Mathematics

3 Ways to Strengthen Your Students’ Critical Thinking Skills

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Psychologist Benjamin Bloom published his now widely spread document in education, “Taxonomy of Educational Objectives,” in 1956. In it, he and his team specify three domains of learning: affective, psychomotor, and cognitive. While the affective domain refers to the emotions, motivations, and attitudes of students, the psychomotor domain focuses on their motor skills.

The cognitive domain—arguably the most influential in a student’s success—covers six categories (according to the revised version of Bloom’s Taxonomy by Anderson & Krathwohl, et al (2001)):

  • Remembering
  • Understanding
  • Applying
  • Analyzing
  • Evaluating
  • Creating

These categories start with memorizing and defining what’s learned in class, build toward drawing connections among different ideas and applying them outside of class, then lead to creating your own work by using what you’ve learned (Armstrong). Building upon these processes develops students’ critical thinking and reasoning skills, which are more important today than ever before. (Pssst! Check out key definitions and verbs to describe each category here from Vanderbilt University’s Center for Teaching.)

So, how can you help students strengthen their critical thinking and reasoning? Below are three ways to incorporate these skills into any curriculum.

1. Allow time within class to brainstorm after asking an open-ended question.

Students need time on their own to think about how to solve a problem, as well as time to talk out their strategies with other students. Problem solving is a key component to critical thinking, and brainstorming gives students the opportunity to explore different perspectives and possible solutions in a low-pressure environment. According to Lee Crockett Watanabe from Global Digital Citizen Foundation, asking a question that can’t simply be answered with a yes or no encourages students to seek out the necessary knowledge on their own (“12 Strong Strategies for Effectively Teaching Critical Thinking Skills.”). Students must use the skills associated with the cognitive domain, such as recalling what they already know about the problem, analyzing different strategies to solve it, and evaluating the quality of each solution.

2. Compare and contrast different ideas. 

Once students learn and understand different approaches to solving a problem, they can evaluate the qualities of each approach. Which one is easier? Which is the most thorough? Which makes the most sense to use in this context? Students need to judge the strengths and weaknesses of varying solutions in order to decide their next steps in solving the problem. Creating a pro/con chart can help, as well as a pro/pro chart, according to instructor Jason Watt. In a pro/pro chart, students see the positives of different perspectives by listing out only the good traits of each, bringing a fresh take to an old decision-making strategy. Watt explains that a pro/pro chart can help students try to find the positives in what they originally thought of as a weakness, allowing them to get creative with their thinking and less intimated to do so (Schwartz).

3. Get them thinking about thinking.

In the revised version of Bloom’s Taxonomy, metacognitive knowledge includes strategy, self-knowledge, and contextual and conditional knowledge (Armstrong). To increase their critical thinking skills, students need to think about how they think. If they pause to reflect upon how they’re studying and learning the class content, they may just improve their grades. Dr. Patricia Chen, a postdoctoral researcher at Stanford, conducted a study in which she asked a group of her students several prompts asking them to think about how they’re studying for an upcoming test and how they could improve their studying. She only reminded a second student group that the test was coming up. The first group outperformed the students who did not reflect on their studying by 1/3 of a letter grade on average (Anderson). Check out more information on the study.

When students analyze their own thinking techniques and visualize how they want to perform on assessments, they develop critical strategies to set goals and determine which resources work best for their unique learning processes. These skills can help students improve their grades, and they’ll transfer over when students are learning in other classes, navigating the workplace, and facing the challenges of daily life.

 

Have other ways to help improve students’ cognitive domains and critical thinking skills? Please share them in the comments below!

 

Anderson, Jenny. “A Stanford researcher’s 15-minute study hack lifts B+ students into the As.” Quartz, 9 May 2017, https://qz.com/978273/a-stanford-professors-15-minute-study-hack-improves-test-grades-by-a-third-of-a-grade/. Accessed 5 June 2017.

Armstrong, Patricia. “Bloom’s Taxonomy.” Vanderbilt University Center for Teaching, https://cft.vanderbilt.edu/guides-sub-pages/blooms-taxonomy/. Accessed 12 June 2017.

Schwartz, Katrina. “Three Tools for Teaching Critical Thinking and Problem Solving Skills.” KQED News, 6 Nov. 2016, https://ww2.kqed.org/mindshift/2016/11/06/three-tools-for-teaching-critical-thinking-and-problem-solving-skills/. Accessed 13 June 2017.

Watanabe Crockett, Lee. “12 Strong Strategies for Effectively Teaching Critical Thinking Skills.” Global Digital Citizen Foundation, 13 March 2017, https://globaldigitalcitizen.org/12-strategies-teaching-critical-thinking-skills. Accessed 12 June 2017.

