Reading and Maths

 An interesting comment by Charlotte Wilkinson, "The Wilkie Way" why we need to see reading as part of mathematics learning.

I would also like to suggest that we help students learn "How to read Maths and Science Material" as it is different and requires different comprehension to the usal prose we use for teaching reading

www.wilkieway.co.nz
Maths and Reading

How many times have I heard teachers saying “we are assessing maths not reading”. The teaching profession has failed to recognise that reading is an essential part of using mathematics in our everyday lives. Principals are now alarmed by the low pass rate in exam trials with year 10 students. They acknowledge the mathematics is at about the right level but the students are unable to access the maths because it is “buried” in words.
This is nothing new and researchers around the world have been saying this for years. For this newsletter I am referencing an article from the Journal of Mathematics Education June 2011 Vol 4 No 1 titled;
Maths Literacy: Are we able to put the mathematics we learn into everyday use?
(Bobby Ojose University of Redlands USA)
Maths literacy is the knowledge to know and apply basic mathematics in our everyday living. An important part of maths literacy is using, doing, and recognizing mathematics in a variety of situations. In dealing with issues that lend themselves to a mathematical treatment, the choice of mathematical methods and representations often depends on the situations in which the problems are presented.
To effectively transfer their knowledge from one area of application to another, students need experience solving problems in many different situations and contexts.
The OECD publication, Measuring Students Knowledge and Skills (OECD 1999) lists the types of texts as part of reading literacy, which in part determines what constitutes mathematics literacy.The publication mentions as examples texts in various formats:

Forms: tax forms, immigrations forms visa forms, application forms, questionnaires;

Information sheets: timetables, pricelists, catalogues, programs

Vouchers: tickets

Certificates: diplomas, contracts etc

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Charts and graphs, iconic representations of data

Diagrams

Tables and matrices
Inorder to comprehend most of these types of text you need to bring knowledge of mathematical skills.
Here is a list (Not exhaustive as knowledge is dynamic and technological advancement is forever changing)
Everyone should:

be able to perform the basic operations of addition, subtraction, multiplication and division with whole numbers, fractions and decimals.

know concepts such as ratios, percentages, roots, square roots, absolute values, reciprocals and exponents.

should know the metric measures of length, area, volume, mass, time and temperature and how to convert between the measures,

understand simple linear equations, plotting graphs of linear equation, slopes,

know operations with positive and negative integers

know the concept of proportional reasoning.

should know the various area and circumference formulae for circles, squares, rectangles and
triangles.
• be familiar with cartesian co-ordinate system in two and three dimensions,
• be able to convert size on a scale model or map to actual dimensional size.
• be able to do basic construction using a compass and straight edge.
• should be familiar with three dimensional shapes in terms of finding volumes and surface areas of
shapes like cone, pyramid, prism, cylinder and sphere
• be able to find the measure of central tendancies when given a set of values
• be able to graph and interpret data as a histogram, pie chart, bar graph and line graph
• know probabilities based on theory and probabilities based on experiment
• compare risk factors in different situations
All of this list falls within levels 1 - 4 of the New Zealand curriculum and many of our students have the
mathematical knowledge so why are they still mathematically illiterate?
This will come down to pedagogical practice and a focus on the competencies required for mathematical Literacy:

Right from year 0 we need to be considering:
Mathematical communication: expressing oneself in a variety of ways - oral, written, pictures, diagrams; understanding someone else’s work.
Representations: Decoding, encoding, translating, distinguishing between, and interpreting different
forms of representations of mathematical objects and situations as well as understanding the relationship
among different representations. (Materials are not because young students are kinesthetic learners)
Symbols: Building an understanding of using symbolic, formal, and technical language and operations
Problem posing and solving: Posing, formulating, defining and solving problems in a variety of ways
Thinking and reasoning: Posing questions characteristic of mathematics, knowing the kind of answers
that mathematics offers, distinguishing among different kinds of statements; understanding and handling
the extent and limits of mathematical concepts.
Mathematical Argumentation: Knowing what proofs are; knowing how proofs differ from other forms of mathematical reasoning, creating and expressing mathematical arguments
Tools and technology: Using aids and tools, including technology when appropriate.
Unless we get back to actually teaching mathematics and not relying on computer programmes or apps for students to teach themselves, we will not solve the problem of mathematical illiteracy.
I sincerely hope the common practice model that will underpin our new curriculum “refresh” will place an emphasis on these competencies as well as a sequence for developing the knowledge and skills.
A sequence for knowledge and skills across the curriculum is relatively easy for mathematics as it is a
fairly hierarchical subject. It is important to make connections between topic areas as seldom does a
mathematical topic exist in isolation in the real world. Any sequence of work must provide the opportunity for explicit teaching, practice, application, discovery, and transfer.
No one resource will provide everything that should be included in your mathematics programme but
having a clear sequence will ensure continuity and progression.
A good resource provides opportunities for explicit teaching, practice, application, discovery and transfer built into the design. Along with a focus on appropriate reading levels to develop reading comprehension including the building of mathematical specific vocabulary.
The Figure it Out series is a greatly under used resource (because it is not user friendly) that will really
focus teachers and students on the need to develop mathematical literacy.
Maths Aotearoa + Wilkie Way + Figure it Out + NZMaths = A great maths curriculum to support the
teaching and learning of mathematics in New Zealand schools.

