Mistakes and Breakout EDU

I had the opportunity to read “Making Room for Inspecting Mistakes” (link: https://drive.google.com/file/d/1PjEE0Y8SqMDVIVrETxgLDWCpX00aeDBO/view?usp=sharing) in this month’s issue of Mathematics Teacher from NCTM. The article discusses using mistakes to help students learn. One of the examples it utilizes is choosing a homework problem that is incorrect to go over for the entire class. Choosing these problems is artful because the teacher must ensure that the problem is going to be useful to the greatest amount of people in the class. There may be people in the class that would make the same mistake, others that get a better understanding of how to complete the problem because they did not know how to originally, and even others that understand where the mistake came from and how to combat it. There are three different contexts for leveraging mistakes that the article discusses: review of homework, during a task example, and during exam preparation. In each of these context, mistakes can be capitalized upon to help students grow in their understanding of the content that is being taught. During review for exams, it is a good check to ensure that students do understand what they have learned throughout the unit/semester/year.

This article prompted me to think about how making mistakes can be useful in the mathematics classroom. We have previously discussed how mistakes can be utilized to help students. Yesterday, and earlier today I had the opportunity to attend the SDEA Student Conference in Mitchell. One of the two breakout sessions utilized Breakout EDU. The concept of a Breakout EDU is similar to an escape room, but students are trying to break into a box. They can be bought online for different content areas. However, they do cost $125 so many teachers write grants to get Breakout boxes. Although escape rooms may just be a fad, Breakout boxes can benefit the classroom. After the activity, I began to think about when I would use Breakout EDU in my own classroom. I believe that these boxes could be useful at the beginning of the year to set a standard for collaboration between students, productive struggle, and making mistakes. Furthermore, during this time at the beginning of the year, a box could be useful as a review from the previous year’s material for the students. We saw in the lesson study that students were reluctant to productively struggle, and using a Breakout box could allow the students to start the year off participating in an activity that calls for productive struggle. Additionally, in the theme of making mistakes in the mathematics classroom, students are bound to make mistakes in their search for the answers to the clues. Using an activity such as Breakout EDU would allow the students to understand that making mistakes is beneficial, especially if they persevere in opening the box. Setting a standard for the benefits listed above of Breakout EDU in the classroom would help establish a particular environment in the classroom for the rest of the school year. This environment is aided in being established because after students complete the Breakout EDU, they discuss what went well for them, what problems they encountered, what did not go well, etc. The reflection is what cements the environment. Overall, there are clear benefits to making and going over mistakes in the mathematics classroom, and Breakout EDU could be used at the beginning of the year to establish an environment that promotes productive struggle, making mistakes, and collaboration between students.


Setting the Stage for Productive Struggle

A growing interest in teaching with productive struggle is a hopeful sign in the math classroom. Productive struggle can lead to deeper understandings, connections, and motivation in students. So, why are some teachers still hesitant to use it?

Recently, I was able to present a lesson over the Law of Sines and Cosines in a local high school. Normally, these students learn in the traditional way of lecture. However, we challenged ourselves to create a lesson that caused students to struggle a little before they came to the solution. When we presented it, many of the students seemed disinterested and unengaged. I was worried about this when creating the lesson, but I did not think it would occur to the extent that it did. So, I have looked into articles that provide ways for a teacher to slowly introduce and transition students into lessons that incorporate productive struggle.

The article, Beyond Growth Mindset: Creating Classroom Opportunities for Meaningful Struggle, gives tips on what to do and what to avoid when teaching with struggle. Years of research has proven that student learning is enhanced when they have to be persistent to reach success–this concept goes all the way back to the educational reformer, John Dewey who “described learning as beginning with a dilemma.” One major key to a successful productive struggle lesson is to have the goal of the lesson be focus on getting students to have a deeper understanding of the material instead of just focusing on creating struggle. One common way to do this is through real world applications. For example, in the lesson we created for the geometry class, there was an activity that involved drones and using them to deliver pizza. Although this is not necessarily happening today, it is something that could take place in the near future. When this activity was introduced towards the end, there seemed to be a switch that flipped. Almost all of the students were working together to figure out the problem. And, they were able to recall previous math knowledge to help them solve the Law of Sines problem. This is another helpful tip when causing productive struggle–having students use math that they are familiar with can help produce productive struggle instead of frustration. Having the knowledge needed for part of the problem gives students a boost of confidence. However, students also need the time to realize these connections. As a teacher, you should be sure to give your students enough time to think about the activity before giving them any hints or guidance. This time is crucial for allowing students to make connections and deepen their understanding of the math at hand.

