# Mathematics concepts, misconceptions, and outcomes

To develop mathematical literacy and critical thinking

*Overview*

- Algebra & patterns
- Attitudinal (disposition & value statements)
- Classification
- Communication
- Connectiona & perspective
- Data Analysis & probability
- Geometry
- Measurement
- Number value
- Operations
- Problem solving
- Reasoning & proof
- Relative Position and Motion
- Variable

*Sample layout for each category*:

# Misconceptions and concepts

## Initial perceptual naive **misconceptions **(all ages)

(Explanations for people's misconceptions: naive understandings & perceptual responses)

- Misconception list

## Beginning **concepts** (preschool - 7 years) ... list

## Literate **concepts** (middle level to adult) ... list

## Advanced **concepts** ... list

# Educator notes

# Algebra and patterns

Misconceptions and concepts

## Initial perceptual naive **misconceptions** (any age)

1(Explanations for people's misconceptions: naive understandings & perceptual responses)

- School math is different than math used outside of school.
- Patterns with numbers are different than patterns with physical objects.

## Beginning **concepts** (preschool - 7 years)

- Number values can be attached to objects and the properties of those objects.
- Number values can be attached to ideas and the properties of those objects.
- Sets can be equal or not equal.
- Sets can be copied.
- Logical patterns exist and occur regularly in mathematics.
- Patterns can be copied ( ordered sets, objects..).
- Patterns can be recognized, extended, or generalized.
- The same pattern can be found in many forms.
- Patterns are found in nature and physical objects as well as numbers.
- Patterns can be described, represented, and communicated in a mathematical way.
- Simple patterns can be recognized in more complicated patterns
- Classification is helpful when searching for patterns

## Literate **concepts** (middle level to adult)

### Algebra: patterns, equations, and functions

- A system can have any number of equations and any number of variables. When there are only two variables it is possible to visualize the relationship with a graph on a coordinate plane.
- Graphs are pictorial representations of information
- Each person in the world could pick a different three points on a line, but since there are infinite ordered pairs as solutions – the line will be graphed the same.
- Simultaneous equations can be represented with graphs
- Systems of equations are dependent, consistent and independent
- Substitution and elimination methods can be used to solve simultaneous equations
- How to graphically represent a system of equations and identify a
- Solution set/feasible region.
- Applying Systems of Equations to application problems
- Choosing the most appropriate method for solving systems of

equations - Use letters, boxes, or other symbols to stand for any number, measured quantity, or object in simple situations to demonstrate the beginning concept of a variable and writing formulas.
- Identify and use various indicators of multiplication (parentheses, X, *) and division (/, Ö).
- Identify, describe, and extend arithmetic patterns, using concrete materials and tables.
- Use input/output or function box to identify and extend patterns.
- Use and interpret variables and mathematical symbols to write and solve one-step equations.
- Functions describe patterns.
- Functions are relationships or rules that uniquely associate elements of one group to elements of another group.
- In a functional relationship a change in one group causes a change in the other group.
- Functions can be used to represent and solve problems.
- Functions that represent an event can be expressed in different ways.
- Variable

### Ratio and proportion

- Ratio is the relationship of one thing to another. Can be used to compare quantities.
- Ratio can be show with words (written and oral), fractional form, and quantities separated with a colon).
- Ratios and proportions result in treating numbers in relation to each other to form new numbers.
- Ratios and proportions result in forming new ways to represent rational amounts and proving that many forms are equivalent even though they don't look alike.
- For equivalence a ratio must be kept constant
- Rates are ratios.
- Rates and ratios can be used to make predictions.
- Ratio tables can be used to make predictions.
- Ratios are used with measurement
- Cross products can be used to determine if two ratios are equivalent.
- Proportion is the relationship of a part to a whole
- Percent is part of one hundred
- Proportion uses the relationship of equality between two ratios.

## Advanced **concepts**

# Educator notes

*Possible outcomes*

- Represent patterns and relationships in a variety of formats and use these representations to predict unknown values and justify solutions.
- Interpret graphs that represent linear and non-linear data and the relationship of the variables.
- Construct and analyze tables and graphs to describe how change in one variable affects a related variable.
- Match visual characteristics of slope with its numerical value by comparing vertical change with horizontal change of two variables.
- Solve problems involving the intersection of two lines on a graph
- Solve and verify simple linear equations algebraically and with representations.
- Create and solve problems, using linear equations.
- Correctly graph a set of equations on the same Cartesian coordinate system
- 1. Set up and label a coordinate system
- 2. Select an appropriate scale
- 3. Represent given equation graphically, accurately and neatly
- 4. Plot points accurately and neatly
- 5. Describe the relationship between the variables
- Identify the solution set from the graph as an ordered pair and explain clearly parallel, identical and intersecting lines
- Solve systems of linear equations using substitution.
- Solve systems of linear equations using elimination/linear combination.
- Identify the most appropriate method for solving a system.

*Solve application problems involving a system of linear equations*.

- Accurately identify the variables before setting up the problem and translating the word problem into an equation.
- Correctly translate word problems into appropriate equations.
- Solve the system accurately using the most appropriate method and labeling the answer using the correct units.
- Solve problems involving equations and inequalities.
- Use appropriate methods to solve linear and quadratic equations.
- Solve problems involving systems of two equations, and systems of two or more inequalities.
- Solve equations using graphing, substitution, elimination, or matrices (x + y = 7, x - y = 1) and (2x - y > 4, x + 3y <= 9)

*Simultaneous linear equations scoring guide*

- Inconsistently solve problems involving two or more equations.
- Solve problems with two or more equations by elimination and graphing
- Solve problems with two or more equations by elimination, graphing, substitution, and matrices as well as create equations from graphs.
- Solve problems with two or more inequalities by creating equations when given a graph.

*Logrithms*

*Common Logrithms* have a base of ten. It is the *log* button on calculators.

- log (100) means.
- log
_{10}(100)=2; Meaning 10^{2}= 100 - log
_{10}(1000)=3; Meaning 10^{3}= 1000

*Natural logarithms* use 2.71828. Euler's Number (e) as the base. It is the *ln* button on a calculator.

- It is how many times e is multiplied to get a specified number.
- Most used logs are common, natural, and base two. Graph for all three.

# Attitudinal (Dispositions and values)

Misconceptions and concepts

## Initial perceptual naive **misconceptions** (any age)

1(Explanations for people's misconceptions: naive understandings & perceptual responses)

- Believe success is based on luck and not on positive attitudes that shape behavior.
- Believe attitudes are set and not changeable.

The following statments can be considered for beginning, literate, and advanced levels. The depth of understanding may increase as students mature with additional experience they can all be understood, exhibited, and valued at all levels.

The are written as concepts and could be used as outcomes if observable evidence was included to substantiate them. Example.

## Literate **concepts** (middle level to adult)

*Acceptant of failure*- Considers failure as an occasional part learning and helpful for redirecting.*Caring, Conscientiousness*- Care for others, the natural world, and human-made objects. Caring includes the ways that people individually and collectively participate for the well being of all things for the present and future.*Cooperative*- Works with others for common goals and shares ideas.*Curious*- Asks and answers questions to understand at deeper levels.*Creative*- Imagines ideas that are original or not ordinarily thought*Disposed to apply knowledge*- Ready to think and apply what they know to current related experiences; all ideas from all dimensions (knowledge, processes, attitudes, and perspectives of science).*Enjoyment*- Expresses pleasure in understanding and pursuing understanding.*Flexibility*- Willing to change with new evidence and/ or explanation.*Grateful -*Thankful and appreciative for others contributions.*Knowledgeable*- Knows many generalizations, concepts, and facts; understands inquiry practices; and understands the history, nature, social, personal, and technological perspectives of different subjects.*Objective -*Makes decisions based on facts.*Open-minded*- Tolerates ideas and opinions of others and the importance of carefully considering ideas that may seem disquieting or at odds with what is generally believed and willing to change ideas in light of new evidence.*Optimistic -*Positive, believe people are caring, helpful, and willing to cooperate. Believe solutions are attainable and learning is never complete.*Passion, zest -*Desire to learn, be involved, take action, and believe learning is infinite.*Persistent*,*grit*- Continues despite obstacles, warnings or setbacks.*Sensitive*- Considers all actions and inactions and the results they might have on all living and nonliving things.*Skeptical*- Doubts, questions, and reconsiders conclusions based on evidence and reasoning.*Tentative*- Hesitant to draw conclusions.*Reflective*- Curious and willingly open-minded to consider new ideas based on evidence and reasoning against previous ideas based on evidence reasoning.*Responsible*- take actions for personal understanding and learning.*Respect for evidence -*Insistent on evidence. Requires evidence to formulate explanations and make decisions and will seek additional evidence and reasons to verify ideas and make decisions.*Self-control*- Choose mastery-oriented behaviors that achieve predictable and reliable outcomes and encourages others to do so also so they may cooperate and be productive.*Self-efficacy*- Believe in their abilities and skills in using different subject practices, processes and knowledge in a useful effective manner to learn and solve problems.*Sociable*, - Values other people and desires to be involved with others to benefit by helping, caring, contributing*Values communication*- Seeks ways to communicate that effectively enable others to accurately conceptualize the ideas wanting to be communicated.

