Making Math Real in the IB

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Hannah Starbuck

In 2019, the International Baccalaureate rewrote their math curriculum, transitioning from three different levels of classes to two. The “newest” class of the two is called Applications and Interpretations, where students consistently look at how math is applied in a real world setting. They are frequently asked to interpret the significance of the different values they calculate and what those values mean in the context of the problem they are solving. While I like the idea of having students explore mathematics through an application based lens, I find that some of the applications are far-fetched and may have little real connection to students.

 

Now in my third year of teaching IB mathematics, I am familiar with the Applications and Interpretations curriculum, both Standard and Higher Levels. I saw an opportunity to make the Standard Level classes more engaging and interesting for students through projects.

 

This school year, I’ve been creating projects that explore the different careers that utilize mathematics. I’ve combed through the Applications and Interpretations textbook and have racked my brain for possible careers that meaningfully apply the math content of the course syllabus. My idea is to ensure each chapter has its own project, or that there be content spread across multiple chapters that can be consolidated into one large project. Here I have to give credit to a former colleague of mine, Rob Barnett, who was kind enough to share his Demographer Project template a few years ago. His work inspired me to start thinking about different careers that use math and that could also be fun and interesting for students to think about.

 

Each project asks for the students to adopt the mindset of a certain career or job. In addition to my own version of the Demographer Project, I have created projects which ask students to think like an artist, a game maker and an architect.

 

As demographers, students were asked to investigate a population they were genuinely interested in and wanted to learn more about. Students had to find reliable and valid data on their population, graph their data and find a suitable model for their data. They looked at whether exponential models or linear models were a better fit and explored a concept called the coefficient of determination, which is an indicator on how well data fits a particular model. Students were also asked to make future predictions about their population, in relation to their models, and discuss the implications of their results. I asked students to take an objective and unbiased approach to interpreting their results and asked them to discuss the implications of their results, as well as how realistic they were.

 

The feedback on this project was very positive. Students were asked to write a reflection about the project and discussed things they liked and things they would change. The majority of my students loved the fact that they got to choose their own population and got to study trends over time, which included past history, the present situation and some speculation for the future. They all said in unison that they became more informed about their particular population. So for me, the interdisciplinary approach was a win, even without the time to collaborate directly with colleagues in other disciplines.

 

The biggest critiques of the project were the difficulties using technology and doing the project mostly remotely. Students had some difficulties understanding the different functions on Desmos and Google

 

Sheets. Though they initially lacked inexperience with these platforms and required a lot of support, after a while, they got the hang of it and understood the significance of each operation.

 

Another project I created based on similar themes asked the students to think like an artist. Students created an artistic piece using Pythagorean Spirals. This allowed students to access their creativity, demonstrate their understanding of the Pythagorean Theorem and create their own original work. This project was generally well-received and students enjoyed watching their art come to life as well as observing the nice patterns that occur within the Pythagorean Spirals.

 

I’ve also recently finished a project that focuses on Probability where students are asked to think like a game maker at a carnival and need to explore the concept of fairness. Students look very analytically at theoretical and experimental probability, different ways of representing/visualizing probability and binomial probability. The culminating activity is a carnival where students have the opportunity to play each other’s games.

 

I’ve also completed a project that focuses on right triangle and non-right triangle trigonometry where students are asked to think like an architect. Their project is to build a school with a small group of people and they have certain conditions they need to respect. This project allows students to explore the applications of the sine and cosine rule, fundamentals of trigonometry and some 3D geometry. Though these concepts have such interesting and abstract proofs, they provide an even richer application for students to construct something that has been a part of their daily lives.

 

While the projects are fairly content heavy, I have done my best to establish the importance of collaboration, studying concepts with practical and realistic application and allowing students to create something that is their own vision. There are also certain skills that I think are valuable for students to take away. These include being able to efficiently use technology, using different math “tools” such as rulers, protractors, and calculators and interpreting the meaning of values and how they relate to the larger problem they are solving.

 

I am continuing to work on projects that focus on being a data analyst, reporter, modeler, urban planner, and more. I find the curriculum planning to be very rewarding and then even more satisfying as I watch my students come alive while engaged in something they’re truly excited about. I am always happy to share materials, you be interested.

