How to Teach Math Using Inquiry-Based Learning
Teaching math has slowly been evolving from a subject that, for many students, feels useless and trivial, to one that helps provide relevance and build important learning skills inside and outside of the classroom. Inquiry learning helps provide students with opportunities to develop critical thinking, teamwork, and resilience – skills that are absolutely essential to be “good” at math.
Teaching math using inquiry-based learning is a powerful way to reinforce the skill of problem-solving for students. In the inquiry math classroom, students are not passive recipients of formulas and facts, but are instead encouraged to ask questions, investigate rules and patterns, and persevere when confronted with challenges. Using an inquiry approach in math helps build important skills such as resilience, problem-solving, and perseverance.
The post below outlines some ways to incorporate an inquiry-based approach into your math classroom (or within your lessons). It also provides some examples of inquiry and PBL projects to help your students gain confidence, build good learning habits, and apply their skills.
Traditional vs. Inquiry-Based Approaches
There’s been a lot of debate about using a “back to basics” traditional approach or an inquiry-based approach when teaching math. Traditional math teaching typically follows a structured curriculum with a focus on rote memorization, guided procedures, and step-by-step instructions, which many would describe as providing a solid foundation for students to build mathematical skills. On the other hand, inquiry-based math emphasizes exploration, critical thinking, and problem-solving through open-ended tasks. Neither of these approaches are wrong.
In my own classrooms, I’ve often opted for a combination of approaches. Recognizing the strengths of each method and using them complementary to each other has resulted in the best outcomes for my students in the past. Traditional math works very well for students who need the reliability of steps and formulas to grasp numerical concepts and skills. Inquiry-based methods work well for students when they need to apply these concepts and skills.
Encouraging students to apply their skills and understanding in the “real world” and helping them learn to make connections between concepts allows them to develop a deeper understanding of math. Combining the two approaches can create a balanced and well-rounded math education, and equip students with not only the procedural knowledge they need, but also the necessary problem-solving skills to apply this knowledge in other contexts.
Why should you use an inquiry approach in math?
For most of us, we’re used to math being presented as a set of rules and procedures we follow to arrive at a correct answer. However, this is usually far from the way math is applied in the real world. A good example is when scientists and climatologists are studying weather patterns – they might ask something like, “how much rain can we expect from this hurricane?”. From there, meteorologists and atmospheric scientists combine data to create models to help approach the question. They apply their understanding of mathematical principles to situations to determine the correct answer, then see if their solution applies in the real world. If not, it’s back to the drawing board.
Framing math in this kind of way, as opposed to the traditional “memorize the formula, practice some questions, then repeat”, can be far more effective. Some of the benefits include:
- Emphasizing the process as opposed to the final product
- Finding several different approaches and choosing the one(s) that work for you
- Seeing math from a more holistic viewpoint
- Teaching students that mistakes are where we learn the most
- Reducing the stigma that math is “hard”, “useless”, or “irrelevant”
- Building critical thinking habits and resilience to persevere through difficulties
- Having the ability to transfer skills to all types of mathematical problems
- Learning more deeply and in a more problem-based way
Below are some unique ways to incorporate an inquiry focus into your math lessons. These ideas can be used as stand-alone projects or as complements to your traditional lessons. They also work well as projects that call for the application of new learning.
Math Inquiry Idea #1 – Urban Planning and Sustainable Cities
For this inquiry idea, students play the role of an urban planner tasked with designing a new layout for a sustainable city. To start, discuss the concepts of urban planning and sustainability.
Ask students how the two are related and ask them what characteristics a sustainable city has. Some characteristics include:
- Reliance on renewable energy (such as wind, solar, and smart grid technologies)
- Plenty of green spaces and biodiversity
- Examples of urban agriculture, such as community gardens
- Climate change adaptation
- Efficient transportation and waste management services
- Mixed-use development
- Community engagement and participation
Spend some time discussing and envisioning what a sustainable city would look like. Allow them to utilize videos, books, and maps to explore their ideas.
- Green City: How One Community Survived a Tornado and Rebuilt for a Sustainable Future (hardcover, ages 5-8)
- Sustainable Development Goals (PDF from the United Nations for your learning wall)
- How Cities Work (Lonely Planet hardcover/interactive book)
- Climate Change and Sustainable Cities (NRDC)
Provide students with resources that highlight real-world examples of sustainable urban planning projects. Guide them to explore how these cities are laid out and where the major roads and buildings are. Are there green spaces? Where are they? Where could they go? Are there clear public transportation links? Could unused space be utilized in a more efficient way?
