If you want to think like a mathematician, you’ll need to work at it. Taking an online course in mathematical thinking is a great way to start.

However, you shouldn’t simply search for the top mathematical thinking courses or popular mathematical thinking courses. Instead, having a resource to show you how the best courses for mathematical thinking compare to one another is far more helpful. That is the purpose of this ranking.

**Featured Programs**

- University of Pennsylvania Computational Thinking for Problem Solving
- Computational Thinking and Big Data
- MIT Probability The Science of Uncertainty and Data

*Disclaimer: Some courses may include an affiliate link. Courses were chosen first based on the methodology with affiliate links only added after the ranking was complete.*

We’ve done the research for you and compiled a list of ten of the best mathematical thinking courses online. Each of the courses in our rankings is evaluated using a three-part methodology:

- Affordability
- Accessibility
- Learning level

By giving you a broader view of each of these courses, you can see which ones hold the most value for you – and why.

So, our rankings have some popular mathematical thinking courses. But with the information provided in the rankings, you can determine the top mathematical thinking courses for your specific needs.

Let’s quickly go over the ranking method we’ve used to compile this list of courses.

## Methodology to Determine the Best Online Courses for Mathematical Thinking

To determine some of the best options for learning mathematical thinking, we looked at online-only courses and scored them based on the three factors listed above.

This enables us to evaluate each course on each of the three individual factors. It also allows us to compare the courses side-by-side according to an overall score.

With these micro-level and macro-level comparisons, you will be able to see how the courses stack up against one another with much greater ease. This should help you make a more informed decision about which classes to take as well.

Below is a summary of each of the three factors we’ve used to rank the top mathematical thinking courses.

### Affordability

The first factor we used in ranking these courses is affordability. The more affordable a course is, the more points it gets:

- Free: 6 points
- Up to $50: 5 points
- $51-$100: 4 points
- $101-$150: 3 points
- $151-$200: 2 points
- $201 and above: 1 point

We assign more points to less expensive courses for one primary reason – most students want the most affordable options.

This doesn’t mean that more expensive courses shouldn’t be considered. But everyone has a budget. If a class looks like a good fit, but it’s too expensive, it’s a moot point to consider taking it.

So, our affordability ranking is the most important of the three simply because it will immediately let you know if a course is within budget or not.

### Accessibility

Online learning is already pretty accessible. But there is still some variation from one class to the next.

For example, some courses are available immediately upon enrollment. This means you could register for a class and start working on the course right away if you like.

But other popular mathematical thinking courses stick to a schedule. Usually, these courses are instructor-led. This means you’ll have set due dates, exam dates, and perhaps even class meeting times. This restricts your access to a certain degree, because you might register for a course and have to wait a few weeks to access it.

Other accessibility features include the number and type of learning materials, how the class can be accessed, and the presence or absence of language support services.

This factor is scored as follows:

- High accessibility: 2 points
- Moderate accessibility: 1 point

### Learning Level

When learning mathematical thinking, you need to take courses that meet you at your learning level. For example, if you’re an advanced student, you don’t need to take a beginner course. This means that knowing the learning level of the class will help you determine which classes are right for you and which ones you can skip.

In terms of awarding points, we give more points to higher-level classes. This isn’t a knock on beginner courses. They are extremely valuable for beginner students. But more advanced courses usually involve deeper, more focused studies. Beginner courses tend to cover many different topics but not that in-depth.

As a result, advanced courses get the most points, as shown below:

- Advanced: 3 points
- Intermediate: 2 points
- Beginner: 1 point

Again, this isn’t to say that beginner courses aren’t worth taking. But since advanced courses offer a more detailed method of study, we give those courses the highest points.

Now let’s get to the rankings!

## Ranking the Best Online Courses for Mathematical Thinking

### #10: Mathematical Thinking in Computer Science

This is an introductory course. It is offered by the University of California at San Diego and is available on Coursera.

This course will help you gain the skills needed to apply mathematical thinking to computer science. There are many different areas in computer science in which mathematical thinking is helpful. This includes:

- Data science
- Machine learning
- Algorithms
- Computer graphics

This online math course walks you through the most important tools used in discrete mathematics. This includes recursion, logic, and invariants. It also includes induction, examples, and optimality.

You will learn how to use these tools to answer questions about solutions, answers, and objects in the context of computer science.

Be aware that you need basic programming knowledge using Python. If you don’t have Python experience, it’s best to take a crash course in it before taking this class.

**Points**: 8

**Affordability**: Free to enroll. A fee (usually around $50) might be required. Other costs are involved for specializations (usually $39-$79 per month). Learn more about Coursera pricing. This is scored as a paid course.

