Math language pdf

Every Content Instructor Should be a Reading to Learn Instructor: !

Including Math Instructors Dan Kesterson

Getting through math and learning math do not necessarily overlap

Why Co-requiste Reading with Math?

Learning math is not just about learning procedures. In fact, procedural memory is not stored in the prefrontal cortex, which limits its usefulness in future application in new contexts. !

Communication Standard !

Mathematics programs should enable students to— !

• organize and consolidate their mathematical thinking through communication: !

• communicate their mathematical thinking coherently and clearly to peers, teachers, and others; !

• analyze and evaluate the mathematical thinking and strategies of others; !

• use the language of mathematics to express mathematical ideas precisely. (Reading and Writing - Edwards)

Some Misconceptions About Learning !

Memorizing math vocabulary lists is an efficient method of learning math language. !

Hearing new math language results in learning new math language. !

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It takes 30 or 40 re-exposures to new vocabulary in different context before before the dendrites and myelination are strong enough to not be easily forgotten. !

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Memorizing is not about if the information will be forgotten, it is a matter of when. !

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Memorizing does not route the information to the prefrontal cortex !

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Memorizing sends the information in a phonological loop and is easily forgotten.

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Some Misconceptions About Learning !

Focusing on reading math is of secondary importance. !

Reading math is of primary importance if one wants to be able to • organize and consolidate their mathematical thinking through communication:

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communicate their mathematical thinking coherently and clearly to peers, teachers, and others;

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analyze and evaluate the mathematical thinking and strategies of others;

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use the language of mathematics to express mathematical ideas precisely. (Reading and Writing - Edwards)

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Some Misconceptions About Learning !

Most of our college learners know how to reflect; it does not have to be learned. !

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Reflection activates the prefrontal cortex. !

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The prefrontal cortex is the seat of executive functioning - planning, problem solving, prioritizing, decision making, etc. !

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Reflecting (stepping back in ones mind and considering how what is being learned relates to prior knowledge, predicting where this is new information can be used, and how the new information is related to the overarching conceptual framework) has to be taught. !

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Most 18 to mid-thirties learners are still developing their prefrontal cortex and have not learned how to route new learning through their prefrontal cortex, which requires reflecting. Years of procedural learning has built neural networks of shallow learning strategies for math that have to be overridden if the learner is not going to default back to less productive learning procedures.

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Some Misconceptions About Learning !

The math learner must learn how to do math procedures before they learn math language. !

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The brain stores meaning - meaning is the result of interconnecting new information with prior knowledge, which only becomes useful when it is understood in the context of a conceptual framework (systematically organized concepts) of prior knowledge. !

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Learning procedures without the language to categorize that being learned is very shallow learning.

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Some Misconceptions About Learning !

There is not enough time to focus on math language and procedures. ! •

Time always is a problem; however, one should not assume that much deep learning is occurring if math language is not taught along with procedures. !

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Language exponentially increases the ability to categorize information and information that is not categorized challenges the limitations of working memory. !

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Learning math language along with procedures makes later related learning easier.

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Some Misconceptions About Learning !

There is not enough time to focus on math language and procedures. ! •

Time always is a problem; however, one should not assume that much deep learning is occurring if math language is not taught along with procedures. !

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Language exponentially increases the ability to categorize information and information that is not categorized challenges the limitations of working memory. !

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Learning math language along with procedures makes later related learning easier.

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Some Misconceptions About Learning !

Students should have learned how to reflect and use math language in high school. ! •

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Many math instructors are not taught most of what follows or how to teach math using what is known about how deep learning occurs nor are most content instructors.

Much of the underpinnings of deep learning are relatively new - such as myelination and what it takes to increase the speed of transmission and processing of information up to 3,000 time faster enabling the learner to have access to information in working memory faster for manipulation.

Most learners find how the brain learns and their role in in it fascinating and are relieved do learn that they have math minds and simply have not learned how to take advantage of how the brain learns when learning math.

Let’s start with working memory. Think of working memory as very limited temporary storage of new information where it can be manipulated.

Working Memory

Working memory is very limited; it can only hold around 4 unrelated items for about 10 - 15 seconds - not enough for critical thinking.

“If we are unable to attend to the information in our working memory, the information lasts only as long as the neurons that hold and maintain their electric charge—a few seconds…. Then it's gone…” (Carr, 2010, p.193) !

If we don’t attend and reflect, we don’t learn anything new in ways that have future application. Learning math language is key. However learning math language alone, just as learning math procedures alone is not sufficient.

Working Memory, Organizing Information, and Language

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1. Working memory -- temporary retention of information, sensory or other, for the performance of a prospective act to solve a problem or to attain a goal. (Fuster; Prefrontal Cortex 138)

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It is in working memory that new information being learned is temporarily stored and manipulated.

Because of the very limited nature of working memory (4 items for 10-15 seconds), the learner must learn how to mentally expand the amount of information and time they have access to in working memory at any given moment or the information is lost. This requires learning how to reflect in the learning process.