Once students have an idea of what features they’d like their city to have, it’s time to draw. Using paper or computers, have students create a map that lays out their city and all of its features. Utilize geography concepts like scale, elevation, and topography so that students understand the connection between math and other subjects. Encourage students to use materials like cardboard, wood, foam, or recyclable materials to bring their maps to life.
Provide students opportunities to reflect on their learning and the challenges they encountered during the design process. Ask questions about the application of mathematical principles and how they helped students with things like scale, measurement, and spacing. Facilitate discussions that encourage problem-solving and allow other students to respectfully ask questions about the sustainability features they see. Prompt students to consider the trade-offs and compromises they had to make in their decisions.
Incorporating Mathematical Concepts
There are a few different ways that math can be applied to this inquiry project:
1. Creating scale models
Determine the scale at which you want to create the model city. Common scales include 1:100, 1:200, or 1:500. Encourage students to plan a layout for the city, including streets, buildings, parks, landmarks, and other features of a sustainable city. Consider the use of grids and geometric shapes for simpler planning. or use pattern blocks to map out the features of the city.
2. Simple calculations
Planning a sustainable city requires the calculation of things like distance, areas, and volume in order to optimize for land use, transportation routes, and building placements. For example, calculating the distance between a major airport and the city centre, or calculating the amount of rainwater that a barrel can hold.
3. Statistical analysis
Through examining articles of pre-existing cities, students gather and analyze demographic data, population trends, and socioeconomic indicators to identify patterns. These can then be used to inform decisions along the way. Students could also conduct surveys to find out what features their classmates would want in a sustainable city and analyze that data too.
4. Graphing and equations
If students conduct surveys or collect forms of data from their peers, they can create formulas for tracking things like future population growth, resource demand, and infrastructure needs. They could also predict greenhouse gas emissions or energy consumption trends too. Encourage students to graph their results so the data can be understood simply.
5. Financial costs
Building a sustainable city isn’t cheap. Discuss the cost of materials, paying workers, and other variables that go into the construction of a city. Older students may choose to calculate the cost-effectiveness of sustainable initiatives and projects by using financial metrics, or conduct a risk assessment to evaluate the probability of things like natural disasters and climate-related events.
Math Inquiry Idea #2 – Applying Mathematics to Cryptography
It’s no surprise that coding and math share a lot of similarities. This inquiry idea is suitable for older students studying computer science and other high-school level mathematics or coding courses.
Cryptography is the science of using math to hide data behind encryption. This helps store, secure, and protect communication. Math and cryptography share a deep connection because math is what provides the foundation for creating secure cryptographic systems.
Most cryptographic systems combine two things:
1. A set of rules that specify the mathematical steps needed to encipher or decipher data (also known as a process or algorithm)
2. A cryptographic key (a string of numbers of characters), or keys
In simpler terms, imagine you had a recipe for baking a cake. In the same way the recipe tells you the steps to follow to bake the cake, a cryptographic algorithm is a set of rules that tells a computer how to turn secret information (plaintext) into scrambled information (ciphertext).
Imagine you put that recipe in a locked box. Having the key to unlock that secret box is like having a cryptographic key, which is a string of numbers or characters that the algorithm uses to scramble and unscramble your data. Without the right keys, you can’t get into the box.
So, when you combine the algorithm with the cryptographic key, it’s like following a secret recipe with a special key to unlock and lock your secret message. The key is what keeps your recipe safe, and the algorithm is the set of rules that keeps everything secure.
Inquiry Projects Involving Cryptography
1. Self-Guided Inquiry
Students can explore the various theories involved in encryption schemes, including:
- Diffie-Hellman key exchange scheme
- Chinese remainder theorem
- Probability theory
- The RSA encryption algorithm
- Complexity theory
- Elliptic curve cyptography
- EIGamal encryption system
- Information theory
Challenge students to not only understand one of the above theories, but to simplify it so that younger students can understand. Similar to the cake example, students use their knowledge and experiment with different analogies to help others understand in an easy-to-understand way. Students can choose to share their information as a presentation, workshop, or through a demonstration.
2. Simple ciphers
Show students simple substitution ciphers, such as:
Caesar cipher – each plaintext letter is replaced with a different one a fixed number of places down the alphabet. For example, a left shift of three places, as seen below:
Pigpen cipher – each plaintext letter is replaced with a simple geometric picture symbol (created using grids and crosses so that each letter is represented by fragments of a grid or cross with or without a dot)
Provide them with decoding charts and encourage them to encode and decode their messages using these ciphers. For example, provide math equations where each answer corresponds to a letter in the alphabet, and challenge students to decode the message by solving the equations. These types of problems utilize deductive reasoning and problem-solving skills as well as helping to solidify basic concepts such as addition, subtraction, multiplication, and division.