**Accessibility**: High. You can start the course instantly upon enrollment. Course materials are available any time. You can access the course via any internet-enabled device.

**Learning Level**: Beginner

### #9: Computational Thinking for Problem Solving

This beginner course is offered by the University of Pennsylvania. It is available on Coursera.

This course combines some of the principles of mathematical thinking with computer science. The result is a class that helps you learn how to problem solve in a systematic manner. You will also learn how to create a solution to a problem that can be carried out by a computer.

To learn how to think computationally, you will:

- Learn about the parts of computational thinking
- Discuss how computer scientists develop algorithms
- Learn how to use Python

As part of this online mathematical thinking class, you will hear from people from many different professions. This goes to show that computational thinking isn’t just limited to computer science.

By the course’s end, you will have a better understanding of analytical thinking. You will also be able to apply your new thinking skills to real-world problems.

**Points**: 8

**Affordability**: Free to enroll. A fee (usually around $50) might be required. Other costs are involved for specializations (usually $39-$79 per month). Learn more about Coursera pricing. This is scored as a paid course.

**Accessibility**: High. This course is 100 percent online. Course materials are available upon enrollment. Video lectures come with subtitles in multiple languages.

**Learning Level**: Beginner

### #8: A Transition to Higher Mathematics

Thinking like a mathematician requires a lot of practice. You can get some of that practice in this intermediate course from Udemy.

You’ll start by learning about logic and how it’s used in math. You’ll also learn how to prove theorems and learn what a proposition is. Essentially, you will gain the skills needed to think like a mathematician, solve problems, and prove theorems.

Other topics include:

- Quantifiers
- Logical connectives
- Mathematical induction
- Equivalence reduction

This is a relatively quick course. It includes four hours of video instruction, so you could complete the class in one sitting if desired. There are online articles and downloadable resources as well. These will help you grasp the topics better.

This short online math course also comes with a certificate of completion.

**Points**: 9

**Affordability**: $11.99

**Accessibility**: High. Easily access the course online anytime of day or night. You get full lifetime access via mobile or TV. There are downloadable resources and online resources to help you as well.

**Learning Level**: Intermediate

### #7: MA111: Introduction to Mathematical Reasoning

This free beginner course is available from Saylor Academy. The course was designed to serve as a bridge from introductory math courses to more advanced math courses. Because some advanced math requires out-of-the-box thinking, this is a good opportunity to use mathematical thinking skills.

Some of the topics you’ll explore in this free mathematical thinking course include:

- Propositional logic
- Elementary set theory
- Principles of counting

More importantly, you’ll learn how to prove a mathematical proposition. Additionally, you’ll work with puzzles and explore how reasoning can be used to help you finish them.

There are eight units in this course. Each unit covers a different topic in mathematics. Most students require about 112 hours of study to complete the course. It is self-paced, though, so you can take as much time as you need.

**Points**: 9

**Affordability**: Free

**Accessibility**: High. This course is completely free and accessible anytime. All learning materials are linked directly in the course for easy access.

**Learning Level**: Beginner

### #6: Algorithmic Thinking (Part 1)

Algorithmic Thinking is a two-part course from Rice University. It is an intermediate class and is available on Coursera.

Since computer scientists work with abstract problems, mathematical thinking is required. In this context, that involves learning about algorithmic thinking. This type of thinking allows you to build simpler solutions to computational problems. Those solutions are also more efficient.

Topics of study in this course include:

- Graph theory
- Graph algorithms
- Python

The biggest part of this top math course is learning how to implement graph algorithms in Python. From there, you will use the algorithms to analyze real-world data sets. The point of this is to help you understand how algorithms and the structures of data sets interact with one another.

You need some experience in Python programming in order to take this course. You should also have a good background in math. This should include algebra and pre-calculus.

**Points**: 9

**Affordability**: Free to enroll. A fee (usually around $50) might be required. Other costs are involved for specializations (usually $39-$79 per month). Learn more about Coursera pricing. This is scored as a paid course.

**Accessibility**: High. You can start this course any time. It comes with video lectures to assist you in learning. The videos are offered with subtitles in multiple languages to increase accessibility for non-English speaking learners.

**Learning Level**: Intermediate

### #5: Algorithmic Thinking (Part 2)

In part two of the Algorithmic Thinking course from Rice University, you will continue to learn how to solve abstract computational problems.

Part two focuses on advanced algorithmic techniques. This includes dynamic programming and divide-and-conquer.

Other topics that you’ll study in this online math course include:

- Algorithmic efficiency
- Dynamic programming
- Python programming

You’ll need some training in Python before taking this class. And while Python is the programming language of choice for this course, the skills you will learn are applicable to other programming languages.