# Educator notes

To write a statement as an attitude, dispositon, and value, a value word must be incuded. For example. To write an attitude, disposition, or value statement about communication it might be something like: Desires to and works to communicate with others so they might better facilitate learning. (value implied). Values the power in communication to facilitate learning and demonstrates it when working with others.

The list includes attitudes that are important. However, to remember or work with all of would benefit by some categorization or organization. If they could be organized with six or seven initial categories so chunking would might assist recall. Possible categories - Personal related, Social or group related, Problem solving, Communication, Creating ideas, or other catgories.

Anyone come up with ideas and want to share, let me know...

# Classification

Organization, order, & systems (cross-cutting)

Misconceptions and concepts

Coordinate with science

## Initial perceptual naive **misconceptions** (any age)

(Explanations for people's misconceptions: naive understandings & perceptual responses)

- Things (objects) just have common characteristics or properties for no important or meaningful reason.
- Objects properties are random and have no use in understanding and explaining the world.
- Objects can only be grouped in one way.
- Order eixsts for no specific benefit (providing understanding or explanation).
- Order is one directional.
- Numbers can't be classified, because they are all different.
- Letters can't be classified, because they are all different.
- Centers on one property, color, shape... to classify when two or more are necessary.
- May believe that since an object was counted once in one group for a property, then that object cant be counted again for another property as doing so would cause a double counting action, which isn't allowed when counting sets, because a one-to-one correpsondence means to count each object only once.
- Does not decenter from the use of only one property and simultaneously use two or more properties, when necessary, to classify objects.

## Beginning **concepts** (preschool - 7 years)

### Classification & organization (see also science concepts)

- Objects, organisms, events, and systems can be organized into groups with similar properties (classification).
- Classification is one way to organize objects, events, and ideas. (organization)
- Objects have properties.
- Objects are identified by their common properties.
- Objects can be grouped by external properties, color. (3.5 years)
- Objects are classified by there common properties.
- Objects have more than one property. (4.5 years)
- Objects in a group can share some characteristics while differing in others. (4.5 years)
- Objects, organisms, events, ideas, and systems can be organized into groups with similar properties.
- Objects are identified by names.
- Objects with similar properties are the same.
- Objects with different properties are different.
- Objects can have properties that are the same and different, but still be the same (triangles - same shape, different size, color)
- Objects can be grouped (classified) into sets/ groups
- Sets can have cardinality. Cardinality of sets can be the same or different.
- Objects with similar properties that change sequentially can be ordered
by that property. See
*order*.

Next classification intermediate concepts

### Order

- Order is created by properties that change sequentially. See number sense and cardinality
- Most of the time certain events happen in a similar manner.
- Some events are more likely to happen than others. See probability
- Some events can be predicted more accurately than others.
- Sometimes people aren't sure what will happen because they don't know everything that might be having an effect on the event.
- Often a person can find out about a group of things by studying just a few of them.

### System

- Parts are related to a whole.
- A whole is related to its parts.
- Parts are related to parts.
- System is a group of related objects, ideas, and events.
- Most things are made of parts.
- Something may not work if a part is missing.
- When parts are put together they can do things they can't do alone.

## Literate** concepts **(middle level to adult)

### Organization & classification Concepts

- Objects, organisms, events, and systems can be organized into groups with similar properties (classification).
- Classification is one way to organize objects, events, and ideas. (organization)
- Objects have properties.
- Objects are identified by their common properties.
- Objects can be grouped by external properties, color. (3.5 years)
- Objects are classified by there common properties.
- Objects have more than one property. (4.5 years)
- Objects in a group can share some characteristics while differing in others. (4.5 years)
- Objects, organisms, events, ideas, and systems can be organized into groups with similar properties.
- Objects are identified by names.
- Objects with similar properties are the same.
- Objects with different properties are different.
- Objects can have properties that are the same and different, but still be the same (triangles - same shape, different size, color)
- Objects can be grouped (classified) into sets/ groups
- Sets can have cardinality. Cardinality of sets can be the same or different.
- Objects with similar properties that change sequentially can be ordered by that property.
- Organization of objects, organisms, events, and systems help people understand similarities and differences that in turn help understand the world.
- Classification is one example of organization.
- Classifications can include
*class inclusion*, a relation between two classes in which all members of one class are included in the other. (8 years) Birds have feathers, there are more birds with black feathers than white. Are there more black feathers than feathers? ... Will answer feathers if have*class inclusion*.

- Objects are classified by their properties.
- Classification by common properties can create similar groups.
- Objects in a group can share some characteristics while differing in others.
- A group of objects may be sub classified in one or more ways.
- A group of objects may be subclassified as members of an
*ascending hierarchy*. (8 years) A cat is a mammal. - Descending hierarchy. (9.5 years) Mammals include cats.
- A group or set can be described and classified by processes as well as properties.
- Thinking about things as groups or sets means looking at how every element relates to other members of the group or set.
- Objects may have properties of two different groups or sets.

### Order Concepts

- Variables affect the order of events.
- Order is required to understand the world and predict events.
- Probability is the relative certainty or uncertainty that people assign to events happening or not happening in a certain place or time.
- Creating knowledge through observation of different variables influence on objects, organisms, populations, communities, and events helps create better explanatory models

### System Concepts

- System is a group of related objects that work together for a particular purpose (machines, organism).
- The parts in a system interact with the other parts to cause the system to work.
- A system may not work if a part is missing, broken, worn out, mismatched, or disconnected.
- Objects can be classified as either natural or of human design,
- Organization of objects, organisms, events, and systems help people understand similarities and differences that in turn help understand the world.
- Sometimes thinking about things as systems improves understanding and sometimes it doesn't.
- System is a group of related objects or components that form a whole.
- Can be concrete objects, groups of objects, processes, or ideas.
- Some systems have boundaries with input and output of resources and feedback.
- Output for one part of a system can be input for another.
- Such feedback is used to control the system.
- Systems are used as units of investigations.
- A system can include processes as well as things.
- Thinking about how a system works means observing and collecting date on each part and how each part interacts with the others.
- Systems can be connected to other systems and thought of as a subsystem.
- Systems may have what appear to be natural boundaries, but are generally arbitrary.
- System ideas are used outside of science. Technology and business use system analysis which looks at systems relationships by their inputs and outputs. Computer programming use these ideas with procedural languages, functions, and object oriented programming.

## Advanced** concepts **

# Educator notes

- Clasification card set (5 different shapes and colors)
- Conservation assessment tasks
- Classification developmental outcomes for scoring guide or rubric

# Communication

Misconceptions and concepts

## Initial perceptual naive **misconceptions** (any age)

(Explanations for people's misconceptions: naive understandings & perceptual responses)

- Communiation is just something we do to talk to each other.
- Doesn't help to study communication because people are either good at it or not.
- Communication is writing.

