# Notes of mathematical knowledge, pedagogical knowledge, & learning activities to develop mathematical literacy

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# Mathematical notes

The purpose of this page is to link to documents to inform educators and other people seeking information to facilitate mathematical literacy. Information is organized in four broad areas:

## Knowledge base & pedagogical tools for mathematical literacy

### Knowledge base

The mathematical knowledge base includes concepts necessary for learners to conceptualize to become mathemtically literate and misconceptions that must be over come. Includes concepts, outcomes, & vocabulary by content, processes, and attitudinal (values & dispositions) dimensions by levels.

### Pedagogical knowledge to facilitate develop professionally and to create learning experiences to facilitate mathematical literacy and many teacher tools to assist

Different fields of mathematics map & explanation video (11:05)

## Math teacher's planning tool box

### General planning

Planning information

Examples to prep intended learnings

Suggestions for selecting problems and investigations

Sample parts and pieces of sequence plans

Activity plans

• Estimating chocolate chip | Has potential, but needs a lot more information for integration of mathematical dimensions and extended activities.

Assorted tools

# Numbers: values and basic operations

## Knowledge base & tools to develop mathematical literacy

### Knowledge base

Mathematical knowledge base with concepts necessary for learners to conceptualize to become mathematically literate and misconceptions that must be over come. Particularly important for how learners develop their understanding of numbers, their values, operations, patterns, and relationships are:

Concepts related to number sense:

## Information to facilitate literacy in numbers (values & basic operations) & tools to achieve it

Numbers & their definitions - Fact sheet of natural, integer, rational, irrational and real numbers. Their definitions, examples, & number line visual representation.

### Place value development & assessment

Development

• Place value - development from age 5 on: ten as big number, group by ten, pre place value, unitize, place value relationships, decimals, exponents, place value & decimal worksheets

Assessment

Development

### Operations

#### Addition & subtraction - whole number development & assessment

Development

Assessment

• Addition and Subtraction - suggestions, directions with problems, scoring guides, & score sheets to assess the development of + & - from four ways: join, separate, part-part-whole, & combine with scoring guide ideas to understanding and use of algorithms

Development

#### Addition, subtraction, multiplication, & division - fractional numbers development & assessment

Division problems represented with squares and rectangles

• Sample problems : 1/2 ÷ 1/2, 1/2 ÷ 1/4 , 1/2 ÷ 3/4, 1/2 ÷ 3/5 (thanks Brant),

## Activities to facilitate number values, & operations literacy

### Number sense - whole numbers

Number relationships --->>> see patterns and algebra

#### Manipulatives for students to use to learn number value

Cards for learners to sort - make a set of cards by combining cards appropriate for a particular learner: numerals, words, ordinal, dot, ten frame with dots. The cards can be sorted into paper bags, a sorting box, or on any flat surface. Different games or activities can be created: sort dots, sort dots and label with the set's cardinality numeral word, sort and label with ordinal word, sort all cards and beat the clock ...

Sample cards to print and cut out :

Dice game - helps students subitize number value and introduction to addition. Roll the dice and flip the values on the die or the sum of the die. Continue to roll until all the numbers (1-12) are flipped (win) or a value rolled has had all it's possible plays, already flipped. Example: roll 2 & 2. Can flip 4, or 1 & 3.

Die roll - simulate a roll of a die .

Number talks: are good tools to show equality of numbers and different ways to represent the same values. Sample number talk outcomes for 16 .

Number line : is a good mathematical tool to represent mathematical values and operations. However, children who are not familiar with them and confident with their use, are not sure what to count: numbers, lines, spaces and where to start (0, 1, ...).

Therefore, before they’re used as a representation to explain and prove mathematical ideas, learners need experiences to develop their understanding and fluency with their use.

Activities, used when counting, such as numbered polka dots organized in a line can provide good background before introducing a number line.

### Really big numbers and infinity

• Olber's Paradox - if the universe is infinite in time and space, stars should occupy every point in space and fill the night sky with light.
• How many grains of sand in the Universe? Archimedes' proof in The Sand Reckoner.

