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.

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Converting Errors into Lessons

This week in Math Methods we were assigned to solve the problem of how many tickets were on the role. We were given an image with a role of tickets, a quarter, and a singular ticket from the role. There were multiple ways of solving this problem (which is the point of it since we are in our Problem Solving Unit), and I chose to take the diameter of the quarter and relate the real life diameter of a quarter and find the radius of the ticket role minus its cardboard tube hole, and divide that by the length of the ticket times the thickness of the ticket (the thickness was given). Here are some images of what I did:

I originally found around 1,000 tickets, which seemed plausible to me. However, upon comparing with my friends (and professor) in class, I found that they all had around 2,000 tickets, and that, in fact, was the correct answer. Where did I go so wrong?

Our professor always allows us to present our work to the class, so after someone with a more correct answer presented, I asked to present mine to see where I went wrong. Upon (quick) further review, we found that all of my work and thought process was correct (the verdict is still up, because we did only look at all of this correctly), and that what probably happened was that enough of my measurements were slightly off enough that it effected the end total dramatically. After trying it again in class, I did get closer to 2,000 tickets.

This event left me with one question, how can I, as a future teacher, turn events like that into deeper learning experiences. It is going to happen, students will mess up in math class. I messed up as a junior math ed major. So, I want to be prepared to counter when it happens. We have talked about productive struggle previously in-class, but I want to find more methods to utilize.

I found an article online called “9 Ways to Help Students Learn Through Mistakes” which lists 9 methods to support student error.

(Link: https://www.teachthought.com/pedagogy/9-ways-help-students-learn-mistakes/ )(Look Professor Reins I figured out how to hyperlink!)

The nine methods are:

  1. See mistakes as a source of understanding: Instead of just correcting students and providing with feedback that they are write or wrong, explain to the students why their answer is wrong. That way deeper understanding is promoted.
  2. Improve motivation and self-esteem by responding to and overcoming mistakes: The sense of pride a student gets from overcoming a mistake is a great feeling. Showing kids the reward for their effort helps students overcome mistakes.
  3. Honor mistakes as guidance for the teacher, too: Students mistakes should not just be left as such. Teachers need to deeply consider the mistake being made whether it is an isolated incident or a trend throughout the classroom. Then take those mistakes to alter lessons and make decisions about the classroom.
  4. Allow mistakes through the learning atmosphere: When the “learning atmosphere” is happening mistakes should be welcome. Instead of reprimanding students for errors during the learning phase of lessons, they should be encouraged to make them and overcome them. Mistakes during the learning phase and the assessment phase should be considered different.
  5. Allow a variety of mistakes: Pose questions such that a variety of mistakes can be made. Encourage mistakes to be made in order to learn.
  6. Provide timely feedback so mistake can be responded to: Make sure that teachers are identifying mistakes as they are doing the lessons, that way students do not get to the end of the unit and realize they do not know how to correctly do something. That is why formal assessment is so important.
  7. Analyze root causes and sources: Identify where the mistake is and how to overcome it.
  8.  Encourage independent mistake correction as a matter of habit: Give students the opportunity to identify and solve their mistakes.
  9. Use technology that supports mistakes and personalized mistake analysis: Technology is an excellent tool that is efficient in identifying mistakes and giving the teacher the feedback on those mistakes.

I think that these are all really good strategies to keep in mind when faced with error in my classroom. My favorite ones are 1, 3, and 4. I think that emphasizing that it is important to make mistakes so that we can learn from them is a really good thing to do. Most mistakes have some form of truth in them, although it may be an altered truth. It is then the job of the teacher to find a way to better teach or correct the errors in order to promote further understanding.

Discourse in the Classroom

The specific section that I had to read about and report about was on discourse in the classroom. This includes discourse between the students and the problems, students and the teacher, and students with other students. The article discussed how this is where real learning occurs and the importance of debating mathematics.

One of the main reasons that I believe this is so important is because it truly gives the students a chance to deepen their knowledge on the topic. Being able to debate specific steps of a problem shows a deeper understanding than knowing a procedure and reproducing it with different numbers. This gives the students a real life way of looking at the mathematics that will help them understand the concept better.

