I. General Information
School Name: Courtenay Elementary School
School District: SD#71 Comox Valley
Inquiry Team Members: Heidi Jungwirth: Heidi.jungwirth@sd71.bc.ca, Alison Walkley: Alison.walkley@sd71.bc.ca. Serina Allison: Serina.allison@sd71.bc.ca
Inquiry Team Contact Email: heidi.jungwirth@sd71.bc.ca
II. Inquiry Project Information
Type of Inquiry: Numeracy & Literacy Project
Grade Levels Addressed Through Inquiry: Intermediate (4-7)
Curricular Areas Addressed: Mathematics / Numeracy
Focus Addressed: Numeracy
In one sentence, what was your focus for the year? Our focus was how to teach problem solving in math.
III. Spirals of Inquiry Details
Scanning: At this point in our inquiry, the 4 questions are the guiding principles of our classrooms. Fortunately, because of a lot of intentional work, they extend to our school as a whole. We asked our students the 4 questions, but in all reality we know that they are engaged in their learning. The second question is a little more challenging, but most students understand what they are learning. Thinking about next steps is the most challenging question. Our emphasis on experiential learning and curiosity means that students are encouraged to ask questions about what they could learn. At our school, there is a strong sense of community, and we know that students feel supported and cared for.
Building on what we have learned in years previous, the First People’s Principles of Learning and the OECD Principles of Learning are embedded in our teaching practice. Our focus this year was to tackle a complex topic, problem solving in math. Having the principles of learning as a guide made it easier for students and teachers to confidently tackle a complex topic.
Focus: Before I talk about the area of inquiry, I will explain that this year we did things differently. Whereas before, Alison and I had both inquired in our classrooms with our own students, this year the focus was on my (gr 5) classroom. Alison and Serina played an important supporting role by listening to my debrief sessions, sometimes observing lessons, and working with me to find connections between problem solving and the other areas of math that we had inquired about. Their input was invaluable, but because it was my class that the inquiry took place in, I will write the rest of this summary as if I had the thoughts and made the decisions.
The area of inquiry this year was problem solving in math. We chose this area because it is the area in math that most students (if not all) struggle with. I (Heidi) had never figured out a way to teach it well, and my attempts over the years had mediocre (and that’s being generous) success.
Hunch: My main hunch was that problem solving is an area of struggle for many reasons. First of all, it is because students are all asked to do the same problem-solving exercises, even if the work is too difficult (or occasionally too easy). This has been our experience in all aspects of our inquiry so far. In other math topics (number sense, fluency) it has been extremely successful to have students working at their “just-right” level, and I suspected that this would also be the case for problem solving.
My second hunch was that problem solving was a complex topic (this is less of a hunch than a proven reality) and that to teach problem solving, I first needed to figure out which skills are used. After I did this, my theory was that I would teach the skills, and the students would learn how to problem solve.
My third hunch was that the work we had done in the previous years of our inquiry (number sense, fluency) would have a positive effect on our students’ ability to tackle problem solving. Of course, each year you get a new group of students, and only 6 of my students had been in my class for the year before. This meant that 6 of my students would have a “heads up” when doing the problem solving, because they had already done an intense year of number-sense and fluency work.
My fourth hunch was the importance of using hands-on materials. This has proven to be extremely important. I will talk about how we did this later in this write-up.
New Professional Learning: At the beginning of the inquiry, I was hoping to find a book called “How to Teach Problem Solving” and I would just have to follow the steps in the book and each student would be successful. Unfortunately, either that book doesn’t exist, or I couldn’t find it.
I found that many resources took the approach that I had used previously. They tried to guide teachers in teaching problem solving, but they didn’t go far enough when breaking down the skills needed. In the end, I didn’t end up using any of these resources and relied heavily on two: First Steps in Math (the operations binder) and the book “Daily Routines to Jump-start Problem-Solving K-8” by John SanGiovanni.