### Number Theory

Number theory is critical for students to do well in algebra and higher levels of mathematics. However, the sad thing is most teachers are unfamiliar with these ideas, or skip them when they occur in their math text or program, as their view of mathematics is limited to basic facts and operations as calculations of numbers.

The following are examples related to number theory.

### Fractions, Decimals, & Percents number values

• Fractions - development of fractional number values, instructional ideas, sample activities, workheets, and assessment. Focus is on the development of fractions as equal parts of a whole & equivalency. Work sheets include: fraction circles, circle disks, paper folding, area models, measurement models, number lines, hundred chart, challenges, & problem suggestions.
• Fractions - Worksheet to represent 1/2, 1/4, 1/8, 1/16, 1/32 on a number line
• Fractions - Worksheet to explore the fraction 3/5 with skip or coral counting & its patterns
• Fractional values compared problems
• Using fractions to find the area of nine shapes in a 17 square unit rectangle - The Pharaoh's problem
• Percentage - sample activities and worksheets
• Ratio and Proportion Problems
• Sequences
• Number line with whole & decimal numbers - place values from billions to billionths
• Numbers for whole & decimals - in order from billions - billionths
• Decimals either end in zero or repeat as nonterminating decimals or rational numbers.
Example: place these numbers on number line 1, 2/3, .5, .3 , 1 1/2, 2, .6 , 1/2, .9 , 1 .1 , 1/3 Misconception .9 doesn’t equal 1. Recognize 1/3 = .3 & 2/3 = .6 . but not .9 = 1.0

### Basic Operations (+-*/)

#### Fractional numbers (+ - * /)

Activities for number sense particularly: fractions, decimals, and percents

Division problems represented with squares and rectangles

• Sample problems : 1/2 ÷ 1/2, 1/2 ÷ 1/4 , 1/2 ÷ 3/4, 1/2 ÷ 3/5 (thanks Brant),

#### Decimal numbers (+ - * /)

Activities for number sense particularly: fractions, decimals, and percents

### Integer values and operations (+ - * /)

Addition and subtraction of positive and negative integers might be conceptualized by representing and controling two properties: position on a number line and direction of body.

The number line can have two halves with the same numbers being both positive and negative. You have two choices for each number + or - as a reference to position. If the number has a value of 4, it could be represented as a positive four or negative four.

You can represent those two numbers by standing on either +4 or -4. Secondly when you are standing on a number line you can face one of two direction, since the number line is bidirectional, you might be facing left or right, forward or backward, in a positive direction or negative direction. or if it is thought of as a thermometer, you can walk hotter or colder.

Problems can be posed as :
Find a starting point, say -3 and if you add a +3, face warmer and walk forward 3.
Find a starting point, say -3 and if + -3, then face warmer and walk backward.
Find a starting point, say -3 and if - - 3, then face colder and walk backward.
Find a starting point, say -3 and if - +3, then face colder and walk forward.

Again both are important and a concrete model can be used to describe a procedural rule, as demonstrated as a plus and a plus are plus, a plus and a minus are a minus, and a minus and a minus are a plus.

Another idea is to record a video with motion. People walking or running forward and backward. They might display signs with forward and backward as they do so. However, it probably will be obvious which direction they moving when it is recorded. When the video is made, then each can be viewed with the movie play forward and backward. A chart of the different results can be made (forward, forward, results in forward; forward, backward, results in backward; backward, forward results in backward; backward, backward results in forward.

After several concrete experience have been experienced explore the this challenge.
How many different ways can two numbers be represented with different a sign and an operation (-2 ++3; -2 -+3; -2 --3; -2 +-3) (2 ++3; 2 -+3; 2 --3; 2 +-3).

# Algebra Patterns and Functions

## Knowledge base & tools to develop mathematical literacy

### Knowledge base

Mathematical knowledge base with concepts necessary for learners to conceptualize to become mathemtically literate and misconceptions that must be over come. Particularly important for how learners develop their understanding for algebra, patterns, and functions are:

Concepts related to:

# Geometry & Spatial reasoning

## Knowledge base & tools for mathematical literacy

Geometry  originates from Greek  ge or gaia meaning earth or land and metria meaning measuring, or measurement of earth or land. Today geometry’s big ideas include the study of spatial properties and relations of points, lines, surfaces, solids, and higher dimensional quantities. Their shapes, variables or invariables, definitions, & proofs.

