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FYMSiC Newsletter 1st Edition, July 2019
Spotlight on Teaching
Milestones
Classroom Resources
Communications
Recent Publications
Upcoming Events

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Welcome !!!

Dear Math and Stats Friends and Colleagues, welcome to the first issue of our First-Year Math and Stats in Canada Newsletter !!!


🇨🇦🇨🇦🇨🇦 Happy Canada Day !!! 🇨🇦🇨🇦🇨🇦

We hope that you will find its content relevant, interesting and informative. Also, we hope that, by contributing to the future issues, you will help this Newsletter grow.

Besides our commitment to our students and our love and passion for math and stats, we, math and stats teaching practitioners from all walks of (academic) life and across Canada, are also sharing multiple concerns about the future of our trade.

In the report from the April 2018 conference, we wrote: “Canada's university mathematical teaching community is facing a number of significant challenges and opportunities: from managing increasingly diverse classes of incoming students to understanding and dealing with the impact of contemporary technology on teaching and delivering courses, to keeping the content of mathematics courses relevant to various academic programs and, most importantly, to efficiently supporting students to achieve their personal, academic, and career goals. Furthermore, the changing landscapes in post-secondary education witness evolving knowledge about best practices for teaching, such as active learning approaches, high failure and withdrawal rates in first-year mathematics classes, as well as students’ uncertainties about the benefits of learning mathematics for their future careers. (D. Barr, A. Burazin, K. Garaschuk, V. Jungic, & M. Lovric, in CMS Notes, September 2018)

The purpose of this Newsletter is to be a forum through which the members of our community can have useful exchanges of information, practices, views, and opinions about the issues listed in the above quote. It is our hope that those exchanges would help us to find means and strategies to better advocate for a major change in the ways our institutions view and value teaching of math and stats at the first-year level.

There are many math and stats teaching-related ideas, initiatives, events (in the broadest sense) taking place all over Canada, and an important goal of this Newsletter is to keep us informed, and to reach out to as many math and stats instructors as possible.

Another important goal is to keep the momentum, namely our conversations at a National level, strong. It is important that we know what our colleagues 3000 km far from us are doing—we must share our knowledge, experience, and expertise. In this sense, the Newsletter will complement what we have been already doing through the First-Year Mathematics and Statistics Courses Repository.

The Newsletter editors will be happy to accept all kinds of contributions related to teaching math and stats. You are organizing an event, or know about one? Let us know—send a short paragraph about the event (name of the event, location, date, website, organizers) for the Upcoming Events. There is a section called Celebrating Milestones ... did you know that this past May, Peter Taylor (Queen’s) has been recognized by his university for 50 years of service (Congratulations, Peter!)? Do you know of anyone who has received an award, a grant or a fellowship, hosting a sabbatical visitor, something else? Let us know. Dying to share a comment, or to tell us about something provoking that you read? We have Short Communications just for these things. As well, we will be happy to include an abstract or a summary of your recent publication in math or stats education. Have a question? Send an email to the Newsletter Editors, Andie Burazin, Lauren DeDieu, and Michael Liut.

Enjoy reading this Newsletter, and thank you for your support!

Veselin Jungic, Simon Fraser University 

Miroslav Lovric, McMaster University 

Thank You !!!

Dianne Ashbourne, an Educational Developer at the University of Toronto Mississauga, has been of great help in guiding and supporting the FYMSiC Newsletter Editors to develop and launch this first edition of the newsletter. She is an integral part of having the newsletter come to life, and the FYMSiC Editors are very appreciative of her input and wisdom.

Thank you Dianne !!! 😃

Spotlight on Teaching

Math is Storytelling

Vincent Bouchard, University of Alberta

What is 13 × 37 equal to? Many of us would not be able to answer this question immediately. Some of us could do it on a piece of paper. Others may need a calculator. In the end, many people would conclude: “Oh, I’m not good at math…” My take is: this has little to do with being good at math. If I asked you to conjugate the French verb “manger” in the imperfect subjunctive, you may not remember how to do it. Would you conclude that “you are not good at storytelling?” Probably not…

We can all tell stories. Some of us are better at storytelling than others: being a good storyteller is an art, which requires skill, practice, and experience. And it requires fluency in a certain language. But the art of storytelling is certainly distinct from technical proficiency in a given language.

