A Metacognitive Approach to

Conceptual Understandingof Math

Systematically organizing math concepts with language in the context of a math conceptual framework through using

reflective mental processes that encourage transfer learning.

Overarching Goal for Reading/Learning Math in RDG 185 The learner will employ thinking reading strategies for developing competence in math.

Research: By learning in ways that develop competence in math, students can more readily apply math knowledge acquired in one context to another context; math becomes useful beyond the math course. If math is not learned using cognitive strategies that develop competence, math procedures are quickly forgotten. This also make learning later related math concepts easier.

The following slides introduce learning strategies for developing competence in math as one is learning.

How to Read to Learn Math

Understanding Mathematics

You understand a piece of mathematics if you can do all of the following:

•Explain mathematical concepts and facts in terms of simpler concepts and facts.

•Easily make logical connections between different facts and concepts

•Recognize the connection when you encounter something new (inside or outside of mathematics) that's close to the mathematics you understand.

•Identify the principles in the given piece of mathematics that make everything work. (i.e., you can see past the clutter.) (Alfeld)Conceptual Understanding (NYSED)

Conceptual understanding consists of those relationships constructed internally and connected to already existing ideas.

It involves the understanding of mathematical ideas and procedures and includes the knowledge of basic arithmetic facts.

Students use conceptual understanding of mathematics when they identify and apply principles, know and apply facts and definitions, and compare and contrast related concepts.

Knowledge learned with understanding provides a foundation for remembering or reconstructing mathematical facts and methods, for solving new and unfamiliar problems, and for generating new knowledge.

•understand facts and ideas in the context of a math conceptual framework

Identify or create the math conceptual framework

Goal: Developing Competence in Math

Construct, relate, and systematically organize the meaning of math concepts in the context of the conceptual framework

Predict: Our brains are structured to remember novel events that are unexpected. Because our brains are encoded to make and respond to predictions, they are particularly stimulated when they predict one effect and experience a different one.

•develop a deep foundation of factual knowledge,

Re-exposure with elaboration:Reflection•writing - summarizing•internal dialogue•inquiry questions

•organize knowledge in ways that facilitate retrieval and application

Mentally Mind Map:•by organizing facts and ideas under the conceptual framework

The Foundation of Reading to Learn Math

Research into human learning has found that in order to learn subject content in ways that

- enable the learner to later apply what they have learned in new situations (transfer) and

- be able to learn related information easier.

The goal of education and reading to learn is Transfer.

Developing competence requires the learner to

- develop a deep foundation of factual knowledge,

- understand facts and ideas in the context of a conceptual framework, and

- organize knowledge in ways that facilitate retrieval and application.

Its more than new dendrites

Working MemoryPrefrontal

Cortex

Application is the goal - transfer

Arithmetic

Changing Mixed Number to Improper fractions

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Common fractions, proper fractions, improper fractions, mixed numbers.How to place a fraction on the number line.How to change an improper fraction to a mixed number. How to change a mixed number to an improper fraction.

Axon

Neuron Ends

Cell Body

Dendrites

Myelin Sheath

Taking Control of Our Brains

Major Mental Processes:

- Construct Meaning - connect new information to prior knowledge

- Strengthen Dendrites and Speed Up Transmission - re-exposure to new facts and ideas with elaboration

Research has shown that, “Practice builds neurological connections and thickens the insulating myelin sheath necessary for fluency, chunking of information, brain efficiency, and deep learning,” (Hill, 2006)

fraction, whole unit,, numerator, denominator, whole number, denominator,, fraction bar, solution, properties of fractions, fractional part of a whole

Prefrontal Cortex

Working Memory

Working Memory: consists of the brain processes used for temporary storage and manipulation of information.

Prefrontal Cortex: executive function - is a set of mental processes that helps connect past experience with present action. People use it to perform activities such as planning, organizing, strategizing, paying attention to and remembering details, and managing time and space.

Reflection

Working MemoryPrefrontal Cortex

Usefulness (transfer): From Working Memory for Learning to Working Memory to Accomplish a Task

Social Inequality Stratification

Systems of Stratification

wealth, prestige, power ranking

slavery, castes, estates, class

income,wealth

Sociology

Social Interaction in Groups

•What do these facts or ideas have to do with the conceptual framework of subject at hand?

•How does this relate to the concepts preceding the new facts, ideas, or concepts?

•What do I already know about these new facts and ideas?

•Do I understand what I just read?

•Where do I think this is going?

•How is this like or different than what I already know?