Apply mathematical concepts to other fields

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Sometimes, getting students excited about math isn’t easy. Nearly every math instructor has heard “When will I use this in real life?” at least once during their teaching career. Many students don’t see right away that they use math just about every day, and you can lose their interest in the subject if you don’t connect your course objectives to their lives outside of class. Thankfully, math applies to more fields than most students realize. Here are just a few ways to connect mathematical concepts to other areas and to get students more motivated to learn.

1. Create art with math.

Not all students see how subjects in STEM connect with the liberal arts. Some people mistakenly think the fields are separate and never the two shall meet. One great way to get rid of this misconception is to show how art can be created by using math. Creative Bloq shows eight examples of beautiful fractal art with suggestions on programs to use in order to create your own fractal masterpieces, such as Mandelbulb 3D and FraxHD.

The co-author of our Single Variable Calculus with Early Transcendentals textbook, Dr. Paul Sisson, used to incorporate art into his math classes when he taught at Louisiana State University – Shreveport. He encouraged students to use software to track complex numbers’ behaviors and create images to which students could assign different colors. Learn more from Math in the Media here.

2. Show students how to be fiscally responsible.

Chances are you have some students who don’t know much about personal finance beyond having a checking and savings account. Teaching them about budgeting, loans, interest, and more will benefit them now and in all the years to come. Students can start with concepts such as calculating tip and figuring out how much money they save when they buy discounted items before moving on to long-term financial decisions, such as putting a down payment on a house and paying a mortgage.

This post from Annenberg Learner summarizes the basics of simple and compound interest that you can incorporate into your class.

3. Calculate sports statistics.

Have students who want to be professional athletes, coaches, sports announcers, agents or just die-hard fans of the game? They’ll benefit from learning how much math goes into any sport. Everything from calculating batting averages in baseball to knowing touchdowns per pass attempt in football to determining the probability of winning a point in tennis can connect the concepts learned in class to some students’ favorite extracurricular activities. Plus, fantasy sports are especially popular, so you may even consider having your class join a fantasy league and see who wins!

Fantasy Sports and Mathematics is a website that includes the latest scores and injuries lists for various sports and sample math problems to use in class. This NYT blog post lists out ways to use sports analytics to teach math and includes additional resources ranging from a video demonstrating what it’s like to return a serve in professional tennis to a graphic showing how often football teams go for the fourth down.

4. Delve into the history of mathematics.

Students gain a deeper appreciation of the subject when they know who’s behind all those theories, formulas, and discoveries. Plus, they just might connect with the subject more when they know that people from similar demographics advanced the field.

A Buzzle article introduces readers to several achievements of African American mathematicians, ranging from those in the 18th century like Benjamin Banneker to the present day like Dr. William A. Massey.

This Smithsonian.com post highlights five influential female mathematicians throughout history, including Ada Lovelace and Emmy Noether. It gives a little background into these women’s lives, explains their accomplishments, and kicks the blatantly false stereotype that women aren’t good at math to the curb!

5. Have students write about how they think they’ll use math in their future careers.

Are your students still not feeling connected with the course content? Dedicate some class time to brainstorming how they’ll use math in the careers they’re planning to pursue. While at first some may assume they won’t use math at all in their chosen professions, they might surprise themselves once they think a little harder and dig deeper into a job’s tasks and expectations. They may want to interview someone in their field via email or phone to get an insider’s perspective into the kind of math skills needed to excel in the workplace.

On the blog Math for Grownups, author Laura Laing interviewed several professionals—including writers, academic advisors, and artists—asking them how they use math in their jobs. Her books Math for Grownups and Math for Writers delve into more detail on these topics and encourage folks who are hesitant about math or think they’re bad at it to rethink their perspective.

What are some lessons you’ve taught that encouraged students to apply math to other subjects and think outside the box? Let us know in the comments!

More simulations, stat!

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Remember that in the appendix of your courseware, great resources for student engagement help you bring the real world into your classroom! Check out two examples:

1. Direct Mail Game

In the game, students assume the role of a direct mail marketing manager for a company that markets inexpensive computer software. Their task is to develop a mailing strategy by finding mailing lists that will produce sufficient sales to be profitable.

The game provides an environment in which students apply statistical concepts while making business decisions.

While students can explore the game on their own, we recommend playing it in class.

Dr. Hawkes created this game when he was a statistics professor. He says that this lesson was always his students’ favorite each semester.

A row of lists 1-3 shows each list's price, number, and how many sampled. Below that, the mailing results are displayed, including the list number, cost per name, sample size, number of responses and their data, the mailing costs, the old balance, profit, and new balance. The cash on hand is displayed below, which is $19,968.50.

2. Name That Distribution

Name that Distribution is a concept builder that strengthens analytical skills in distribution recognition and data analysis.

Students view the histogram, frequency, statistics, and sample data of a distribution. They can increase the number of intervals and choose to view a hint if they’re unsure of the answer. As they play with the different options to analyze the data, students combine that information to make an educated guess about the distribution type.