When we learn a foreign language along with its grammar and idioms, we learn how the language is written and spoken. We become technically proficient at understanding and using the language for which we can then use to tell stories. And if we do not speak, write, or read the language for many years, we forget its subtleties. We make mistakes. But that doesn’t mean that we have lost the skill of storytelling. For instance, I am not a native English speaker. I make grammatical mistakes. My vocabulary is limited. But I can tell stories (some of them even somewhat interesting according to my kids!).

Mathematics is storytelling, but in a different language. It is storytelling about symmetries. About patterns. About shapes. About numbers. It is storytelling about how various patterns that are common in different contexts can be unified neatly in terms of abstract structures. It is storytelling, coated in the abstract language of mathematics.

I truly believe that all of us can do mathematics, just as all of us can tell stories. Some of us are certainly better at mathematics than others, and being a good mathematician requires skill, practice, and experience. And it does require technical proficiency in the mathematical language. It does require being able to correctly manipulate abstract structures, such as numbers. But this is not what mathematics is really about. Just as storytelling is certainly not about conjugating verbs.

What makes a good mathematical story? This a notoriously difficult question, just as it would be rather complex to pinpoint what makes a written story good. But there are a few important facts. A mathematical proof (which is a fancy word used by mathematicians to describe a mathematical story) needs to be correct, according to the rules of the mathematical language. The mathematical language is based on logic. Thus, a mathematical story must also be logically consistent.

In the end, what makes a mathematical story good is very subjective. A lot of it is aesthetic: in the eyes of a mathematician, it is beautiful. It may shed light on new connections between various abstract structures that appear a priori unrelated. For instance, Euler’s identity, which relates five of the fundamental constants in mathematics, namely the base of the natural logarithm, the imaginary unit, the constant, and the numbers 0 and 1 is a truly beautiful mathematical story.

Ultimately, a good mathematical story requires an interesting plot. Which is where the art of mathematics lies. As the famous mathematician Georg Cantor once said: “The essence of mathematics lies entirely in its freedom.” Mathematics is different from science, inasmuch as we are free to write our own mathematical stories. We are only bound by the limits of the mathematical language, which is constantly evolving. But we need good ideas. We need inspiration. It turns out that Nature, as described by physics, is an extremely fertile source of good mathematical ideas. Just as Nature is often the inspiration for many good written stories!

Going back to 13 × 37, it is not so surprising that many of us have forgotten how to find the answer off the top of our heads. If you learned a foreign language many years ago, but did not use it much at all in the meantime, you may also have forgotten how to conjugate verbs correctly. But that does not mean that we are not all able to tell mathematical stories. With some training and practice, we can construct logically consistent arguments. We can abstract the essence of a concept. And being able to tell mathematical stories is immensely important in all spheres of society. For instance, knowing when a true statement implies another one, or when it does not, is absolutely crucial to construct sound political arguments. Mathematical stories are everywhere.

To the question of calculating 13 × 37, instead of the usual “I’m not good at math…”, perhaps a better answer would have been the following: “Hmm, I don’t remember how to calculate this off the top of my head. But let me tell you a good mathematical story. Let us assume that if I knew the answer, than I would be good at math. Now suppose that I don’t know the answer. Does that make me bad at math?”

Celebrating Milestones

Congratulations to all !!! 😃
The FYMSiC community is proud of your amazing accomplishments !!!

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Recognitions


Queen's University Professor, Peter Taylor, from the Department of Mathematics and Statistics is being Honoured for 50 Years of Service.

He has given direction and guidance to many. Thank you for being such a great mentor and inspiration to all and an integral part of the Canadian math and stats community !!!