Guide to Cryptography Mathematics
Explorer Academy: Code-Breaking Activity Adventure (National Geographic)
Code Cracking for Kids (Codes and Ciphers)
Cryptography as a Teaching Tool
Math Inquiry Idea #3 – Sports and Statistics
While not uncommon in the math classroom, sports and statistics can help foster critical thinking skills in a way that appeals to a lot of students. By encouraging students to explore the connection between math and sports, they get the chance to see how mathematical concepts apply to familiar, real-life situations.
Of course, not every student is interested in sports. In these cases, it helps to point out the fact that sports aren’t just about the rules, plays, and competition. Instead, merging sports and mathematics offers students new and interesting ways to apply their learning in strands such as geometry, statistics, algebra, and measurement.
To get students thinking about the intersection of math and sports, consider starting off by discussing the importance of statistics and how they are used in sports such as basketball, soccer, baseball, or football.
Students may suggest the following points:
- Statistics help track player performance and evaluation (helpful for managers and scouts to assess a players’ strengths and weaknesses)
- They allow coaches and managers to notice trends and patterns in team dynamics (which helps identify areas for improvement or adjustment)
- The use of statistics helps coaches and team members make strategic and informed decisions (helpful in making lineup changes, substitutions, and set goals)
- Statistics also help enhance fan engagement by providing valuable insight into the dynamics of the game (which fuels discussions and debates within the sports community)
Encourage students to generate questions and inquiries related to sports statistics. Prompt them to think about what types of data are collected in sports, and what they’re curious about in general. Provide them with time and opportunities to research statistics for players or sports they’re interested in. Statistics can come from a variety of courses, such as online databases, sports websites, and historical data archives. Guide students in collecting and organizing data on player and team stats, game scores, and other relevant metrics.
From here, students can take their learning in several directions. Some suggestions are below:
1. Drawing comparisons
Students love making comparisons, whether between each other, celebrities, sports stars, or their favourite sports team. In a statistics inquiry, encourage students to make comparisons between different players, teams, or seasons using measures such as averages, percentages, ratios, and rates. For example, students can compare the shooting percentages of basketball players, betting averages of baseball players, or the win-loss records of sports teams.
Guide students in exploring mathematical concepts and principles embedded in these statistics. Prompt them to ask inquiry questions to guide their own learning. One example might be, “how did Tom Brady’s passing yards change from when he played for New England vs. when he played for Tampa Bay?” Here are some other examples:
- How does a player’s shooting percentage affect their team’s overall performance?
- What factors influence a team’s success in a particular sport?
- How do player statistics change over the course of a season?
Facilitate investigations by helping students analyze their data, generate hypotheses, conduct experiments, and draw conclusions based on their findings. Encourage students to share their learning and insights through various inquiry-based presentation formats, including traditional presentations, discussions, the use of charts or graphs, and multimedia presentations. Ensure they are explaining their mathematical reasoning and encourage peer feedback.
2. Creating a “Statistics Palette”
Have you ever heard of “movie palettes”? Basically, it’s a set of specific colours used throughout a particular movie or film. They’re carefully curated to evoke specific moods and feelings, convey thematic elements, and enhance storytelling. Movie palettes are relatively recent and can be purchased as a unique piece to showcase your favourite movie. During the process, the dominant colour from each scene of your favourite movie is turned into a vertical strip, and arranged chronologically, side-by-side, onto a canvas. This results in a unique piece of artwork that represents your favourite movie as a colour palette.
Students can apply this idea to a sports statistics inquiry in math by assigning colours to specific data categories. For example, examine Tom Brady’s passing yards when he played for the New England Patriots and compare them to his passing yards when he played for the Tampa Bay Buccaneers. Group his statistics by number of yards (for example <1000 per season, 1000-1500 per season, etc.) and assign each group a colour, shown below:
Math Inquiry Idea #4 – Geometry in Architecture
This inquiry idea is great because it allows teachers to facilitate learning in a cross-curricular way, often incorporating several subjects. To start, give students the opportunity to explore the history and significance of geometry in architecture, from ancient civilizations to the contemporary world. Students can examine great architectural wonders of the past, including:
- The Great Pyramid of Giza (Egypt, 2560 BCE)
- The Hanging Gardens of Babylon (Iraq, 600 BCE)
- The Colossus of Rhodes (Greece, 280 BCE)
- Chichen Itza (Mexico, 700 AD)
- The Taj Mahal (India, 1648 AD)
By examining architectural masterpieces from the past, students will gain an appreciation for the role of geometry in creating these structures. As facilitator, help students to identify geometric shapes and principles found within these structures and explore how proportions, patterns, and culture influence architectural design and construction processes.
