Investigative Learning and Investigative Mathematics – when curiosity drives learning

Curiosity as the engine of learning

In today's school, it is easy to get caught up in standardized methods, ready-made answers and measurable performance. Students are often given assignments where the goal has already been set, and the teaching is guided by the results rather than questions. But what happens if we instead turn the perspective around? If students are allowed to discover, investigate and create their own solution strategies, the result is something more than just a correct answer – it is understanding and genuine learning.

Investigative Learning is about putting curiosity and inquiry at the center. Here, the students get to formulate their own questions, collect data, test hypotheses and construct reasoning. It is a learning where process and thought development are just as important as the answer itself. Investigative Mathematics takes this further to the world of mathematics, where students create strategies, discover patterns and build mathematical models, without throwing away basic skills. On the contrary, they become even more important as tools for exploration and analysis.

I intend to show how Investigative Learning and Investigative Mathematics work in practice, why they are relevant in today's school environment, and how they differ from other pedagogical methods that are often confused with investigative learning.

1. What is Investigative Learning?

Investigative Learning is a form of learning in which students not only receive knowledge but also create it through active inquiry. The core of the method consists of students:

1.              Formulating his own questions

2.              Investigating phenomena and situations

3.              Collecting and interpreting data

4.              Combining observations with hypotheses

5.              Building reasoning and models

The teacher functions here as a supervisor and co-investigator, not as one who delivers answers. The focus is on process and thinking rather than on ready-made solutions.

Investigation is central – students get to start with the phenomenon first, and formulate questions that then lead them on to mathematical or scientific concepts. In other words, students get to experience real-world principles before they are given a formal definition, which increases understanding and creates motivation.

2. The difference from other pedagogical "based" methods

There are several teaching methods that are sometimes confused with Investigative Learning, but there are clear differences:

2.1 Project-Based Learning (PBL)

1.              Focus: a project or end result

2.              Students work towards a finished goal

3.              Investigation is often guided towards the project's requirements

In Investigative Learning, the process is more important than the result, and the students' own questions steer the direction.

2.2 Inquiry-Based Learning

1.              Focus: the student's questions, but often structured by the teacher

2.              The teacher often plans the questions in advance

Investigative Learning differs in that the students' actual curiosity is the starting point, not the teacher's finished question.

2.3 Problem-Based Learning

1.              Focus: solving a defined problem

2.              The problem is often already defined by the teacher

Investigative Learning begins with a phenomenon or idea that students themselves explore to formulate problems and hypotheses.

2.4 Evidence-Based Teaching

1.              Focus: methods with documented effect

2.              Prioritizing measurable results

Investigative Learning focuses on thinking and understanding, not just on what can be measured.

In summary, where other methods steer towards goals, problems or results, Investigative Learning steers towards mindset and curiosity.

3. Why is Investigative Learning needed?

Three main reasons justify the use of this pedagogy:

3.1 The brain learns through investigation

Neurological research shows that knowledge is better stored when students themselves construct understanding. Passive reception of information often leads to short-term learning, while active inquiry creates deep, lasting understanding.

3.2 The modern world requires critical thinking

At a time when facts are easily accessible, the ability to analyze, reason, and model is becoming  more important than just being able to memorize information.

3.3 Curiosity is a strong driving force

Children are born scientists: they test, compare, observe and question. Schools often risk dampening this natural desire to investigate. Investigative Learning restores the right of students to follow their questions, which in itself increases motivation and the quality of learning.

4. Investigative Mathematics – mathematics as investigation

Investigative Mathematics means that students don't just learn mathematics as a skill, but create mathematics through inquiry. Here, procedures and formulas are not at the centre, but strategies, reasoning and model creation.

Students get to discover for themselves:

1.              connection

2.              pattern

3.              Generalizations

4.              Features and models

5.              Mathematical concepts

Instead of memorizing algorithms, they learn to think mathematically, which strengthens understanding on a deeper level.

5. Basic knowledge is key

There is a misconception that investigative mathematics means abandoning basic skills. On the contrary: basic knowledge is the tools that make investigation possible.

Students need to be able to:

1.              Count correctly

2.              Interpreting data and tables

3.              follow logical chains

4.              express reasoning clearly

Basic knowledge is the language of the investigation, while the investigative process is the story that language expresses.