Social Inequality

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Start here

Mind Mapping Math Knowledge, Procedures and Concepts Mind Mapping for a Procedure The name of the math procedure should be the center of the map (ex. Writing Mixed Numbers (2 ½) as an Improper Fraction).Each main branch off the center of the map should have printed on it a step in the procedure being learned using math language. (Ex. Multiply the denominator of the fraction by the whole number. Hint: use abbreviations)Off each main branch should be examples of the numbers and symbols representing the step being learned. (ex. 2 ½, write 2 X 2 = 4)Also, off the main branch should be a drawing of a concrete example representing the concept being learned. (ex. Draw 3 cookies being cut in half) Mind Mapping for a Concept The name of the main concepts in the reading selection should be in the center of the map (ex. Proper Fractions, Improper Fractions and Mixed Numbers)Each main branch off the center of the map should have printed on it new terminology (ex. Proper fraction)Off of each main branch should be examples of number representing new words. (ex. Proper fraction 2/3; Improper fraction 7/5; Mixed number 2 ¼Also off each branch should be a drawing of a concrete example representing the new terminology. (ex. For Mixed number, draw three pizzas and one ¼th slices of pizza for 3 ¼.

Strategy for developing a deep foundation of factual knowledge.

Create a mind map that organizes the math concepts (math vocabulary) with illustrations of each concept.

Example: Under “Definition of Fractions”, we encounter the following math vocabulary: whole number, fraction, numerator, denominator, fraction bar,

Definition of a

Fraction

Whole number

Number with no parts

Ex. 1 whole apple

fractionwhole divided in to equal parts

Ex. circle divided into 4 partsnumerator

denominator

fraction bar1/4

Contrary to popular belief, learning basic facts is not a prerequisite for creativethinking and problem solving-- it's the other way around.

Once you grasp the big concepts around a subject, good thinking will lead you to the important facts. (John Bransford)

Example: Under “Changing Mixed Number to an Improper Fraction”, we encounter the following math vocabulary (concepts), and operations (multiplication and addition) for the steps in procedure of making the change for 3 1/2; Bob has three and a half grapefruits and wants to invite friends over for a party. He wants to know how many friends he can invite if everyone including himself get a half a grapefruit?

Strategy for understanding facts and ideas in the context of a conceptual framework and organizing knowledge in ways that

facilitate retrieval and application.

Create a mind map that organizes the math concepts (math vocabulary) with illustrations of each concept.

Changing Mixed

Number to Improper Fraction

1. Multiply whole number by

denominator3 x 2 = 6 (6 is the product)

2. Add the result (product)

to the numerator

6 + 1 = 7 (7 is the sum)

3. Place the result

(sum) of step 2 over

the denominator

7/2

Using Rules of Consolidation When New Math Information is Found 1. Deliberately re-expose yourself to the information if you want to retrieve it later. 2. Deliberately re-expose yourself to the information more elaborately if you want the retrieval to be of higher quality. “More elaborately” means thinking, talking or writing about what was just read. Any mental activity in which the reader slows down and mentally tries to connect what they are reading to what they already know is elaboration. (It is very important to try and find real life examples in the text at this time.) 3. Deliberately re-expose yourself to the information more elaborately, and in fixed intervals, if you want the retrieval to be the most vivid it can be. (Medina) Fixed Time Intervals for Re-exposing and Elaborating •As the reader identifies what is important while reading, stop re-expose yourself to the information and elaborate on the it (have an internal dialogue, what do you already know about what you are reading, write about it (take notes in your own words), explain it to yourself out loud.•When you come to a new topic or paragraph, explain to yourself what you have just read; this is re-exposure to the information.•When you finish studying, take a few minutes to re-expose yourself to the information and elaborate.•Within 90 minutes to 2 hours, re-expose yourself to the information and elaborate.•Review again the next day as soon as you can

Internal Dialogue with Math Concepts and Procedures It is important that math students learn to have an ongoing internal dialogue (mental conversation with themselves) as they are learning mew math concepts and procedures. It is common for math students to passively watch instructors work problems on the board and mimic what they saw while doing their homework. There is almost no way for these students to actually learn math in any way that ensures application in the future.

Re-expose elaborately

in Time Intervals

1. When you have read a new topic or paragraph, explain to

yourself what you have just

read.

2. When you finish studying,

take a few minutes to re-

expose yourself to the

information and elaborate.

3. Within 90 minutes to 2

hours, re-expose yourself to the

information and elaborate.

Assuming that the learners know the arithmetic operations, re-exposure of math language that strengthens dendrites and myelination (learning and speed of transmission) for procedures.

Instead of working 30 homework problems by focusing on procedures (which is needed, but not sufficient). For example for changing 3 1/2 to an improper fraction, the learner usually multiplies 2 time 3 and adds the answer to 1 and places it over 2 to get the improper fraction. Instead, take advantage of what we know about how the brain learns and strengthens what it has learned. Since the learner already know how to multiply and add, work every problem by using the language:

“I have three and a half grapefruits; how many half are there? I am going to change a mixed number (3 1/2) to an improper fraction. I am first going to multiply the denominator by the whole number to see how many halves there are in the three grapefruits and I am going to add the product to the numerator to see how many halves I have in all. If I place the sum over the denominator, I will have an improper fraction.”

Do this for every problem to learn the language.