The article provides the following list of key elements for providing productive struggle:

  • Determine timing and placement for productive struggle within the unit or curriculum—lessons that are “preparing students to hear something really important.”
  • Align struggle activities with clear, specific learning goals.
  • Design struggle tasks based on assessment of students’ prior knowledge and skills.
  • Foster a safe environment that encourages student inquiry and exploration of important ideas.
  • Use probing questions to solicit student thinking and provide strategic assistance to nudge students through their zone of proximal development–the zone of students’ thinking just beyond the level they can do completely on their own.
  • Follow-up each struggle episode with carefully structured lessons that build on students’ ideas, address misconceptions, and help students forge new understandings.
  • Assist students to reflect and articulate what they learned as a result of productive persistence.

In the future, these tips will definitely help to transition into a classroom with productive struggle much better than the geometry class I just experienced. For instance, I think that the students would have gotten much more out of our discovery activity for the Law of Sines and Cosines if we gave them more thinking time and created an environment that allowed students to talk about their ideas without fear of being wrong.

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Productive Struggle in the Classroom

Last week I discussed nine ways to motivate students, as found in an article linked in that post. Early this week, Dr. Reins sent out an article about a 10th way to motivate students. The tenth way being promoting productive struggle (link: https://www.edutopia.org/article/promoting-productive-struggle-math) through bellringers. In the article, the author (a teacher) discusses different problems that he presents his students that have minimal words so the students have to figure out what the question is asking, and how to answer it. Within the article was a link to a TEDx Talk called “Teaching Without Words,” presented by Matthew Peterson. He talked about taking away words from math problems, and instead creating a computer platform full of games to teach math. A platform that contains no words. At first, I was skeptical of how this may work, mostly because all of the students’ learning then becomes on a screen and there is no tactile component. However, Peterson cited a study that they conducted about the impact of removing words from the mathematics classroom and replacing them with these games. It was conducted in five cities, and across the board the math proficiency rates almost tripled.

After reading the article that was sent out and watching the TEDx Talk, I began to think about how I can promote productive struggle in my own classroom, besides removing words from instruction. Although I believe that removing words from problems is beneficial for students, I think that in my future classroom I would like to implement removing words without turning to solely technology. Incorporating technology into the classroom is beneficial, but I believe that there are benefits to using tactile learning, so I would modify the original idea in that. Using ideas similar to what Giardi implemented in his school would be interesting for students, and a good way to begin a lesson with the students. Considering removing words from my instruction (of which there are many in mathematics textbooks), I began to search on NCTM’s website, I found an article (link: https://drive.google.com/file/d/1MolRN-jEtJPBiYVMJWWxX3g0tixHSObS/view?usp=sharing) that focuses on supporting productive struggle through communication with students. Although I have seen it before, the content within is still applicable and helpful. Specifically, the article discusses different communication “moves” a teacher can make to help provide productive struggle in the classroom. It provides three communication moves: assessing questions, advancing questions, and judicious telling. Assessing and advancing questions are clear in their implications, but judicious telling is subtly encouraging students to get onto the right path. I could implement these communication strategies in my future classroom to align with NCTM’s publications on promoting productive struggle in the classroom.

In terms of inciting productive struggle in a classroom, as we saw this past week in the lesson study, it can be much more difficult than we hope. The students in the class were unused to a teacher asking them to struggle in finding the answer to questions asked. Promoting productive struggle in the classroom is challenging for teachers, but the end result is students who have a deep understanding of mathematics, have developed applicable skills, and more.