This is a short course. Most students complete it in about 12 hours. If you need to quickly learn about mathematical thinking as it’s applied to computer science, this is a good option.

**Points**: 9

**Affordability**: Free to enroll. A fee (usually around $50) might be required. Other costs are involved for specializations (usually $39-$79 per month). Learn more about Coursera pricing. This is scored as a paid course.

**Accessibility**: High. Begin your studies at any time. The course is available 24/7 as well. Video lectures are available with subtitles in 12 languages.

**Learning Level**: Intermediate

### #4: Introduction to Computational Thinking with Julia

This beginner course is from the Massachusetts Institute of Technology. It revolves around computational thinking. You will learn how to apply computational thinking principles using the Julia programming language.

Computational thinking involves a lot of mathematical thinking principles. This includes using conjecturing, generalizing, explaining, and other tools to solve problems.

Specifically, this free mathematical thinking course will cover:

- Data science
- Mathematical models
- Artificial intelligence

You will also explore ways to apply computational thinking for responding to COVID-19.

Since this is an introductory course, there are no prerequisites. However, you must have access to a computer with the Julia programming language in order to complete class assignments.

**Points**: 9

**Affordability**: Free

**Accessibility**: High. This course is available to anyone, anytime. The course comes with multiple resources for learning, all of which are available online.

**Learning Level**: Beginner

### #3: Computational Thinking and Big Data

Mathematical thinking helps you solve novel problems. In computer science and other industries, it can be used to develop computational thinking skills as well. This beginner course from edX and the University of Adelaide will help you do just that.

Computational thinking helps you devise a problem and express a solution in a manner that a computer can effectively solve it. In this course, you will learn how this method of thinking can be applied to data science.

Some of the concepts you will study in this free online mathematical thinking course include:

- Pattern recognition
- Abstraction
- Decomposition

You will also learn what algorithmic thinking is. You’ll learn how to apply that type of thinking to solve problems as well.

This course requires you to use tools like Java and R. Some familiarity with them before taking this course will be very helpful.

**Points**: 9

**Affordability**: Free

**Accessibility**: High. This course is accessible immediately upon enrollment. All course materials are available online. There are no official prerequisites, so you can take this course without any experience in computational thinking or big data.

**Learning Level**: Beginner

### #2: Introduction to Mathematical Thinking

This is a beginner course from Stanford University. The primary goal of the class is to help you improve your ability to think mathematically.

To achieve this goal, you will learn to think outside the box. This is done through lectures and problem-based assignments. These lectures and assignments will challenge the way you think about math and enable you to develop critical problem solving skills.

The topics of study in this class include:

- Logical Combinators
- Implications
- Quantifiers
- Proofs

Additionally, you’ll learn about elements of number theory. There is also a unit on real analysis.

This top math online class can be taken as an eight-week basic course. There is also a ten-week extended course available. The basic course is intended for people who are interested in boosting their analytical thinking abilities. The extended course is designed for first-year college students that are majoring in math or a math-related subject.

**Points**: 9

**Affordability**: Free

**Accessibility**: High. This course is available anytime day or night. Video transcripts are available in English and Portuguese. The course is offered from Stanford as well as Coursera.

**Learning Level**: Beginner

### #1: Probability – The Science of Uncertainty and Data

This is an advanced course from the Massachusetts Institute of Technology. It is available on edX.

This best class for mathematical thinking focuses on the uncertainty of data. You will learn how to apply principles of mathematical thinking (like solving novel problems) to analyze data and make predictions.

One of the most important parts of this course is probabilistic modeling. This involves using the language of math (another mathematical thinking tool) to create models.

You will also learn about:

- Laws of large numbers
- Random processes
- Multiple discrete variables

This best free math course can be taken by itself. It is also offered as part of the MITx Micromasters Program in Statistics and Data Science.

This is an instructor-led course. This means there is a course schedule with specific due dates. Additionally, there are prerequisites that have to be met. This includes having taken college-level calculus.

**Points**: 11

**Affordability**: Free

**Accessibility**: Moderate. This course is available 24/7 online. It’s accessible from computers, smartphones, and other internet-enabled devices.

**Learning Level**: Advanced

## Frequently Asked Questions

**What is mathematical thinking?**

Perhaps the easiest way to explain mathematical thinking is by talking about what you do with it.

Mathematical thinking is a process. Unlike the type of math you learn in school, mathematical thinking seeks to help you identify ways you can use math to solve out-of-the-box problems.