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Awards

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Grants

  • UofT Learning & Education Advancement Fund (LEAF) Seed Grant, Implementing Computational Assignments in Calculus
    Andie Burazin, Assistant Professor, Teaching Stream, Robert Gillespie Academic Skills Centre & Department of Mathematical and Computational Sciences, University of Toronto Mississauga
    Tyler Holden, Assistant Professor, Teaching Stream, Department of Mathematical and Computational Sciences, University of Toronto Mississauga

Featured Classroom Resources

Want to learn about a new and cool resource to incorporate in the Classroom? This is a fantastic space to share that classroom resource with the FYMSiC community !!!

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An Active Learning Lesson Idea: 
(Re)Discovering Varignon's Theorem

Gary Au, University of Saskatchewan
 

Setting

Introductory Linear Algebra course, 70 students (from diverse programs); has also worked well with bright high-school students.

 

Motivation

This activity involves a result that is somewhat surprising yet rather simple to discover. There are also many different ways to prove this result, which leads to many natural follow-up questions that are mathematically rich yet very accessible.

 

Activity

The class before (in which we discussed vector addition and dot products) I simply told my students that we would be doing something different the following class, and the only preparation they need was to come to class with a pencil, a ruler and an open mind. On the day of the activity, I handed out worksheets that prompted each student to draw two different looking quadrilaterals. After giving them a minute to do so, further prompts were displayed on the projection screen, asking them to connect the midpoints of the 4 sides of each of their quadrilaterals to form two new, inscribed quadrilaterals. The students were then encouraged to discuss their creations with their peers, and suggest any patterns they notice. Here are samples of the drawings they came up with:

Figure: Sample student drawings

 

The students quickly got the hunch that the midpoint quadrilateral seems to always be a parallelogram, and translated the conjecture into multiple statements involving vectors. Among them, they chose the simplest looking statement (that required showing two certain vectors are equal) for our first attempt of a proof. I started a “What We Know” panel on the board and solicited potentially useful ideas to put there. Once we gathered enough observations, the students were able to piece together a proof. During the whole process I focused on facilitating the discussion, scribing their ideas on the board, while constantly suppressing my “instructor” urge to give ideas away when they were momentarily not progressing. I tried my best to make sure the joy and excitement of discovery truly belonged to my students.

By the time we had a completely crowdsourced proof on the board, we were about 30 minutes into our 50-minute class. We then explored some follow-up questions, which are detailed on the slides below. (We were only able to discuss parts of the first two questions due to time constraints.)

Towards the end of the class, I reminded my students that, having already gone from millennia-old grade school arithmetic to mathematical subjects that are “only” a few centuries old such as Calculus and Linear Algebra, they are slowly marching towards the frontier of Mathematics, where new knowledge is still being continuously discovered in the present day. I revealed to the class about the attribution of the result we proved (it's Varignon's Theorem), suggested some further reading, and ended on the note that they can generate knowledge that is new to them --- or even to the world --- by simply being thoughtful and inquisitive.

Materials

Slides
Worksheet

Short Communications

Quick write-ups to inform and inspire the FYMSiC community about opinions, research studies, motivational pieces, outreach activities, and much more !!! 😃

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"What I Wish My Students Knew" - A Survey

Veselin Jungic, Simon Fraser University


What would you, as a university course instructor, say to your former, current, and/or future students? Would that be advice? A list of principles that determine your teaching philosophy? Your approach to teaching? A moment that changed your teaching perspective? Your opinion about students in general? A description of your "ideal" student? A "Thank You"? A "Sorry"? A memory from your own days as a student? An anecdote? Or a piece of personal information that you wish that your students knew but never had an opportunity to tell? Or anything else ...

The goal of this survey is to collect reflections about teaching and being a teacher that may be shared (anonymously, if you choose) with other members of the teaching community.

