Empowered Problem Solving Workshop: My Takeaways

My takeaways from #epsworkshop (April Soo)

I love it when professional development is purposeful and practical.  I’ve been following Robert Kaplinsky for some time now and finally decided to enrol in his Empowered Problem Solving Workshop.

My reflection post in the last module of the empowered problem solving workshop.
My reflection post in the last module of the workshop. Sad it’s over…

I don’t have time for problem solving in my classroom.”

TRUST me, I’ve been there. The first time I ever taught Calculus, my talk time during an 80-minute block was probably at 50-80%. It was awful, I was so dehydrated. It also didn’t help that I did not have a strong enough grasp of the material that I could deliver problem-based lessons with any sort of confidence. I was simultaneously teaching and relearning the material myself so how could I expect my students to develop these deep understandings when I was barely keeping my head above water?

No bueno…

Looking back, I realize that trying something is always better than nothing.  Problem solving isn’t something you do “if you have time for it,” like at the end of a unit. Because you know what? You’re never going to have time for it. You’ll always feel like the time could be better used for review, a project, to reinforce a skill…etc. Problem solving is not something you should “make time for”, it needs to be integrated into the content we teach. I would argue that the heart of mathematics is problem solving. The sooner we realize that math isn’t just about getting the right answer, passing a test, or even getting into university, the sooner we can teach in a way that honours what doing mathematics is truly about.

Why Problem Solving?

My students lack the basic skills and understanding to do these types of problems.”

If that is what you’re thinking, know that I too, have had this same thought. Herein lies the beauty of problem-based lessons: students don’t need to be pre-taught skills or content, they can learn them along the way.

“I’ve tried problem solving before and it doesn’t work. Students just want to be told what to do.”

Guilty. I’ve been there too. It’s not going to be perfect the first time you do this. Students WILL resist, and you need to persist. If you don’t genuinely believe that problem solving is worth the time and effort, your students won’t buy into it either.

When I first tried problem-based lessons, I did not spend enough time anticipating student responses and was taken off guard by solutions or strategies I hadn’t thought of. I tried to lead meaningful discussions about student work, but because I wasn’t getting the engagement I wanted, sometimes ended up making the connections for the students (I’m still working on scaling back my “rescue the student” instincts). Success, however, comes in small doses, like getting a student who normally never raised their hand to try a problem on the board, or maybe just seeing a decrease in off task behaviour.

Teaching problem-based lessons takes effort, from the student AND the teacher, but that is precisely why its so awesome. Students aren’t just passive recipients of knowledge, and teachers don’t need to spoon feed their students.

My Biggest Takeaways

  1. Problem-solving: Just DO IT!

2. Be deliberate about how to facilitate meaningful discussions in math. Often, we get to an answer and that’s it. Full stop. Getting to the last act of a 3-Act Math Task doesn’t mean that the learning stops there. Here is a wonderful opportunity to discuss various approaches to the problem, potential sources of error, limitations of our mathematical models, and to make connections between different solutions. This is an area where I feel I need the most practice, and it is also most difficult to implement during this time of online learning due to COVID-19. I’m limited by the fact that I cannot circulate the classroom or peep over students’ shoulders to see where they are at, but I am trying to find alternative ways to connect.

5 Practices for Orchestrating Discussion

Here’s a snapshot of me working out a selection strategy for sharing student work, and anticipating questions that might be helpful to ask:

My rough notes as I thought about how I might lead a discussion about student work.
Which responses would you pick to share? In what order would you share them?

3. You can always add information, but you can’t take it back. Dan Meyer refers to this as turning up the Math Dial. Robert Kaplinsky talks about “undercooking” our students (like you would a steak). Ask questions in a way that ranges from least helpful to most helpful to give your students a chance to make connections for themselves.

4. Ask yourself “Why” more often. Why am I doing this problem? To introduce a new concept? Get my students used to productive struggle? Problem completion?  Be intentional about the purpose of your lesson and what can be realistically achieved.

5. Ask better questions. Shallow questions tend to lead to false positives. A student may appear to have procedural knowledge, fluency, and conceptual understanding, when in reality they are just good at replicating the work that you do (me in school…). You might be asking, “How do I really know if my students have the components of mathematical rigor?” Check out Robert Kaplinsky’s Open Middle problems and Depth of Knowledge Matrices.

Depth of knowledge matrix (credit to Robert Kaplinsky)

What do you do when students submit low quality/low effort work?

I’ve been taking an online workshop to learn more about practical ways we can implement problem-based lessons in our math classrooms called Empowered Problem Solving by Robert Kaplinsky (#mathhero #teachercrush). In one of the workshop modules we troubleshoot various issues that may arise throughout the process of teaching a problem based lesson, for instance:

  • What happens if students don’t ask for information that they need to solve the problem?
  • What do you do if a student shares a strategy that you don’t understand or did not anticipate?
  • What do you do when students submit low quality or low effort work?