 

As each school year passes, it becomes more and more apparent to me that teaching mathematics with some sort of meaningful and interesting application is crucial to student learning and engagement. Packer (2022), a writer for The Atlantic, asks, “What is school for?” A former teacher responded, “The original thinkers of public education were concerned almost to a point of paranoia about creating self-governing citizens.” The word self-governing is powerful, especially in the context of education. Self-governing is defined as “having control or rule over oneself”. By allowing students to have choice over what they learn in their different classes, we are empowering them to have control over their learning and hopefully communicating the message of self-governing. Karakoc and Cengiz (2015) discuss a couple approaches of realistic mathematics education that include “developing instruction based in experimentally real contexts” and “designing activities to promote pedagogical strategies that support students’ collective

 

investigation of reality”. These ideas confirm the importance of application-based mathematics to scenarios that students care about and show a genuine interest in. Howe (2018) writes “students benefit from learning experiences that are meaningful, relevant and well-connected to their own experiences. For that to happen, the people teaching those students must be prepared to take on new attitudes of reflectiveness and inquisitiveness”. This is a great opportunity for those of us in education, specifically math education, to create projects and high ceiling math problems that directly relate to students. It also gives them a chance to research topics and ideas that they are truly passionate and interested in and allows them to find authentic and genuine connections to mathematics.

 

If you would like to discuss project ideas or have questions/comments, please feel free to email me at hstarbuck@fcaq.k12.ec.

 

ABOUT THE AUTHOR

 

 

Hannah Starbuck earned her Undergraduate Degree in Mathematics from Fort Lewis College and her Master’s Degree in Math Education from University of Colorado – Denver. She has taught math in Colorado, Switzerland and Ecuador. She currently lives and teaches in Quito, Ecuador and is writing projects that align with both IB DP and MYP curriculum.

 

BIBLIOGRAPHY

 

Chang Wathall, Jennifer, et. al. Mathematics: Applications and Interpretation Standard Level. Oxford University Press, 2019.

 

Packer, G. (2022, March 10) The Grown-Ups Are Losing It. The Atlantic. https://www.theatlantic.com/magazine/archive/2022/04/pandemic-politics-public-schools/622824/

 

Karakoc, G & Alacaci, C. (2015). Real World Connections in High School Mathematics Curriculum and Teaching. Turkish Journal of Mathematics and Computer Education. https://www.researchgate.net/publication/276172693_Real_World_Connections_in_High_School_Mathe matics_Curriculum_and_Teaching

 

Howe, E.R. (2018, September 11) Let’s Teach Students Why Math Matters in the Real World. The Conversation.

https://theconversation.com/lets-teach-students-why-math-matters-in-the-real-world-102316

Changes in maths pedagogy: Making maths accessible for all

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Karen Morrison and Lisa Greenstein

 

According to Oxford University Press’ 2021 survey of maths educators, a massive 84% of teachers say they have changed how they teach maths in the last year or more. This situation has been exacerbated by the impact of the global pandemic, but maths pedagogy is continually shifting as learner needs and resources evolve. Here, we reflect on the changes in maths pedagogy over the last 10 years and explore how educators can now make maths more accessible for all.

 

Ten years ago, the buzzword in maths teaching was problem-solving. Teachers already had a sense of wanting to move away from drill and practice. They wanted to move towards giving students the tools to solve problems in creative ways, but there wasn’t a lot of commonly-used vocabulary around how to do that. This vocabulary issue was complicated further recently by the COVID-19 pandemic as 75% of educational professionals stated that school closures had impacted children’s understanding of mathematical language (Oxford University Press, 2021).

 

Some students struggle to visualise and communicate ideas in written or oral forms, and there are some students who don’t understand the task instructions (Lyttle, 2021). This lack of understanding is more profound with EAL students or those who are neruodiverse, such as with Attention-Deficit / Hyperactivity Disorder (ADHD). Therefore, there is a need to not only plan maths lessons, but the specific language within those lessons, so we say what is mathematically meaningful (Prescott et. al., 2020).

 

Today, we (teachers) have so much more awareness of what it means to think mathematically. We talk about questions with an open middle. We talk about exploring mistakes. We’ve come such a long way from agitating over whether kids could remember the 7 times table.

 

The biggest change in the way we approach maths now is the shift towards asking big, open-ended questions from the beginning – before we teach kids a given algorithm or formula. You don’t want to start off by telling them how to do something, and then ask them to do it. You want to start off by posing a question that they can genuinely grapple with.