## Beginning **concepts** (preschool - 7 years)

- Communication helps us learn from other people.
- Pictures can be used to represent objects and events.
- Communication helps us explain evidence and reasoning to each other.
- Communication requires a message being sent and received.
- Information can be communicated in many different ways each of which have advantages and disadvantages.
- Objects can be described and compared by properties.
- Before and after pictures can be used to represent change.
- Clear communication gives other people information about your discoveries and ideas.
- Explanations are better when specific evidence is provided.
- Communication allows other people to agree or disagree with a person's findings.
- People have always tried to communicate with one another.
- People have invented devices to communicate (paper, ink, radio, telephone, telecommunications, computer disks) Errors can occur when communicating.
- Repeating messages is a way to avoid miscommunication.
- Directions can be written so other people can try procedures.
- Sketches can be used to explain procedures, events, or ideas to the creator and other people.
- Messages can be carried by many different media (light, electricity, sound, objects, glass fibers).
- The ability to code messages has allowed faster communication.
- Numerical data can be used to describe and compare objects and events to the creator and other people.
- Mathematical ideas can be communicated.
- There are different ways to communicate mathematically.
- Mathematical ideas can be communicated six ways: written narrative, spoken words, pictures, manipulatives, symbols, and movements.
- Combining different ways of communication can make for more efficient or better communication.
- Tables, charts and graphs can be used to communcate relationships.
- Tables, charts, and graphs can be used to represent objects and events.
- Graphs can be used to identify and represent relationships.
- Accurate data keeping and openness are essential to assure an investigator's credibility.
- Graphs can be used to recognize, represent and predict future relationships to the creator and other people.
- Other kinds of tables, matrices, diagrams, webs, symbols, maps can be used to interpret and communicate information.
- Regular and polar coordinates can be used to locate objects.

# Educator notes

Most activities can include communication of mathematics concepts.

See examples of outcomes related to communication in data analysis educator notes.

*Six ways to communicate mathematically*.

- Pictures - Four ways to make 25 cents
- Manipulatives - use of - cubes Writing after Average student height
- Writing - Percentage - Sale problem
- Acting it out - The Door Bell Rang - from the book,
*The Door Bell Rang*by Pat Hutchins - Symbols - Pigs Go to Dinner - from the book,
*Pigs Will Be Pigs: Fun with Math and Money*. by Amy Axelrod, 1994 - Orally - talk it out.

# Connections and perspective

Misconceptions and concepts

## Initial perceptual naive **misconceptions** (any age)

(Explanations for people's misconceptions: naive understandings & perceptual responses)

- Mathematics is only for numbers making other numbers.

**Concepts** apply to all levels. Adapt according to level.

Beginning (preschool - 7 years), Literate **concepts** (middle level to adult) & Advanced

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

# Educator notes

# Data Analysis and Probability

Misconceptions and concepts

# Data Analysis

## Initial perceptual naive **misconceptions **(any age)

(Explanations for people's misconceptions: naive understandings & perceptual responses)

- Data is only related to one thing. Height is related to each person.

## Beginning **concepts** (preschool - 7 years)

- Number values can be attached to objects, ideas, and their associated properties.
- Information about objects can be collected, compared, interpreted, organized, displayed, and used to explain something about the objects.
- Data can be arranged in different ways.
- Organization of data can help interpretation (graphs, charts, tables, Venn diagrams...)
- Samples can be used to make predictions.
- Range of data is the spread or difference between the least and most set of quantities (smallest and largest quantities) Range is affect most by an extreem quantity.
- Mean (arithmetic mean or average) is the sum of the value of each item divided by the number of items.
- Measure of central tendency includes average, mean, mode, median...
- Mode is the item represented the most in a set of data.
- Median is the middle set of data. Mean is equal to the number of cases plus one divided by 2. It would be the middle value in an odd numbers set of cases or the mid point between the two middle sets of cases in an even number of cases.
- Different data measures may be appropriate to describe different sets of data.
- Table
- Bar graphs are used with categorical data.
- Scatter plot is
- Stem and leaf plot... A stem and leaf plot can be used to find mean, median, or mode.
- Line graph is

## Literate **concepts** (middle level to adult)

- Data can be arranged in different ways.
- Data can be represented differently in different forms.
- Some representations are better than others.
- Organization of data can help interpretation (graphs, charts, tables, Venn diagrams...)
- Different data measures may be appropriate to describe different sets of data.
- Bar graphs are... and are used with categorical data.
- Samples can be used to make predictions.
- Range of data is the spread or difference between the least and most set of quantities (smallest and largest quantities) Range is affect most by an extreem quantity.
- Mean is the the quotient of the sum of data points divided by the number of data points; (arithmetic mean or average) is the sum of the value of each item divided by the number of items.
- Median is the middle point, or the middle set of data, when all data points or measurements are ordered from smallest to largest.
- Median is the average of the middle points of a sequence of data points arranged by value.
- Median is the value at the midpoint of a frequency distribution such that there is an equal probability of falling above or below it.
- Median is equal to the number of cases plus one divided by 2. It would be the middle value in an odd number set of cases or the mid point between the two middle sets of cases in an even number of cases.
- Mean is the averaged of a set of data or mesurements.
- Mode is the data point that occurs most frequently in a set of data or measurements.
- Measure of central tendency includes average, mean, mode, median...
- Scatter plot
- Cluster is a group of data points close together separated by gaps between data.
- Outlier is an isolated data point, or cluster of data, some distance away from the rest of the data.
- Stem and leaf plot... A stem and leaf plot can be used to find mean, median, or mode.
- Line graph
- Pie chart
- Categorical data is
- Continuous data is
- Different scales on a graph creates different impressions of the same data.
- Symmetric, A distribution where one haave looks like the other half (mirror images).
- Skewed is a distribution that has a long thin tail stretching in one direction.
- Normal curve of distribution is also know as the Gaussian curve.

## Advanced **concepts**

- Sigma (SD, standard deviation) represents the measure of the amount of variation or dispersion of a set of data values.

# Educator notes

Many activities that include data analysis concepts will relate well to communication of mathematics concepts.

### Outcomes related to data analysis and communication

*Beginning*

- Organize concrete objects according to related properties for quantities from least to most.
- Identify properties of objects and events and use them to organize and describe data by quantities: least, most, range, and mid values.
- Collect, record, organize, and interpret data in line plots, tables, charts, and graphs (pie graphs, bar graphs, and pictographs).
- Use pictures or symbols to represent properties of quantity of objects to describe those objects or events by range, least, most ...
- Describe like properties with different values as variables.
- Describe relationships with other properties and variables.
- Draw valid conclusions from data displayed in line plots, tables, charts, and graphs (pie graphs, bar graphs, and pictographs).

*Literate*

- Collect, construct, and interpret data displays and compute mean, median, and mode.
- Make predictions from data and explain reasoning and informal measures of central tendency.
- Analyze statistical claims and design experiments, and they may use simulations to model real-world situations.
- Some understanding of sampling and make predictions based on experiments or data.
- Comfortable using various graphs to represent different types of data in different situations.
- Represent patterns and relationships in a variety of formats and use these representations to predict unknown values and justify solutions.
- Interpret graphs that represent linear and non-linear data and the relationship of the variables.
- Construct and analyze tables and graphs to describe how change in one variable affects a related variable.
- Match visual characteristics of slope with its numerical value by comparing vertical change with horizontal change of two variables.
- Solve problems involving the intersection of two lines on a graph
- Solve and verify simple linear equations algebraically and with representations.
- Create and solve problems, using linear equations.
- Describe a graph by identifying intercepts, slopes, maximum, minimum, increasing, decreasing, parallel, and perpendicular.
- Use families of curves to describe the effect of changing coefficients of an equation.

