Breaking the hierarchies in mathematics education

FYMSiC Conference 2024

Johanna Rämö, johanna.ramo@uef.fi

University of Eastern Finland

Image credit: Veikko Somerpuro

Why are we teaching mathematics today?

There are several good answers to this question. Here is one I’ll be focusing on:

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To teach students skills that are useful in life.

The Finnish university education

• Students have a major subject and take one or several minor subjects
• Students who take mathematic courses are typically
• mathematics students
• pre-service teachers
• computer science students
• statistics students
• physics students
• economy students

The Finnish university education

• First courses in mathematics
• Calculus / Real analysis (depending on the university)
• Introduction to proof
• Linear algebra

The Finnish university education

• The responsible teacher for the course has full autonomy regarding the teaching and the assessment method
• Assessment is usually low-stakes
• students can retake exams multiple times
• grades do not play a big role in future studies or in getting a job

Traditional mathematics teaching in universities

• Lectures
• approximately two 2-hour lectures per week
• Homework
• students solve problems after the lectures
• Homework sessions
• students take turns in presenting their solutions on the blackboard
• teaching assistant supervises
• Assessment
• exam + bonus points from homework

Why are we teaching mathematics today?

To teach students skills that are useful in life

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Why are we teaching mathematics today?

To teach students skills that are useful in life.

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What is useful?

Most important skills in working life, according to Finnish alumni

Most important skills in working life, according to Finnish alumni

Ability to learn new things

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Most important skills in working life, according to Finnish alumni

Ability to regulate own actions

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Most important skills in working life, according to Finnish alumni

Cooperation skills

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Most important skills in working life, according to Finnish alumni

Creativity

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Most important skills in working life, according to Finnish alumni

Theoretical knowledge in one’s own field

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Traditional teaching in Finland

• What skills students learn?
• Some problem solving but also lot of memorising
• Hardly any communication or cooperation skills
• What practices students take part in? Are they authentic in any way?
• Sitting in a lecture for several hours
• Working alone

Traditional mathematics teaching in universities

• Lectures
• approximately 2 lectures per week
• Homework
• students solve problems after the lectures
• Homework sessions
• students take turns in presenting their solutions on the blackboard
• teaching assistant supervises
• Assessment
• exam + bonus points from homework

Hierarchies

Hierarchies

• We can improve teaching by disrupting some of the harmful and unnecessary hierarchies
• Students need an access to the community of practice of mathematicians (Lave & Wenger, 1991)

Extreme Apprenticeship (XA)

• Students participate in activities that resemble those of professional mathematicians
• Based on situated view on learning (Lave & Wenger, 1991) and Cognitive Apprenticeship (Collins et al., 1991)
• Form of Inquiry-Based Mathematics Education (Laursen & Rasmussen, 2019)

Image credit: Veikko Somerpuro

Extreme Apprenticeship (XA)

• The methods scales up to courses with hundreds of students
• Since 2011; used in linear algebra, logic, probability…

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Rämö, J., Lahdenperä, J., & Häsä, J. (2020). The Extreme Apprenticeship method. PRIMUS, 31(10), 1106-1120.

Image credit: Veikko Somerpuro

The learning cycle of the XA method

Rämö et al., 2020

Tutor

Students can come and go as they wish

Drop-in sessions several days a week

Main method of teaching

Image credit: Veikko Somerpuro

Tutors are undergraduate/graduate students

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Model how mathematicians work

Teach study skills

Guide by listening

Image credit: Veikko Somerpuro

Tutor

The learning space is in the main hallway of the department

Professor

Group of students

You can draw on all surfaces

Mathematical community made visible and accessible to both students and faculty

Image credit: Veikko Somerpuro

Tutors

• Are easier to approach to than professors
• Create a sense of community
• Learn communication and teaching skills
• Deepen their understanding of mathematics
• Need support

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Image credit: Susanna Oksanen

Professional development of the tutors

• Weekly meetings, in which pedagogical issues are reflected in small groups
• Every meeting has an overarching theme
• It’s OK to do mistakes
• Listening to the students, encouraging students to talk
• Letting students do and discover
• Being active, approaching shy students
• Teaching study skills and helping in reading course material

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Image credit: Veikko Somerpuro

Results

• Compared to traditional lecture-based teaching, the XA method supports a better quality of learning
• Supports students self-efficacy (how competent they feel)
• Supports deep approach to learning (students try to understand what they study)
• Prevents surface approach to learning (memorisation, fragmented learning)

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Lahdenperä, J., Postareff, L., & Rämö, J. (2019). Supporting Quality of Learning in University Mathematics: a Comparison of Two Instructional Designs. International Journal of Research in Undergraduate Mathematics Education, 5(1), 75-96.

Results

• Compared to traditional lecture-based teaching, the XA method supports female students
• Lecture-based group
• Female students had lower performance and self-efficacy than male students
• XA group
• Female students did as well as males

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Lahdenperä, J. (2018). Comparing male and female students’ self-efficacy and self-regulation skills in two undergraduate mathematics course contexts. In V. Durand-Guerrier, R. Hochmuth, S. Goodchild, & N. M. Hogstad (Eds.), Proceedings of INDRUM 2018.