Analytic geometry is the combination of geometry and algebra.

Relative position is used to locate objects in space with distance and directions.

### Knowledge base

Mathematical knowledge base with concepts necessary for learners to conceptualize to become mathemtically literate and misconceptions that must be over come. Particularly important for geometry and spatial reasoning are:

### Tools to facilitate literacy in geometry & spatial reasoning & tools to achieve it

Background information for learning and teaching.

## Activities to facilitate geometry & visual spatial literacy

### Puzzles & challenges, with work sheets

• Pythagoras theorem - Explanation sheet to justify a 2 + b 2 = c with 1. Area model & 2. lengths of sides model
• Relative position & topology - Work sheet to identify: in, on, out, for 3d & 2d pictures, & draw the 5 shapes. Age: preschool - first grade
• Tessellation - Work sheet: How to make a tessellation
• Triangles - properties, kinds, angles, sides, and relationships
• Ruler and compass constructions
• Cross sections

# Data Analysis & Probability

## Knowledge base & tools to develop mathematical literacy

### Knowledge base

The mathematical knowledge base includes concepts necessary for learners to conceptualize to become mathemtically literate and misconceptions that must be over come to be literate in geometry and spatial reasoning.

Data analysis

Probability

# Measurement

## Knowledge base & tools to develop mathematical literacy

### Knowledge base

The mathematical knowledge base includes concepts necessary for learners to conceptualize to become mathemtically literate and misconceptions that must be over come. Particularly important for measurement include:

### Information to facilitate literacy in  measurement & tools to achieve it (pedagogy)

Historical weights found around the world suggest traders recognized the need for standard measurements.

### Activities to facilitate measurement literacy

• Introductory activities should have learners directly compare objects as big and little. Compare hands, clothes, bowls, eating utensils, toys, crayons, blocks, and more. Then have them directly compare and order three objects as big, medium, & small. Work towards sequencing larger sets of objects.
• Use non standard units to measure all sorts of objects.
• Measurement unit - primary level introduction to measurement with first straw activity to develop a concept of standard units of linear measurement , then similar hands on measurement activities for volume, mass, & temperature.
• Measuremen unit - middle level review of linear, mass, volume, measurement of matter, with an introduction to density with directed inquiry activities & blank learner lab notes
• Metric fact sheet - Fact sheet with metric units: meter, liter, gram, prefixes on a number line, diagram of dust particle & human hair, metric prefixes with symbols to + & - powers of thirty.
• Observation & measurement unit - How do we observe? properties to observe, change, measurement, properties to measure, & measurement procedure work sheet
• Density unit - learning cycle inquiry activities for middle grade to apply measurement: linear, mass, volume, of matter to calculate density of solids, liquids, & gases. Includes 14 activities with lesson plans & lab sheets.
• Cartography map making, angle measurement, compass, clinometer, measure height, measure distance, locating objects with distance & angle combinations, contour maps,
• Measuring heights of trees & other objects - clinometer, shadows, scale (Biltmore) stick
• 21 Centimeter card game : make a deck with 45 cards. Draw a slanted straight line (at different degrees on each card) with the following lengths, on cards as listed below:
• 4 cards each with a - 1 cm line,
• 4 cards each with a - 2 cm line,
• 4 cards each with a - 3 cm line,
• 4 cards each with a - 4 cm line,
• 4 cards each with a - 5 cm line,
• 5 cards each with a - 6 cm line,
• 5 cards each with a - 7cm line,
• 5 cards each with a - 8 cm line,
• 5 cards each with a - 9 cm line, &
• 5 cards each with a - 10cm line.