That last question really had me thinking a lot about assessment. When students submit low quality work it is often because they don’t know what the expectations are. Something I do quite often in my classes is share student work samples after an assignment or test to address common errors or mathematical practices. Here’s a brief overview of my journey in providing feedback for my students:

“What Should My Answer Look Like” Posters from MathEqualsLove,
Examples are from my class 🙂

I don’t make enough time for level 3 work, and I should. Within a single semester, my goal is to give students at least two opportunities to do meaningful peer assessment. Of course, I anticipate this to be a gradual process, and it might take some time to get to a point where students can comfortably and confidently do peer assessment.

Assessment is difficult; even with a simplified assessment scheme one two teachers may assess the same student work slightly differently depending on their interpretation of what is “correct” or what qualifies as “sufficient reasoning.” Unfortunately these discrepancies will arise no matter what, but I think there is a lot of value in putting the students in our shoes and giving them opportunities to assess each other’s work.

Inspired by the Empowered Problem Solving Workshop, I’ve created a Mathematical Peer Editing Checklist and Feedback Form with practices I value and that I think is general enough to be used with most and/or all problem-based lessons. I’ve also incorporated an “overall feedback” section in the form based on Kaplinsky’s Levels of Convincing (originally inspired by Jo Boaler #mathhero #teachercrush) that asks students to rate each other’s mathematical writing based on how convincing they think their argument/work is.

  • Do you think this framework would work with your students?
  • How would you modify it to make it better?
  • Anyone have suggestions for a more concise title, as opposed to “Mathematical Peer Editing Checklist and Feedback Form”?
  • Thoughts on my use of the word “writer” to describe the student who’s work is being critiqued?
  • Other thoughts?

Make It Stick

This blog post is about how the math department at my school in Suzhou, China implemented changes to the way we taught Math 10 and 11 to incorporate data and research from cognitive science to help our students learn better. I include a summary of what we learned, and some ideas for improvement.

 

It began last summer, at math camp. Yes, I attended math camp as a fully-fledged adult! Yes, there were other adults present. And yes, it was awesome! (Officially named the “Summer Math Conference for Teachers” but let’s not get into the nitty gritty). One of my favourite sessions was the one led by Sheri Hill, Arian Rawle, and Lindsay Kueh on the grade 10 course redesign they have implemented in at their school in Ontario. The course redesign is based on research and best pedagogical practices outlined in the book Make It Stick, The Science of Successful Learning by Peter C. Brown, Henry L. Roediger III, and Mark A. McDaniel.

Book Synopsis: Why is it that students seem to understand what is being taught in class but end up failing when it comes to test day? How does one progress from fluency to mastery over challenging content? Many common study habits like re-reading and highlighting text create illusions of mastery but are in fact completely ineffective. This books explores insights from research in cognitive science on learning, memory, and the brain, as well its implications on teaching and learning. 


THE WHAT

​After the session, I couldn’t wait to bring these ideas back to the math team at my school in Suzhou, China so we could start putting them in action too! We began by looking at issues we noticed our students faced:

 

  • Not knowing, understanding, or practicing math vocabulary
  • Low retention rates of material from one year to the next (in some cases from one week to the next!)
  • Lack of basic skills (algebra, numeracy)
  • Low perseverance
  • Low completion rates for homework

We made it our goal to address some of the issues above, taking many ideas directly from the session presented by Hill, Rawle, and Kueh.

Like Hill, Rawl, and Kueh, we removed unit tests, which freed up a significant amount time for other topics and activities. Instead, we moved to weekly cumulative quizzes that held students accountable to everything they have learned in class up to the Friday before quiz day (no skills expire!).

The weekly schedule looks as follows:

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Our school runs on 80 minute blocks, with Fridays being half days with 40 minute blocks.

 


​The HOW and WHY

​Fast Fours. Four warm up questions printed front and back on half a sheet of A4 paper that’s ready for students as they walk into class. Each question relates to a different math topic that may be review from previous years, numeracy focused, or review of current material. By mixing up the problem types, we are introducing interleaving to students, the idea that we learn better when multiple topics or subjects are woven into the same learning session.

 

If you are interested in redesigning your course or looking for ideas on where to begin, I would say the Fast Fours are the easiest to implement. They work well because in all likelihood, every student is able to answer at least one question out of four, it is low stakes (not graded), gives you time to check in with students at the beginning of the class, check homework, answer questions, and it also gives students an opportunity to collaborate and help each other. (Scroll to the bottom of this post to see examples of these documents).