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2. Chunking (categorizing information ) — information that can be coherently organized can be treated as a single element (one of the four items) in working memory. (Baars; Essential Sources, 322)

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In other words, by coherently organizing new information being learned (interconnected with related prior knowledge), the learner has access to that coherently organized information in working memory, thus overriding the item and time limitations of working memory. This vastly increases the amount of information the learner has access to in working memory when mentally manipulating information.

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3. One reason human cognition is so powerful is because we have language in our brains, which exponentially increases the ability to categorize information. (LeDoux; Synaptic Self, 177)

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Since coherently organizing new information greatly increases the amount of information the learner has access to to manipulate at any given moment and language in our brains exponentially increases the ability to categorize information, learning math language while learning math procedures is essential to deep learning of math concepts.

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Working Memory, Reflection, and the Prefrontal Cortex

1. Higher cognitive functions for which the prefrontal cortex is essential -- language, intelligence, and reasoning do not reach full maturity until the third decade of life.

(Fuster; Prefrontal Cortex)

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Math learners from ages 18 to mid thirties are still developing their prefrontal cortex, which is essential for higher functioning such as planning, prioritizing, decision making, problem solving, etc.

2. Planning and decision making are two of the major executive functions of the prefrontal cortex.(Fuster; Prefrontal Cortex 4) 3. Procedural memory is stored in other structures, not in prefrontal cortex. (Fuster; Prefrontal Cortex 344)

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4. Organism can produce no sustained goal-directed action, except the most automatic and routine, without the temporal integration functions of the prefrontal cortex. (Fuster; Prefrontal Cortex 335)

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Learning procedures alone does not route that being learned through the prefrontal cortex nor when being retrieved does it pass through the prefrontal cortex and its temporal integration functions for planning, prioritizing, decision making, problem solving, etc. (executive functions).

It is reflection, an internal conversation with oneself or others, that activates and routes new learning through the prefrontal cortex. Without learning math language, working memory is limited and the executive functions of the prefrontal cortex do not come into play.

(Never underestimate the power or writing to clarify and organize information as a reflection process.)

Core Mental Processes for Co-requisite Transfer Learning of Math Course Content and Procedures ! ! ! ! ! ! ! ! ! !

•calibration

of risk and reward,

•problem-solving,

3. reflecting; an internal conversation

•prioritizing, •thinking

ahead,

•self-evaluation, •long-term

planning

First, the learner needs to be !

•holding

2. stepping back mentally

the new information in mind,

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•stepping

back mentally,

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•and

reflecting.

Working Memory 1. holding the new information in mind

Fraction is a way of representing division of a 'whole' into ‘parts'.

It has the form

where the

frac tion

ol h W

b ar

Numerator is the number of parts chosen

and the

Denominator is the total number of parts.

d

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calibration of risk and reward, problem-solving, prioritizing, thinking ahead, self-evaluation, long-term planning

ator n i m eno

er parts b o n h t i w um Number n e Ex. 1 whole apple

ti frac

on

whole divided in to equal Ex. c parts ir into cle divid 4 pa ed r ts

Definition of a Fraction numerator

er b y b m u n e l o wh 1. Multiply tor 3 x 2 = 6 (6 is the denomina produ ct)

result e h t e c 3. Pla ver o 2 p e t fs (sum) o inator m o n e d the 7/2

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calibration of risk and reward, problem-solving, prioritizing, thinking ahead, self-evaluation, long-term planning

Changing Mixed Number to Improper Fraction

ct) u d ro

(p t l u r es r e h t a to d r e d 2. A e num h to t 6 + 1 = 7 (7 is the sum)

It is important that the information being learned is routed through the hippocampus and on to the prefrontal cortex if the learner wants to be able to use the information for critical thinking and problem solving. This requires that the learner stop, step back mentally and reflect. This seemingly simple mental task is the most challenging of the processes for learning in ways that transfer.

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calibration of risk and reward, problem-solving, prioritizing, thinking ahead, self-evaluation, long-term planning

Reflecting is the foundation of deep learning. The brain can store information all over the place; therefore, it is crucial that the learner route new information through the prefrontal cortex (seat of executive thinking). It is in the prefrontal cortex that planning, prioritizing, decision-making, problem-solving, and application in new situations takes place.

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calibration of risk and reward, problem-solving, prioritizing, thinking ahead, self-evaluation, long-term planning

Learners 18 years old to their midthirties have not fully developed their prefrontal cortex and these reflecting processes help to develop thinking with the prefrontal cortex.

So far, for transfer learning to occur, the learner has to create and over learn a conceptual framework for the subject at hand; they have deliberately connect new information with prior knowledge through the proess of reflection - stopping, stepping back, and reflecting on what they are learning if the information is to route through the prefrontal cortex.

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calibration of risk and reward, problem-solving, prioritizing, thinking ahead, self-evaluation, long-term planning

Reflecting is also important because how new information is encoded (neural pathways it takes is also the neural pathways for retrieval.

Questioning in Reflection

KEY: Does this fact, idea, concept fit within the conceptual framework?

Can I predict where this is going?

What do I already know?

Internal Dialogue Questions

Are there examples, explanations , or illustrations of the concepts?

How is what I am reading like or different that what I already know?

Conceptual Understanding !