So, the process of mathematical thinking involves using resources of mathematical knowledge. Combined with that is the use of heuristic strategies that can be leaned on to solve problems. Some mathematicians explain that the control you exert over the math problem and your beliefs about math also play a part in mathematical thinking.

In other words, it’s not just about what you know about math. Mathematical thinking is largely dependent upon what you do with that knowledge and what you believe you can do with that knowledge.

Mathematical knowledge and heuristic strategies are things we learn in math courses throughout our educations. But the control and beliefs parts are tougher to teach.

Control refers to one’s ability to use math knowledge to guide the problem solving process in the right direction. So not only do you need math knowledge, but you also need to know how to use it to achieve the desired end.

But exerting control isn’t all you need to fully harness the power of mathematical thinking. You also must believe in your abilities and in the value of math. That is, if you believe that math is useful, you’ll be more likely to use the knowledge you have about math to solve math problems.

Other mathematicians explain that mathematical thinking involves other components as well. For example, mathematical thinking relies on effective communication. You need to be able to write out the problem or show your work. You might also need to be able to verbally explain the process you used to solve the problem.

Additionally, possessing good organizational skills is a huge part of mathematical thinking. After all, it’s hard to solve complex, novel math problems if you can’t organize the knowledge you have about math. You must be able to organize the work you do to solve the problem as well.

As part of mathematical thinking, there are several processes that mathematicians utilize. This includes:

- Generalizing
- Specializing
- Convincing
- Conjecturing

When you think mathematically, you might generalize to look for patterns or relationships that could help you solve the problem at hand. But other times getting specific or specialized, and trying special cases will be more helpful for solving the problem.

Convincing is another part of mathematical thinking. There are times when you need to find reasons why something is true. You then must communicate those findings to others to try to prove that that thing is true. Yet, mathematical thinking also requires that you be able to conjecture. Predict relationships. Guess about the results. These are also key parts of learning how to think mathematically.

Another way of thinking about “what is mathematical thinking?” is to understand that the math you learn in school focuses more on these processes. What they don’t emphasize is how to use those processes to solve unfamiliar problems.

For example, top mathematical thinking courses might ask you to use the following steps to solve a novel problem:

- Break down the problem into smaller parts
- Identify problems that are similar that might help you solve the current problem
- Identify appropriate skills and knowledge needed to solve the problem
- Identify assumptions that can be make
- Select a strategy for problem-solving that is appropriate
- Think about other methods that could help you solve the problem
- Look for patterns
- Create examples

So, as you can see, the process of mathematical thinking involves a lot. On the one hand, there’s the math knowledge we discussed earlier. But on the other hand, it also involves a lot of thinking, considering, analysis, and so forth.

**What are the types of mathematical thinking?**

In the previous question and answer, we discussed some of the processes involved in mathematical thinking. The types of mathematical thinking are built upon these processes.

There are many different types of mathematical thinking. There are many different applications of mathematical thinking as well. But we can narrow these types down to three primary types of mathematical thinking.

The first type is spatial reasoning. This type of mathematical thinking involves the ability to imagine objects in different ways. For example, spatial reasoning might allow you to imagine a trapezoid from various angles. This is a valuable type of thinking to have for studies in geometry.

The second common type of mathematical thinking is computational reasoning. This type of thinking pairs math knowledge with logic. Together, these skills are used to consider various options for problem-solving. The result is the ability to map out possible methods for solving a problem and maximizing the likelihood of reaching the desired outcome.

Of course, mathematical thinking involves a lot of logic. Likewise, scientific reasoning is a type of mathematical thinking. Having the ability to observe and classify data is necessary. Being able to hypothesize about different outcomes and test those hypotheses in experiments is crucial as well.

The ability to carry out scientific reasoning stems from mathematical thinking. You use logic. You learn how to investigate and explore different options. You learn how to observe and make “if-then” analyses. All of this results in improved inductive and deductive reasoning. These types of reasoning aren’t just important for mathematical thinking. They are also important life skills to possess.

**Are there free online courses for mathematical thinking?**

Yes. As noted in our rankings, there are many different free online courses. In fact, the majority of the top mathematical thinking courses on our list are free. Likewise, many of the most popular mathematical thinking courses are free. This is why free courses earn more points in our rankings.

However, as explained in the methodology section of this article, the affordability of a course is just one of several important factors that can be used to evaluate a course. While saving money and taking free courses is a good thing, you also have to be sure that the course you’re taking will help you meet your goals.

For example, if you’re a high school math student that wants to learn how to think mathematically, a free course might help you do that. But let’s say you’re a math teacher. Let’s also say that you want to learn how to teach others how to think mathematically. If there are no free courses to help you achieve that goal, you’ll need to look at paid course options.