• Game play : Deal three cards to each player and play like 21. Player that line lengths add closest to 21 cm, without going over is the winner. After the deal each player can choose to be dealt another card or hold. Player closest to 21 cm, with out going over wins. Winner measures each card with a cm ruler to verify their score.
Alternative version : Play as above and record the lengths of their cards, if they don't go over 21. First player to 121 wins.
• Linear & volume -
• Solar system - big distances & volume activities (5E learning cycle) to explore the size and relative position of the solar system aligned to the Common Core Standards, Next Generation Science Standards, National Science Education Standards, & 21st Century Skills: critical thinking, communications and measurement.
• Temperature How does temperature in centigrade Celsius compare to temperature in Fahrenheit? Facts: Water freezes at 32 degrees Fahrenheit and 0 degrees Celsius. Water boils at 212 degrees Fahrenheit and 100 degrees Celsius. Find a thermometer that has both scales and read from one scale to the other. C = (F - 32) * 5/9; C = (F - 32) / 9/5; F = (C * 9/5) + 32 Verify or prove the formulas.
• Money - Sequence of outcomes to develop with activities to learn to count change:
• Know people use money to buy goods and services.
• Sort money by its appearance.
• Draw pictures of coins and paper money.
• Select coins on their desk (penny, quarters, dimes and nickel) as called.
• Identify and state the values of money (penny = 1 cent , quarter = 25 cents, dime = 10 cents, nickel = 5 cents, ... ).
• Sort money by its value.
• Order money by its value.
• Recognize different coins can have similar values.
• Represent coins and their values: draw a circle for each coin (penny, quarter, dime, nickel) with the value inside: (25, 25, 10, 10, 5).
• Identify a coin, its value, and point to its value represented on a number dot sequence, trail, or line . Select a second coin. Identify the coin, its value, and show how to add and represent its value and sum of both.
• Identify a coin, its value, and point to its value represented on a labeled hundreds chart . Select a second coin. Identify the coin, its value, and show how to add and represent its value and sum of both. Draw an arrow from 1 to the value of the first coin. Circle the value and draw an arrow for the value of the next coin and circle the sum. (arrow math)
• Use arrow math and a labeled hundreds chart chart to solve coin problems with two addends.
• Use arrow math and a labeled hundreds chart chart to solve coin problems with more than two addends.
• Given an assorted collection of coins (that add to less than 100 cents), Use arrow math and a labeled hundreds chart chart to solve coin problems with more than two addends.
• Identify different orders to count change more efficiently. (Start with the largest coins, group similar coins, group by easy numbers (10, 50, 100, & 25).
• Repeat the above procedures on a blank hundreds chart .
• Repeat the above procedures with paper money.

Can use the counting money strategy for counting decimal numbers. Write the starting value (1.89), select a count (.01), write it, write the sum (1.90), write the next coin (.10), then the sum (2.00), and continue as before.

# Process dimensions of mathematics

Information to plan and facilitate the learning and use of skills in the process dimensions of mathematics.

## Problem solving Tools to facilitate mathematical problem solving

• Problem solving oncepts, misconceptions, & outcomes by levels in the mathematical knowledge base

## Reasoning and proof

• Reasoning & proof Concepts, misconceptions, & outcomes by levels in the mathematical knowledge base

## Representation

Representation Concepts, misconceptions, & outcomes by levels in the mathematical knowledge base

### Area representations

Lynne Outhred and Michael Mitchelmore asked learners to illustrate the area of a rectangular shape. They found learners demonstrated four different strategies: incomplete covering, visual covering, concrete covering, and measurement. The drawings below show samples from each.

### Representation of relative position of objects

Illustration of Internal Representation

Actual object, real world, or external representation

### Representation of external objects with points on an other external object

What does this have to do with mathematical representations?