When students debate over topics and procedures, they see the importance of certain steps and really connect each step to the next.

It is important as teachers that we allow this discourse to happen and even make it happen. This happens by setting up classroom situations so that the students feel comfortable to debate and be wrong. Debating in front of the whole class makes the student vulnerable and so, as teachers, we need to be aware of how we react to this discourse and how we encourage it.

I found a source that specifically discusses the importance of discourse and how as teachers we can create this in our classrooms. The source comes from: http://www.nctm.org/Publications/mathematics-teacher/2007/Vol101/Issue4/Let_s-Talk_-Promoting-Mathematical-Discourse-in-the-Classroom/

This article talks about the two main factors that teachers have to remember when facilitating discourse are: cognitive discourse and motivational discourse.

“Cognitive discourse refers to what the teacher says to promote conceptual understanding of the mathematics itself” (286). A common misconception that many teachers have is that they now should sit back and let the students make the connections and figure out the mathematics on their own; however, teachers still have a very important role. This being that they have to ask questions to ensure that their students are fully understanding and they have to lead them to the correct answer without giving it away.

A large problem with discourse is that it can be difficult to get all of your students to share their ideas and announce their opinions. This is why teachers need motivational discourse as well. “Motivational discourse refers not only to praise offered to students but also to supportive and non-supportive statements teachers make that encourage or discourage participation in mathematics classroom discussions” (287). A big way that teachers can provide this is by emphasizing that we learnt through our mistakes and everyone makes mistakes. However, these exchanges need to be supportive. A teacher needs to have the students explain how they got to their answer as opposed to focusing on whether it is correct or not.

I know, as a student, the best way a teacher could get me to not participate was to embarrass me in front of the class. If a teacher would explain to the class why my answer was wrong and tell me to correct it, I most certainly would not put my hand up the next time in fear that it would happen again. Teachers need to create a comfort and support in their classroom where students feel that they can be wrong and it will be okay.

What are some best practices for teaching high school mathematics?

By Kevin Reins

This week I read the responses of a two part series in Ed Week by Larry Ferlazzo. The question of the week was, “What are some best practices for teaching high school mathematics?” This intrigued me as I was preparing the 18th revision of my secondary mathematics methods course here at USD.

“…there are a zillion different instructional strategies and practices that math teachers can use in high school.”

The focus of part 1 was on these ‘instructional strategies and practices,’

  • teach to big ideas (see image above), it allows students to have interconnected schema
  • focus on the processes and connections between different processes
  • use instructional routines (see Why instructional routines?)
  • keep a record of conversations when you orchestrate full group discussions
  • be selective and cautious in your use of technology
  • incorporate high leverage long-term strategies
    • -David Wees
  • embrace mistakes, normalizing mistakes, safe space for discussion and correction, utilize error analysis
  • formal error analysis through test corrections
    • -Jillian Henry
  •  provide relevance and contexts for the mathematics
  • engage students in a variety of practices and strategies
  • provide scaffolds for those who need the extra support when working with challenging content
  • plan intentionally and deliberately so your instruction is impactful, consistent, and effective
  • develop a community of learners where group participation and interaction is expected
  • employ student-centered teaching and learning
  • provide opportunities for students to develop and strengthen their skills of mathematical communication (including vocabulary)
  • make the development of a variety of problem-solving techniques a priority
  • eliminate the blank paper; require students to write (1) determine a strategy that could be used to solve the problem, (2) write a question that you have about the problem, (3) record everything you know about the content related to the problem.
  • develop their ability to ask good questions during problem solving phases, Entry (getting started), Moving (when stuck), Reflection (thinking about thinking), and Extension (deeper thinking).
  • utilize graphic organizers to help them employ processes independently
    • -Tammy Jones & Leslie Texas
  •  What works at elementary or middle level works for high school
  • pose interesting problems or set the stage for students to pose interesting questions/problems about the situation
  • encourage investigations, experiments, collaboration, and discourse as students explore problems
  • expect representations or models for the problems being investigated
  • engage students in discourse, creating mathematical arguments and critiquing the reasoning of others
  • proving their work with both formal and informal proofs 
    • Anne Collins

Part 2: Students must ‘engage in math problem-solving’ and not just ‘follow procedures.’