The survey is accessible at the link: http://websurvey.sfu.ca/survey/269496490

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A Blended Model and its Benefits

Fok-Shuen Leung, University of British Columbia
Vanessa Radzimski, University of the Fraser Valley

As instruments of first year mathematics teaching, lectures do many things adequately and one thing very well. In a recent assessment, French and Kennedy (2017) list seven qualities of lectures. Six of them are pedagogical, ranging from the modest—lectures “potentially promote skills in listening”—to the major—lectures “can provide context and structure”. The importance of the seventh quality is underlined by its being stated without hedging (potentially promote, can provide) as well as by its very inclusion in the list, despite the authors’ concession that it is not a pedagogical quality. They write, simply: “Lectures are a cost-effective and efficient method for teaching at scale.”

When it comes to defending lectures, scale is the closer. Sustainable methods must account for sheer numbers.

There are thousands of students each year who take first year calculus at UBC. They are taught by faculty members, postdoctoral fellows and graduate students. Since the number of students is growing faster than the number of instructors, the class sizes have also been steadily increasing. A graduate student teaching for the first time can expect a class of around 80 students. Faculty sections can be as large as 450.

For the past five years, a “blended model” has been used in a number of first year calculus classes at UBC. The model was designed with three goals in mind:

  1. to benefit students by having them learn new mathematics in a small, active setting;
  2. to benefit graduate students by having them teach in a small, active setting; and
  3. to retain the efficiencies of lectures.

The blended model replaces the standard three 50-minute lectures per week with one 50-minute lecture and two 50-minute “small classes”.

The lecture is taught by a faculty member. As French and Kennedy suggest, it provides context and structure. It might include a small number of canonical examples and some technical tools.

The small classes are in groups of 25-35 students facilitated by one graduate TA and one undergraduate TA, and they are where the real learning takes place. They are active: the teaching team designs the classes around a series of problems for students to work on, mainly in groups. The guideline is that, 80% of the time, students in small classes should be actively solving problems. And the concepts covered in them are new: these are not workshops where students do practice questions; they are guided problem-solving sessions that lead students to encounter and articulate new material. For example, students might learn about the Chain Rule in their lecture, but see and be asked to describe implicit differentiation for the first time in their small class.

Does this blended model achieve its goals?

Students appear to benefit. The written feedback that they provide is overwhelmingly positive. We also measured the final exam performance of students in a blended section of a recent integral calculus course. The final exam was common to all 10 sections of the course, and constructed by the instructor-in-charge, who did not teach the blended section. Grades were adjusted according to students’ final grade in the prerequisite differential calculus course, using a linear regression model. Students in the blended section scored, on average, 5.6% higher on the final exam, with a p-value of 0.013. We ran the same comparison using every other section as the experimental section (including in the previous year, when no section was blended), and in no other case found an average difference with p<0.05. Likewise, head-to-head comparisons between individual sections confirmed that the blended model has a significant positive effect.

Graduate students also appear to benefit. As a training ground, small classes are much more forgiving than an entire section of a course. In weekly team meetings, graduate student instructors help design the curriculum, discuss how to run their classroom, and shore up their pedagogical content knowledge (Shulman, 1986). The small class environment provides a space for graduate students to experiment with student-centred pedagogies and develop their identities as instructors, along with a community of other graduate student instructors. As we write in a recent paper describing the model: instructors “are being trained, not overwhelmed” (Radzimski et al., 2019). In contrast, in a standard lecture model, first time graduate student instructors are often expected to handle most of the logistics that come with teaching an entire course, leaving little room for innovation, reflection, and self-discovery.

Finally, the efficiencies of lectures are retained. Faculty members are generally comfortable, and often good, at giving lectures. In the blended model, the role of the lecture is circumscribed but not diminished. Faculty members spend less time preparing and giving lectures, and more time organizing the course and mentoring junior instructors. This seems to be a sensible deployment of resources.

The importance of resource management is the blended model’s underlying theme. However, we acknowledge that it may not be a particularly appealing one for the readers of this newsletter. After all, if you are reading this, you are likely a very successful instructor—one who uses those three hours of lectures in creative, effective, memorable ways. Why give that up?

This is a fair criticism, and we have no decisive response. We suggest only that the frame of creativity be widened. The driver of innovation might not always be the experienced lecturer who makes a hall filled with hundreds of students crackle with ideas. Every so often, it might be the new graduate student instructor, standing in her small class of 30 students, figuring things out for the first time.