Weekly Quizzes. The quizzes themselves are one-page, double-sided documents comprised of three main sections. Part A focuses on vocabulary where we ask students to match key terms or fill in the blanks. Part B is review of previously learned material, and may include basic algebra questions from previous grade levels. Part C is new material that was covered the week before.

The quizzes only take up half a block and we drop the lowest two quizzes at the end of the year. If students are away for a quiz, they do not write the quiz and instead it counts as one of their dropped quizzes. 

Problem Solving/Project Days. At the start of the semester, we wanted to implement problem solving days that helped students dive deeper into the content they learned throughout the week and try some more challenging problems. Alternatively, our vision was to use these days as project days.

Fun Fridays. Our Friday blocks are shortened and many teachers find this time unproductive for teaching new material, which made having Fun Fridays built into our schedule a good fit. During this time, we may play a fun review game with students, have them explore an activity on Desmos, or one might even teach them something outside of the prescribed curricular content like coding.

Homework. In their original course redesign, Hill, Rawle and Kueh wrote customized homework assignments that introduced the ideas of interleaving and spaced practice to their students. That is, their homework assignments would begin a set of ten mandatory questions: five questions from previous material, and five questions from the lesson, as well as one or two challenge questions.

Unfortunately, our team was unable to implement so many changes at once, so we simply kept homework the same, and instead implemented randomized homework checks. Our hopes were to emphasize the importance of practice, and keep students accountable for it. 


​RESULTS

Some of my takeaways from this semester:
​

 

  1. Fast Fours. I’m definitely keeping the fast fours in my classes. In their of year reflections, students mentioned that this was one of their favourite things to do because it helped them remember content they had not practiced for a while, and they were able to get immediate feedback on it. One student suggested having a balance of easy and more difficult questions for those who finish early (perhaps a 3:1 easy to challenging ratio).
  2. Weekly quizzes. Since the weekly quizzes introduce interleaved and spaced practice, they reduce the need for large blocks of class time devoted to final exam review as we were continuously reviewing content throughout the entire semester. In terms of grading, however, it is important to keep up with it to ensure that students get feedback before the next quiz. An outcome we did not expect was that despite seeing several iterations of the same types of questions, students continued to struggle with the finance unit and were unable to identify the correct formula to use for the question.
  3. Vocabulary. As a department, we found it extremely valuable to teach, review, repeat, and practice math specific vocabulary to help students increase fluency and be better equipped to answer difficult problems. Many Chinese students arrive in our classes already having much of the essential background knowledge in math but lack the English skills to succeed, so we have found this to be a fruitful endeavor. We plan to begin our Math 10 classes with a mini vocabulary unit to give students started with some common terminology and foundational knowledge for the upcoming semester.
  4. Problem Solving/Project Days. Problem solving was a lot harder to implement, and we did not have a clear structure for it. As a result, Thursdays were mainly used for projects or as additional lesson days.

Overall, the teachers in my department felt the changes were worthwhile to implement and will continue with the same program for semester two, with a few new projects that we’ll be adding to some units that did not have one. In the future, I’d like to rethink how we might implement problem solving days in a more structured way. 

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Select responses from student feedback regarding Fast Fours in Pre-Calculus 11 (Nov 2019)

 

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Select responses from student feedback regarding Weekly Quizzes in Pre-Calculus 11 (Nov 2019)

 

AREAS OF IMPROVEMENT and NEXT STEPS

Finance Unit. I’m unhappy with the way finance is currently being taught to our students, and I think we can do better. I remember very little in the way of learning about finance when I was in high school. This was usually the topic my teachers skimmed over, and hence my dislike for it as a teacher now myself. In textbooks, it is usually presented as a series of formulas and how to apply those formulas, which is, I think, an area where we are doing our students a disservice. I’m a strong believer in getting students to first understanding the math behind the formulas, and some of these formulas (like the loan formula, for instance), does not lend itself well to building students’ conceptual understanding of it at the grade 10 level.

 

A useful analogy from Barbara Oakley’s course Learning How to Learn goes like this: a formula is like a summary, it describes several important ideas that mathematicians have packaged into a simple and elegant mathematical statement. Take Newton’s second law of motion for example, which is stated formally as, “The acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object” (physicsclassroom.com). Simply put, the relationship can be condensed into the mathematical formula f = ma.  As such, we must understand the meaning behind each symbol and look at how they work together to tell a story.

My plan is to put together a rough plan for how we can revamp the finance unit, pitch these ideas to my team before the start of semester two, and see if we can collectively find a way to improve the way we teach this unit to our students (more to come in a later post).  

