This week we were asked to explore an aspect of online/blended learning that we are interested in. As I am working on building a blended Calculus 30 course, I felt this would be a great time to read into how others are structuring blended learning of the mathematics variety. I should also warn you, this is a long post but, if you get through it, you will find my favourite definition of blended learning (thus far anyways).

In my searching, I found a series of articles written by Birgit Loch, Rosy Borland and collaborative authors Liam McManus and Nadesda Sukhorukova for the annual ASCILITE conference. Their papers discuss a variety of ideas, implications and challenges around creating a blended mathematics course. In their first article, *The transition from traditional face-to-face teaching to blended learning – implications and challenges from a mathematics discipline perspective (*Loch & Borland, 2014), the authors discuss how mathematics is often overlooked for moving to a blended format for three reasons:

- Limited resources and cramped curricula.
- Instructor reluctance to move from the traditional “chalk and talk” as they have never experienced an alternative method of learning.
- The belief that mathematics is
*different*from other disciplines and it doesn’t need to be re-invented through another method of instruction.

I have to admit, I am not going to argue any of these points but I may use them to help plead the case for blended learning. We have discussed in class about the cramped curricula and, to me, that is the ultimate reason why we SHOULD be moving to blended learning, allowing our students a better opportunity to have access to the teachers to help them with their learning. What better reason to try something new as “I’ve never seen anything else done”, I hardly think that “this is how it is has always been done” is a good enough excuse to not move to blended learning. And mathematics is definitely different than many other disciplines, Loch and Borland mention that mathematics is “complex due to the visual nature of the discipline” and recognize that the digital typesetting of mathematics can be difficult for students and instructors to communicate in short response times. With applications such as SeeSaw and FreshGrade where students are able to post pictures of their work, I’m not sure that this is completely valid, although I definitely think that timely feedback is still a challenge digitally. They also discuss the technology requirements for both students and instructors as well as a fear that by focusing on the online submission of assessments, “the development of deeper mathematical understanding that occurs during practice may be impacted as students may be ‘doing’ less mathematics because they no longer write it out” (Loch & Borland, 2014).

Loch and Borland go on to discuss the use of the flipped classroom and how it has developed a more active classroom, where students are able to “do” mathematics with the support of the instructor as opposed to this time being used for lecture. This allows for the concepts to be developed deeper. Within the flipped classroom, they recommend using audience response systems (such as Kahoot, Mentimeter, or Plickers) to help gauge student understanding and misconceptions but they question whether students with low prerequisite knowledge are truly capable of learning in this manner. Interestingly, they found that students that were in active learning classrooms were 1.5 times less likely to fail than those in traditional lectures (Freeman et al, 2014 via Loch and Borland, 2014).

While in the classroom, Loch and Borland discuss “board tutorials” where students work a problem together on a whiteboard, effectively “doing” mathematics together and collaboratively. This comment made me think back to my undergrad and Math 223 where we would often go to office hours and have our professor, Douglas Farenick, find us a large chalkboard to work the problem as a group. This course was one that challenged my thinking and caused me to struggle in mathematics (something that was new to me at the time) but when asked for help from a student, I look back at that and try to emulate it, as I find it was one of the most useful exercises I have done (thank you Doug if you read this).

The article is summed up with seven questions that the authors feel need to be further researched:

- What can we do to ensure students engage with both online content and classroom activities?
- How can we encourage school leavers enrolled in first year mathematics units to self-regulate their learning?
- How can we build in redundancies, eg. enable students to recover if they have not watched a video beforehand or have not attended class?
- What technology is needed to enable effective online communication and collaboration to support learning in Mathematics?
- What technology is needed to support deep learning of mathematics? What new technologies might be on the horizon? What impact can learning spaces have on student engagement?
- On a departmental level, what is the best approach for supporting teaching staff (including sessional staff) to develop and implement innovative pedagogy approaches, promote digital content creation and use technology to enhance learning and teaching outcomes?
- How do we measure the success of a flipped classroom?

The second article by these authors, *Implementing blended learning at faculty level: Supporting staff, and the ‘ripple effect’* by Borland, Loch and McManus (2015) discusses question #6 above and the supports needed to implement blended learning at an institutional level and discusses many of the common themes that come up in our #eci834 discussions such as cost, accessibility, and professional development. What really jumped out at me from this article was the definition of blended learning that they chose to follow:

“an understanding of blended learning as being an approach which increases opportunities for students to engage with content and resources online in order to make more time available in face-to-face classes for active learning” (Borland, Loch, & McManus, 2015)

This definition really resonates with me, I value the face-to-face connections that I make with students and I like that their focus was to increase the effectiveness of this time, taking the lecture out of the face-to-face sessions and focusing on the student and their needs.

The third article, *How to engage students in blended learning in a mathematics course: The students’ views* by Loch, Borland and Sukhorukova (2016) addresses questions #1 and #3 from the above list, and do so from the perspective of the student. They state that students in blended learning courses need to be self-directed and self-regulated learners (they could use the skills from Twana’s post on online learning success strategies). Students in this study stated they like the face-to-face sessions because they were able to ask questions and gain further clarification on topics and it was found that students reacted positively to interactive and technology-enhanced classrooms where they were able to contribute in discussions with their peers (Donovan & Loch, 2013 via Loch, Borland, & Sukhorukova, 2016). Students were also honest, stating that they do not always watch as many recorded lectures as intended or even never watch them at all, students cannot be forced to engage in teaching activities of any sort if they do not want to. I appreciate this very open, honest, and abnormal statement in the article as too often we focus on being able to reach every child when, we know deep down, some are just not ready to be reached.