I also realized that being in our 4th year of inquiring about math means that Alison and I have built up a strong foundation ourselves, and that for this problem-solving inquiry we could use the skills, intuition, and pedagogy that we had developed from becoming knowledgeable about number sense and fluency. Breaking down problem solving into individual skills and teaching them in a systematic cumulative way using formative feedback proved to be as successful as it had when teaching number sense and fluency.
It is a significant step for teachers to recognize that they have built up the skills, knowledge, and experience necessary to make bold decisions regarding curriculum development and pedagogy. Participating in this 4-year supportive inquiry process has been fundamental for Alison and I to get to this point.
Taking Action: Problem Solving in Math is complex and it will take more than one year of inquiry to fully understand and implement an effective pedagogy. I will outline for you the major understandings that we have made this year. I will also outline for you the steps we are taking this summer to develop a “program” to further field test next year. The goal of this is to create a curriculum (K-5) for problem solving.
The first thing we had to figure out was what exactly are the steps involved in math problem solving? This is what we came up with:
1. Setting: What is the situation of the problem?
2. Number/Quantity: What are the numbers or quantities involved in the problem?
3. Units: What do the numbers mean? What are we talking about? How many of what?
4. Problem: What is the math problem asking? What is the story? What will the answer tell you?
5. Strategy/Operation: What mathematical strategy or operation will solve the problem?
6. Steps: What steps do you need to take to solve the problem?
7. Resolution: What is the final answer or solution to the problem?
8. Check: Does the answer make sense?
9. Reflection: What are you proud of? What would you like your teacher to notice?
Once the steps were identified, then the next thing I did was teach these skills (1 hour each Friday) in a systematic/cumulative way. This differs from traditional math instruction where problem solving would be a “unit of instruction” that might last a few weeks to a month.
Teaching in a systematic/cumulative way means that you teach:
1. Explore skill 1
2. Teach skill 1
3. Explore skill 2
4. Review skill 1, teach skill 2
5. Explore skill 3
6. Mention (briefly) skill 1, review skill 2, teach skill 3,
7. Explore skill 4
8. Assume they can do skill 1, mention skill 2, review skill 3, teach skill 4,
9. Explore skill 5
10. Continue in this way until all skills are learned.
This continues with new skills being explored, then taught, then reviewed, then mentioned. This of course does not happen quickly. It might take a few weeks at any one step. The amount of time between going on to the next step will depend on many things, and as a professional the teacher makes that judgement.
One important thing that I have learned is that after practicing a skill for an amount of time, I can expect that a student has learned it. (it’s ok to require some peer support). If, after a reasonable amount of time, a student has not learned a skill, then it is important that I take the time to figure out why. What is going on for that student? What key understanding does that student not have? What skills does that student not know how to do?
An important resource for these investigations is the First Steps in Mathematics assessments. These skill specific assessments can help you pinpoint where the gaps are in a student’s skills and understandings. You can then do the recommended interventions and help a student get back on track. In the past, I thought that students would just catch up if I repeated things enough. This rarely happens. If a student cannot do one of the problem-solving sub skills, they will most likely continue to struggle.
After spending a year teaching problem solving in a systematic/cumulative way, there are some key learnings I have made.
1. It is crucial to connect problem solving to physical objects, which helps all students make better connections. Eventually, some students choose not to use the physical objects, but I always made them available. We did this by making a problem-solving kit called “Dog Math”. Next year, I plan to use “Dog Math” plus 3 new kits for problem solving. “Road Trip Math”, “Nature Math”, and “Shopping Math”
2. Teach the sub-skills of problem solving in a systematic/cumulative way. I wrote about this earlier, so I won’t go into detail. Expect that students will learn the skills, and if they don’t learn them, make sure to find out why.
3. Think of a math problem as a story. We spent a great deal of time on this, and it was another way to connect problem solving to real-life happenings.
4. It works best to have longer problem-solving math lessons (we spent 1 hour each Friday) and to teach problem solving throughout the year, rather than teach it as a unit of instruction.