 

Rather than simply focusing on questions and answers, it’s now important to have wider conversations about maths. For example, by asking learners to explain how they arrived at an answer and discuss this with them (Sylva et. Al 2020). We need to create opportunities for rich interactions that involve lasting activities where children work together to solve problems, giving learners the thinking time they need to develop their own ideas and discuss them openly (Williams, 2021).

 

Today, we need to pose questions which include mixed units, reasoning and thinking, justifying your answer, and solving a problem that can have different solutions – and yet it doesn’t seem frightening or off-putting. This allows for everyone to feel included and encourages them to share their ideas.

 

Teachers are talking about growth mindset more than ever, and embracing this attitude in classrooms is slowly becoming the norrm. A growth mindset, put simply, is the realisation that there is no such thing as “good at maths”.

 

Not knowing the answer is part of the learning process and in fact, research shows that the brain can only make new connections when it experiences challenge (Wathall,2021). The human brain has incredible plasticity and learning is a process of stretching ourselves through the struggle of doing something that seems difficult – impossible even – at first. Children learn that this sense of struggle is ok. That it’s fine (and necessary even) to find maths difficult. The difficulty is not a sign of failure; it’s a sign of learning.

 

In order to encourage a growth mindset in pupils, teachers must utilize inquiry-based learning to promote debate, problem-solving and critical thinking to help build understanding. Allowing learners to work through problems with their friends can also make them more engaged and help to remove any stigma about struggling. Furthermore, high-ceiling, low-threshold activities allow every learner to demonstrate what they can do, without worrying about what they can’t do (Wathall, 2021).

 

Put simply, “Individuals who believe their talents can be developed (through hard work, good strategies and input from others) have a growth mindset” (Dweck, 2016).

 

The interventions carried out with struggling students in disadvantaged communities have shown that a more flexible approach and questions that genuinely make students think increase engagement, and lead to improved academic performance (Boaler, 2020).

 

When educational authors develop a primary maths course for children, they are now thinking about how best to start every child (and level of learner) on a lifelong journey with maths. They cover traditional bases of number sense, sorting, measuring, identifying shapes. Making the link between numerals and quantity is still, and always will be, essential for young learners’ understanding and they need access to many opportunities to experience and explore this (Williams, 2021). But authors also celebrate exploration and investigation; developing a sense of playfulness and fun, as well as a willingness to struggle when things get tricky. These things are often the intention of today’s mathematical education authors – to introduce challenge and struggle.

 

Another wonderful aspect of developing maths content today is social media. Just following a few hashtags – #mathteachersofinstagram, #mathsadventure, #numberchat, or #iteachmath – can turn up such a wealth of information and knowledge.

 

Teachers are having conversations in these spaces on how to cultivate a growth mindset in the classroom (Wathall, 2021); they are sharing moments of challenge or success from their own classroom experience; educators are sharing online resources and conferences and courses. Social media has its detractors, but ten years ago, teachers just didn’t have access to all this wonderful shared knowledge and experience.

 

Maths teaching in international schools is continually changing. But now, more than ever, teachers are focused on challenging traditional approaches to teaching maths, by changing perceptions of it in order to build curiosity, joy, and wonder in order to celebrate growth mindset and reduce anxiety. Developing and practicing behaviours such as problem solving, collaboration, and resilience will only continue to gain importance and focus in the years to come (Neale, 2021).

 

Introducing Nelson Maths

International Education experts, Karen Morrison and Lisa Greenstein have combined their knowledge and experience in the creation of the new edition of Nelson Maths – a rigorous, whole-school programme for teaching and learning maths from early years through to the end of primary education, from Oxford University Press. Written for learners across the world, it enables all children to start and sustain a lifelong journey with maths. The new edition includes a brand-new look and feel, vocabulary support and activities that prompt engagement with the latest mathematical thinking, such as problem solving and growth mindset. Find out more about the new Nelson Maths at: www.oxfordprimary.com/nelsonmaths

 

Bibliography & further reading

 

Boaler, J. (2016) Mathematical Mindsets, San Francisco, Jossey-Bass.

 

Dweck, C. (2015). Carol Dweck revisits the growth mindset. Education Week, 35(5), 20-24.