The acquisition of best practices for teaching high school mathematics is necessary for student academic success.

The focus of part 2 was on engaging students in problem-solving. The following was said by the experts interviewed,

  • you must have as your guiding philosophical principle the belief that all students can learn
  • you must provide opportunities for them to fall in love with learning
  • Standards for Mathematical Practices can serve as a guide for the ways students need to be engaged in mathematics
  • choose open-ended problems
  • focus more on the process rather than the correct answers
  • challenge them with mathematically rigorous tasks, choosing a Higher-Level Demand Task
  • learn how to anticipate student responses and misconceptions for tasks
  • ask students to find multiple strategies to the tasks you present
  • learn how to help students learn from mistakes
    • -Wendy Monroy, LA math coach
  • math learning should be developing conceptual understandings of the mathematics
  • focus on the conceptual relationships
  • create a synergy between the lower levels and higher levels of thinking through inquiry
  • create a social environment that promotes team work and collaboration
  • provide an open, secure environment that allows for mistakes as a part of the learning process
  •  use an inductive teaching approach (vs deductive)
  • reduce teacher talk time (increase productive mathematical discourse)
  • differentiate by content, product, and affect (Tomlinson)
  • use all types of assessment; visible thinking routines, “I use to think… Now I think…” (Harvard University’s Project Zero)
  • use a flexible fronts layout of the classroom which encourages more collaboration
    • -Jennifer Chang Wathall, educational consultant in concept-based mathematics/curriculum
  •  give challenging problems that build patience and persistence in their maturing problem solving skills
  • then spend ample time in joyful struggle
  • create rich mathematical dialogue that leaves the building
  • 12 challenging problems that 5 of which will appear on the final, and give them time in class to work on them (e.g., A point P, inside a square, was 3, 4, and 5 units away from three of the corners. Find the length of the side of the square.)
    • -Sunil Singh, author of Pi of Life: The Hidden Happiness of Mathematics
  • sufficient time to make sure that students know how to solve problems using different methods
  • look for opportunities for students to have multiple entry points or strategies for solving a problem
  • take time to discuss strategic choices
  • find flaws in short cuts and when certain methods won’t work
  • open their mind to new and different approaches
    • -Matthew Beyranevand, author of Teach Math Like This, Not Like That: Four Critical Areas to Improve Student Learning.

So after bolding all of the big ideas of the laundry list of instructional strategies and practices that were provided I compared it to the content that I normally would teach in my secondary mathematics methods course. The result was two ideas, one new, and one that could use a deeper focus. I would like to incorporate more ideas on how to utilize math mistakes in the classroom as learning opportunities. I also would like to explore instructional routines a bit more.

To think more deeply about embracing mistakes, normalizing mistakes, and creating a safe space for discussion and correction, I think it is important to start off knowing what some common math mistakes in high school are. I found Math Mistakes website that does just that. This should be a good start for a discussion on how to utilize some of these mistakes when they pop up as a learning opportunity.

With respect to instructional routines, I read Why instructional routines? It turns out I know what they are and how one should utilize tasks in teaching. One instructional routine that David Wees talks about is, Contemplate then Calculate, as a tool for learning how to use the 5 Practices for Orchestrating Productive Mathematical Discussions. The high level goals of Contemplate then Calculate are to support students in surfacing and naming mathematical structure, more broadly in pausing to think about the mathematics present in a task before attempting a solution strategy, and even more broadly in learning from other students’ different strategies for solving the same problem.

“Instructional [routines] are tasks enacted in classrooms that structure the relationship between the teacher and the students around content in ways that consistently maintain high expectations of student learning while adapting to the contingencies of particular instructional interactions.” Kazemi, E., Franke, M., & Lampert, M. (2009)

I’m looking forward to expanding my teaching and learning opportunities to include both instructional routines and normalizing mistakes.