Summary

• We tried to make teaching as student-centred as possible
• Low hierarchies
• Accessibility
• Learning skills relevant to real-life
• Both students and tutors learn valuable skills

Summary

• We tried to make teaching as student-centred as possible
• Low hierarchies
• Accessibility
• Learning skills relevant to real-life
• Both students and tutors learn valuable skills

But…

Assessment

One thing remained – the examination

Argh, why is changing assessment so difficult!

• I had 400 students in my course
• I felt that the exam was a guarantee that students knew enough

Argh, why is changing assessment so difficult!

• I had 400 students in my course
• I felt that the exam was a guarantee that students knew enough

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At the same time I knew that “Assessment drives learning” (George Miller)

Argh, why is changing assessment so difficult!

• I had 400 students in my course
• I felt that the exam was a guarantee that students knew enough

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At the same time I knew that “Assessment drives learning” (George Miller)

Hierarchy

Argh, why is changing assessment so difficult!

• I had 400 students in my course
• I felt that the exam was a guarantee that students knew enough

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At the same time I knew that “Assessment drives learning” (George Miller)

Assessment of learning

Assessment

• Traditional assessment has many drawbacks
• Exam stress
• Bulimic learning
• “Exam strategies”
• Challenges in producing fair exams
• Assessment has a huge impact on how students study

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Au (2022). Unequal by design: High-stakes testing and the standardization of inequality

Räisäinen et al. (2016). Students' and Teacher's Experiences of the Validity and Reliability of Assessment in a Bioscience Course

“If tobacco smoking was invented today, it would not be permitted.”

Norman H Tiffin, 2015

Canadian Journal of Respiratory Therapy

If our current assessment regime was invented today, would it be permitted?

Assessment for learning

Assessment is conducted with the intention of improving learning (Wiliam, 2001)

Assessment as learning

• Emphasises the dual role of assessment tasks as learning opportunities as well as instances of evaluation
• Long-term, sustainable learning gains that stem from students’ active and reflective role in the assessment process

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Z. Yan, & L. Yang (Eds.), 2021. Assessment as Learning : Maximising Opportunities for Student Learning and Achievement

Our solution: self-assessment

For life-long learning, students should learn to assess their current skills with respect to their goals (see Boud et al., 2010)

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DISA, Digital Self-Assessment

• Detailed learning objectives
• Students practice self-assessment during the course and receive automated feedback on their assessments
• A digital tool compared students’ self-assessments to their coursework

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Häsä, J. I. A., Rämö, J., & Nieminen, J. H. (2021). Supporting quality of learning by letting students give their own grades: An innovative self-assessment model in university mathematics. In Z. Yan, & L. Yang (Eds.), Assessment as Learning : Maximising Opportunities for Student Learning and Achievement.

Part of a rubrik

Also generic skills can be learning objectives

Example of a self-assessment exercise

DISA, Digital Self-Assessment

• No exam. Students give themselves their own grades
• The teacher could intervene if students self-assessment did not align at all with the work they had done

Results on DISA self-assessment

• Self-assessment promotes self-efficacy and deep approach to learning better than an exam
• Students feel that they study for themselves (

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Nieminen, J. H., Asikainen, H., & Rämö, J. (2021). Promoting deep approach to learning and self-efficacy by changing the purpose of self-assessment: a comparison of summative and formative models. Studies in Higher Education, 46(7), 1296-1311.

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Nieminen, J. H., & Tuohilampi, L. (2020). ‘Finally studying for myself’–examining student agency in summative and formative self-assessment models. Assessment & Evaluation in Higher Education, 1-15.

Students reflections on self-assessment

"Now I didn’t focus on memorising things. Instead, I focused on understanding the topics, so that in the future, if necessary, I can use them / re-learn them quickly.”

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"[...] I think that there wasn’t any bulimic learning. I remember the content better. “

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“[…] Success in the exams isn’t only a matter of preparation. On some days, it’s more difficult to concentrate than on other days. In this kind of a course [...] the final grade doesn’t depend so much on a single performance, but every prepared solution and also the self-assessment count for the final grade.”

“[…] the exam measures just content knowledge. In self-assessment, also other kinds of skills are considered.

Different versions of self-assessment in the XA courses

• No self-grading, but bonus points from rich self-assessments
• In small courses self-assessment can been accompanied with a one-on-one assessment discussion with the teacher. Digital tools are not needed

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More information and concrete examples:

https://digicampus.fi/course/view.php?id=4440

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-> Access as a guest

Should we ban exams altogether?

Self-evident…

If I am taken ill I want to see a doctor who has taken exams and passed them

… but also:

I want doctors to be able to assess very well what they know and don’t know

Photo: Petr Kratochvil

Self-assessment vs. Exam

• Self-assessment skills are important in life. Students need to practise them
• In addition, student’s have to have ownership over their assessment
• If we only use exams in evaluation, it solidifies students’ belief that they are not capable of assessing their own learning

Life-long learning

• After school, grades lose their significance
• Reflection skills become crucial
• Being able to describe one’s skills is important (Tuononen et al. 2019)

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Tuononen, T., Parpala, A., & Lindblom-Ylänne, S. (2019). Graduates' evaluations of usefulness of university education, and early career success - a longitudinal study of the transition to working life. Assessment & Evaluation in Higher Education, 44(4), 581-595.

Don’t good grades matter for getting a good job?

The most important factors in getting employed, according to Finnish alumni

The most important factors in getting employed, according to Finnish alumni

ability to verbalise one’s own skills

Thank you!

johanna.ramo@uef.fi

Image credit: Veikko Somerpuro