Do, Say, Write & Wilkie Way thoughts and Problems

 Too often we forget the 3 major, for both concepts and skills, that most students need to go through for understanding and application.

1.   They need to explore with materials (all types)                 the "DO" Stage

2.   When fluent with exploring and seeing patterns etc with materials they need to explain verbally what they have found out                                                                  the "Say" Stage

3.    Now that they are fluent in talking about the relationships and patterns, they should be encouraged to write what they have been saying (in both their own language and maths language)      Dont go straight to "6 + 4 = 10", but also have "six and four more make ten" as this is what they will be saying

Charlotte Wilkinson, Wilkie Way is saying something similar in her latest Newsletter

 

PATTERNS: Using Crosses and coloured pens to create patterns

 Patterns are the basis of mathematics, if we help students find and see patterns, they are often able to use this knowledge to apply to other situations.

The multiplication array is full of patterns and we often do not ask students to find the many patterns. When you take another step there are hidden patterns in the Digital Roots of the numbers(The Digital Root is found by adding the numbers successively until a single digit is obtained. (127   -   1 + 2 + 7 = 10   -   1  + 0 = 1 1 is the digital root of 127)  The nine, three and six times tables are worth exploring with their digital roots.

For younger students we need to get them to make and draw patterns and then see if they can find underlying patterns.(Using Numerals/Numbers all the time is often too abstract for some students)

Crosses

Teachers: instead of making crosses students could colour in squares on grid paper)

    With a red and green pen you can make 4 sets of 2 crosses:  (perhaps red and green tiles/counters or blocks for some students would be better)

        XX    XX   XX   XX

    How many different sets of 3 crosses can you make?

                                                                        XXX    XXX    XXX ……….

    How many different sets of 4 crosses can you make?

    Can you see a pattern?  making a Table/list may help

    Could you predict how many sets using 5 crosses?

 

 

Now using a Blue Pen with the Red and Green Pens

        How many sets of 2 crosses can you make now?
                                                (A starter:  XX    XX    XX    XX ………….)

        How many sets of three crosses with 3 colours?
                                                ( XXX    XXX    XXX …..)

        How many sets of four crosses with three colours?
                                                (XXXX        XXXX        XXXX …….)

    Can you see a Pattern, perhaps a table will help?  

    Are you able to write your patterns in sentences?

 

 Adapted from "Bounce To It" Gillian Hatch 1984

 

Stepping Off Points #4 COUNTING TRIANGLES

 If the vertices if a square are joined in every possible way with straight lines, then 8 triangles and formed.

Four small triangles and four larger ones.

Can you find the 8 triangles?

How many triangles are formed when you join the vertices of a Regular Pentagon in every way?  

 

Make sure you get all of them some small some large and some in between

What about a Regular Hexagon? 

 

 Can you see a pattern between the number of sides and the number of triangles?

Making a table/chart sometimes helps to see patterns

Name of Shape	Number of Sides	Number of Triangles
Square	4	8
Pentagon	5
Hexagon	6
Octagon	8

 Explain your pattern in words.

Can you predict the number of triangles for a Decagon(10-sided) Icosagon(20-sided)?

Now explore the number of triangles for irregular polygons

This activity adapted from "Points of Departure" Association of Teachers of Mathematics 1989.

 

 

 

 

Stepping Off Points: #3 PROBLEMS ON A CHESSBOARD

 Problems on a Chess Board

A regular problem is to place 8 Queens on a chessboard so that no queen threatens another queen.  How can this be done (Remember that a Queen can move in all directions)

 

 

Now:  What is the greatest number of Bishops that can be placed on a chessboard so that no Bishop threatens another Bishop? (Bishops move along Diagonals)

What is the minimum number of Knights that can be placed on a chessboard so as to occupy or threaten every square?(Knights move along horizontal and vertical lines)

What is the minimum number of squares that eight Queens can command(that is to occupy or threaten?)

Change the size of the board to say a 4 x 4, 6, x 6, 7 x 7, 10 x 10, and see if there is a pattern for each of the pieces for each sized board.

 

 

 Dominoes and the Chessboard

It is easy to put dominoes (where each domino covers two squares) on a chessboard and cover all the squares.

But what happens if you have a chess board where two corner squares have been cut off.  Can you now place Dominoes to cover all squares?

 

 

 

Adapted and extended from Points of Departure by Association of Teachers Mathematics  1989