1.1: Thinking Mathematically

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Thinking Like Mathematicians

Mathematical thinking is more than just memorizing formulas or getting the right answer—it's about exploring ideas, understanding why things work, and making connections. As future elementary school teachers, you'll help young children learn what it means to think mathematically, and that goes beyond simply following steps. You'll help them discover that math is about curiosity, patterns, problem-solving, and creativity (CDE, 2013; NCTM, 2020; Seda and Brown, 2021).

Imagine your students asking, "What is math really about?" or "What does it mean to be good at math?" Mathematical thinking involves reasoning, making sense of numbers, and understanding the relationships between different ideas. It’s about encouraging children to explore, take risks, and see mistakes as opportunities to learn—all while focusing on the why behind each concept (Boaler, 2022)

As teachers, your role will be to foster this kind of thinking in your classrooms. This means shifting away from just teaching procedures to helping students develop a deep understanding of concepts—what we call moving from procedural to conceptual mathematics. By encouraging young learners to engage deeply, to make sense of math, and to see it as a tool for understanding the world around them, you’re not only teaching math—you’re teaching them how to think, play, wonder, and explore.

To embody these ideas in our class, you need to be the kind of student you want to have. Cultivate the math mindset that is needed not only to be successful in learning math but also in teaching it. Embrace curiosity, be open to making and learning from mistakes, and engage deeply with the concepts we explore. By modeling a positive attitude toward math and a willingness to take risks, you are preparing yourselves to inspire the same mindset in your future students.

Think About and Discuss

As we dive into this class, I encourage you to consider what your mathematical experiences have been. What emotions or memories surface when you think about learning math as a child? Were you encouraged to ask questions, to make mistakes, to explore? Or were you expected to memorize and follow procedures?

What can we do to make math feel less like a set of rigid rules and more like a powerful tool for understanding and interacting with the world?

How did your past experiences shape your current beliefs about what it means to be “good at math”?
In what ways can you help your future students develop a growth mindset toward mathematics, seeing struggle and mistakes as natural and essential parts of learning?
What does it mean to create an equitable math classroom—where every student feels seen, heard, and capable?
How might mathematical thinking help students make sense of the world beyond the classroom?

Let’s start this journey not just by thinking about how we’ll teach math, but why—and for whom.

Before reading what the key aspects of mathematical thinking are, let's do some math.

Wonder, Play, Grow

Count the dots and be prepared to share how you counted them.

$"A pattern of 15 evenly spaced blue circles arranged in a grid-like formation with a missing section in the middle, creating an abstract design."$

🧠Let's Listen to Learn❤️

Now Add 17 + 18, how did you do it?

Take a look at the Video below from Math Celebrity Howie Hua. Compare your ideas with his and see how all of this connects to what we mean by mathematical thinking.

Mathematical Thinking

Now that you have had a chance to play with mathematics in a flexible way, here are some key aspects of what it means to think mathematically:

1. Reasoning and Sense-Making

Mathematical thinking involves making sense of problems and persevering in solving them. This means being able to:

Analyze problems to understand what is being asked.
Explore different strategies for solving a problem.
Justify and explain why a solution makes sense.
Question assumptions and evaluate the reasonableness of results.

For example, if a student encounters a word problem, thinking mathematically means first understanding the relationships in the problem (not just looking for key words to "grab numbers") (CDE, 2013).

2. Recognizing Patterns and Structures

Mathematical thinking involves identifying patterns, structures, and relationships in numbers, shapes, and data. It includes:

Seeing connections between ideas (e.g., recognizing that division and fractions are related).
Noticing repeated reasoning (e.g., understanding why the distributive property works across many examples).
Using structure to simplify problems (e.g., factoring expressions or breaking numbers into smaller, more manageable parts for mental math).

