When I do this in person with teachers, I have them close their eyes. I can't ask you to do that, however. You won't be able to read what I'm writing! :) Instead, just take a moment to reflect on the question before moving on.
I am going to give you a word to think about. This is not meant to be in a specific context. You don't have to think about how it relates to math. When you think about this word, focus on the feelings it invokes in you. There is no right or wrong answer or feeling. Just be honest.
The word: Problems
Reflect. Think about it. Don't read the next paragraph until you are ready.
How did you feel about the word "problems"? Was it overall a positive or negative emotion for you? In my experience, many people (but not all) think about problems negatively. These people see problems every day as negative things. Maybe they are unsolvable. Maybe I'm having money trouble or my child is misbehaving or there are issues with my boss. Some people see problems as positive, as a challenge to be solved.
Let's look at another word. Complete the same reflection. What emotions does this word invoke in you?
The word: Puzzlers
Reflect. Think about it. Don't read the next paragraph until you are ready.
How did you feel about the word "puzzlers"? Was this one a positive or negative emotion for you? In my experience, more people (but not all) felt a positive connection to this word. Puzzlers can usually be solved. They are challenging. You can ask others for help but often you feel more satisfaction when you can solve it on your own.
We are going to come back to the term "puzzlers" later. Let's move on for now.
In Alberta, there are 7 mathematical processes attached to our mathematics curriculum. The following is an exact quote from the Alberta Mathematics 2016 Program of Studies.
|Mathematical Processes||There are critical components that students must encounter in a mathematics program in order to achieve the goals of mathematics education and embrace lifelong learning in mathematics.|
|Communication [C]||communication in order to learn and express their understanding|
|Connections[CN]||connect mathematical ideas to other concepts in mathematics, to everyday experiences and to other disciplines|
|Mental Mathematics and Estimation [ME]||demonstrate fluency with mental mathematics and estimation|
|Problem Solving [PS]||develop and apply new mathematical knowledge through problem solving|
|Reasoning [R]||develop mathematical reasoning|
|Technology [T]||select and use technologies as tools for learning and for solving problems|
|Visualization||develop visualization skills to assist in processing information, making connections and solving problems|
The mathematics curriculum is meant to be taught through these 7 processes. I find that the most misunderstood process is Problem Solving. In my experience it has been used synonymously with the term "word problems" in many classrooms. However, the typical way that word problems are used in the classroom is not as problem solving.
What is Problem Solving?
Just because it is a word problem, does not necessarily mean it is problem solving.
“Learning through problem solving should be the focus of mathematics at all grade levels. When students encounter new situations and respond to questions of the type How would you …? or How could you …?, the problem-solving approach is being modelled. Students develop their own problem-solving strategies by listening to, discussing and trying different strategies.
A problem-solving activity must ask students to determine a way to get from what is known to what is sought. If students have already been given ways to solve the problem, it is not a problem, but practice. A true problem requires students to use prior learnings in new ways and contexts. Problem solving requires and builds depth of conceptual understanding and student engagement.” (Alberta Education)
Is this problem solving?
My students have learned how to add two digit numbers with regrouping. I give them this question: Johnny has 25 cents. He finds 35 cents. How much money does he have now?
This is just a word problem. Remember, "If students have already been given ways to solve the problem, it is not a problem, but practice." Of course, students need practice. There is nothing wrong with the question as long as we understand its purpose - to practice a skill that has already been learned/taught. However, we need to find ways to help students learn through problem solving as well.
Why use Problem Solving?
“Problem solving is a powerful teaching tool that fosters multiple, creative and innovative solutions. Creating an environment where students openly look for, and engage in, finding a variety of strategies for solving problems empowers students to explore alternatives and develops confident, cognitive mathematical risk takers.” (Alberta Education)
We need our students to develop problem solving skills. Not just in math class but every where. The problem solving skills they learn in math class can be used in many non math contexts. It's about perseverance. It's about knowing that you can figure it out given enough time and the right access to resources when needed. I truly believe that there are many missed opportunities to teach math through problem solving.
What do I do differently?
Let's return to the idea of Puzzlers that we touched on in the beginning. I use the term puzzlers instead of the word "problems"or "problem solving" because of the emotional connections students have with the words "puzzlers" and "problems". I use Puzzlers in my class all the time. Every day.
