would be possible to get expressions to evaluate themselves, in the same way binary operations between elementary value types should be defined, it is feesible that expression valued binary operations (that exist inside a single expression haha) can also be defined. Thus rather than call simplify on an expression of 2x + 3x, we will evaluate the binary operation.

Mathematical Objects Model