For quite some time now our high school English Language Arts teacher has had students do reading logs with metacognition questions. They were required to read for a required amount of time and then asked to write about what they were thinking about. Even if what they were thinking about was not what they were reading. They were asked to write where they got “stuck” on words, sentences, etc. Two years ago, when the students were deeply immersed into this idea I started thinking about what this would look like in math. What is the metacognitive process doing in our minds? For math, I thought this process entailed planning out how to solve a problem, getting the required items to solve this problem, going through the plan and checking the reasonableness of the answer and lastly learning where you are getting stuck so that you can fix misconceptions in order to be successful. With that said I present to you the Math Metacognition that I have come up with.
Here is link to download PDF document of these questions
Two years ago when I first started implementing this I used it as a way for students to get started and I also used it as a way for students to get some points back on their tests. They had to do all 4 Metacog questions for EACH incorrect question on their test. The more detailed the metacognition was (especially #2) the more points back the students received. Note: They couldn’t get all points back to earn an A but they could raise their test score a good amount. I realized that these steps really helped students get started on problems especially when they don’t know how or where to start. Later I figured out that these are the steps I think about in my brain BEFORE I solve any problem. We go through these steps (somewhat) to solve most math problems and to think that students are able to think about this intrinsically is CRAZY! I don’t know how we mathematicians/teachers/good problems solvers learned it but we did it and most of my students aren’t learning it (at least not before they get to me). I forgot about these steps last year but I am bringing them back and plan to really push students to refer to these steps if they get stuck or need help to get started.
Below is a breakdown of the reasoning behind each question and the reason for including it based on its importance.
1: The first question is to get the students to re-read the question, look for key words inside the word problem and to pull out that “hook” question that is normally seen at the end of given math problems. The second and third questions are to get student’s brains thinking about how they know they are done with a problem. Most of the time students dive into a problem without even knowing what it will look like when they are done. Then they end up asking me “Am I done?” which I respond “I don’t know, are you?” So I wanted students to start thinking about what their goal is AND WHAT WILL IT LOOK LIKE!?? How will they know they are done with the problem. This then leads into the ability to check if the answer they get, when complete, is even reasonable. This also starts the brain process of estimation which we all know is quite a downfall in most students.
2: These questions were made for students who were looking at problems they had done incorrectly. I sometimes take this question and modify it so they just try to answer where they got stuck. These questions are the ones that you as the teacher can start honing in on where a student is getting their thinking mixed up or where there are misconceptions in their thought process. When we (teachers) can figure out this gap/pothole in student’s thinking we will be able to re-calibrate and get them thinking on the “right track”. We can clear up the misconceptions and get them feeling successful again.
3: This is a typical question you see in most strategies. List all the information you know from the problem. I realized that this should not only be listing stuff they know FROM the problem but instead list stuff they know about the problem because if some kids memorize formulas that would be a good place for them to write down formulas or sentences that they might remember their teacher stating about concepts, etc. This is also for students to realize that they “DON’T KNOW NOTHING” about the problem.
4: Here is their plan of attack. So this again is another of those typical question math teachers see. How are you going to do it? What is your road map? Now because in question 1 they talked about what the answer will look like they are aware of what they are looking for as they are planning their attack. They are also trying to figure out how they are going to manipulate math to “get what they want” (x by itself, area of circle, etc.).
Now I do realize that these steps don’t actually ask you to DO the problem and follow through with your plan of attack which is something I am very aware of. If a student is unable to finish a problem but is at least able to do these four steps it is more work than not even trying to attempt the problem. Going through the metacognition is a lot of work and most of the time once students take the time to really think about these questions in order to get started they are most likely than not able to solve the problem. If they are still unable to answer the question then at least they got started by answering the metacog questions. You as a teacher (and the students) are able to see where they get stuck and what they need to work on in order to be successful (question 2).
Feel free to use these questions but if you modify them I would love to see because I am sure there are better versions that people are able to make via modification.