One way you can do this is to provide students with a question that is slightly above their skill set. For example, you are working on 2 digit addition. You start with a simple question such as 21 + 33. This is a question without regrouping. You notice that students have mastered this so you tell them you are giving them a “Puzzler”. A “Puzzler” is a bit more challenging. A “Puzzler” is going to make you think. And, you may not finish the “Puzzler” in the amount of time you are given and that’s ok as long as you persevere for the entire time you are given. You give students the new question 28 + 33. This question has regrouping. Students are provided time to work in a Think-Pair-Share process where they work on their own (typically no more than 2 minutes so that frustration doesn't have time to set in), share their strategy with a partner and then some students share with the class. You can talk about how they solved it. How is it similar to what they've done? What makes it a puzzler? Then give them an opportunity to practice by providing another question that involves simple regrouping. This allows the students who didn't figure it out an opportunity to try the strategies that they heard. When students are comfortable, you can give them a new puzzler such as 48 + 95 or 138 + 26. You decide what the next more complex question would be. By using this processes, I only have to "teach" the first easy step. After that, they learn to puzzle their way through it.
I love this method because it teaches students that they can figure things out on their own. They build perseverance. They learn that it's ok to struggle in the beginning and to just keep plugging away at it. They learn that sometimes they may have to get outside help to understand and that's ok. Students can also work at their own rate on these. Often I will have several puzzlers written out on the board in different colors. They will try the first one, self assess and tell me if they are ready to move up in complexity or if they would like me to give them another of the same color type to try again. I've never had a student keep working on the same color if they should be moving up to the next color. They want to try a harder puzzler. They want to get up to the super duper ooper puzzler (that is always above grade level). When a student is confidently working above grade level, I just get them to create their own puzzlers. They solve it and trade with other students who are working at that level as well. This gives me time to support all of the students who need more help.
I created this simple poster to put up in my classroom. Download it and feel free to use it as is or edit it.
Problem Solving is NOT a unit of study...
Problem Solving is not a unit of study you do in June. It has to be embedded every day all year. You don't develop problem solving skills by limiting problem solving time to solving a word problem of the week or as a unit at the end of the year or even by doing a project at the end of each unit. It has to be developed EVERY SINGLE DAY ALL YEAR.
Problem Solving does not have to be a word problem...
Is this problem solving? 25 x 14
It depends. Did you already teach your students how to multiply two 2 digit numbers? If so, then it is not problem solving. It's practice. OR, have they just learned the distributive property as it relates to 2 digit by 1 digit multipliction? Then, yes it is problem solving because they are taking that understanding and applying it to a new context. Keep in mind, though, you may have students who knew how to do this before they came to you so for those students, it is practice.
Is this problem solving? 2 x 6
It depends. Have you already been working on these facts? If so, it is not problem solving. Even if they don't know the answer, they still have learned the skills to figure it out. OR, have you learned only up to 2x5 and students have to figure out how to use the facts they do know in order to figure it out. In this case, it's problem solving.
Where do I get the ideas for puzzlers?
The same place you get all your questions...your brain, the internet, textbooks, etc. Just stop pre-teaching everything. Help them understand the lowest level (21 + 7) and then give the rest as puzzlers. Start with the word problem that you would have given them at the end of learning.
Another simple method I've used...
When I taught high school math, I used to teach the steps for everything. Do this first. Do this next. Rinse and repeat. One day, in the middle of a lesson, I looked at the next question and thought to my self - this question is similar to what we have been doing but a little bit different - so I tried something new. This is how the activity progressed:
T: Ok class. We are going to try something different. Put your pencils down. Don't touch them. Now, I'm going to give you a minute to read the next question and think about what you might try first to solve this. Don't do anything though. Don't touch your pencil. Just think about it. Don't worry about doing the question. I'm not looking for an answer...just a possible first step to try.
I gave students a minute or two to think.
T: Turn to your partner, tell them what idea you had for a first step and why you think that would be a good first step. It's ok if you're not confident that it will work or that it's right. Just share.
Partners share ideas.
T: Who would like to share either their first step or their partner's first step?
Students share different first steps.
T: You've heard several possibilities for a first step. Let's decide on one to try first.
I had them vote. If I am worried that they are leaning towards one that won't work and this is the first time they are playing in a problem solving context, I will choose for them.
T: We are all going to do this first step together. Walk me through it.
I wrote on the board as students helped walk me through it. Many students also wrote it down but I'm a big fan of focusing on the process rather than copying when we are learning.
T: Now that we have the first step, take a moment to think individually about what our next step could be.
Repeat the Think Pair Share process until the problem is complete.
I loved this process because it taught my students what they needed to do in order to solve a question that appeared overwhelming in the beginning. Focus on the first step. Try it. Now what? Try it. Now what? Keep going.
After students were comfortable with this, I would make sure that we tried a recommendation that would not lead to a successful solution. This way, they learned how to deal with that situation. Oops. That didn't work. What did we learn from that path? Ok. Let's go back and try something different.
The biggest change I noticed? When students were given a question to try on their own, I stopped seeing hands going up immediately followed by "Mrs. Berg. I don't know what to do! You didn't teach us how to do this!" They had built the confidence to just try...something....anything...and see where it led.
All students are problem solvers.
Our job is to help them become better problem solvers.
Self-reflection question for you: What strategies do you use to build a problem-solving environment in your classroom?