5. Using formative feedback in math is guaranteed to boost student skills and confidence. Instead of students doing their work and “handing it in for marking”, I circulate and make comments/guide student work/check that it is being done. Having a conversation about the work means that the feedback is immediate, and students love having conversations about what they are doing.
6. Students learn best when there are check-ins during the lesson. In a 1-hour lesson, I would have time to introduce and teach the skill, then have the students go back to their tables and work on it. Part way through the working time, I would bring the students back to the carpet and have students share where they were at and what they had done. After this short check-in, students went back to their tables and kept working. Having these check-ins helps those who are struggling to get ideas about where to go, and builds confidence in those who are on track.
7. Approach problems from both sides. Sometimes students read a problem and are asked to find the solution. Sometimes students are given the solution and asked to make up the problem.
8. Use units always. Always use units. Every step of every problem must have the units included. This has been a major learning for me. I didn’t realize that students will use numbers and not know what the number represents. Using units means that it is clear what is being talked about. I now require that each step of problem solving must have the units written in, and every conversation includes the units of what we are talking about. (are we talking about dogs or cups of dog food?)
9. Leaving space for exploration develops curiosity and creates engagement. For example, the first lesson with our “Dog Math Kit” went like this: the students took a kit each and tried to figure out what math they could make with it. They wrote this math in their Math Journals and then we came together and shared what we had discovered. The results were amazing. The students were 100% engaged and the math that came out of that lesson was amazing. This excitement about the dog math continued on for the entire year, and I believe that is directly tied into them first being able to explore and wonder about what they could do with the kits.
10. Take time to make sure that students understand what they are doing when they use an operation. This was another major learning for me. Students might know how to add or subtract, or even divide, but many do not understand what this means in real life, or in practical terms. If students don’t understand what they are doing, then they will struggle with problem solving. Taking time to teach this and to check-in with students at each lesson is invaluable.
Checking: This year long deep dive into problem solving has made a tremendous difference for all my students and has made a tremendous difference for me as a teacher. When we began this inquiry, there were 2 or 3 students in my class who would persevere through solving a math problem. Most students tried, but they didn’t know what to do, got stuck, and soon gave up. At the end of the year, all but 1 of my students were able to work industriously for 45 minutes to solve a math problem! Many students started to complain that the problems were too easy, so I built in ways that those students could challenge themselves.
At the end of the year, as a summative assessment, I gave my students a problem to solve (using our dog math kit). I wrote out the story problem and gave them the problem solving template that we developed. (and revised throughout the year as we learned things). Out of 22 students, 21 completed the assessment. Here is a sampling of student responses.
I have uploaded some examples of the problem-solving summative assessment problem (click to enlarge image):
Reflections/Advice: This inquiry has been overwhelmingly successful, but it has also been deeply challenging and at times frustrating. I am grateful to have had an amazing, supportive team to pick me up and dust me off when things were at their most challenging. Here is what I have learned.
1. When you are tackling a difficult subject, seek out colleagues who will support you. It’s especially important that they can listen.
2. Equally important to listening and supporting is listening and challenging. You want inquiry partners who will (gently) point out where things went wrong and offer ideas of what to try next.
3. Expect to make mistakes. Lots of them.
4. When the lesson doesn’t go well, take some time to try and figure out what went wrong. They say you learn from a mistake, but you really only learn when you reflect on a mistake.
5. In the end, my class had a lot of fun doing problem solving! This is what I had hoped for, but many times during the year I wondered if we could ever get there.
6. Students love to challenge themselves when they feel confident that they have the skills, and when they know that they are working in a supportive environment. This is the gift of formative feedback. I now teach all math using immediate formative feedback, and I am working towards teaching all subject areas in this way.
Perhaps the best summation of this year’s work is demonstrated by this (unsolicited) card that I received at the end of this year: (I can’t upload an image because I’m at my max). The card says, “Thank-you for teaching me this year. I definitely got better at problem solving this year”.