In other words, don’t take popular mathematical thinking courses simply because they’re free! Instead, make sure that the courses you take fit your specific needs. This means evaluating factors other than affordability, including learning level and accessibility.

**How can I improve my mathematical thinking?**

Ask any mathematician or math teacher, and they’ll tell you that one of the best things you can do to improve your mathematical thinking is to work on your general problem solving skills.

More specifically, learning how to solve non-routine problems is key. While many students in school are taught math as a way to solve common problems, the true task of mathematical thinking is to think outside the box. It’s important to develop basic math skills. But understanding how to apply what you know to solve problems you’ve never encountered before is the real goal.

So, how do you improve your problem solving skills?

For starters, it takes practice. Getting better at problem solving isn’t a skill that randomly develops. You need to learn how to problem solve first. Then you need to practice using those skills in a variety of situations – math problems and in life in general.

Modeling is a big part of this. If you’re a student, having a math teacher walk you through the process of problem solving and learning to think mathematically is critically important.

So, for example, let’s say you’re in a math class and your teacher is working with you on building mathematical thinking skills.

The first thing they might do is walk you through a new problem. This is an instructor-led task. They might read the problem aloud to the class. They might go over any math-related vocabulary in the question, that way you know what the words mean. They might also identify the different types of math skills needed to solve the problem. It’s a good idea to think about certain questions as well, like:

- What is the question asking?
- What information do they want?
- What information do I already have?

Then when it’s time to work through the problem, your teacher might model how they would solve it. Doing so requires them to show their work. This can be done with words, pictures, and numbers, of course!

This step-by-step process is used by teachers all over the world to help their students develop mathematical thinking skills. Again, it’s a process, and by modeling that process, teachers get their students to recognize what it looks and feels like to think mathematically.

If you aren’t a student, you can follow this same process on your own. Carefully read the question. Then identify any terms in the question that you don’t know and look them up. Lastly, think about what skills you need to work through the problem. Show your work, and analyze your work to revisit how you were able to solve the problem.

Of course, mathematical thinking can be difficult to develop. If you’re learning on your own, you’ll likely need more direction. That’s where online courses for mathematical thinking come in.

Taking courses like the ones listed in our rankings is a great way to build your mathematical thinking skills. These classes give you a chance to work at your own pace (in most cases) yet still have guidance from an instructor to help you along.

More than that, the top mathematical thinking courses provide an opportunity to practice these skills. Like developing any skill, mathematical thinking requires practice. Lots of practice.

While some people are very good about practicing a new skill on their own, others need some structure. If you’re one such person, even a self-paced course might give you the structure you need to successfully become a mathematical thinker.

**What degrees and certificates have mathematical thinking courses?**

Just about any degree or certificate in math should include some studies in mathematical thinking. The emphasis on mathematical thinking will vary from one school to the next. So will the number of mathematical thinking courses that are offered.

However, if you pursue a degree in math, chances are that you’ll have an opportunity to take at least one course on mathematical thinking.

The most common undergraduate degree in math is a Bachelor of Arts. Some schools offer a Bachelor of Science degree in math. The difference between the two is that a Bachelor of Arts requires a few less STEM-related classes. That is, science, technology, engineering, and math classes. It is more focused on a liberal arts curriculum. A Bachelor of Science degree is also usually more research-oriented.

Of course, there are Master of Arts and Master of Science options. Ph.D. programs in math are also quite common.

Regarding specific math degrees that have mathematical thinking courses, you might consider the following as options:

*Applied Mathematics*– This degree is common for students that want to pursue a job in science, computer technology, or engineering. The focus is on learning how to apply mathematical principles to real-life problems.*Statistics*– Statistics is all about quantitative reasoning. A major in statistics will teach you how to collect data, analyze it, and interpret it. Possessing mathematical thinking skills will allow you to do so with greater ease.*Computer Science*– Computer science degree programs focus on using computers and computer-based tools to solve problems. Mathematical thinking courses are valuable for this degree because mathematical thinking helps you work through novel problems.

Much like degrees in math, there are many opportunities to get a certificate in math. Many certificate programs include some level of training in mathematical thinking.

Again, there are undergraduate and graduate certificate options for math. There are also teaching certification options. These programs provide you with the necessary knowledge of math, but also provide you with opportunities to learn how to be a teacher.

Mathematical thinking courses in teacher education programs are becoming more and more common. If you want to learn how to think mathematically and you have a desire to teach others, this could be a good fit for you.

*Disclaimer: Some courses may include an affiliate link. Courses were chosen first based on the methodology with affiliate links only added after the ranking was complete.*

OCR Staff

February 2021

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