An aspect of a blended course that I did not think of until reading this article was the ability for the instructor to incorporate additional information on the “why” we are learning this, with students in the study stating that they enjoyed being able to get deeper into why we focus on a specific concept and its general use outside of the grading scheme for the course. This is a common question in my classroom and I like that blended learning provides a non-mandatory platform for students to pursue their interests in this way to gain a deeper understanding.

In discussing the motivation for watching videos ahead of class time, a variety of ideas were provided. The one I disliked the most: providing marks for watching the videos. Signing in to watch the videos does not mean that students are actually “watching” the videos, they can hit play, mute, and walk away, never gaining the understanding they should and receiving grades that do not reflect their understanding accurately. The one I like the most: recap the content at the beginning of class, using one or maybe two examples, and provide a plan for each class so that students know what they are missing if they have to. Recapping with a couple of examples helps instructors know where their students are in their understanding and allows for further instruction, if necessary, before the daily task is started.

One major fault in blended learning as addressed in this article is that math is very hierarchical, constantly building on the content previously learned. If, when watching a video example, a student does not understand an early step, the entire video is lost, as is the time dedicated to it. This is something to consider in making videos for a blended learning course as you need to ensure you are being detailed enough that the weakest student in the class will be able to follow along, an perhaps may need to provide a review of prerequisite knowledge to help all students ensure they are confident in their solutions.

If you are still with me, congratulations, this has been a very long post and I am not quite done yet. In reading this, I am inspired to change not just my Calculus 30 course but by other mathematics courses as well, although I will wait until the fall semester to ensure I have had the time to adequately prep and organize my ideas in a meaningful and effective manner. In general, this is how I think I would like to set up my classes:

- Flipped classroom: Students are required to watch video of the examples with notes to follow along with before coming to class (such as how Ashley has described she runs her flipped classroom).
- At beginning of class, have a Plickers activity to determine how students are doing. This will require them to tell me if they are confident, need some clarification, or have no idea what happened.
- One to two examples on the board, so I can gauge where my students are. This will be followed by students asking specific questions on their current misconceptions.
- Group “Board Work” where students will be given a enrichment question and will need to come up with a solution collaboratively. I would like to play around with a presentation method similar to the interactive notebooks that Andy uses in his class, with some modifications.
- Work time for an practice questions. Or more time for enrichment. I am a firm believer in not everyone needs the same amount of practice to understand a concept and therefore do not require them to do all assigned textbook questions.

Well, I think that is it that I have to share, what do you think about this set-up for high school mathematics classes? Have you done something similar to any of the parts? What do you think of the findings of the articles? Let me know in the comments!

Glad you’ve found others who can guide your Calc.30 course! It didn’t occur to me to try to find examples of blended classes for my Music 9. I’ve been looking at it too closely, and need to take a step back. Thanks! Great post!

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I love the idea of groups of students working around whiteboards! I let my students do this, but after reading your blog I just have a vision now of extra whiteboards in my classroom for all students to have access…thanks!!

When you mentioned about the math videos being hierarchical and if they cannot get past a certain step the rest of the video is lost, it reminded me of how many Crash Course videos will refer to previous videos to brush up on a certain concept that is important for the current video’s topic and anyone can just click on the link that pops up to take viewers to the previous video. Perhaps a system like this can be instituted to make the path to learn the necessary concepts easy for students.

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I love the idea of white boards as well. I purchased small personal sized ones for my class and the students love them, not just for math but for formulating thoughts on writing assignment and projects.

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I also really like the idea of students working together on enrichment problems on the board. I have heard other teachers talk about painting desks with whiteboard paint so that they can use their desks as whiteboards. I love the idea but I’m hesitant to ask for permission to paint the desks in my classroom haha.

I like the set-up you would have for your flipped class, but I have to add a comment from my own personal experience. The reason I decided to flip my classroom was so that it opened up more time to work with students 1 on 1 or in groups within the classroom. Fitting the plickers, examples, enrichment and work time into each class would be hard to accomplish. Once again that’s just my personal opinion based on experience and I suppose based on my preferences. A good idea might be to break those activities up throughout the week, for example, “Math Madness Monday’s” might be the days you dedicate to a work period, while Wednesdays might be called “Work It Out Wednesdays” in which students work in groups to work through the enrichment problems.

I have bell work daily in which I ask students to complete an example or ask them to explain a concept to me in their own words. Then we have a brief discussion about it and they can jump into the questions afterwards. I really like the idea of plickers asking students about their confidence level though! I feel like my bell work could be replaced by an interactive tool like plickers or socrative to have a better idea of who watched the videos and is understanding the concepts. Great post Kara!

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I love the use of mini whiteboards in my classroom, there is something very safe about using a whiteboard.

I also struggle to see students use videos to learn new instruction if the pre-skills are not there. In my math class, I provide students with videos and practice on their pre-skills that will be needed in order to engage in new instruction. In addition, another teacher and myself also work with a daily rotation of students in small groups (a max of 5 per group). If students aren’t working with a teacher then they are working on Mathletics or math games. I’m not sure how this would look in a high school room but just a thought.

I also like the idea of a flipped classroom, I think the value in Math is the ability to share one’s thinking. Your enrichment tasks would be a great option for this. Another tool you could use is explain everything. In addition, you could challenge some of your students to make a video explaining the concepts.

Great post Kara!

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