1. Retrieval Practice by Delano P. Wegener, Ph.D. August 2015
Copyright 2015 by Delano P. Wegener, Ph.D. All Rights Reserved.

2. Interrupted Forgetting Curve
“Periodic practice arrests forgetting, strengthens retrieval routes, and is essential for hanging onto the knowledge you want to gain.” (DrDel) Notice that after each successive interruption forgetting “tails off” at a higher percentage.
Uninterrupted Forgetting Curve
Interrupted Forgetting Curve
1. Brown, Peter C. ( ). Make It Stick (p.4). Harvard University Press. Kindle Edition.

3. When you space out practice of a task and get a little rusty between sessions, or you interleave the practice of two or more types of tasks, retrieval is harder and feels less productive, but the effort produces longer lasting learning and enables more versatile application of it in later settings. (DrDel) Spaced practice is a very powerful learning tool. Spaced practice is easy to implement with proper use of flash cards. The advantage of spaced practice was first observed by Ebbinghaus in 1885 and no study has ever been able to refute it. My students have successfully use spaced practice since I started touting its benefits around We all want longer lasting results even if only to get through the comprehensive final exam.
Adapted from: Brown, Peter C. ( ). Make It Stick (p.4). Harvard University Press. Kindle Edition.

4. (DrDel) The preferred method of interrupting the forgetting curve is with spaced retrieval practice.
Such practice is achieved with flashcards, interleaved practice exercises, and frequent low-stakes quizzes.

5. “All new learning requires a foundation of prior knowledge
“All new learning requires a foundation of prior knowledge.” 1 (DrDel) Memorization of stipulative mathematics definitions provides a form of foundational knowledge upon which to build a complete concept. An understanding of place value is necessary for a real understanding of polynomials. A knowledge of the set of rational numbers is required to understand many results in algebra. (e.g. Rational Zeros Theorem). A knowledge of the concept of set is prerequisite to learning about solution sets.
1. Brown, Peter C. ( ). Make It Stick (p.5). Harvard University Press. Kindle Edition.

6. when learning is harder, it’s stronger and lasts longer.
Many teachers believe that if they can make learning easier and faster, the learning will be better. Much research turns this belief on its head: It is apparent from research that Cramming is a form of massed practice. Math textbooks are a form of massed practice. Conventional assignment of exercises is a form of massed practice.
when learning is harder,
it’s stronger and lasts longer.
gains achieved during massed practice are transitory and melt away quickly.
Adapted from: Brown, Peter C. ( ). Make It Stick (p.9-10). Harvard University Press. Kindle Edition.

7. “The fact that you can repeat the phrases in a text or your lecture notes is no indication that you understand the significance of the precepts they describe, their application, or how they relate to what you already know about the subject.” 1 (DrDel) Memorization of stipulative mathematics definitions is essential because of their stipulative nature and because they are the core element around which we construct complete concepts. The above claim from “Make It Stick” does not preclude the necessity, in some instances, to begin the study of an abstract concept with rote memorization of key definitions.
1. Brown, Peter C. ( ). Make It Stick (p.16). Harvard University Press. Kindle Edition.

8. A baseball consists of a central core tightly wrapped in several layers of different materials. Construction of a golf ball is similar. As a matter of fact many familiar objects, including the earth, have similar constructions.
On a more abstract level, simple mathematics concepts are similarly constructed. It is convenient to think of them as a central core (definition) tightly wrapped in several layers of illuminating information.
Complex mathematics concepts consist of many simple mathematics concepts joined together.

9. Even the most diligent students are often hobbled by two liabilities:
a failure to know the areas where their learning is weak
a preference for study methods that create a false sense of mastery.
Adapted from: Brown, Peter C. ( ). Make It Stick (p.17). Harvard University Press. Kindle Edition.

10. The idea that individuals have distinct learning styles has been around long enough to become part of the folklore of educational practice and an integral part of how many people perceive themselves. We acknowledge that everyone has learning preferences, but we are not persuaded that you learn better when the manner of instruction fits those preferences. Yet there are other kinds of differences in how people learn that do matter.
Adapted from: Brown, Peter C. ( ). Make It Stick (pp ). Harvard University Press. Kindle Edition.

11. In 2008 the cognitive psychologists Harold Pashler, Mark McDaniel, Doug Rohrer, and Bob Bjork were commissioned to conduct a review … Their conclusion: It is more important that the mode of instruction match the nature of the subject being taught: visual instruction for geometry and geography, verbal instruction for poetry, and so on.
Adapted from: Brown, Peter C. ( ). Make It Stick (pp ). Harvard University Press. Kindle Edition.