This habit of seeing structure allows mathematicians to generalize and extend their understanding to new situations (Boaler, 2022)

3. Abstract and Quantitative Thinking

Mathematical thinking requires students to move between concrete examples and abstract ideas. It involves:

Representing situations symbolically, such as using equations, graphs, or diagrams.
Understanding what quantities mean in context and interpreting their relationships (quantitative reasoning).
Making sense of abstract ideas such as infinity, variables, or probability.

For example, when analyzing a real-world situation, students should be able to translate it into a mathematical model and interpret their results back in context (CDE, 2013).

4. Problem-Solving and Creativity

Mathematical thinking is inherently creative. It involves:

Exploring multiple strategies to solve a problem.
Trying new approaches if the first attempt doesn’t work.
Applying knowledge in novel or unfamiliar situations.

Students should feel empowered to take risks, make mistakes, and learn from them as part of the process of developing deeper understanding (Boaler, 2022).

5. Communication and Justification

A crucial aspect of mathematical thinking is being able to articulate your reasoning. This includes:

Explaining why a particular solution is correct.
Sharing and discussing ideas with peers.
Writing arguments using precise mathematical language and notation.

For example, a student solving an equation should not only find the solution but also explain why their process works (NCTM, 2020)

6. Applying Mathematics to the Real World

Thinking mathematically includes applying math concepts to interpret and solve real-world problems. This involves:

Modeling situations using appropriate tools (e.g., graphs, equations, simulations).
Making decisions based on data and quantitative reasoning.
Understanding the relevance of math in fields like science, technology, and finance.

For instance, understanding percentages and proportional reasoning helps students analyze topics like interest rates, population growth, or voting statistics (CDE, 2013).

7. Embracing Productive Struggle

Mathematical thinking requires a growth mindset—believing that ability in math develops through effort and learning. It involves:

Persevering when problems seem challenging.
Viewing mistakes as opportunities for learning.
Building confidence in tackling complex ideas over time.

Students who think mathematically recognize that struggle is part of the learning process and a key to growth. (CDE, 2013; NCTM, 2020)

8. Using Tools and Technology Strategically

Mathematical thinking also includes selecting and using tools appropriately, such as calculators, graphs, manipulatives, or software, to support understanding and problem-solving (CDE, 2013) This means:

Deciding when to use a tool and interpreting the results meaningfully.
Using tools to experiment and explore concepts.

Wonder, Play, Grow

Use any method you can think of to solve this problem.

Math $\PageIndex{1}$

Think About and Discuss

Now that you have a process to solve the problem, discuss with your peers how they solved it. Then, think about how a 5th grader might approach the problem. Once you're ready, play the video below to see how it went and explore the aspects of mathematical thinking involved in her solution.

5th Grader Solving the Hearts Problem

https://vimeo.com/user7358057/review...961/a5004a5990

pw: cmlvideos

How to Foster Mathematical Thinking

As we can see, students have the extraordinary ability to think mathematically. As future teachers and caregivers for children, what can we do to help students develop this mathematical mindset? How can we foster productive mathematical thinking so that students begin to develop a positive identity about their ability to do math mathematics.

Imagine what this would look like, sound like, and feel like in a classroom. The California Mathematics Framework emphasizes creating classrooms where students are encouraged to:

Ask questions like "Why?" and "How do you know?"
Share their thinking and learn from peers.
Engage in rich tasks that require reasoning and creativity.
Build connections between different mathematical ideas and real-world contexts.

Mathematical thinking is not just about getting the "right answer"; it’s about cultivating habits of exploration, reasoning, and reflection that enable students to approach challenges with curiosity and confidence (CDE, 2013)

Doing Mathematics - Standards of Mathematical Practice

Future public school teachers need guidance as to what productive mathematical thinking looks like and what it means for students TO DO mathematics. This guidance comes in the form of The Standards for Mathematical Practice. This guidance describes varieties of expertise that mathematics educators at all levels should seek to develop in their students. Take a look below. But do not be overwhelmed, we will highlight these practices as we go in our class together. The goal is for you to recognize when you are doing these things and start to imagine what this looks like when you have your own students.

Math $\PageIndex{2}$

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