*Correctly graph a set of equations on the same Cartesian coordinate system*

- Set up and label a coordinate system
- Select an appropriate scale
- Represent given equation graphically, accurately and neatly
- Plot points accurately and neatly
- Describe the relationship between the variables
- Identify the solution set from the graph as an ordered pair and explain clearly parallel, identical and intersecting lines

# Probability

## Initial perceptual naive **misconceptions **(any age)

(Explanations for people's misconceptions: naive understandings & perceptual responses)

- I can cause a certain tile to be selected. (think real hard, pick a certain way...).
- One object (event) always gets chosen more often.
- It is magic.
- Students will select the second statement, with multiple possibilities, as more probable (conjunction fallacy).
- A survey was given to a representatvie sample of students in grades 1-12. J. T. was included in the sample and selected by chance. Which of the followiong is more probable? 1) J. T. missed five days or more of school. 2) J. T. missed five days or more of school and is below sixth grade.

## Beginning **concepts** (preschool - 7 years)

- An event has a set of outcomes. The smallest set of outcomes is the empty set or impossibility.
- Predictions are derived from possibilities.
- A prediction about a set (population) can be made from one or more samples.
- Some possibilities have a better chance of occurring than others.
- Some possibilities are not likely to happen.
- Some possibilities are impossible to happen.
- Events are impossible, possible, or certain.
- Probability is how likely an outcome is to occur.
- A fair game is one in which every player has an equal opportunity to be successful.
- A tree diagram can be used to identify all outcomes of events.
- Each selection increases the possibility of selecting all possibilities at least once.
- Accuracy of prediction may be increased with more trials.

## Literate **concepts** (middle level to adult)

- An event has a set of outcomes. The smallest set of outcomes is the empty set or impossibility.
- Events are impossible, possible, or certain.
- Probability is how likely an outcome is to occur.
- The probability of an outcome is the number of specific outcomes out of the total number of all possible outcomes of one event.
- Probability can be used to make predictions.
- Probability can be determined
*theoretically*(Use reasoning to find the total possible outcomes, the total of each specific outcome, and the proportion of the specific to the total). - Probability can be determined
*experimentally*(Experimentally collect data for each specific outcome and use it to calculate the proportion of the specific to the total). - Probabilities can be written in the form of a fraction, decimal, and percent.
- Probability is between 0 and 1.
- A fair game is one where all participants have equal chance of success.
- A tree diagram can be used to identify all outcomes of events.
- Conjunction rule: the probability of two events occurring (in conjunction) is always less than or equal to the probability of either one occurring alone.

## Advanced **concepts**

- Chirality is the property of a structure that its mirror image can not be superimposed on each other. Example. Left hand is chiral to a right hand. Left hand image can not be superimposed on the image of a right hand. Used mostly to describe molecules, atoms, ions, and other particles.

# Educator notes

Children can accurately predict relative probabilities with the use of perception for probability devices that are scaled (spinners with different colored sections of different areas, different amounts of colored objects in a container).

Conduct experiments or simulations to demonstrate experimental probability, theoretical probability and relative frequency.

Students will select the second statement as more probable (conjunction fallacy).

- A survey was given to a representatvie sample of students in grades 1-12. J. T. was included in the sample and selected by chance. Which of the following is more probable? 1) J. T. missed five days or more of school. 2) J. T. missed five days or more of school and is below sixth grade.

Probability [(JT in 5 grade) & (JT less than 6g)] < or = probaility (JT in 5 grade)

Students who make these errors may not understand probability, may use intuition to connect what seems more normal, not understand intersection, inclusion, or subsets. Venn diagrams can provide learning and instructional support to represent these ideas.

# Geometry

Misconceptions and concepts

## Initial perceptual naive **misconceptions** (any age)

(Explanations for people's misconceptions: naive understandings & perceptual responses)

- Shapes are defined by their outline not properties.

## Beginning **concepts** (preschool - 7 years)

- Shapes are identified by their properties.
- Shapes are categorized by their properties. (classification)

## Literate **concepts** (middle level to adult)

*Area*is the surface space of a two-dimenional space measured in square measurements.is made where two lines intersect.*Angle**Circle*is a two dimensional closed figure with everypoint being equal distance from its center.- Circle properties include: circumference, diameter, radius, sector, chord, area, and infinite lines of symmetry

*Congruence*is when two figures have the same shape and size.*Coordinate system*is a spatial reference system that uses vertical and horizontal references from a point of origin.*Line*is- A line has different properties (line, line segments, ray)

*Patterns*can be described mathematically.*Parallel lines*are lines in the same plane that do not intersect.-
*Perimeter*is the distance around a shape measured in linear measurements. *Perpendicular lines*intersect so they form a right angles or 90 degree angle. Can be made on a piece of paper holding the points on either end of the line together and folding at the mid point.*Pi*is the relationship between the diameter of a circle and its circumference.*Plane*is a flat surface such that if a straight line is draw from from on point on the line to another, the line would be entirely on the surface.*Point*is a place in space that has a location, but no size. It is impossible to draw, because if it is, then it has size.*Polar coordinate system*is a spatial reference system that uses and angle and distance for reference from a point of origin.*Polygon*is a plane figure whose sides are line segments. They can be convex or non convex. Regular polygons have angles and sides all of equal measure.*Symmetry*- Symmetry or line symmetry is an identical relationship on two sides of a line drawn through a figure or parts of a figure. A figure has line symmetry if a part can be flipped about a line so both parts coincides. A mirror can be used to illustrate symmetry if placing the mirror on the line of symmetry the reflection is the same as the figure in front of the mirror. Symmetric parts are unchanged when reflected, rotated, or scaled.

*Tessellation*is a 2-dimensional pattern made from plane figures that fit together to fill the plane, leaving no spaces.*Triangle*is a two dimensional closed figure with three sides.- Perpendicular bisectors can be made by joining the vertices and folding between them.
- Midpoint can be found by folding all three perpendicular bisectors and finding their union, where they cross.

*Two-dimensional geometric figures*include: circle, square, triangle, rectangle, polygon, ...*Volume*is the amount of space an object occupies.- Pythagorean theorem a
^{2}+ b^{2}= c^{2}

## Advanced **concepts**

# Educator notes

See also Teacher resources

The Pythagorean theorem was known long before Pythagoras, but he may well have been the first to prove it. Chinese knew of it 600 years earlier than Pythagoras.

Measuring the hypotenuse of a right triangle led to the idea of square roots as a right triangle with two sides, each 1 unit will make the hypotenuse the square root of two. Which created a problem of how to solve for it. 1 2/5 , not good enough, 1 41/100 closer, 1 207/500 closer, but can't really get there. Which led to irrational numbers.

# Measurement

Misconceptions and concepts

## Initial perceptual naive **misconceptions** (any age)

(Explanations for people's misconceptions: naive understandings & perceptual responses)

- Units change in their value.
- Other people will know what unit I reference. (No understanding of a standard unit).
- Any unit is okay.
- Move unit and count. (No understanding of accuracy needed in placing unit(s) of measurement.
- It's possible for a measurement to be perfectly accurate.

## Beginning **concepts** (preschool - 7 years)

- Properties can be counted.
- Conservation of number - the number of objects does not change with the position of the objects.
- Conservation of length - length of an object does not change when its position is changed or its shape is altered by bending.
- Objects can be used to compare other objects.
- Measurement is a way of detecting change.
- Linear measurement is the distance between two points.
- Volume is the measurement of space an object occupies.
- Area measures the surface of an object.
- A standard unit of measurement helps communication.
- When the thermometer goes up the temperature is hotter.
- Measurement helps in making better observations.
- Scales measure mass and weight.
- Measuring cups measure volume
- Measurements can be compared.
- Measurement is used in everyday life (recipes, plans, designing, building)
- Time is the measurement of years divided into seconds, minutes, hours, days, weeks, months, decades, centuries.
- Mathematics can be used to solve problems with time.
- Time is communicated in standard units.
- The duration of an event from the beginning to the end is measured in time.
- Time represents past, present, and future events.