 

Greenstein and Morrison. (2022) “Making maths accessible for all: Behind the scenes with our writers”. Oxford University press [blog article] via: https://educationblog.oup.com/international/nelson-behind-the-scenes [Accessed 18/02/22].

 

Lyttle, D. (2021) “Lessons from the pandemic: Putting our findings into practice”. [Oxford University Press: Online Article] via www.oxfordprimary.com/mathswhitepaper [Accessed 18/02/2022]

 

Mitra, S. (2012). Beyond the Hole in the Wall. Ted books.

 

Mitra, S., & Crawley, E. (2014). Effectiveness of self-organised learning by children: Gateshead experiments. Journal of Education and Human Development, 3(3), 79-88.

 

Oxford University Press. (2021) “Maths and the impact of Covid-19: survey of international teachers”, Maths survey of UK teachers, Oxford University Press 2021.

 

Oxford University Press. (2021) “Preparing for the future: Using curiosity and creativity to boost confidence in maths. A whitepaper for international educators”. [Whitepaper] via www.oxfordprimary.com/mathswhitepaper [Accessed 18/02/2022]

 

Rowland, T (2008). The purpose, design and use of examples in the teaching of elementary mathematics. Educational studies in mathematics 69/2, 149-163.

 

Williams, H. (2021) “Building solid foundations”. [Oxford University Press: Online Article] via www.oxfordprimary.com/mathswhitepaper [Accessed 18/02/2022]

 

Wathall, J. (2016). Concept-Based Mathematics: Teaching for Deep Understanding in Secondary Classrooms (1 edition). Thousan Oaks California, Corwin.

 

Wathall, J. (2021). “Develop a growth mindset”. [Oxford University Press: Online Article] via www.oxfordprimary.com/mathswhitepaper [Accessed 18/02/2022]

 

Watson, A. (2021). Care in Mathematics Education: Alternative Educational Spaces and Practices. Springer Nature.

 

What does a modern mathematics classroom look like?

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Hannah Starbuck, Mathematics Teacher, Leysin American School.

 

The Mathematics department goal this year was communication. This was perfect because it tied into a large part of my vision of a modern classroom. The areas I mainly focused on this year were Problems of the Week, Socratic Seminars, and reading articles.

 

A Problem of the Week (POW), is a large, fairly open-ended maths problem, with an elegant solution or solutions. These questions encourage multiple pathways of thinking and provide students ample opportunity to look for and identify patterns and trends, a useful habit for mathematicians. I’ve given POWs the last three years I’ve been teaching. Students tend to enjoy them and enjoy collaborating with one another. The biggest pushback I get is my expectation that students produce a formal write-up of the problem.

 

Students are asked to explain in detail the problem and their process of solving the problem, convincing readers of their methods and solution(s) and reflecting on their learning and the task. The question I get: “Why do we have to write in maths class?”.

 

From the student’s perspective, the purpose of maths class is to “solve problems and get right answers.” But now I tell my students that we write in maths class because it is good for our communication skills, that mathematical writing is just as important as writing an essay on a novel or a lab report AND that it helps to articulate mathematical thinking that might be difficult to express orally.

 

Students will sometimes go out of their way to complain about writing, but their reflections and mathematical writing always amaze me. For starters, their reflections are insightful and honest. They provide evidence and defend their claims as to why they deserve a certain mark, or what should be changed about the problem. Their thoughts are generally mature, well thought-out, and genuine. Students go into detail as they describe how hard they worked on this problem, and discuss the different maths concepts they applied. It’s important students embrace the idea of convincing a skeptic. Students are very thoughtful in defending their processes and answers and have detailed work and proof to support their thoughts. Even if the solution isn’t correct, I don’t penalize students for trying to defend their work.

 

Additionally rewarding for me is to watch students interact with one another as they solve these questions. My role is to provide clarity or define words that might be confusing. The problem solving is left to the students. Naturally they want to work together and bounce ideas off each other. The resulting mathematical discussions are truly rich and exciting to listen to. My role is to be the skeptic, asking a lot of how and why questions, encouraging students to be more convincing, and to make strong arguments.

 

The resulting discussions are impressive. That is the exciting part of POWs. You have opportunities to listen to students from all over the world discuss and debate maths to solve these problems, as well as watch their confidence grow when they are successful.

