Metacognition
Math is like an atomic-powered prosthesis that you attach to your common sense, vastly multiplying its reach and strength. -- Jordan Ellenberg, How Not To Be Wrong
Metacognition
The ability to quickly estimate a quantity is an art and an excellent way to sharpen your quantitative intuition. An important part of mastering this technique is the ability to diagnose your own understanding. This allows you to place your efforts and questions into the parts of your abilities that need practice.
How do you know when you have learned something well?
When you study, how do you know when you have studied enough?
How do you set aside time for focus and learning?
Much of your success will depend on your beliefs about your ability to learn and improve through effort. Research has shown that those who believe that they can improve their abilities through practice are able to persevere through tough times and get better at tough tasks like mathematics.
For example, what does it mean to think mathematically? How is applied mathematics -- using math to tackle problems -- different from mathematics as you understand it?
Study Strategies
What does science tell us about how to study for your midterm?
Sleep
Retrieval Practice
Elaborative Interrogation
Spaced Practice
Stress can help your performance
Study Strategies
This science on learning and psychology is still emerging
What are the costs and benefits of changing your behavior based on this science?
Stress Reappraisal
An experiment showed exam improvement based on student interpretation of stress
Sleep
Sleep is correlated with academic performance
Retrieval Practice
Studying is the practice of storing ideas in your memory
Testing requires you to retrieve ideas from your memory
Retrieval Practice
Rereading materials trains the brain in storing information, not retrieving it
Asking and answering questions, flashcards, and other techniques practice retrieval
Elaborative Interrogation
This is a method of generating deeper questions about the material and answering them
This can be done well with other students
We will expect you to understand subtle points of our concepts
Spaced Practice
Your learning will be deeper if you space your study over the next few days
You will be better served by
This requires discipline and time management
Study Strategies
Science, evidence, and our decisions revisited
This science on learning and psychology is still emerging
What are the costs and benefits of changing your behavior based on this science?
Links
Do you believe you are a "math person"
Are you comfortable with mathematics?
Do you think your abilities can improve with effort and practice?
Do you think that improving your mathematical abilities will help you in the future?
When you make important financial decisions, will you try to understand the details?
Grit, Mindset, Effort
This will not always be easy or pleasurable
You should be prepared to fail
Some of your efforts should look like this
I frequently struggle to learn new branches of mathematics
Bloom's Taxonomy
There are different levels of cognitive skills that we sharpen in higher education. Your goal should be to exercise the highest levels of cognitive skills. Education scholars created a list of these cognitive processes called Bloom's Taxonomy.
Remembering "retrieving relevant knowledge from long-term memory"
Understanding "determining the meaning of instructional messages"
Applying "carrying out or using a procedure in a given situation"
Analyzing "breaking material into its constituent pants and detecting how the parts relate to one another and to an overall structure or purpose"
Evaluating "making judgements based on criteria and standards"
Creating "putting elements together to form a novel, coherent, whole or make an original product"
You have probably mastered mathematical rules and computation tools at the level of remembering, understanding, and applying. You remember the rules and ways to use mathematical formulas as well as the procedures to use computational tools. You understand the relationships within a mathematical formula. When it comes time to apply formulas, you are likely comfortable with a "plug-and-chug" approach to getting a numerical answer from a formula.
This class aims to develop your skills analyzing situations to break them down into parts so you can choose which models to apply. You will learn how to evaluate your own work and that of others to judge whether it correctly follows the rules of mathematics, computation, and evidence. You will create your own estimations and form your own questions to produce new written estimations and quantitative arguments.
You may initially have difficulty analyzing problems, evaluating the claims of others, and creating your own models, but this will get easier with practice. It is important to recognize that since computers are able to perform many tasks that fall into the lower levels of this taxonomy, higher-order skills are more valuable and less likely to be automated.
Learning and Effort
To get do the desired outcome of learning you need to devote time and attention to learning. There are two main contributions to your learning, the learning in class, and your study outside of class. The efficiency with which you approach each of these maximizes the amount of learning from each. At the risk of oversimplifying a complex topic
desired learning = class learning + study learning
We can expand both of these and look at how efficiently we use our time
desired learning = class time x class focus + study time x study focus
Using algebra to isolate study time, which most students want to minimize,
study time = (desired learning - class time * class focus) / study focus
Crudely, for a desired learning outcome, you can decrease the time you must spend studying by increasing your focus during class and during study.
Scheduling your time
What works better for you when you work on assignments?
Finishing in a single four-hour block?
Breaking up an assignment over four separate one-hour blocks?
Do you find yourself working when you are not at your cognitive peak?
Motivation
Why is the study of mathematics and its application important? What will you be able to do if you develop your quantitative skills that you cannot do now?
Why are you in college?
What is important about your degree?
Is the diploma important?
Is your choice of major important?
Are the abilities you will develop important?
One popular reason is to have better employment
To get better employment one needs better skills
Applied mathematics skills are valuable for many jobs and problems
Why is math important to you?
Do you see yourself using math in the future?
What kinds of math do you think are most important?
The power of mathematics
Mathematics sharpens your common sense.
Many critical questions about the environment, government, and social
justice are essentially mathematics questions.
We will demonstrate the power of basic mathematical concepts to gain
insight into these sorts of problems.
The power of mathematics
Mathematics allows you to see the similar structure in many different situations. It is similar to the way a knowledge of music allows you to see the similarity between many popular songs.
Knowledge and Skills
You will gain very little knowledge in this class since we will be using mathematics you have already learned
You will gain many new skills in applying your existing mathematics to problems
Last updated