## Literate **concepts** (middle level to adult)

- Conservation of volume - volume of an object or substance does not change when the position and shape of the object or substance change position.
- Conservation of mass - mass of an object or substance does not change when the postion or shape of the object or substance changes.
- Conservation of area - area of a surface does not change when the position of the surface changes position.
- Properties and change of properties can be quantified.
- All measurement is relative to a unit, usually a standard unit.
- Scale is proportional.
- Measurement helps in making more accurate observations.
- Quantitative estimates of familiar lengths, weights, and time intervals can be confirmed by measurement.
- Measurement helps in making better observations.
- Rulers are used to measure linear measurement.
- Scales measure mass and weight.
- Measuring cups measure volume.
- Measurements can be compared.
- Measurement is used in everyday life (recipes, plans, designing, building).
- Rate is based on time.
- Standard units include:
- Linear measurement standard units are cm, m, km, inch, foot, yard, mile
- Volume- standard units include liter, ml, cup, pint, quart, gallon
- Time - measure in units of seconds, minutes, hours, days, weeks, months, years, decades, centuries. Time is used to order events. Twenty-four hours in a day, about 30 days in a month, 365 days in a year, 52 weeks in a year, 12 months in a year. Calendar is used to measure time. Clocks are used to measure time. Clocks can be analog and digital. A day is divided into daytime and nighttime. Time is determined by Earth's movement. Time is cyclic (seasons, days of weeks, months).
- Temperature measures hot and cold. Degrees in Celsius and Fahrenheit are standard units of temperature
- Mass - standard units are g, kg, pounds, ounces, tons

- Money is represented in increments of .01, .05, .1, .25, .5, 1.00, 2.00, 5.00 10.00, 20.00, 50.00, 100.00 (beats me from here)
- Interest is a percentage paid for the use of money.
- All measurement has error.

- All measurement is inaccurate and therfore an estimation.
- People choose how to estimate by the accuracy required.
- Estimation is an approximate selection of a unit of measurement and the number value used for the actual measurement (... .001, .01, .1, 0, 1, 10, 100, 1ooo, ...).

## Advanced **concepts**

*Scale*is a proportional relationship of characteristics, properties, or relationships within a system as its dimensions are increased or decreased.*Rate*involves a measure of change for a part relative to a whole (birth rate as part of population growth and comparing one measured quantity to another measured quantity (km per hour).

# Educator notes

See also teacher resources.

# Number value:

Whole numbers, Place value, fractions, decimals, percent, integers

# Whole number value

Misconceptions and concept

## Initial perceptual naive **misconceptions** (any age)

(Explanations for people's misconceptions: naive understandings & perceptual responses)

- Number words are like a rhyme onetwothreefourfivesix...
- Counting is saying the number words onetwothreefourfivesix...
- Counting is saying the number words and pointing

## Beginning **concepts** (preschool - 7 years)

- Certain words are used for counting.
- One-to-one correspondence
- Cardinality
- Conservation of number
- Hierarchical inclusion for counting
- Addends as cardinal values
- Counting on
- Ordinality
- Total hierarchical inclusion
- Count back

## Literate **concepts** (middle level to adult)

- Number notation
- Number relationships
- Simple one-to-one relationships
- One to many relationships
- Hierarchical inclusion for number value as addition
- Nominal numbers
- Infinity

## Advanced **concepts**

# Educator notes

- Prenumber sense development ( Classification, Counting, Synchrony, One-to-one correspondence, Counting systematically, Subitizing, Conservation of numbers, Number value or Cardinality, Zero, & numbers beyond 10)
- Number sense development (Cardinality, Ordinality, Number notation, Number relationships, Simple one-to-one relationships, One to many relationships, Hierarchical inclusion for counting, Hierarchical inclusion for number value as addition, Number value and operations, Zero, Beyond ten, Nominal numbers, Infinity)
- Counting (Includes same ideas as in prenumber sense and number sense development in one document)
- Place value development
- Activities
- Vocabulary
- Conservation and development of reasoning tasks

# Place value

Misconceptions and concept

## Initial perceptual naive **misconceptions** (any age)

(Explanations for people's misconceptions: naive understandings & perceptual responses)

## Beginning **concepts** (preschool - 7 years)

- Ten is a big number.
- Ten is a bigger number than 1, 2, 3, ... 9.
- Ten is less than 11, 12, ....
- Objects can be grouped by tens.
- Groups of ten are equivalent.
- Grouping by tens makes it easier to count.
- Objects can be grouped into tens and leftovers.
- Ten is a group of ten ones.
- Objects in groups of ten can be combined by skip counting by tens.
- Counting by tens is helpful.
- Counting by tens is like counting by ones with zeros.
- Objects can be grouped into tens when thinking of number value and to solve problems. (significance of ten)
- Numbers can be grouped as ten and more.

## Literate **concepts** (middle level to adult)

- Unitize - groups of ten are simultaneously both: equal groups of ten and the equivalent number of units or ones. For example thirty is simultaneously three equal groups of ten and thirty ones. In 30 the three means three groups of ten or thirty ones (plus zero additional ones represented by the zero). In 32 the three means three groups of ten or thirty ones plus two additional ones represented by the two.
- Place value unitizes by tens, hundreds, thousands, ...
- Numbers can be located as nearest to a selected number value (measuring, estimation, rounding).
- Rounding is an approximate estimation to a selected value (... .001, .01, .1, 0, 1, 10, 100, 1ooo, ...).
- Rounding is not a property of numbers but a property of a system of numbers used to represent them.
- Ten has a significant role in our base ten number system. Ten, hundred, and thousand are important multiples of this number system.
- A number system is a code that uses a set of symbols (numerals) and rules for combining the symbols to represent number values.
- All numbers can be represented with only ten symbols or numerals.
- Ten numbers can represent any unit (ones, tens, hundreds, …).
- Zero has an important function in writing numbers.
- Position of a digit can represent different values.
- The sum of the value of all digits determines the value of a numeral (cardinality) The total value of a numeral is the sum of all digits values in its particular place. A digits value is a function of its number value and it's place value.
- Numbers can be composed and decomposed into different equivalent groups of tens and more (taken apart (decomposed) and put together in different order (composed) or regrouped and written in expanded notation.
- Numbers of groups can be operated on without regard to the value of the group (62 - 41), (60 - 40) can be operated as (6 - 4) and (2 - 1) and write, think, or say 21 without really (multiplying by ten). This thinking about and working with groups is a result of unitizing.
- Addition facts can often use ten as an anchor (32 = 30 +2) for addition and subtraction. This is helpful when adding (27 + 35), (27 + 30) + 5 = (57 + 3) + 2 = 62 both for adding or subtracting to make tens, and adding and subtracting in leaps of ten.
- Place value is exponential: 10
^{0},10^{1},10^{2}, 10^{3}...

(50 = 5 * 10^{1}+0*10^{0},

51 = 5 * 10^{1}+1*10^{0},

52 = 5 * 10^{1}+2*10^{0, }200= 2*10^{2}+0*10^{1}+ 0*10^{0},

201=2*10^{2}+0*10^{1}+ 1*10^{0},

234=2*10^{2}+ 3*10^{1}+4*10^{0}) - Decimals as exponential: 10
^{-1},10^{-2}, 10^{-3} - Unitizing place values of decimal numbers, less than and greater than one, as an exponential progressions.
- Hindus are credited with the invention of a place value system.
- The Arabs are credited with applying and spreading its use.
- Translations of Fibonacci's work introduced it to Europe.

## Advanced **concepts**

# Educator notes

# Fractions

Misconceptions and concepts

## Initial perceptual naive **misconceptions** (any age)

(Explanations for people's misconceptions: naive understandings & perceptual responses)

- Fractions are pieces of a whole. They don't have to be equal size ...
- Fractions are pieces of things.
- All fractions are smaller than one.
- The bigger the denominator the bigger the fraction.

## Beginning **concepts** (preschool - 7 years)

- Fractions are equal parts.