References

French, S., & Kennedy, G. (2017). Reassessing the value of university lectures. Teaching in Higher Education, 22(6), 639-654.

Radzimski, V., Leung, F.S., Sargent, P., & Prat, A. (2019). Small scale learning in a large scale class: A blended model for team teaching in mathematics. PRIMUS(in press).

Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational researcher, 15(2), 4-14.

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You're Teaching Counting Wrong!

Shannon Ezzat, University of Winnipeg
 

One topic that is taught in a whole lot of first year math and stats courses is the idea of counting with permutations, combinations, and multipermutations.

In most, if not all, resources I've come across, the same basic treatment is given (see Khan Academy for an example): after fundamentals like counting independent events and factorial notation, permutations are covered with examples like “How many ways can 8 runners receive a gold, silver, and bronze medal?” and the formula n!(n-r)! is introduced; the denominator term is usually explained away as a “convenient” way of writing n×(n-1) × ... × (n-r+1).

Immediately after, combinations are introduced with examples like “How many different 3-person committees can you form from ten people?”. Students are shown the formula n!r!(n-r)! and are told to take the permutation formula and divide by r! since “order of the choices doesn't matter”. Then, multipermutations are taught with examples like “How many unique arrangements of the letters STATISTICS are there?”. This is taught in a number of different ways.

However, I believe this order is backwards. Everything should be viewed as an instance of an arrangement of letters problem, and thus a multipermutation. If you can convince students that the number of arrangements of ABCDEF is 6!, CANADA is 6!3! (where we divide by 3! since the order of A's doesn't matter), and then STATISTICS is 10!3!2!3! then they can use this idea to solve any arrangement-style counting question (combinations, permutations, and multipermutations).

Consider the permutation question above. If we assign G,S,B to the medal winners in the obvious way, and X to the five non-winners, then any arrangement of the letters GSBXXXXX gives us one instance of medal winners. Using the idea from multipermutations, we can count these arrangement as 8!5! where the 5! now has meaning; it comes from the order of the X's (non-winners) in your arrangement not mattering.

For the combination problem above, we can assign C to the 3 committee members, and X to those 7 not selected. Similarly to the above, the number of arrangements of CCCXXXXXXX is 10!3!7! which is, of course, the number of committees. Again, both denominator factorials have meaning now.

Note that formulas are not even discussed; they don't need to be if approached like this. In my experience students find this much easier than remembering the formula and which goes with the correct situation, and rather than blindly following a formula, they are considering what each factorial in the expression means and thus understanding how the counting is happening. I have had success with this, and if anyone else decides to give it a try I'd love to hear how it goes for you.

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2019 High School Computer Science and Mathematics Teacher Workshop: A Dialogue on CS and Math Education

Tyler Holden, University of Toronto Mississauga

Michael Liut, University of Toronto Mississauga
 

Every year, the Department of Mathematical and Computational Sciences at the University of Toronto Mississauga (UTM) hosts a High School Teacher Workshop with computer science (CS) and mathematics teachers. Teachers hail from across the Greater Toronto Area and its surrounding municipalities, with the aim to collaborate about the ways in which first-year CS and mathematics courses at UTM have evolved to adapt to the changes in the local Ontario high school curriculum. It is an opportunity to openly discuss the teachers’ insights and experiences with grade 11 and 12 computer science and mathematics courses.

In early May 2019, we hosted the workshop under the title “A Dialogue on CS and Math Education.” The objective was to develop resources that would better help university instructors, high school teachers, and incoming students through the transitional period from grade 12 high school to first-year university.  As we brainstormed and created resources, participants continually engaged in an on-going conversation on how to better support and guide students for success in their academic endeavours. This year we invited fellow colleagues from other universities and colleges to participate in the community.

The workshop was a two-day event. On the first day, university instructors and high school teachers brainstormed ideas for helpful resources to provide to students. To better understand what elements were to be included in the resources, undergraduate students shared their experiences taking first-year courses in a Q&A format. Ending the day, a plenary speaker, Kevin Browne (Professor, Mohawk College and Lecturer, McMaster University) suggested recommendations for new approaches to computer science pedagogy and curriculum, as well as some of his outreach efforts outside of the classroom.