12. determine how much a student knows.
One of the most striking recent research findings is the power of active retrieval to strengthen memory. Moreover, the more effortful the retrieval, the stronger the benefit. Testing is a form of active retrieval. If a test is used to practice retrieval of knowledge from memory then that test is a tool for learning. Therefore frequent spaced self-tests and other low-stakes tests are powerful learning tools. (DrDel) Retrieval practice is also critical to making knowledge accessible to us when we need it.
Tests can be used to:
determine how much a student knows.
determining what a student must learn.
practice retrieval of knowledge from memory.
Adapted from: Brown, Peter C. ( ). Make It Stick (p.19). Harvard University Press. Kindle Edition.

13. Interleaving practice different topics related topics or skills
As a result of recent research we know simple and practical strategies that anybody can use, at any point in life, to learn better and remember longer:
Retrieval practice
low-stakes quizzes
self-tests
Spaced practice
Interleaving practice
different topics
related topics or skills
Trying to solve a problem before being taught the solution.
Distilling the underlying principles or rules that differentiate types of problems, and so on.
Brown, Peter C. ( ). Make It Stick (p.19-21). Harvard University Press. Kindle Edition.

14. Take charge of your education by implementing the following.
Practice Retrieving New Learning from Memory
Space Out Retrieval Practice
Interleave the Study of Different Problem Types
Other Effective Study Strategies
Elaboration
Generation
Reflection
Calibration
Mnemonics
Avoid massed practice
Avoid excessive rereading
Avoid excessive underlining
Brown, Peter C. ( ). Make It Stick (p ). Harvard University Press. Kindle Edition.

15. Practice Retrieving
First, as we recode and consolidate new material from short-term memory into long-term memory, we must anchor it there securely.
Second, we must associate the material with a diverse set of cues that will make us adept at recalling the knowledge later. Having effective retrieval cues is an aspect of learning that often goes overlooked.
Adapted from: Brown, Peter C. ( ). Make It Stick (p.75). Harvard University Press. Kindle Edition.

16. Practice Retrieving
Aristotle wrote: “exercise in repeatedly recalling a thing strengthens the memory.”
Practicing retrieval makes learning stick far better than re-exposure to the original material does.
The act of retrieving a memory changes the memory, making it easier to retrieve again later.
Repeated retrieval not only makes memories more durable but produces knowledge that can be retrieved more readily, in more varied settings, and applied to a wider variety of problems.
Adapted from: Brown, Peter C. ( ). Make It Stick (p.28,41,43). Harvard University Press. Kindle Edition.

17. Spaced Retrieval Practice
When retrieval practice is spaced, allowing some forgetting to occur between tests, it leads to stronger long-term retention than when it is massed.
The advantage of spaced practice was first observed by Ebbinghaus in 1885 and no study has ever been able to refute it.
Delaying subsequent retrieval practice is more potent for reinforcing retention than immediate practice, because delayed retrieval requires more effort.
(DrDel) It has always baffled me that for 130 years we have known the disadvantage of massed practice and the advantage of spaced and yet all math textbooks and other study materials are based on massed practice.
Adapted from: Brown, Peter C. ( ). Make It Stick (p.32,43). Harvard University Press. Kindle Edition.

18. Avoid Massed Practice
“Faith in focused, repetitive practice of one thing at a time until we’ve got it nailed is pervasive among classroom teachers, athletes, corporate trainers, and students. Researchers call this kind of practice “massed,” and our faith rests in large part on the simple fact that when we do it, we can see it making a difference. Nevertheless, despite what our eyes tell us, this faith is misplaced.” 1
“Cramming for exams is a form of massed practice. It feels like a productive strategy, and it may get you through the next day’s midterm, but most of the material will be long forgotten by the time you sit down for the final. “ 1
Working many problems of the same type at the end of a section is massed practice and is quite ineffective. It is more like training than learning.
1. Brown, Peter C. ( ). Make It Stick (p.47,48). Harvard University Press. Kindle Edition.

19. Avoid Massed Practice
“In math education, massing is embedded in the textbook: each chapter is dedicated to a particular kind of problem, which you study in class and then practice by working, say, twenty examples for homework before you move on. The next chapter has a different type of problem, and you dive into the same kind of concentrated learning and practice of that solution. On you march, chapter by chapter, through the semester. But then, on the final exam, lo and behold, the problems are all mixed up: you’re staring at each one in turn, asking yourself Which algorithm do I use? Was it in chapter 5, 6, or 7? When you have learned under conditions of massed or blocked repetition, you have had no practice on that critical sorting process.” 1
1. Brown, Peter C. ( ). Make It Stick (p.53). Harvard University Press. Kindle Edition.