## Literate **concepts** (middle level to adult)

- Fractions are part/whole relations
- Fraction is equal parts of a whole (one) or
- Fraction is equal parts of a set or group (of wholes or fractional parts) they are considered one whole.
- Human beings constructed fractions to deal with fair sharing situations.
- Fractional parts must be equivalent and they must be equivalent in relationship to the whole.
- The whole matters, fractions are relations.
- To compare fractions the whole must be the same
- Fractional parts don't need to be congruent to be equivalent.
- Can compensate: remove here, add there, as long as every loss has an equal gain.
- Fractions are division (three dollars shared with four people)
- Multiplication is connected to fractions (3/4 = 3 * 1/4)
- Multiplication and division or rational numbers are relations on relations
- If numerators are common only denominators matter when comparing
- If denominators are common only the numerators matter when comparing
- The bigger the denominator the smaller the piece
- Fractions represent division Fractions are equal parts of a whole, part, group, or set
- Equivalent fractions are equal representations of the same whole, part, group, or set
- Equivalent fractions represent different ways of describing the same amount using different sized fractional parts
- Fractions can represent parts and wholes
- Fractional names tell how many parts of equal size are needed at make a whole
- The more parts needed to make a whole, the smaller the parts
- The numerator tells the how many parts are represented and the denominator the kind or number of parts the numerator counts.
- Fractions can be added, subtracted, multiplied, and divided.
- Fractions, decimals, percent have a unique relationship.
- Interest is a percentage paid for the use of money.

## Advanced **concepts**

# Educator notes

*Fractions* as *equal parts* or a whole can be learned visually by students who haven't conserved area when the areas are identical in shape and area. Activities where students explore different shapes and discuss those which are identical and those which are not can be extended to shapes that are divided into parts. Squares, rectangles, and triangles and students are asked which are the same size and which are not. If the shapes are on paper they can be cut out and compared to see if the are equal. Those that are and can be used to complete a larger shape entirely can be identified as fractional parts (1/2, 1/3, & 1/4).

# Decimals

Misconceptions and concepts

## Initial perceptual naive **misconceptions** (any age)

(Explanations for people's misconceptions: naive understandings & perceptual responses)

## Beginning **concepts** (preschool - 7 years)

- Money is a decimal number

## Literate **concepts** (middle level to adult)

- Decimals can represent fractions
- Decimals can represent percents
- Decimals can represent ratios
- Decimals represent division
- Decimals are based on place value
- Fractions, decimals, percent have a unique relationship.
- Whenever a fraction has a denominator of ten the numerator represents tenths. 3/10 = .3, 53/10= 5.3…
- All 1/2 fractions are equal to .5
- Adding a zero to the divisor moves the decimal one place like it does when you divide or multiply whole numbers by 10 or 100…
- Adding decimals is like adding fractions with common denominators
- Dividing by 10, 100, 1000, gives an answer that has as many digits to the right of the decimal as zeros on the number you are dividing.
- It happens because you are dividing by ten…
- Base ten is infinite both for number values that increase and number values that decrease.
- The decimal point can reperesent "and" when reading decimal numbers.
- Decimal point locates the unit (ones) position.

## Advanced **concepts**

# Educator notes

# Percent

Misconceptions and concepts

## Initial perceptual naive **misconceptions** (any age)

(Explanations for people's misconceptions: naive understandings & perceptual responses)

## Beginning **concepts** (preschool - 7 years)

- Percent represents hundredths.
- Fractions, decimals, and percents can be changed back and forth to make problems easier to solve. (1/2, .5, .50, 50%; 10%, .1, .10, 1/10; 1/4, .25, 25%;…)
- Fractions, decimals, percent have a unique relationship.

## Literate **concepts** (middle level to adult)

## Advanced **concepts**

# Educator notes

# Integers

Misconceptions and concepts

## Initial perceptual naive **misconceptions** (any age)

(Explanations for people's misconceptions: naive understandings & perceptual responses)

## Beginning **concepts** (preschool - 7 years)

## Literate **concepts** (middle level to adult)

## Advanced **concepts**

# Educator notes

# Operations (+, -, *, /)

Misconceptions and concepts

## Initial perceptual naive **misconceptions **(any age)

(Explanations for people's misconceptions: naive understandings & perceptual responses)

*Beginning Concepts*

*Computation estimation concepts*

- Numbers can be taken apart and put together in a variety of ways.
- Estimation can be done by replacing or substituting difficult numbers with close or nice numbers and worked with mentally.
- There is no right way to estimate or compute mentally.
- Estimation is used not only for estimation but also when we use mental calculation, paper - pencil calculation, a computer, and a calculator.

## Literate concepts (middle level to adult)

*Computation estimation concepts*

- Some mental computation starts with the large digits and works toward the smaller (left to right). Paper and pencil computation is digit oriented and right handed.
- Rounding is not a property of numbers but a property of a system of numbers used to represent them.
- Rounding is an approximate estimation to a selected value (... .001, .01, .1, 0, 1, 10, 100, 1ooo, ...).

# Addition and subtraction

## Initial perceptual naive **misconceptions **(any age)

(Explanations for people's misconceptions: naive understandings & perceptual responses)

## Beginning **concepts** (preschool - 7 years)

### Whole Numbers

- There are four ways to apply addition and subtraction
- Combination of number values
- Separation of number values
- Part-part-whole relationships of number values
- Comparing or equalizing number values

- Different values of objects can be recognized by a pattern of their position (subitize).
- One more is addition.
- One less is subtraction.
- Two more is addition.
- Two less is subtraction.
- Can add by counting two separate sets of objects, sliding them together into one group, and then count the new group to find how many altogether.
- Two different groups of objects can be added by counting on from the total number in one group.
- Dice can be added by subitizing the first die and counting on.
- How many objects are left in a set of objects can be found by counting back from the total number in an initial group.
- Numbers can be decomposed into smaller groups (addends) and added together in different orders (commute them and compose them) to find the sums.
- Numbers greater than 20 can be added and subtracted by working left to right - decomposing into tens and ones, adding or subtracting tens, then adding or subtracting ones, and then adding or subtracting the tens and ones.

## Literate **concepts** (middle level to adult)

### Whole Numbers

- Addition is the joining of groups (sets).
- Addition can be used to solve joining problems.
- Addition can be used to solve separation problems.
- Addition can be used to solve part - part - whole problems.
- Addition can be used to solve comparison or equivalent problems.
- Subtraction is separation of a set from another set.
- Subtraction is the removal of a group.
- Subtraction is the joining of groups.
- Subtraction can be used to solve part-part-whole problems.
- Subtraction can be used to solve comparison and equalization problems.
- Subtraction is how much is left.
- Subtraction is how much is missing.
- Subtraction is how much more.
- A number can be represented as a difference with different groups of numbers (subtrahends and minuends)
- Addition and subtraction are related.
- Numbers can be operated on with addition and subtraction in different groups of numbers.
- Addition has the commutative property (3+4) = (4+3). The operation or addition has the same value regardelss of the order in which the numbers are added. Or if the values commute or change
*order*the addition operation will result in the same sum. - Addition has the
*associative property*, the sum of a set of numbers is independent of how they are*grouped or associated*. (1 + 2) + 3 = 1 + (2 + 3). - Addition has the
*identity property*for 0, zero added to any number is the value of any number, or the number itself. - Knowing the addition facts of ten helps to use ten as an anchor for addition and subtraction.
- Addition, subtraction, and comparing operations can be thought of as measurement or using a ruler.
- One more and one less is the value before and after a counting sequence.
- Two more and two less is the second value before and after a counting sequence.
- All numbers in a counting sequence can be thought of a distances relative to the amounts being added or subtracted.
- Two separate sets of objects can be added together by counting each set, sliding them together into one group, and then counting the new group to find how many altogether.
- The total number of objects in two separate sets of objects can be found by using the value of one set and counting on from that total number.
- The total number of objects left in a set of objects can be found by counting back from the total number in the initial group.
- Numbers can be added by decomposing numbers into smaller addends, commuting them and composing them with the different numbers. Example - two die with a roll of sixes, decompose them into 5, 5, 1, 1, commute and compose them like 5 + 5 = 10; 10 + 2 = 12.
- Values greater than 20 can be added or subtracted from left to right by decomposing into tens and ones, adding or subtracting tens, then adding or subtracting ones, and then adding or subtracting the tens and ones. For example: 46 + 23 by deconstructing 46 and 23 into 40 + 6 and 20 + 3, then adding the 40 and 20 and then the 6 and 3 and then the 60 and the 9 getting 69.
- Values greater than 20 can be added or subtracted from left to right by starting with an initial number: 46, deconstruct a second 23 to 20 + 3. Then add on from: 46 + 20 to get 66 and then add on the 3 to get 69.
- All addition and subtraction must account for the place value of each number as it is composed or decomposed to arrive at a sum or difference. This can be shown if numbers are decomposed into expanded notation, based on place value, and then each expanded number can be added or subtracted with another number with the same place value. If numbers can't be operated on, then the numbers can be regrouped into different multiples of ten as needed using expanded notation to check for accuracy.
- Addition and subtraction requires numbers be operated on with respect to their place values.
- There are many strategies to add and subtract.
- Making nice numbers (368 + 204 = 368 + 200 + 4 = 568 + 4=572).
- Keeping the whole (71 - 36 = {subtract 1 from both to get} 70 - 35 = 35) (342 - 37 = {add 3 to both to get} 345 - 40).