The Power Behind High Expectations

While discussing the question of “What can I do to help make math accessible to all students?,” there were many answers that were good. However, the one that stuck out to me the most was having clear, high expectations for each student. Personally, this was my main support throughout high school and even college. There was always someone pushing me to do my best and even aim for a little above that. I was fortunate to have my parents and sister as part of these people, however, I know that not all students are able to have this kind of support. As a teacher and role model, it is important to show your students you care about them and their success, not only in school but in life as a whole. From a previous course, I heard about a man named Daniel Kish.

Since then, I have done further research on this man and his interesting life. I have watched a ted talk, listened to podcasts, and read many articles. Daniel Kish was a man born with ‘retinal cancer.’ Throughout his childhood, Daniel was not given special treatment by his mother, family, teachers, or peers. This was not done out of hostility, but instead on purpose. In doing this, he was forced to learn how to adapt to his surroundings with the abilities he does have. He began to use sound to “see” his surroundings; this is known as echolocation. Kish would make a clicking noise with his tongue and listen for the sound waves to bounce back to him, making him aware of the surrounding space. Had he been given special treatment such as a personal assistant or a wheelchair that wouldn’t allow him to run into anything, Daniel would not have acquired the skills he has today.

This situation can be compared to many common cases we will face as teachers such as students of another race, financial status, or popularity level. No matter the background of the students, they should all be given high expectations that they are able to rise and work towards. I know that if I had not been given such high expectations to strive for, I would have fallen short of my capability. In order to execute this in your classroom, here are some common rules to keep in your mind:

  1. Avoid labeling your students, even if it is just categorizing them in your head.
  2. Learn about their career path goal and discuss it as though they will make it happen. Avoid using “if” statements.
  3. Hold them to expectations and discuss with them what happened if they fall short and how they might fix it next time.

Along with these three things to keep in mind, I think the most important thing for a teacher to do is truly believe that ALL students are capable of learning and succeeding with your highest expectations for them.


Productive Struggle in Mathematics

I discovered the following article on productive struggle in a mathematics classroom:


I came across these two articles while learning more about this subject and how to promote it in the classroom. In our reading of Principles to Actions, the portion on productive struggle talks about moving away from fixed mindsets and gives actions that can be taken to promote a growth mindset that is necessary for productive struggle to begin.

According to the Interactive STEM Research Brief, it is important to “ignite” productive struggle by using problems that students can see themselves in. Problems in traditional textbooks do not necessarily interest students because it appears to be “of another world”. Imagine trying to fly a helicopter the first time you saw one. (No, the foot pedals are not the brake and accelerator. I wonder what these two handles do?)

This article gives four strategies for promoting productive struggle :

Ask questions that help students focus on their thinking and identify the source of their struggle, then encourage students to look at other ways to approach the problem.
Encourage students to reflect on their work and support student struggle in their effort and not just in getting the correct answers.
Give time and help students manage their struggles through adversity and failure by not stepping in too soon or helping too much and thus take the intellectual work away from the students.
Acknowledge that struggle is an important part of learning and doing mathematics.

This made me think about how we view resiliency. Is this a quality or an action? If someone told me to be resilient, do I strain more when solving problems?

I believe resiliency should be a descriptor of competency similar to that of the word smart. The path to resiliency is a continued effort using the things we know on the journey to solving a problem.

In the book “Toughness” by Jay Bilas, Dr. Henry Friedman, oncologist at Duke University, states,”The foundation of toughness is hope.” Those two words do not seem to be synonyms, but it is the overriding principle to consistently engage students in the productive struggle of mathematics. Hope in mathematics is being able to identify something you know within a problem and using it in order to solve it.

In Principles to Actions, an example is given about productive struggle with a problem including fractions. The teacher in the example asks students to write down two things they know about the problem and one thing they wish they knew. This allows to identify what they know and build strategies from it to discover what they want to know.

I have spoken to several people in various occupations, especially construction, that have the following complaint: “Most of the guys that we hire out of high school can’t even read a tape measure.” This is a multi-fold problem that promoting productive struggle in the classroom can solve. By promoting these strategies, student can become independent problem-solvers. Even if they haven’t been given an explicit lesson on how to read a tape measure, they can observe the lines and see how they are distinct. Through this method of discovery, students can make sense of new things they interact with when they enter “a whole new world” of society.