Active Retrieval. Next semester, I plan to pause frequently during lessons to quiz my students on material. I’ll ask them to put their away notes, and engage in some simple recall exercises. A useful analogy to think about this is described in Make It Stick; Dr. Wenderoth, a biology professor at the University of Washington tells her students to “Think of your minds as a forest, and the answer is in there somewhere… The more times you make a path to find it, the stronger that path will become.” This is exactly what happens in your brain as you engage in active retrieval to strengthen new neural connections as you gain new knowledge or learn new skills. 

Elaboration. Once a week, I will ask students to complete a written reflection or summary of ideas learned throughout the week in their own words, with added connections and extensions if applicable. It will be a five sentence summary of concepts learned, with enough detail to help recall important ideas when it is read it at a later date, not too much detail that students end up reciting the entire lesson. 

 

Sample Fast Fours
File Size: 38 kb
File Type: pdf

Download File

 


 

Sample Weekly Quiz
File Size: 891 kb
File Type: pdf

Download File

 


Ms. April

I don’t teach for the vanity, and let’s face it – teachers do not exactly have the best reputation these days. Something stupid about money-hungry fiends who take up way too much of the taxpayer’s money, blah blah blah. It’s not like we are educating the future citizens of the world or whatever, so no big deal. For the record, it is a huge burden to teach kids stuff they will never use in real life. Even textbooks have to work extra hard with their fancy graphics and enticing fonts to convince children that modeling the shoulder height of a male African elephant is an example of using cube roots in “real life.”

​Note that the problem states “a male African elephant,” as in (singular) male African elephant. Why the hell does nobody ever care about the female population? What made this particular male elephant so special that he can have his shoulder height modeled by a mathematical function? Won’t the other elephants feel left out? 

Some days I feel as if I am teaching from inside of a five foot thick cinder block that’s been buried ten feet underground. A few sympathetic students will strain their ears and squint their eyes, but no one is really listening. 

Other times my lessons go so horribly that I wish I could morph into a bird and fly away. At least that will be more exciting than what my students will have to endure. There are moments when I forget that teaching is not the same thing as learning, and there are instances when I  knowingly commit the heinous crime of giving my students the “I taught it so you should know it” attitude. I know, I’m awful. 

I have been told that it can take years to make a difference in someone’s life, and most of us do not have the privilege of witnessing that change. I have also been told that making a difference in somebody’s life can be as simple as handing out a lollipop.  

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My personal “lollipop” moment.
 My “lollipop” moment happened on my graduation day. 

Four years ago I was an orientation leader for the incoming class of con-ed 2015, Queen’s University. A couple of us volunteered to write letters to future members of the con-ed family that year. I had a lot of fun with those letters and wrote them on hello kitty paper and decorated the margins with stickers from my personal sticker collection (of which I am very proud of). Only one person out of five responded to my invitation to email any questions or concerns they had to me. Orientation week came and went, and for a while, that was that.

In all honesty, I had forgotten all about those letters. But one of those letters had been sent to a young man named Mike. Mike went on to become the Rector of Queen’s University in 2014. On the day I received my Bachelors of Education, I walked across the stage of Grant Hall to shake the Rector’s hand. He leaned in and said to me, “April I just wanted to tell you that you were the one who wrote the letter to me. The one with the hello kitty stationary and all. I wanted to tell you what a difference it made.” I was so shocked I nearly pushed him off the stage (okay, it was a gentle nudge, but my family members who were watching from the balcony swore that it looked like I punched him in the shoulder). 

That story still gives me warm fuzzies every time I think about it. Who would have thought that a letter I wrote, and purposefully sprinkled with tacky looking gold-trimmed stickers would have been something that could ever have an impact? I mean, okay, I doubt I drastically altered the course of his life by sending him that letter, but I will be forever grateful to Mike for showing me what I difference I have made. 

To quote Drew Dudley who quoted Marianne Williamson, “Our greatest fear is not that we are inadequate. Our greatest fear is that we are powerful beyond measure. It is our light, and not our darkness, that frightens us.”

Sometimes our students have small ways of telling us we matter, and they will come at moments when we least expect it. On Friday, I complimented a student for the cute stickers she had on her notebook. Surprisingly, stickers are not easy to come by in Kazakhstan, and if you know me, you will know that I am a proud owner of a shoe-box full of stickers that I have been hoarding since I was seven. Today, that student came to school with a pack of happy face stickers. She gave them to me.  
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So, to my fellow teachers who may feel discouraged, worn out, or overworked, I say – teach on! Follow your guts and stick to your principles. Teach because you matter more than you know. Teach because you are powerful beyond measure. Teach because you have the courage to teach.  

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And most importantly, teach because there might be stickers involved! (Ok, not really, that was a bad joke).