 

The second focus area for me was Socratic Seminars centered on math education. I chose to do this because I had 12th graders who were experienced with debate, had good English levels, and I had small enough classes to make sure everyone’s voice was heard. Some of the ideas we discussed were student visions of a modern maths classroom, the necessity of homework, how students are assessed, theoretical and application maths, and mental maths, and many more. Students were expected to prepare questions and opinions to contribute to the discussion, demonstrate respectful and polite body language during the discussion, and reflect on the seminar after it concluded. The majority of the time, students were prepared and contributed interesting questions and thoughts. Students also demonstrated a genuine interest in what others had to say. They maintained eye contact, sat up straight, and showed appropriate signs when they were ready to contribute to the discussion.

 

What these seminars taught me is that our students are a lot more able than we sometimes give them credit for. The 12th graders I taught had been in school for 13-14 years; they’ve seen just about everything. “As an IB student, I find homework to be very burdensome”, “I strongly dislike routine”, “It is an effective way for students to make projects based on the topic passed, or to make a quiz right after you finished the topic”. These are direct quotes from my students discussing the seminar themes. Their language is honest and shows they’ve played the game of school for a long time and KNOW how to play the game. I try to incorporate a lot of their ideas. To me, this is an obvious motion to listen to students more than we do. Teachers definitely possess the training AND the experience of being both a student and a teacher, but let’s not forget we also had ideas on how to make school more fun for us. When given the opportunity for students to use their voices, they clearly can make educated and insightful observations.

 

The third item I focused on was having students read various articles on global maths education. This seemed appropriate because my students were from all around the world and could provide insight and perspective on each topic we read about. Some of the topics we read about were: counting on fingers, math anxiety, mathematical geniuses, and teaching math as a language. Similar to Socratic Seminars, I asked students to write their initial opinions based on the headline of the article and give an explanation on why they felt this way. Then they read the article and shared their opinions, touching on whether their previous ideas had changed, why or why not, and tried to connect the article back to their home country. Again, their responses are incredible to read. Students provide so much detail and perspective, and what’s even more interesting is that a lot of them had never thought about some of the topics we read about.

 

For example, we discussed the benefits of counting on one’s fingers and encouraging teachers to allow it. “Many people, especially children, are using the fingers for counting. The brain structure is working by counting fingers, it is such an instinct. I think that using fingers before school is fine, but step by step you need to get rid of it. I think that taking notes is more efficient”, “Most people, myself included, are visual and tactile learners to some extent. So using our fingers to count quickly is a good way to do mental calculations”. These quotes from my students confirm many things for me. The primary one is each student comes from a unique background and maths has its own identity and culture in their homeland.

 

Maths takes many shapes and forms depending on location and culture. And once again, students have shown me their strong abilities to read, interpret, and form a thoughtful and relevant opinion on a concept that might be unfamiliar. This is the power of giving students the opportunity to use their voices: they provide insight, perspective, and ideas to those around them, including their teachers.

 

Where do I go from here? There are a few areas I’d like to experiment with. The first is using a portfolio as a way to assess students. Ideally the portfolio would contain projects, reflections, assignments, and other pieces of “evidence” that students would use to show their development throughout the year. The second is to buy a class set of books and incorporate literature into maths class. I have a few titles I’d be interested in using. The next step is to discuss best practices with an English Literature teacher. I’ll also change some of the topics for the Socratic Seminars to be more mathematical. Zero is a popular choice, prime numbers, calculus, etc, all have potential for an interesting discussion. And finally, I’d like to incorporate more projects into maths class. Maths is not a one size fits all subject, it has so much room for creativity, problem-solving, and growth. Projects give students more choice and authenticity in their learning and what students create is truly interesting! The best thing we can do for our students is to empower them by giving them opportunities to speak, write, listen, read, and create pieces of work that demonstrate learning and also have meaning and personality.

 

BIBLIOGRAPHY

Fendel, D., Resek, D., Aler, L., Fraser, W. (1997). Integrated Mathematics Program (IMP) Curriculum. Integrated Mathematics Program (IMP) Curriculum, Key Curriculum Press: Emeryville, CA.

 

ABOUT THE AUTHOR

Hannah Starbuck is a maths teacher, volleyball coach and resident scholar at Leysin American School. She is originally from Colorado and is finishing her sixth year of teaching.