### Activities to facilitate representation

• Representations of 21 / 7
• Area & perimeter - pools and sidewalks - Work sheet patterns, relationships, & functions with area and perimeter
• Growing squares challenge - Growing squares challenge slides for the challenge above. Slides step through adding toothpicks to make growing squares, record data in a table, and ind the a pattern of increasing squares and increasing number of toothpicks. v
• Growing squares worksheet - Use toothpicks to find a pattern & formulas to determine the number of toothpicks added to make larger squares. Toothpicks in the perimeter, toothpicks to make the squares in the squares, and the number of squares in the growing squares.
• Toothpicks in a rectangular pattern - Worksheet with hints to write equations for the sums of toothpick patterns in a rectangle and discussion ideas for what is an equation, function, and can they be both.
• Pattern activities - Investigatons to develop patterns & formulas. Starting with number strips, dot patterns & other growing patterns: v, w, pyramid, prism, tower, triangle, rectangle, log stack, can stack, hand shake, dancers, step pyramid, & disc ...

## Communication

Communication Concepts, misconceptions, & outcomes by levels in the mathematical knowledge base

## Connections & perspective

Connections & perspective Concepts, misconceptions, & outcomes by levels in the mathematical knowledge base

### Information to facilitate literacy in connections & perspective & tools to achieve it

• Mathematical ideas build upon each other.
• Mathematical ideas are connected to other mathematical ideas.
• Mathematical ideas are connected to the world.

## Technology, computer programming, software, 3D printing

--- Coding software ---

Mozilla free

Technology

• History of the cell phone Read ... Cell phone mathematics . The story of how Brad Parkinson used creativity and mathematics to enable the use of cell phones.

Math software

Coding resources

Pedagogical resources

3 D printing

Top

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Page Overview

Development, planning, activities , problems, learning units & packets to develop mathematical literacy by dimension

Process dimensions:

Content : dimensions

• Technology - calculator, programming, & other math related tech ideas

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There are 10 kinds of people in the world. Those who understand binary and those who don't.

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Why is it whenever students don't do well on a national test the news agencies report one or a few particular items that were on the test by stating the test questions, then they believe it is necessary to provide the answers.

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If their listeners need the answers, why should we expect students to be able to answer it correctly?

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The Jibaro Indians of the Amazon rain forest express the number five by the phrase:
"wehe amukei,"
"I have finished one hand,"

and the number ten by
"mai wehe amukei,"
"I have finished both hands."

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"These landmarks must not be treated as objectives to be marked off. The goal is to mathematize not pass landmarks. Enjoy the journey create systems to do mathematics with the knowledge that students have. One should recognize the fallacy of thinking there can be one activity or sequence of activities for all students."

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One of the universal customs that man has successfully established on earth is the
Hindu - Arabic numerals
to record numbers.

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Math is meditation for thinking about the world.

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Geometry is the science of correct reasoning on incorrect figures.

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Galileo was put under house arrest and allowed visitors, but no mathematicians .

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### Common sense & knowledge

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### Counting in Winnebago

1. WI n
2. NO n BA
3. THABTHI n
4. DUBA
5. SATO n
6. SHAPE
7. PETHO n BA
8. PETHABTHI n
9. SHO n KA
10. GTHEBO n

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### Interesting book

The Power of Logical Thinking: Easy Lessons in the Art of Reasoning... and Hard Facts About Its Absence in Our Lives, Marilyn Vos Savant (1996) ISBN 0-312-13985-3 Saint Martin's Press: New York.

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If the universe is infinite in time and space, stars should occupy every point in space and fill the night sky with light.

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Mathmaticians think about more than math.

Earth will run out of the basic resources, and we cannot predict what will happen after that. We will run out of water, air, soil, rare metals, not to mention oil. Everything will essentially come to an end within fifty years. What will happen after that? I am scared. It may be okay if we find solutions, but if we don’t then everything may come to an end very quickly. Mathematics may help to solve the problem, but if we are not successful, there will not be any mathematics left, I am afraid!

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U.S. Students

Score in the top tier on
the PISA 2012 Math Creative Problem SolvingAssessment

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Measurement is never exact .

It is only as accurate as who and with what is doing the measurement.

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Shape with sides in a pattern that never repeats!

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The story of how Brad Parkinson used creativity & mathematics to enable the use of cell phones.

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Data is just numbers. They need an explanation or theory to make it data .

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Algebra was greatly advanced by al-Khwarizmi’s book:
Hisab al-jabr w'al-muqabala From which the word algebra came.

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