Check out a hint:
A text box called Hints is shown, which more information on uniform distributions. It displays the theoretical variance, and explains how to identify this distribution type.

Below is an example of a distribution type:

A histogram is shown. Below it is the question "What type of distribution is described by the sample?" with the multiple choice options of uniform, normal, exponential, chi-square, Poisson, and mystery. An arrow points to the answer uniform, and another arrow points to a submit button.
Students new to this kind of data analysis will begin to understand how they can apply this knowledge to the many real-world scenarios that they can evaluate through detecting typical or skewed data.

The Direct Mail lesson is available in:

  • Discovering Statistics Appendix A.9
  • Discovering Business Statistics Appendix A.10
  • Beginning Statistics Appendix A.5

The Name that Distribution lesson is available in Appendix A.3 in all three statistics courses.

Learning assessment expert and author Dr. Nolting interviews AMATYC president Jane Tanner

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AMATYC is just around the corner, and we can’t wait for that educational, fun-filled conference! Before we head out for the special event, we wanted to let you know that our friend and national expert in assessing math learning problems and developing solutions, Dr. Paul Nolting, interviewed AMATYC President Jane Tanner on his blog, http://www.academicsuccessblog.com/.

Dr. Nolting assesses math learning problems, develops effective student-learning strategies, and assesses institutional variables that affect math success and math study skills. Over the last 25 years, he has consulted with over 100 colleges, universities, and high schools campuses to improve success in the math classroom. He is the author of Winning at Math, which is the only math-specific study skills book to offer statistical evidence demonstrating an improvement in students’ ability to learn math and make better grades.

In his interview, Dr. Nolting asks the AMATYC president questions that strike a chord with all developmental math instructors today. He touches on key topics and starts out by asking Professor Tanner how she sees the current state of developmental mathematics at the national level. Professor Tanner replied:

My opinion is that it is in a state of flux. That is my opinion, not necessarily that of AMATYC or anyone else. A lot of colleges out there know we need to change what is currently being done, because the current success rate in developmental mathematics is not very great for students. These schools know something needs to be done—these are the forward thinkers that are willing to try new things and take risks. There are others out there who want to continue to do the same old things, because that is what they are used to, and they are not as willing to take risks. My opinion is that you need to be willing to try something different. You need to keep in mind what is best for your school and students, not what is easiest for you…

Later in the interview, Dr. Nolting asked, “How do you think institutions should go about choosing a new design, or, for that matter, what should institutions do if they are torn between different designs? How do we avoid chaos as pride and conviction inevitably seep into this process?”

Below is an excerpt of Professor Tanner’s response:

You need to research what is out there. You can visit other schools that are using a certain method that might work for you, or attend the AMATYC and NADE conferences where there are other people going through things that you may be going through. There are a lot of different models out there, all in addition to the pathways focus. What needs to be done is that you spend enough time investigating so that you choose the best thing for your college—but you can’t necessarily take forever to do it, because then you aren’t accomplishing anything either.

Read part one of the interview here!

 

Interested in learning more about math study skills? Check out the webinar from Dr. Nolting and Hawkes’s own Emily Judy for tips and resources.

 

Nolting, Paul. “Dr. Nolting Interviews Jane Tanner, president of AMATYC: Part One.” Academic Success Blog,  www.academicsuccessblog.com/blog/interview-with-jane-tanner-president-of-amatyc-part-one. Accessed 10 Nov. 2016.

College algebra students plagued by scattered notes will benefit from this!

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Do your college algebra students have difficulty keeping track of the concepts you’re teaching? Are they having trouble organizing their notes to prepare for tests? We have the solution:

Cover of College Algebra Guided Notebook by Christopher Schroeder at Morehead State University

College Algebra Guided Notebook

Dr. Christopher Schroeder 
Morehead State University

This guided notebook complements Hawkes Learning’s online courseware because it:

  • offers students writing and thinking space to work out algebra problems
  • walks students through Hawkes’s online Learn mode with detailed explanations
  • provides additional practice problems and links to online videos
  • relates math to students in a conversational tone
  • keeps notes in one centralized location

If you’re interested in learning more, contact us at 1-800-426-9538 or sales@hawkeslearning.com.

An InteGREATed Course for College Algebra

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Here at Hawkes Learning, we’re excited about developing our new course offering, College Algebra Plus Integrated Review! Target specific remediation needs for just-in-time supplementation of foundational concepts in college algebra with these materials.

This new integrated course enhances curriculum-level math with applicable review skills to shorten the prerequisite sequence without compromising competency. If you teach a college algebra corequisite course, these materials are for you!

Below is the table of contents.

