# Mathematical thinking: A big idea for professional development.

Mathematical thinking is central to deep and sustainable learning of mathematics. Taught ideas that are understood deeply are not just 'received' passively but worked on by the student. They need to be thought about, reasoned with and discussed.

Mathematical thinking involves:

Looking for patterns

Looking for relationships

Reasoning logically, explaining, hypothesising and proving/disproving

In this blog post, we will explore these big ideas in more detail and provide some examples of how they can be implemented in the classroom. Download our free guide on how to implement mathematical thinking in your classroom!

### Looking for Patterns

One of the most important skills for mathematical thinking is the ability to look for patterns. Patterns can be found in all aspects of mathematics, from number sequences to geometric shapes. When students are able to identify patterns, they can begin to make predictions and generalizations about mathematical concepts.

For example, students might be asked to look for patterns in a number sequence. The sequence might be 1, 2, 3, 5, 8, 13, 21, 34, ... Students can begin to make predictions about the next number in the sequence by looking for patterns. They might notice that the numbers are increasing by adding the previous two numbers together. This allows them to predict that the next number in the sequence is 55.

### Looking for Relationships

Another important skill for mathematical thinking is the ability to look for relationships. Relationships can be found between numbers, shapes, and other mathematical concepts. When students are able to identify relationships, they can begin to understand how different mathematical concepts are connected.

For example, students might be asked to look for relationships between different shapes. They might notice that all triangles have three sides and three angles. They might also notice that all squares have four sides and four right angles. This allows them to understand that there is a relationship between triangles and squares.

### Reasoning Logically

Reasoning logically is another important skill for mathematical thinking. When students are able to reason logically, they can use their knowledge of mathematical concepts to solve problems. They can also use their reasoning skills to explain their thinking to others.

For example, students might be asked to solve the following problem:

There are 12 apples in a basket. If you take away 3 apples, how many apples are left in the basket?

Students can use their knowledge of subtraction to solve this problem. They can also use their reasoning skills to explain their thinking to others. They might say something like, "I started with 12 apples. I took away 3 apples. That means there are 9 apples left in the basket."

### Explaining, Conjecturing, and Proving

Explaining, conjecturing, and proving are all important skills for mathematical thinking. When students are able to explain their thinking, they are able to communicate their ideas to others. When they are able to conjecture, they are able to make predictions about mathematical concepts. When they are able to prove, they are able to provide evidence to support their claims.

For example, students might be asked to explain why the sum of the interior angles in a triangle is always 180 degrees. They might say something like, "I can prove that the sum of the interior angles in a triangle is always 180 degrees by drawing a triangle and using my knowledge of angles."

### Implementing Mathematical Thinking in the Classroom

There are many ways to implement mathematical thinking in the classroom. One way is to use open-ended problems. Open-ended problems are problems that do not have a single correct answer. They allow students to explore different mathematical concepts and to think creatively about how to solve problems.

Another way to implement mathematical thinking in the classroom is to use manipulatives. Manipulatives are objects that students can use to represent mathematical concepts. They can help students to visualize mathematical concepts and to make connections between different mathematical ideas.

Finally, it is important to create a classroom environment that encourages mathematical thinking. This means providing students with opportunities to discuss their thinking, to share their ideas, and to make mistakes. It also means providing students with feedback on their work and helping them to improve their mathematical thinking skills.

### Conclusion

Mathematical thinking is an essential skill for all students. It is important for students to be able to look for patterns, look for relationships, reason logically, explain, conjecture, and prove. These skills can be implemented in the classroom by using open-ended problems, manipulatives, and by creating a classroom environment that encourages mathematical thinking.

Download our free staff meeting / professional development guide on how to implement mathematical thinking in your classroom!