AI News, BOOK REVIEW: Learning mathematics of Machine Learning: bridging the gap

Learning mathematics of Machine Learning: bridging the gap

Image source: Glenfinnan Viaduct – aka “The Harry Potter Bridge” source Wikipedia – an apt analogy bridging the known to the unknown!

To put this idea into some more context: The maths behind Machine Learning comprises of four key areas: So, the question then is: How can we start with high school maths and use that knowledge to bridge the gap with maths for AI and Machine Learning?

We work with the following guiding principles The approach is based on working with only seven concepts: 1)  Process as a black box model - Any process can be modelled as a black box ...

We start with linear regression because it is taught in schools (y = mx +c ) 3)  From the linear equation, you can understand the workings of a Perceptron and hence the basics of a neural network 4)  We then consider the ways of finding the best solution using techniques like closed-form and optimization leading to the idea of gradient descent (introducing the concept of defining a loss and minimizing it iteratively).

5)  Linear regression can be extended to logistic regression using the General Linear Model (and hence to classification) 6)  Evaluating a model – ROC curve and other techniques 7)  Wider classification of algorithms based on flach Logical models – use a logical expression to divide the instance space into segments and construct a logical model.

Concrete Pictorial Abstract

Unlike traditional maths teaching methods where teachers demonstrate how to solve a problem, the CPA approach brings concepts to life by allowing children to experience and handle physical (concrete) objects.

This stage encourages children to make a mental connection between the physical object they just handled and the abstract pictures, diagrams or models that represent the objects from the problem.

Children are introduced to the concept at a symbolic level, using only numbers, notation, and mathematical symbols (for example, +, –, x, /) to indicate addition, multiplication or division.

Although we’ve presented CPA as three distinct stages, a skilled teacher will go back and forth between each stage to reinforce concepts.

By systematically varying the apparatus and methods used to solve a problem, children can craft powerful mental connections between the concrete, pictorial, and abstract phases.

However, concrete materials are frequently shelved by the time children reach KS2 — many teachers believe them to be too childish or distracting.

By the end of KS1, children need to be able to go beyond the use of concrete equipment to access learning using either pictorial representations or abstract understanding.

Bar Modelling

The bar model method draws on the the concrete, pictorial, and abstract (CPA) approach — an essential maths mastery concept.

On one hand, the Singapore maths model method — bar modelling — provides pupils with a powerful tool for solving word problems.

However, the lasting power of bar modelling is that once pupils master the approach, they can easily use bar models year after year across many maths topics.

Maths models using concrete or pictorial rectangles allow pupils to understand complex formulas (for example, algebra) on an intuitive, conceptual level.

Instead of simply following the steps of any given formula, students will possess a strong understanding of what is actually happening when applying or working with formulas.

Singapore math

Singapore math (or Singapore maths in British English[1]) is a teaching method based on the national mathematics curriculum used for kindergarten through sixth grade in Singapore.[2][3]

to describe an approach originally developed in Singapore to teach students to learn and master fewer mathematical concepts at greater detail as well as having them learn these concepts using a three-step learning process: concrete, pictorial, and abstract.[2][3]

In the concrete step, students engage in hands-on learning experiences using concrete objects such as chips, dice, or paper clips.[5]

The development of Singapore math began in the 1980s when the country's Ministry of Education developed its own mathematics textbooks that focused on problem solving and heuristic model drawing.[3][7]

The CDIS developed and distributed a textbook series for elementary schools in Singapore called Primary Mathematics, which was first published in 1982 and subsequently revised in 1992 to emphasize problem solving.[13][14]

In the late 1990s, the country's Ministry of Education opened the elementary school textbook market to private companies, and Marshall Cavendish, a local and private publisher of educational materials, began to publish and market the Primary Mathematics textbooks.[1][14][15]

TIMSS, an international assessment for math and science among fourth and eighth graders, ranked Singapore's fourth and eighth grade students first in mathematics four times (1995, 1999, 2003, and 2015) among participating nations.[10][13][11]

In the U.S., it was found that Singapore math emphasizes the essential math skills recommended in the 2006 Focal Points publication by the National Council of Teachers of Mathematics (NCTM), the 2008 final report by the National Mathematics Advisory Panel, and the proposed Common Core State Standards, though it generally progresses to topics at an earlier grade level compared to U.S. standards.[21][22]

By visualizing the difference between the two bars, students learn to solve problems of addition by adding one bar to the other, which will, in this instance, produce an answer of fifteen paper clips.

Once students have learned to solve mathematical problems using bar modeling, they begin to solve mathematical problems with exclusively abstract tools: numbers and symbols.

By visualizing these two parts, students would simply solve the above word problem by adding both parts together to build a whole bar of 100.

Conversely, a student could use whole-part model to solve a subtraction problem such as 100 - 70, by having the longer part be 70 and the whole bar be 100.

The student could solve this multiplication problem by drawing one bar to represent the unknown answer, and subdivide that bar into four equal parts, with each part representing $30.

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