1. Co-requisite Reading in Math The goal of co-requisite reading in math is to enable the learner to develop competence in math by learning mental processes and habits of mind that result in learning that is transferable to new situations and future learning.

2. 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

3. Axon Neuron Ends Cell Body Dendrites Myelin Sheath Working Memory Prefrontal Cortex Taking Control of Our Brains

4. Working Memory First, you read or hear a new rule, MULTIPLE PROCESSES RULE When solving for an unknown that involves more than one process, do the addition and subtraction before the multiplication and division. Strategies 1. Think about it, stop and reflect on it immediately. 2. Repeat it. 3. Think elaborately; wrap your experience with linear equations around the new information to make connections, which makes the learning more meaningful. 4. Image these new connections 5. Organize these connections

5. Working Memory As you read or hear a new rule, MULTIPLE PROCESSES RULE When solving for an unknown that involves more than one process, do the addition and subtraction before the multiplication and division. Strategies 1. Think about the rule, stop and reflect on it immediately - NOT LATER! Note: Reflecting moves the information to the prefrontal cortex, where it can be used in the thinking needed to solve problems. First reflection, What does the multiple process rule mean? What does solving for an unknown mean in a linear equation? What is an unknown? What is a linear equation with more than one process? What are the processes? What is the order of processes in multiple process rule?

6. Working Memory As you read or hear a new rule, MULTIPLE PROCESSES RULE When solving for an unknown that involves more than one process, do the addition and subtraction before the multiplication and division. Strategies 2. Repeat it Note: Repeating it strengthens dendrites and builds myelin on the axon making future transmission and processing faster (up to 300 times). Second repeat reflection, What does the multiple process rule mean? What does solving for an unknown mean in a linear equation? What is an unknown? What is a linear equation with more than one process? What are the processes? What is the order of processes in multiple process rule?

7. Working Memory As you read or hear a new rule, MULTIPLE PROCESSES RULE When solving for an unknown that involves more than one process, do the addition and subtraction before the multiplication and division. Strategies 3. Think elaborately; wrap your experience with linear equations around the new information to make connections, which makes the learning more meaningful. Note: Hopefully, you have been developing a conceptual framework for what you have been learning about linear equation, such as solving for an unknown with a single power, Third, bring all the interrelated concepts you have been learning about linear equations to the new information (multiple process rule), such as solving for the unknown rule, opposite process rule, and etc. Continue to have an internal conversation about what you are learning.

8. Working Memory As you read or hear a new rule, MULTIPLE PROCESSES RULE When solving for an unknown that involves more than one process, do the addition and subtraction before the multiplication and division. Strategies 4. Image these new connections We are built for images. We need to take advantage of that. Think about things in images, write things down that way. Note: We are built for images. We need to take advantage of that. Think about things in images, write things down that way. Fourth, image in you head a simple linear equation with multiple processes and do the processes in your head with imagery.

9. Working Memory As you read or hear a new rule, MULTIPLE PROCESSES RULE When solving for an unknown that involves more than one process, do the addition and subtraction before the multiplication and division. Strategies 5. Organize what you are learning. Add to the conceptual framework that you have been building by systematically organizing the the concepts you have been learning about linear equations. Note: Add to the conceptual framework that you have been building by systematically organizing the the concepts you have been learning about linear equations. Fifth, try mind mapping new and related concepts at first to get used to systematically organizing new information.

10. Axon Neuron Ends Cell Body Dendrites Myelin Sheath Learning: growing new dendrites. 1st Fact: In order to understand and learn anything, new learning must interconnect with what the learner already knows. If this occurs, learning also occurs and new dendrites grow.

11. Axon Neuron Ends Cell Body Dendrites Myelin Sheath Learning: strengthening newly grown dendrites. 2nd Fact: Re-exposure to new learning strengthens the newly grown dendrites. If they are not strengthened, they will be reabsorbed - forgetting.

12. Axon Neuron Ends Cell Body Dendrites Myelin Sheath Learning: speeding up the transmission of newly learned information 3rd Fact: The yellow fat around the axon is insulation. Every time the learner re-exposes themselves to the newly learned information, the myelin gets thicker and the speed of transmission can get over 300 times faster with re-exposure.

13. Prefrontal Cortex Learning: It makes a difference where new information is stored in the brain. 4th Fact: Newly learned information must be stored in the prefrontal cortex if it is to become useful. When encountering new information that needs to be learned, it is imperative that the learner stop and reflect on what they are learning.