6. Example: Investigation of linear functions

Traditional teaching often starts with the formula y = kx + m, and students practice calculations. Investigative Mathematics instead begins with a phenomenon:

"A bus ticket costs €2 to get on and €0.15 per kilometre. How does the price change?"

Students:

1.              Testing different distances and prices

2.              Creating tables and graphs

3.              Analysing the rate of change

4.              Discusses start-up costs and slope

Gradually, they discover the concepts of slope (k), intersection (m), and proportionality, not as abstract formulas, but as something that describes reality.

7. Example: Quadratic function investigation

Quadratic functions open up for more creative investigations. A concrete example:

"How high does a ball bounce?"

Students film the ball, measure height and time, and draw diagrams. They quickly discover that the movement is not linear, but curved. Questions that arise:

1.              Where is the vertex?

2.              Why is the curve symmetrical?

3.              How do different factors affect the shape of the curve?

Students discover the parabola, vertex and symmetry line through active investigation. Strategies and models are created by the students themselves, which strengthens both understanding and creativity.

8. AI as a co-investigator

In today's classrooms, AI can become a powerful co-investigator:

1.              AI can ask questions that students have not thought of themselves

2.              Visualize data, simulate scenarios, and generate graphs

3.              Suggest alternative strategies and methods

AI does not replace thinking – it reinforces and stimulates. By using AI, students can focus on analysis, reflection, and strategic reasoning instead of routine calculations.

9. The Role of the Teacher in Investigative Learning

In this pedagogy, the role of the teacher changes:

1.              The teacher becomes a supervisor and co-investigator

2.              Creates situations where questions arise

3.              Follow the students' tracks and listen to their thoughts

4.              Helps formulate models and hypotheses

5.              Encourages students to reflect and motivate their ideas

The teacher is present and guiding, but not the one who dictates answers.

10. Basic principles for success

In order for Investigative Learning to work, the school needs:

1.              Allow time and space for investigation

2.              Encourage mistakes as learning

3.              Support students' questions and hypotheses

4.              Integrate foundational knowledge as tools

5.              Using technology such as AI as support, not compensation

When these principles are in place, curiosity, creativity, and critical thinking grow in the students.

11. Curiosity as a driving force

Curiosity is the strongest motivator in learning. Children are natural scientists; They test, observe and experiment. Investigative Learning gives students the right to follow their questions, making learning both meaningful and lasting.

When students investigate for themselves:

1.              Deepening understanding

2.              Motivation and commitment increase

3.              Creativity and problem-solving skills are developed

4.              Basic knowledge is actively used and strengthened

12. The Role of Inquiry in Mathematics

In mathematics, it is not enough to know procedures. Investigative Mathematics teaches students to:

1.              Interpreting and analyzing data

2.              Discovering patterns and correlations

3.              Building models

4.              Formulate strategies

5.              Reason and explain mathematical relationships

This is the key to deep mathematical understanding, far beyond memorization.

13. Examples of Investigative Questions

In the classroom, questions can be open-ended and exploratory:

1.              What happens if the starting value changes?

2.              How do different parameters affect the shape of the graph?

3.              Are there situations where the model does not work?

4.              How can we test our hypotheses in reality?

Such questions drive inquiry, reasoning, and creative thinking.

14. Exploratory Learning in Practice

Practical steps in teaching may include:

1.              Present a phenomenon or problem

2.              Let the students formulate their own questions

3.              Collect and analyze data

4.              Building models and strategies

5.              Test, revise and reflect

6.              Discuss conclusions and reasoning

This cycle encourages active thinking and makes learning meaningful and relevant.

15. Conclusion

Investigative Learning and Investigative Mathematics reintroduce curiosity into the classroom. Students become active thinkers who:

1.              Formulating questions

2.              Creating strategies

3.              Analyzing data

4.              Building models

5.              Using AI as a co-investigator

6.              Deepens and applies basic knowledge

This pedagogy shows that learning can be exploratory, creative and engaging, far beyond answers and ready-made answers. Curiosity is not just a tool – it is an engine and a goal in itself.

In a world where information is easily accessible, the ability to think, analyse and create understanding is  the greatest competence students can develop. Investigative Learning gives students just that – and gives the classroom a human, creative and meaningful dimension.