College Algebra Plus Integrated Review

Chapter 1.R: Integrated Review
1.R.1 Exponents, Prime Numbers, and LCM
1.R.2 Reducing Fractions
1.R.3 Decimals and Percents
1.R.4 Simplifying Radicals
Chapter 1: Number Systems and Fundamental Concepts of Algebra
1.1 The Real Number System
1.2 The Arithmetic of Algebraic Expressions
1.3a Properties of Exponents
1.3b Scientific Notation and Geometric Problems Using Exponents
1.4a Properties of Radicals
1.4b Rational Number Exponents
1.5 Polynomials and Factoring
1.6 The Complex Number System
Chapter 1 Review
Chapter 2.R: Integrated Review
2.R.1 Multiplication and Division with Fractions
2.R.2 Addition and Subtraction with Fractions
2.R.3 Applications: Number Problems and Consecutive Integers
2.R.4 Solving Equations: Ratios and Proportions
Chapter 2: Equations and Inequalities of One Variable
2.1a Linear Equations in One Variable
2.1b Applications of Linear Equations in One Variable
2.2 Linear Inequalities in One Variable
2.3 Quadratic Equations in One Variable
2.4 Higher Degree Polynomial Equations
2.5 Rational Expressions and Equations
2.6 Radical Equations
Chapter 2 Review
Chapter 3: Linear Equations and Inequalities of Two Variables
3.1 The Cartesian Coordinate System
3.2 Linear Equations in Two Variables
3.3 Forms of Linear Equations
3.4 Parallel and Perpendicular Lines
3.5 Linear Inequalities in Two Variables
3.6 Introduction to Circles
Chapter 3 Review
Chapter 4.R: Integrated Review
4.R.1 Order of Operations
4.R.2 Variables and Algebraic Expressions
4.R.3 Simplifying Expressions
4.R.4 Translating Phrases into Algebraic Expressions
Chapter 4: Relations, Functions, and Their Graphs
4.1 Relations and Functions
4.2a Linear and Quadratic Functions
4.2b Max/Min Applications of Quadratic Functions
4.3a Other Common Functions
4.3b Direct and Inverse Variation
4.4 Transformations of Functions
4.5 Combining Functions
4.6 Inverses of Functions
Chapter 4 Review
Chapter 5.R: Integrated Review
5.R.1 Greatest Common Factor of Two or More Terms
5.R.2 Factoring Trinomials by Grouping
5.R.3 Additional Factoring Practice
Chapter 5: Polynomial Functions
5.1 Introduction to Polynomial Equations and Graphs
5.2 Polynomial Division and the Division Algorithm
5.3 Locating Real Zeros of Polynomials
5.4 The Fundamental Theorem of Algebra
Chapter 5 Review
Chapter 6.R: Integrated Review
6.R.1 Defining Rational Expressions
6.R.2 Special Products
6.R.3 Special Factorizations – Squares
Chapter 6: Rational Functions and Conic Sections
6.1a Rational Functions
6.1b Rational Inequalities
6.2 The Ellipse
6.3 The Parabola
6.4 The Hyperbola
Chapter 6 Review
Chapter 7.R: Integrated Review
7.R.1 Simplifying Integer Exponents I
7.R.2 Simplifying Integer Exponents II
7.R.3 Rational Exponents
Chapter 7: Exponential and Logarithmic Functions
7.1 Exponential Functions and their Graphs
7.2 Applications of Exponential Functions
7.3 Logarithmic Functions and their Graphs
7.4 Properties and Applications of Logarithms
7.5 Exponential and Logarithmic Equations
Chapter 7 Review
Chapter 8.R: Integrated Review
8.R.1 Solving Systems of Linear Equations by Graphing
8.R.2 Systems of Linear Inequalities
Chapter 8: Systems of Equations
8.1 Solving Systems by Substitution and Elimination
8.2 Matrix Notation and Gaussian Elimination
8.3 Determinants and Cramer’s Rule
8.4 The Algebra of Matrices
8.5 Inverses of Matrices
8.6 Linear Programming
8.7 Nonlinear Systems of Equations
Chapter 8 Review
Chapter 9: An Introduction to Sequences, Series, Combinatorics, and Probability
9.1 Sequences and Series
9.2 Arithmetic Sequences and Series
9.3 Geometric Sequences and Series
9.4 Mathematical Induction
9.5a An Introduction to Combinatorics – Counting, Permutations, and Combinations
9.5b An Introduction to Combinatorics – The Binomial and Multinomial Theorems
9.6 An Introduction to Probability
Chapter 9 Review
Chapter A: Appendix
A.1 Introduction to Polynomial Equations and Graphs (excluding complex numbers)
A.2 Polynomial Division and the Division Algorithm (excluding complex numbers)
A.3 Locating Real Zeros of Polynomials (excluding complex numbers)
A.4 The Fundamental Theorem of Algebra (excluding complex numbers)