Demonstrating Competence by Making Mistakes

  • Demonstrating Competence by Making Mistakes

    By Dan Teague, posted October 26, 2015 —

    Common advice for new teachers is to be sure to do all the homework problems before you assign them. This is good advice. Much of what is possible in our classrooms comes from our reputation among students (and their parents). When students trust you, you have leverage and leeway in trying new things. A solid reputation allows for creativity in your teaching, which is often rewarded with creativity in student work. Everyone wins.

    Building a reputation takes some time. The first requirement from parents and students is mathematical competence—knowing your stuff so that the inevitable errors at the board we all make are shrugged off as just that, instead of being viewed as a warning sign that “my teacher can’t do the problems either.”

    So think carefully about your homework assignments and the example problems you use for class and make sure they move student understanding forward. Be judicious in the number of problems you assign. There is no good way for reasonable students to do twenty-five problems a night other than for them to close their minds, put their heads down, and grind. No one can do twenty-five problems thoughtfully, so choose rich problems that are not repetitive. As Jo Boaler noted in her NCTM presentation in April 2015, there is almost no brain activity when doing repetitive problems, each one like the previous one. Choose problems that make students’ brains spark.

    “Do all the homework problems before you assign them” is good advice . . . for a while.

    Wowing students with your ability to immediately solve every problem without making any false steps is nice and builds the reputation so fundamental to your success, but it gives students a very odd (and quite false) sense of what doing mathematics is about and how real mathematical problems are done. Students can easily come to believe that, to be good at mathematics, they must be able to do every problem without error and without thought. So, if they are like most students and have to work at it, they can come to believe they must not be very good. Moreover, they believe that mathematics is done by remembering how, and it is only a small step from there to believing that mathematics should be done by remembering how.

    This means that they believe it is not possible for them to do something they haven’t been taught.

    Students need to see their teachers figuring things out. You should be prepared to be unprepared. That is, to work a problem for the first time, thinking out loud as you go, so students see that working out a challenging problem often involves missteps. And corrections. And playing around with the problem. Students need to learn how to decide whether the approach they are trying is making progress and when to abandon it and begin again. Most important, they need to learn how to look carefully at their errors and use them to find a correct approach. All these skills are essential to creatively using mathematics to model and understand the world, and all need to be taught as a natural part of learning and doing mathematics.

    Moreover, if we assign only problems we have done before, then we will assign only problems we can do, and our ability becomes a limitation for our students. How will they learn to do things we can’t? How will they become better than us? After all, if our students aren’t better than their teachers, then we are moving backward. One of my prime directives in teaching is to not let the limitations of my talents be limitations on theirs.

    2015-09 Teague    

    DAN TEAGUE,, teaches at the North Carolina School of Science and Mathematics in Durham. He is interested in mathematical modeling and finding problems that connect concepts from different areas of mathematics.


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    Caitlin DeMers - 3/18/2016 1:02:21 PM
    I am currently a math tutor and studying to be a middle level math educator. I tutor my peers and although it is not exactly the same thing, a lot of the time I feel as though I am their teacher. I feel so flustered and nervous when I do not know something or I have to look in their book to spark my memory. I just do not want them to think I am unfit or under-qualified to be a math tutor or a math teacher. This blog definitely shed some light on this subject for me. Just like most pre-service teachers, I am nervous about letting my students down, but this blog really showed me that teachers do not have to be perfect. When I make mistakes, it is okay, because mistakes are a part of the learning process. I think teaching students to accept that they will make mistakes and that mistakes are a part of the learning process will impact students in an immeasurable way.

    Dylan Breton - 2/29/2016 8:38:20 AM
    Hey Dan, first of all I enjoyed reading your blog post and agree with you 100%! Growing up through the American education system myself, I often found teachers simply using the back of the book to provide students with answers and corrections on their homework assignments. I HATED THIS! It was until high school where I finally had a teacher who actually did the homework assignments previously to the day's lesson of correcting the homework. I find that it makes it easier for the teacher to relate to problems and trouble that students may be having. Completing the homework as a teacher also allows the teachers to recognize different levels of difficulties with problems which may help in correlation to the class's overall difficulty they may be having on a lesson or chapter. Very informing and will use in my own teachings.

    Kimberley Astle - 12/19/2015 11:50:57 AM
    Dan, we have been exploring the concept of productive struggle and growth mindset at my elementary school, especially in math. Some of the things we have learned is that the brain literally makes more neuron connections when we are experiencing productive struggle (making mistakes), than when we do work that our brain finds too easy. So, when we make mistakes and work to correct those mistakes, we learn far more than we do when we get the answer right the first time. It's like weight lifting. Lifting a pencil ten times will not help up to develop our lifting ability, but lifting a weight that challenges us helps us to grow more muscle mass, acquire more confidence, and gain more stamina. Math is the same way. If we want our students to acquire this mindset, then they definitely need to see us engaged in productive struggle ourselves as well as their peers. Sharing the challenges we had and how we overcame them is a great way to develop a class culture that celebrates making and learning from mistakes. I also completely agree with you that assigning more than a few problems for homework serves no purpose other than to make children see math as a burden and a chore. I assign 3-5 problems at different levels so my students can choose the level of challenge that is right for them. If we assign just a few well-designed and intentional problems, then our students will gain the benefits of independent practice, without developing a negative viewpoint. Negativity towards math is already an obstacle for many students, we don't need to contribute to this problem.

    John Benson - 12/8/2015 8:14:56 AM
    Dan, this is the most articulate comment I have read on the subject of the teacher's relationship with homework. I agree totally with what you have written. It is essential that the teacher's community know that the teacher is a good mathematician. But it then is essential for the students to experience what a good mathematician does when faced with an unfamiliar situation. Thank you so much for pointing this out.

    John Kolapo - 12/2/2015 3:56:06 AM
    Hell Dan, I would like to admit that what you said about doing all the work first before giving them was really a helpful way of showing that you know your job as a new teacher. I actually did that during my first year of teaching. As the years went by, I discovered common mistakes that students make when solving problems and I used this also in explanation to how students can work better. Later on, when I saw that the students had a lot of confidence in my capacity as a teacher, I began to bring in more problems that were not in the textbook. This helped me and the students to work together and figure out problems. When we get stuck, we try to take a pause and look at it all over again to figure it out. It motivated the students to take up challenges on problem solving. Now, I don't review problems before I assign them to the students because I want them to see the confidence I have in facing a challenging question and the way I go about solving it, also motivates the student towards learning.

    Laurie Callbeck - 10/26/2015 8:07:26 AM
    When I first started to read this post I found myself disagree almost immediately with the concept that teachers should do all the assigned problems ahead of time. However, as I continued to read, I realized that the point was for beginning teachers to do so to establish they street cred. I wholeheartedly agree with Dan that it becomes very important for the teacher to not have all the answers and show proficiency with problem solving. My students started to show great improvement with problem solving when I began modeling problems that I didn't already have worked out. When I teach the same course from year to year, I don't review my problems before I teach so that I can enter into the questions cold and make them as "fresh" as possible for me and my students. This process takes confidence in oneself. The hard part is when your answer doesn't match the one in the textbook. I turn this into a great learning opportunity and ask students to find my error. If they can't and I can't (which has happened especially as we implement new curriculum), students get anxious - the teacher needs to have the confidence to state when the book is wrong ... it can happen.