Plenary speaker Peter Taylor (Professor, Queen’s University) shared his work at the high school level in his math9-12.ca project, which helps address the need for students to learn how to computationally think and to better prepare for university. Before the creation of the resources began, both computer science and mathematics university instructors discussed the expectations entering university through first-year assessments and standard baseline knowledge. The resource development began and the results of our work can be found in this GitHub Repository.

At the end of the workshop, we asked all participants to fill out an online survey. The responses will be used for the next iteration of the workshop in the following academic year. Some ideas that high school teachers presented include: inviting senior administration and school board representatives to attend; having media attend; having high school students, and creating tools collaboratively with both high school teachers and university instructors to encourage more students to take computer science and mathematics courses; and to better prepare students for their first-year studies at a post secondary institution. The feedback received from the workshop was very positive, as one high school teacher said, ‘[the workshop] felt like a support group for cs and math educators who actually care → very ‘therapeutic’.

For further details and questions about the 2019 High School Computer Science and Mathematics Teacher Workshop: A Dialogue on CS and Math Education, please visit the link or contact directly via email.

Recent Publications

Providing the FYMSiC community with
recently published work by our members !!!

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Burazin, A., Jungic, V., & Lovric, M. (2019). First Year Mathematics Repository Workshop, Banff International Research Station, Banff, AB, February 8-10, 2019. CMESG Newsletter/Bulletin, 35(2), 2-5.
http://www.cmesg.org/wp-content/uploads/2019/06/CMESG352.pdf

 

Burazin, A. & Lovric, M. (2019). Rethinking Teaching First-year Mathematics in University. In H. Holm, S. Mathieu-Soucy (Eds.), Proceedings of the 2018 Annual Meeting of the Canadian Mathematics Education Study Group pp. 123-130. Squamish, N.C.: Quest University.
http://www.cmesg.org/wp-content/uploads/2019/05/CMESG-2018.pdf

 

Dawson, R. (2019). Why Do We Teach What We Teach?, CMS Notes, 51(3), 2. https://cms.math.ca/notes/v51/n3/Notesv51n3.pdf

 

DeDieu, L. (2019). Incorporating Mathematical Writing into a Second Year Differential Equations Course, Proceedings of the 2018 Annual Meeting of the Canadian Mathematics Education Study Group, pp. 153-155.
http://www.cmesg.org/wp-content/uploads/2019/05/CMESG-2018.pdf

 

Jungic, V. (2018). Students and Instructors in Large Classes: Building Dialogue and Mutual Trust, In Golding, J. M., Kern, K., & Rawn, C. (Eds.). Strategies for teaching large classes effectively in higher education.

 

Upcoming Events

Spreading the word to the FYMSiC community about fun and 
educational events to possibly attend !!! 😃

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Conferences

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Workshops

  • PIMS Workshop on Open Educational Resources and Technologies in Mathematics
    Location: Banff International Research Station in Banff, Alberta
    Date: Friday, July 26 to Sunday, July 28, 2019

    Description:

    The workshop, to be held July 26-28, 2019 at BIRS, will be a gathering of experts and enthusiasts in open educational resources (OER) in mathematics, and associated technologies. Particular areas of interest include the PreTeXt open textbook language, open homework development, including WeBWorK and Moodle, and Jupyter notebooks for teaching. A goal of the workshop is to build an OER community, especially in Alberta, to promote increased pooling of resources and effort.

    Confirmed participants include active members of the OER community in western Canada, as well as experts from the PreTeXt community from northwest USA. At this time, there are no spaces remaining for additional participants.

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Thank you for reading !!! 😃

Do you have something you want published?

If you wish to contribute in the next edition or have any questions or comments,
please email the Editors at newsletter@firstyearmath.ca

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Editors: Andie Burazin, Lauren DeDieu, and Michael Liut
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