Recognizing Historical Successes
A great example is the use of geometry by the Inca at Machu Picchu, where they built intricate terraces and staircases, both for agricultural purposes and in order to navigate the steep Andean terrain. In this case, geometry helped them design and construct these structures with precision. Show students a photo of the staircases and terraces, and ask students to identify their characteristics; for example, the terraces follow the contours of the mountainside, and the staircases were engineered to conform to the natural topography of the landscape.
Once students feel comfortable identifying these characteristics, have them explore the use of water irrigation systems and aqueducts. Encourage them to make inferences about how the architectural design of these systems resulted in optimization water distribution, minimal erosion, and the conservation of natural resources.
As students continue to gain confidence in exploring these architectural features, encourage them to use more precise mathematical language. For example, instead of saying that the Inca “made steep paths”, they can say that the Inca “constructed steep slopes”.
Solving Modern-Day Problems
Once students understand the link between geometry and construction, and have had some time to explore the impact of mathematics on construction, challenge them to apply their understanding to real-world design challenges and problems. This is where a wonderful intersection of subjects – math, design, engineering, physics, and environmental science – occurs. Students may be tasked with designing and construction scale models of popular landmarks, exploring principles of stability and load-bearing, or analyzing the benefits of using specific shapes such as arches, domes, and columns.
A good application of these skills is to tackle the challenge of how to create safe and stable wildlife crossings over busy roads and highways. This is a problem that a lot of students show interest in because they want to see animals be able to coexist safely in our modern world. The use of bridges, tunnels, overpasses to allow animals to safely cross roads and highways requires problem-solving skills and spatial thinking in addition to mathematical thinking. A good place to start is by getting familiar with the Banff Wildlife Crossings Project, which has reduced animal-vehicle collisions in the province of Alberta by more than 80% since its construction.
The use of manipulatives such as pattern tiles, tile boards, and geoboards (available in a set of 6 here) help students to connect the building process with geometric concepts. Encourage them to explore how shape, symmetry, and proportion work together to create stunning buildings, homes, and other architectural wonders. Challenge students to sketch their ideas, incorporating geometric elements such as a variety of shapes, angles, patterns, tessellations, and symmetry in their designs.
Teachers can prompt students to consider other elements, such as:
- Proportions and measurements of windows, doors, and other features
- Sustainable and eco-friendly features, such as solar panels, wind turbines, rainwater harvesting, living walls, green roods, and natural ventilation
- Aesthetics (for example, creating building that mesh nicely with the culture and traditions of a particular place)
Once students have visualized their ideas, they are encouraged to display their learning in a way that is most meaningful to them. For example, if they sketched a scale-drawing of a wildlife bridge, the next natural step would be to create a physical model or prototype to demonstrate how it would look in real-life. Some may choose to conduct a gallery walk, showing their peers and others their idea progress through rough sketches and blueprints. Regardless of their presentation method, students should be encourage to provide feedback on both their own work and their peers’ work. Review sessions can be conducted to facilitate the exchange of constructive feedback and suggestions for improvement, which are a natural part of the inquiry process.
Fusing math and inquiry learning in the classroom is important in all 21st century classrooms. An inquiry-based approach encourages students to ask questions, explore mathematical principles, and see how math relates to many different aspects of their lives. By allowing students to make connections and participate in authentic learning, they will hopefully develop an appreciation for the relevance and applicability of mathematics in everyday life, even if they are not keen to pursue a job that requires extensive math knowledge.
(1) Teaching math using inquiry-based learning is a powerful way to reinforce the skill of problem-solving for students
(2) Applying inquiry-based learning in math doesn’t have to be a “one or the other” approach. Rather, you can combine elements of traditional math teaching and incorporate tenets of inquiry along the way to solidify concepts and apply learning in new contexts
(3) Using scenario-based learning or delving into real-life problems that involve math helps build problem-solving and communication skills, as well as perseverance in students of all ages
(4) Employing a cross-curricular approach allows students to appreciate the applicability of math in everyday life
(5) Inquiry-based learning in math empowers students to take ownership of their learning and draw conclusions independently or collaboratively
Cover photo by Alena Darmel