14. Prefrontal Cortex Learning: Reflection is important for later critical thinking: Making info useful. 5th Fact: Reflection isn’t just looking back at the information; it is having and internal conversation or dialogue with ourself about what is being learned.

15. Axon Neuron Ends Cell Body Dendrites Myelin Sheath Prefrontal Cortex 6th Fact: Reflection not only moves the information to the prefrontal cortex where it becomes useful, it also taps a lot of prior knowledge from all over the brain building neural networks of interrelated information. Learning: Building neural networks of related information.

16. Working Memory Learning: Working memory is a system in the brain for holding & manipulating what is in conscious awareness while reading. 7th Fact: As the learner is reading, working memory only holds 4 unrelated items for about 20 secs before it starts to forget.

17. Working Memory Learning: Working memory is a system in the brain for holding & manipulating what is in conscious awareness while reading. 7th Fact: As the learner is reading, working memory only holds 4 unrelated items for about 20 secs before it starts to forget.

18. Summing Up Facts 1-8, 1st Fact: In order to understand and learn anything, new learning must interconnect with what the learner already knows. If this occurs, learning also occurs and new dendrites grow. 2nd Fact: Re-exposure to new learning strengthens the newly grown dendrites. If they are not strengthened, they will be reabsorbed - forgetting. 3rd Fact: The yellow fat around the axon is insulation. Every time the learner re- exposes themselves to the newly learned information, the myelin gets thicker and the speed of transmission can get over 300 times faster with re-exposure. 4th Fact: Newly learned information must be stored in the prefrontal cortex if it is to become useful. When encountering new information that needs to be learned, it is imperative that the learner stop and reflect on what they are learning.

19. Summing Up Facts 1-8, 5th Fact: Reflection isn’t just looking back at the information; it is having and internal conversation or dialogue with ourself about what is being learned. 6th Fact: Reflection not only moves the information to the prefrontal cortex where it becomes useful, it also taps a lot of prior knowledge from all over the brain building neural networks of interrelated information. 7th Fact: As the learner is reading, working memory only holds 4 unrelated items for about 20 secs before it starts to forget. 8th Fact: Building neural networks by reflecting increases what can be brought to working memory.

20. Contrary to popular belief, 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)

21. Challenge: math instruction tends to focus on basic facts and procedures; the shift needed is a focus on the big concepts and interrelate the facts and procedures to the big concepts.

22. The goal is to connect equation in one unknown, concept being learned, to prior knowledge (equation and variable) and grown a new dendrite (learning) The black dendrites represent prior knowledge New Concept: equation in one unknown equation variable

23. Re-exposure build myelin in the axon and speeds processing transmission The black dendrites represent prior knowledge The newly grown dendrite represents learning.

24. Working Memory Learning: Working memory is a system in the brain for holding & manipulating what is in conscious awareness while reading. As the learner is reading, working memory only holds 4 unrelated items for about 20 secs before it starts to forget.

25. Working Memory As the learner is reading, working memory only holds 4 unrelated items for about 15 secs before it starts to be forgotten. To solve an equation, you must find a number that can replace the unknown in the equation. Prior Knowledge New Knowledge equation solve an equation variable; one variable unknown is represented by a variable

26. Consolidate and organize prior knowledge in long-term memory equation If an equation contains only one variable, it is called an equation in one unknown. If this unknown is in the first power, the equation is called a linear equation. Prior Knowledge New Knowledge equation equation in one unknown variable linear equation unknown is represented by a variable first power variable unknown solve an equation

27. Working Memory As the learner is reading, working memory only holds 4 unrelated items for about 15 secs before it starts to be forgotten. If an equation contains only one variable, it is called an equation in one unknown. If this unknown is in the first power, the equation is called a linear equation. Prior Knowledge New Knowledge equation equation in one unknown variable; one variable linear equation unknown is represented by a variable first power

28. Consolidate and organize prior knowledge in long-term memory equation If an equation contains only one variable, it is called an equation in one unknown. If this unknown is in the first power, the equation is called a linear equation. Prior Knowledge New Knowledge equation equation in one unknown variable linear equation unknown is represented by a variable first power variable unknown first power equation in one unknown linear equation

29. Consolidate and organize prior knowledge in long-term memory Equation addition property of equation multiplication property of equation subtract both sides divide both sides What am I solving for? How do I get what I am solving for on the left side? Look for addition, then subtraction, then multiplication, then division? Perform the inverse operation on both sides? Learning and Consolidating new information that must become strong and rapidly accessible prior knowledge requires re-exposure with elaboration within a conceptual framework. Examples: the properties of an equation; basic steps for solving an equation. Solving for the Unknown Rule Whatever you do to one side of an equation, you must do to the other side.