- Numbers in expanded notation can be added or subtracted if they have the same exponents. If exponents are different, numbers needed to be changed so they do have the same exponent value (place values).

### Fractions and decimals

## Advanced **concepts**

- There are no advanced concepts for addition and subtraction as they are operations of arithmetic. Not advanced mathematics.

# Educator notes

- Development of addition and subtraction

# Multiplication and division

## Initial perceptual naive **misconceptions **(any age)

(Explanations for people's misconceptions: naive understandings & perceptual responses)

## Beginning **concepts** (preschool - 7 years)

### Whole Numbers

- Multiplication is a kind of repeated addition.
- Skip counting is repeated addition.
- Multiplication is repeated addition of equal groups.

## Literate **concepts** (middle level to adult)

### Whole Numbers

*Multiplication*

- Multiples are products of unitized groups. Number of groups and units in the groups.
- A group of objects can be thought of as both the number of objects and as one group or unit.
- The parts (the number of objects in the group) and the whole (the group) can be considered simultaneously. A bag of six cookies can be both six cookies and one bag of cookies. Three threes = three groups of three. Or a number of groups with a number of objects. This same relationship is necessary to understand place value and exponents. If a person counts equal groups as units, then thinking of multiplication. If they skip count, then thinking as repeated addition.
- Multiplication is counting groups of like sizes and determining how many in all the groups.
- One number (factor) represents the number of groups and the other the number (factor) represents the number of objects in each group, and another is the total (product).
- Measurement, area, volume, The product is something different than the factors (square feet is the product of linear feet, cubic cm...). arrays, matrices.
- When the number of groups (factor) is known and how many objects (factor) are in each group is known, then the operation to find the total (product) is multiplication.
- Multiplication names a product in terms of two known factors.
- Digits of numbers must be multiplied according to their place values and for all places.

*Division*

- Division is can be thought of as repeated subtraction
- When either the number of groups or how many objects are in each group is known and the product of these numbers is known, then the operation is unknown as division.
- Fair - sharing or partition problems is when the size of the sets is unknown and the total is partitioned or shared with the known number of groups.
- Division is grouping in equal groups.
- Remainders can be represented by being discarded leaving the smallest whole, forced up to the next whole, round to the nearest whole for an approximate result, or written as a fraction.
- Measurement problem is when the number of sets is unknown and the size of the equal sets is known. The whole is measured into sets of the known size.
- Division names a missing factor in terms of the known factor and the product.
- Digits of numbers must be divided according to their place values.

*Properties related to multiplication and division*

- One number (factor) represents the number of groups and the other the number (factor) represents the number of objects in each group, and another is the total (product).
- Multiplication and division are related.
- Identity property of multiplication. Multiplication by one.
- Commutative property
- Associative property
- Distributable property
- Factors are divisors of their product
- Primes have only factors of one and the number
- Composites have more factors than one and the number

### Fractions and decimals

## Advanced **concepts**

# Educator notes

Development of Multiplication and Division Understanding and Strategies

# Problem solving

Misconceptions and concepts

## Initial perceptual naive **misconceptions** (any age)

(Explanations for people's misconceptions: naive understandings & perceptual responses)

- Smart people just know how solve problems.

**Concepts** apply to all levels. Adapt according to level.

Beginning (preschool - 7 years), Literate **concepts** (middle level to adult) & Advanced

- Problems are solved with a heuristic, a repertoire of strategies, metacognition or reflection, and persistence.
- A heuristic is a series of generalized steps.
- It is helpful to know and use a
*problem solving heuristic*to solve problems, think about what you know or have done and what you need to find out or do. - Problems can be solved with
*different strategies*. - Use
*manipulatives*to represent objects and actions in the problem. - Work a
*simpler*problem. *Trial and error*,*guess and check*.- Work
*backwards* - Use
*smaller*numbers - Use
*systematic*steps. - Look for, recognize and describe
*patterns*: quantity, AB/AB, ABBA/ABBA, size, area, volume, rotation, shading, shape, position, subtraction, addition, reflection, multiplication, analogy, and recursive - Break a problem into two related problems and solve the original problem in
*two steps*: one for each problem. *Act out*the problem. Physically or mentally.- Use a
*pictures*,*graphical**representation*-*model*, drawing picture or*diagram* - Problems can be solved with
*models*and*equations*. *Categorize*information to find relationships and patterns that will assist reasoning and proof.*Organize*data to look for patterns sequence, chart, table, making a graph, Venn diagrams, and dichotomous key.- Process of
*elimination*or process of*identification* *Write*an open sentence- Use
*algebraic reasoning* - Use
*logical reasoning*: matrices, deductive, inductive, truth tables *Brainstorming*- Use
*equivalent numbers*3/5, 6/10, 60/100, .6, 60% - It's good to monitor and reflect on the process of mathematical problem solving and regulate our actions.
- It's good to develop the habit and ability to
*monitor and regulate*our thinking processes and actions when solving problems. *Metacogniton*(self talk) and*group discussion*is helpful to talk through a problem to solve it and*reflect*on the accuracy of the process and solution.- The more problems I solve (
*persistence*) the easier it is to solve problems and use mathematics. - Some
*habits of mind*are more conducive for solving problems than others. *Algorithm*is a step by step list of instructions to solve a type of problems or task.

### Strategies for solving problems

# Educator notes

# Reasoning and Proof

Misconceptions and concepts

## Initial perceptual naive **misconceptions** (any age)

(Explanations for people's misconceptions: naive understandings & perceptual responses)

- Ideas just pop into your mind.
- Reasoning in math is knowing to +, -, *, or /.

**Concepts** apply to all levels. Adapt according to level.

Beginning (preschool - 7 years), Literate **concepts** (middle level to adult) & Advanced

- Reasoning begins with examination, comparison, and evaluation.
- Examination can include discovering similarities and differences to organize, categorize, seriate, make connections, create correspondences and create other structures. All which can be connected for deeper understanding and to solve problems.
- Organizing similar problems with their solutions may provide examples to use to generalize patterns, explanations, expressions to describe conditionw in which the mathematical ideas are valid.
- When expressions, equations, or procedures are deemed valid in enough specific situations, usually the next step is to see if the method is valid for all instances (infinite).
- A procedure or equation may be verified or proven if a reason can be given for every possible solution.
- Conjecture is a statement, which may be proven true or false. It may be based on an idea or conclusion with insufficient evidence or reason to prove. As such it can be a starting point for a proof.
- One counter example will disprove a conjecture.
- When explaining a proof a reason needs to be given for every change made.
- Pictures and charts can be used to help support reasoning and proof.
- Numbers can be compared by by thinking of their values on a balance (equal arm balance) and changing the numbers on either side until it is balanced.
- Numbers can be made equal by thinking of their values on a balance (equal arm balance) and changing the numbers on either side until it is balanced.
- Changes to those numbers are created with the mathematical operations.