Developing Competence in Math and Transfer Learning !

KEY: Math rules and procedures, facts and ideas must be understood in the context of a conceptual framework if they are to become useful - applicable in new situations and make related learning easier. !

It is with math language that the overarching conceptual frameworks of math topics must be constructed because math language exponentially increases the ability to categorize information. !

Math rules and procedures which are not categorized by the brain limits the amount of information working memory can temporarily store and manipulate.

Transfer Learning

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Procedural memory is stored in other structures, not in prefrontal cortex.

(Fuster; Prefrontal Cortex 344) !

If the learner wants to learn in ways that transfer to new contexts, the learner needs !

• Develop a deep foundation of factual knowledge; in order to do so, the learner must !

• Understand facts and ideas, rules and procedures in the context of a conceptual framework and !

• Organize knowledge in ways that facilitate retrieval and application.

(Bransfrod, 2001) !

When the mental processes for deep and transfer learning that are present in this presentation, the learner increase their chances of being able to apply what they are learning in new situations, retrieve and construct new meaning easier. !

Higher cognitive functions for which the prefrontal cortex is essential -language, intelligence, and reasoning do not reach full maturity until the third decade of life. (Fuster; Prefrontal Cortex)

Second, within the process of reflecting, the learner needs to ensure that they are applying mental processes that result in !

- Understanding facts and ideas in the context of a conceptual framework !

- Organizing knowledge in ways that facilitate retrieval and application !

- Developing a deep foundation of factual knowledge

“A key finding in the learning and transfer literature is that organizing information into a conceptual framework allows for greater “transfer”; that is, it allows the student to apply what was learned in new situations and to learn related information more quickly.” (Bransford)

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Working Memory

Math Language and Transfer Learning

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Understanding the Language of Math and Reflection is but Part of Conceptual Understanding of Math

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Contrary to popular belief, John Bransford, learning basic facts is not a prerequisite for creative thinking 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) One can also add - learning procedures alone for working math problems are not a prerequisite for creative thinking or 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).

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For transfer learning (apply that being learned in new contexts) the learner needs to understand language, rules, and procedures in the context of a conceptual framework.

A conceptual framework is made up of systematically organized concepts.

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Third, re-expose oneself with elaboration by stepping back and reexposing ones self to the relationship between the new information and the the conceptual framework.

Every time the learner steps back and re-exposes themselves to the relationship between the new information and the the conceptual framework, a new layer of myelin wraps around the axon increasing the speed of transmission, thereby bringing more to working memory and faster.

v v

Working Memory

Faster and Easier Retrieval is Vital to Application !

The yellow below on the brain cell’s axon is myelin - a thin layer of fat. It acts as insulation for the electrical signal that is carrying the information being learned.

New Facts and Ideas

Dendrites of Prior Knowledge

New Learning

Faster and Easier Retrieval Vital to Application !

Once the learner has reflected on what they already know and predicted what is coming next and especially figured out how new information is related to the conceptual framework, the real work of increasing the myelin around the axon to increase the transmission, processing and retrieval of that learned.

New Facts and Ideas product

profit

Dendrites of Prior Knowledge

owner/ finance satisfying customer needs

New Learning

marketing/ customers

Goal of Business management/ employees

Faster and Easier Retrieval Vital to Application !

Once the learner has reflected on what they already know, predicted what is coming next, and especially figured out how new information is related to the conceptual framework, the real work of increasing the myelin around the axon to strengthen the learning, as well as increasing the transmission, processing and retrieval of that learned.

Re-exposure with Elaboration product

profit

Dendrites of Prior Knowledge

owner/ finance satisfying customer needs

New Learning

marketing/ customers

Goal of Business management/ employees

Faster and Easier Retrieval Vital to Application !

Every time the learner steps back and re-exposes themselves to the relationship between the new information and the the conceptual framework, a new layer of myelin wraps around the axon increasing the speed of transmission. Consider this: With repeated re-exposure with elaboration, the electrical signal can get 100 times faster.

Re-exposure with elaboration

With repeated re-exposure with elaboration, the rest period at the synapse is 35% shorter.

This results in a 3,000 times faster transmission of the electrical signal.

100 times faster 3,000 times faster transmission

profit 35% shorter satisfying customer needs

New Learning

marketing/ customers

Goal of Business

owner/ finance

product management/ employees

Faster and Easier Retrieval Vital to Application !

With repeated re-exposure with elaboration, the electrical signal can get 100 times faster.

Re-exposure with elaboration

With repeated re-exposure with elaboration, the rest period at the synapse is 35% shorter.

This results in a 3,000 times faster transmission of the electrical signal.

100 times faster 3,000 times faster transmission

profit 35% shorter satisfying customer needs

New Learning

marketing/ customers

Goal of Business

owner/ finance

product management/ employees

Think of it this way: when the learner needs to recall the information or make a decision, or solve a problem with the information, the information gets to working memory 3,000 times faster, as well as brings the neural network of information that has been learned in the context of the conceptual framework. This is huge for transfer learning - becoming useful and increasing potential for application.

Working memory is not limited to 4 unrelated items if the information is organized - then it is limitless.