30. ReflectionNo Reflection The prefrontal cortex is the seat of “executive function” - problem-solving, decision-making, planning, application Prefrontal cortex Note: reflection takes place when the learner stops and has in internal conversation about what is being learned. Note: the deepest reflection involves elaboration and understanding in the context of a conceptual framework

31. Why learn the language of math? Language labels concepts so that more can be brought to working memory at any given moment. Math terminology is a holder of related math concepts. Math terminology help the brain organize facts and ideas. Working memory is limited to 4 unrelated ideas fro 10 to 15 seconds. Working memory is unlimited for organized facts and ideas. Language is like a holder for those realted concepts.

32. Working Memory Prefrontal Cortex Reflection to understand in the context of a conceptual framework Equation addition property of equation multiplication property of equation subtract both sides Solve equation: isolate variable What am I solving for? How do I get what I am solving for on the left side? Look for addition, then subtraction, then multiplication, then division? Perform the inverse operation on both sides? Solve the formula P = 2L + 2W for L Start here divide both sides What am I solving for? How do I get what I am solving for on the left side? Look for addition, then subtraction, then multiplication, then division? Perform the inverse operation on both sides?

33. The MAT 55/65 Challenge After assessment, the assumption is that the borderline MAT 55 learner doesn’t now some of the 55 concepts and/or procedures and that they can learn these concepts before and as needed which taking MAT 65. THE CHALLENGE: Learners tend to learn math concepts and procedures as separate or isolates items - hinders transfer learning. For reading, the challenge is helping the learner learn concepts and procedures deeply enough using mental processes and habits of mind that it becomes readily assessable prior knowledge that has been understood in the context of a conceptual framework and organized in ways that facilitate retrieval and application.

34. Working Memory Prefrontal Cortex Reflection to understand in the context of a conceptual framework Equation addition property of equation multiplication property of equation subtract both sides Solve equation: isolate variable What am I solving for? How do I get what I am solving for on the left side? Look for addition, then subtraction, then multiplication, then division? Perform the inverse operation on both sides? Solving for the Unknown Rule Start here divide both sides What am I solving for? How do I get what I am solving for on the left side? Look for addition, then subtraction, then multiplication, then division? Perform the inverse operation on both sides? SOLVING FOR THE UNKNOWN RULE Whatever you do to one side of an equation, you must do to the other side.

35. Working Memory Prefrontal Cortex Reflection to understand in the context of a conceptual framework Equation addition property of equation multiplication property of equation subtract both sides Solve equation: isolate variable What am I solving for? How do I get what I am solving for on the left side? Look for addition, then subtraction, then multiplication, then division? Perform the inverse operation on both sides? OPPOSITE PROCESS RULE Start here divide both sides What am I solving for? How do I get what I am solving for on the left side? Look for addition, then subtraction, then multiplication, then division? Perform the inverse operation on both sides? OPPOSITE PROCESS RULE If an equation indicates a process such as addition, subtraction, multiplication, or division, solve for the unknown or variable by using the opposite process.

36. Working Memory Prefrontal Cortex Reflection to understand in the context of a conceptual framework Equation addition property of equation multiplication property of equation subtract both sides Solve equation: isolate variable What am I solving for? How do I get what I am solving for on the left side? Look for addition, then subtraction, then multiplication, then division? Perform the inverse operation on both sides? MULTIPLE PROCESSES RULE Start here divide both sides What am I solving for? How do I get what I am solving for on the left side? Look for addition, then subtraction, then multiplication, then division? Perform the inverse operation on both sides? MULTIPLE PROCESSES RULE When solving for an unknown that involves more than one process, do the addition and subtraction before the multiplication and division.

37. 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.

38. 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.

39. develop a deep foundation of factual knowledge, Re-exposure with elaboration: Reflection writing - summarizing internal dialogue inquiry questions

40. organize knowledge in ways that facilitate retrieval and application Mentally Mind Map: by organizing facts and ideas under the conceptual framework

41. 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 ¼ Also 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.

42. 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 fraction whole divided in to equal parts Ex. circle divided into 4 parts numerator denominator fraction bar 1/4

43. 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 denominator 3 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

44. 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

45. Working Memory Prefrontal Cortex Application is the goal - transfer Arithmetic Changing Mixed Number to Improper fractions restaurant 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.

46. 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

47. 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

48. 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.

49. 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.