# Educator notes

## Big Ideas to unpack & map - Source Science for All Americans

- Some aspects of reasoning have clear logical rules, others have only guidelines, and still others have almost unlimited room for creativity (and, of course, error). A convincing argument requires both true statements and valid connections among them. Yet formal logic concerns the validity of the connections among statements, not whether the statements are actually true. It is logically correct to argue that if all birds can fly and penguins are birds, then penguins can fly. But the conclusion is not true unless the premises are true: Do all birds really fly, and are penguins really birds?
*Examination of the truth of premises*is as important to good reasoning as the logic that operates on them is. In this case, because the logic is correct but the conclusion is false (penguins cannot fly), one or both of the premises must be false (not all birds can fly, and/or penguins are not birds). - Very complex logical arguments can be built from a small number of logical steps, which hang on precise use of the basic terms "if," "and," "or," and "not." For example, medical diagnosis involves branching chains of logic such as "If the patient has disease X or disease Y and also has laboratory result B, but does not have a history of C, then he or she should get treatment D." Such logical problem solving may require expert knowledge of many relationships, access to much data to feed into the relationships, and skill in deducing branching chains of logical operations. Because computers can store and retrieve large numbers of relationships and data and can rapidly perform long series of logical steps, they are being used increasingly to help experts solve complex problems that would otherwise be very difficult or impossible to solve. Not all logical problems, however, can be solved by computers.
- Logical connections can easily be distorted. For example, the proposition that all birds can fly does not imply logically that all creatures that can fly are birds. As obvious as this simple example may seem, distortion often occurs, particularly in emotionally charged situations. For example: All guilty prisoners refuse to testify against themselves; prisoner Smith refuses to testify against himself; therefore, prisoner Smith is guilty.
- Distortions in logic often result from not distinguishing between necessary conditions and sufficient conditions. A condition that is necessary for a consequence is always required but may not be enough in itself—being a U.S. citizen is necessary to be elected president, for example, but not sufficient. A condition that is sufficient for a consequence is enough by itself, but there may be other ways to arrive at the same consequence—winning the state lottery is sufficient for becoming rich, but there are other ways. A condition, however, may be both necessary and sufficient; for example, receiving a majority of the electoral vote is both necessary for becoming president and sufficient for doing so, because it is the only way.
- Logic has limited usefulness in finding solutions to many problems. Outside of abstract models, we often cannot establish with confidence either the truth of the premises or the logical connections between them. Precise logic requires that we can make declarations such as "If X is true, then Y is true also" (a barking dog does not bite), and "X is true" (Spot barks). Typically, however, all we know is that "if X is true, then Y is often true also" (a barking dog usually does not bite) and "X seems to be approximately true a lot of the time" (Spot usually barks). Commonly, therefore, strict logic has to be replaced by probabilities or other kinds of reasoning that lead to much less certain results—for example, to the claim that on average, rain will fall before evening on 70 percent of days that have morning weather conditions similar to today's.
- If we apply logical deduction to a general rule (all feathered creatures fly), we can produce a conclusion about a particular instance or class of instances (penguins fly). But where do the general rules come from? Often they are generalizations made from observations—finding a number of similar instances and guessing that what is true of them is true of all their class ("every feathered creature I have seen can fly, so perhaps all can"). Or a general rule may spring from the imagination, by no traceable means, with the hope that some observable aspects of phenomena can be shown to follow logically from it (example: "What if it were true that the sun is the center of motion for all the planets, including the earth? Could such a system produce the apparent motions in the sky?").
- Once a general rule has been hypothesized, by whatever means, logic serves in checking its validity. If a contrary instance is found (a feathered creature that cannot fly), the hypothesis is not true. On the other hand, the only way to prove logically that a general hypothesis about a class is true is to examine all possible instances (all birds), which is difficult in practice and sometimes impossible even in principle. So it is usually much easier to prove general hypotheses to be logically false than to prove them to be true. Computers now sometimes make it possible to demonstrate the truth of questionable mathematical generalizations convincingly, even if not to prove them, by testing enormous numbers of particular cases.
- Science can use deductive logic if general principles about phenomena have been hypothesized, but such logic cannot lead to those general principles. Scientific principles are usually arrived at by generalizing from a limited number of experiences—for instance, if all observed feathered creatures hatch from eggs, then perhaps all feathered creatures do. This is a very important kind of reasoning even if the number of observations is small (for example, being burned once by fire may be enough to make someone wary of fire for life). However, our natural tendency to generalize can also lead us astray. Getting sick the day after breaking a mirror may be enough to make someone afraid of broken mirrors for life. On a more sophisticated level, finding that several patients having the same symptoms recover after using a new drug may lead a doctor to generalize that all similar patients will recover by using it, even though recovery might have occurred just by chance.
- The human tendency to generalize has some subtle aspects. Once formed, generalities tend to influence people's perception and interpretation of events. Having made the generalization that the drug will help all patients having certain symptoms, for example, the doctor may be likely to interpret a patient's condition as having improved after taking the drug, even if that is doubtful. To prevent such biases in research, scientists commonly use a "blind" procedure in which the person observing or interpreting results is not the same person who controls the conditions (for instance, the doctor who judges the patient's condition does not know what specific treatment that patient received).
- Much of reasoning, and perhaps most of creative thought, involves not only logic but analogies. When one situation seems to resemble another in some way, we may believe that it resembles it in other ways too. For example, light spreads away from a source much as water waves spread from a disturbance, so perhaps light acts like water waves in other ways, such as producing interference patterns where waves cross (it does). Or, the sun is like a fire in that it produces heat and light, so perhaps it too involves burning fuel (in fact, it does not). The important point is that reasoning by analogy can suggest conclusions, but it can never prove them to be true.

# Representation

relates to
Relative position amd motion (science cross-cutting)

Misconceptions and concepts

## Initial perceptual naive **misconceptions **(any age)

(Explanations for people's misconceptions: naive understandings & perceptual responses)

- Objects are always relative to me.
- It doesn't matter if a person changes position, because people understand what I mean.

**Concepts** apply to all levels. Adapt according to level.

Beginning (preschool - 7 years), Literate **concepts** (middle level to adult) & Advanced

### Representation (also communication)

- Representations help organize, record, and communicate mathematical ideas.
- Representations help solve problems.
- Mathematical ideas are represented externally and internally.
- Mathematical ideas are represented with object and actions with those objects.
- The representation, null and zero are special representations of the lack of objects or ideas.
- All mathematical representations are connected to physical entities.
- When representing values graphically the use of a scale and units helps to visualize that representation and to do so more accurately and proportionally. For example the difference between six feet and ten feet is four. The difference between 72 inches and 120 inches is 48. How each are represented makes a difference as to communicating the equality or not.
- Tables, charts, and graphs can be used to represent objects and events.
- Graphs can be used to identify and represent relationships.

### Relative position

- An object's position can change.
- An object is located relative to a reference object.
- Objects can be located with different combinations of distances and directions from a singular point or multiple points. (one point as a reference object can be used to locate another point or object with a distance and direction.) (An object or point can be located from two know points with a distance and direction from one point and either a distance or a direction from the second point.)

### Relative motion

- Objects move in different ways (straight, crocked, circular, and back and forth).
- Objects move fast and slow.
- An object's motion can be described by tracing and measuring its position over time.
- Motion is relative to a reference point.
- Motion can be too fast or slow for people to see.
- Objects move steadily or change direction.
- Objects that vibrate is motion that is relative to itself. See sound.
- Motion of an object can be described by its position, direction of motion, and speed.
- Motion can be measured and represented on a graph.
- Motion and force are often related. See force.

# Educator notes

# Variables

Misconceptions and concepts

## Initial perceptual naive **misconceptions **(any age)

(Explanations for people's misconceptions: naive understandings & perceptual responses)

- Objects change because they want to or people wish they would.

## Beginning **concepts** (preschool - 7 years)

- Variable is a property that changes.
- Results can be changed by changing what and how objects interact.

## Literate **concepts** (middle level to adult)

- Properties can be made into variables by determining a range through which they can vary.
- Variables are conditions that change.
- Variables need to be controlled for an experiment to be a fair comparison.
- A controlled experiment is one with all the conditions (variables) the same except the one that is being tested.
- Identifying, selecting, controlling, and manipulating variables is central to experimentation.

## Advanced **concepts**

# Educator notes

Focus questiona.

- If x has only one solution, is it a variable?
- Are variables nouns or what?