20. Interleaving
“Practice that’s spaced out, interleaved with other learning, and varied produces better mastery, longer retention, and more versatility.” 1
“The learning from interleaved practice feels slower than learning from massed practice. Teachers and students sense the difference. They can see that their grasp of each element is coming more slowly, and the compensating long-term advantage is not apparent to them. As a result, interleaving is unpopular and seldom used.” 1
“A significant advantage of interleaving and variation is that they help us learn better how to assess context and discriminate between problems, selecting and applying the correct solution from a range of possibilities.” 1
1. Brown, Peter C. ( ). Make It Stick (p.47,50). Harvard University Press. Kindle Edition.

21. Elaboration
“Elaboration is the process of finding additional layers of meaning in new material.” 1
Elaboration may include illustrative examples, relating it to knowledge you already know, or a visual image.
The more you can elaborate on how new learning relates to what you already know, the stronger your grasp of the new learning will be, and the more connections you create to remember it later.
Elaboration is very important in mathematics. Titles of courses, textbooks, chapters, sections, and subsections are deliberately chosen to promote this last form of elaboration.
1. Brown, Peter C. ( ). Make It Stick (p.207). Harvard University Press. Kindle Edition.

22. Reflection
“Reflection is the act of taking a few minutes to review what has been learned in a recent class or experience and asking yourself questions.” 1
Reflection adds layers to learning and strengthens skills.
Questions you might ask after a math class or study session.
Is this an object, operation, or relation?
How does this material relate to previous knowledge?
Did the examples illustrate the concepts?
Can I create additional examples?
What kind of problems will this help solve?
What makes this concept important?
Is this a generalization of a known concept?
1. Brown, Peter C. ( ). Make It Stick (p.209). Harvard University Press. Kindle Edition.

23. Generation
Generation is an attempt to answer a question or solve a problem before being shown the answer or the solution.
Fill in the blank test questions are simple examples of generation.
Using known concepts to devise a solution to an unfamiliar problem without being told how to solve the problem is an advanced form of generation we expect students to achieve in math classes.
Adapted from: Brown, Peter C. ( ). Make It Stick (p.208). Harvard University Press. Kindle Edition.

24. Calibration
“Calibration is the act of aligning your judgments of what you know and don’t know with objective feedback so as to avoid being carried off by the illusions of mastery that catch many learners by surprise at test time.” 1
“Calibration is simply the act of using an objective instrument to clear away illusions and adjust your judgment to better reflect reality.” 1
Flashcards provide a ready quiz. Read side one and then WRITE (without looking) precisely what is on side two.
In mathematics if you cannot WRITE a definition, property, rule, or procedure precisely, then you do not know it.
Brown, Peter C. ( ). Make It Stick (p ). Harvard University Press. Kindle Edition.

25. Calibration
Use quizzes and practice tests to see whether you know as much as you think you do.
It is very important that you write the answers to questions in a self-quiz.
It is natural to look at a question on a practice test and say: Yup, I know that, and move on to the next question without making the effort to write the answer.
If you don’t write the answer, you may be giving in to the illusion of knowing, when in fact you would have difficulty rendering an accurate or complete response. This is extremely important in mathematics because mathematics demands precision and completeness.
Adapted from: Brown, Peter C. ( ). Make It Stick (p.210). Harvard University Press. Kindle Edition.

26. Calibration
Treat practice tests as tests, check your answers. Be a harsh grader.
Focus your studying effort on the areas where you are not up to snuff.
Adapted from: Brown, Peter C. ( ). Make It Stick (p.210). Harvard University Press. Kindle Edition.

27. Mnemonics
“Mnemonic devices help you to retrieve what you have learned and to hold arbitrary information in memory.” 1
I do not recommend mnemonic devices for mathematics because:
What you are trying to remember is not arbitrary.
The mnemonic device obscures the math concept.
The mnemonic device obscures connections and context.
The mnemonic is harder to remember than the math.
1. Brown, Peter C. ( ). Make It Stick (p.211). Harvard University Press. Kindle Edition.

28. Current research has yielded at least three keystone and five supporting study strategies which can be implemented for significant benefit. That same research identifies at least three strategies to avoid.
Practice Retrieving New Learning from Memory
Space Out Retrieval Practice
Interleave the Study of Different Problem Types
Other Effective Study Strategies
Elaboration
Generation
Reflection
Calibration
Mnemonics
Avoid massed practice
Avoid excessive rereading
Avoid excessive underlining
Brown, Peter C. ( ). Make It Stick (p ). Harvard University Press. Kindle Edition.