Embracing Difficulties

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I just finished up chapter four in Making it Stick.  Parts of the chapter involve the topic of challenge and how it impacts memory.  Looking back at my K-12 experience, what I remember is often associated to how I felt during the experience.  The best experiences for me required an extensive amount of effort and perseverance that eventually led to a productive outcome.  Some of the more challenging experiences were also memorable.  I learned from both those positive and negative outcomes. It’s interesting that the experiences that I remember were either positive or negative.  I don’t have many so-so memories during school – they don’t stand out.

Chapter four emphasizes how difficulty can help students retain information for longer periods of time.  I’m going to interchange the terms difficulty and challenge for this post. Challenge triggers retrieval processes and encourages students to make connections to find a solution.  This is often termed “desirable difficulties” by the Bjorks.  Chapter four discusses the importance of generative learning.  Basically, generative learning places students in a situation where they solve problems without being explicit taught how to solve them.  Students are required to make connections and generate answers without repeating a process that was clearly taught by a teacher.  The responsibility is on the students to generate a solution.  When I first read this I wasn’t exactly sure about this idea.  I work with mostly elementary math students and some want to know exactly what and how to complete a task.  If they’re unsure students might say “you never taught us ______.” It takes a shift in mindset to take a risk and generate solutions based on prior knowledge.  In the end students might be absolutely right or wrong, but they took a risk and came up with a solution.  Praising the effort involved and reflecting on the journey is important.   When coming across open-ended tasks students need to understand that learning is a journey and challenge is part of that process.

Next year I’m planning on incorporating more opportunities for students to participate in generative learning.  I believe it first starts with creating an environment where students aren’t “spoon-fed the solution” and they have to think critically about the situation.  I find that students are more likely to check their answer for reasonableness with tasks like this.  That environment should encourage students to speak up, offer their ideas, use trial-and-error, make connections, and become aware that learning is a journey.  This culture and mindset takes time to build, but the dividends it pays throughout the year benefits all involved.

I’m staring to to take a look at next years plans. Currently there’s one task for each unit that’s designed for generative learning.  Sometimes I have students work on these tasks in groups, while other times it’s independent work.  These types of tasks are often open-ended and may have multiple solutions.  They also involve a hefty time commitment and can reach multiple math standards within one tasks.  Over the summer I’m planning on finding additional ideas using MARS and Illustrative Mathematics resources.

Next steps: At the end of each task I’d like to have a class conversation about the task.  Have a regular reflection component can bring additional connections.  I’m planning on continuing to have students journal about these experiences throughout the year.  I’m also hoping that these types of tasks translate into students being more willing to take additional ownership for creating and monitoring their math identities.

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Reflection and Math Goals

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Two of my classes took assessments this week.  These are considered unit assessments and are related to math skills that the class has been working on over the past 1-2 months.  My fourth grade class just finished up a fraction unit, while fifth graders ended a unit on equations. I tend to grade the tests and then pass them back in the next day or two.  Seeing that it takes so much class time to give these tests (and the grading) I want students to be able to use these assessments.  By using them, I mean that students should be able to look at them with formative lens and purposefully reflect on the results.  Usually the assessment process looks like this:

Stage 1

  • Assessments are passed back to students
  • Students review their score and are excited or disappointed
  • Students try to figure out how everyone else did

Stage 2

  • Teacher reviews the assessment solutions with the class
  • Students ask questions about why or maybe how they can get additional credit
  • Students see where fixable mistakes exist

Stage 3

  • Students receive their math journals
  • Students fill out a reflection sheet looking at skill strengths and areas to improve
  • Students indicate the most memorable activity and why
  • The teacher and student meet and sign-off on the test analysis and reflection portion

 

Okay, so stages 1-3 have been happening in my classroom for the past seven or so years.  It’s become part of my classroom’s math routine.  I see benefits in having students reflect on their progress on assessments, but I also want students to look at an assessment beyond the grade itself.  I’ve blogged about this evolution before. I stopped putting actual letter grades on assessments because of this.  I also considered taking off the point totals as well, but ended up keeping them since it was on the grade report anyway.

I see value in the student reflection component.  I believe students feel empowered when they’re given more control, choice, and access in the classroom.  This year I’ve added my own stage 4.  I’ve added this for a couple different reasons.  One, I’ve noticed that students that don’t necessarily meet their own expectations are really hard on themselves.  They often react negatively on the reflection component and I don’t want students to feel worse after reflecting on their performance.  I want this to be a valuable experience and growth opportunity.  Two, my students have kept their math journal for multiple years.  Some of them are jam packed with notes, reflections, and foldables.  You’d be surprised at how much is in some of these journals.  One thing that students continually tell me is that they love going back in their journal and looking at what they completed over the past few years.  They see that their mathematical writing has changed as well as the concepts that they’ve encountered.  It’s similar to a math yearbook to many of my students.  My third reason is that I’ve always been interested in how students perceive themselves as math students.  Over the years, I’ve emphasized that creating an individual math identity is important. I emphasize this at my school’s back to school session. This math identity shouldn’t come from a parent, but instilled within.  Being able to see students for multiple years allows me more of an opportunity to do this.  Also,  I’m excited to share this at NCTM and learn with other educators about the goal setting and monitoring process. This has been an area of growth for me as I’m continually refining the student math reflection process.

So, here’s stage four:

Stage 4

  • Students review and rate their perceived effort level and attention to detail
  • Students provide an example of where their effort level increased
  • Students create a math goal that will be achieved by the end of the year
  • Student indicate how they know that the goal will be met
  • The teacher and student sign-off on the reflection sheet

 


 

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Don’t get me wrong, this type of reflection is time consuming.  Whenever I discuss this process with other teachers I get quite a few questions about how to find the time.   Meeting 1:1 with kids to discuss their goal takes time and usually the other students are in stations or working on something independently. I can usually finish up meeting with the kids over 1-2 classes.  Instruction still occurs during this time, it’s just not a whole-group model.

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I’ve attempted many strategies to move kids away from comparing their score with others.  One strategy that seemed to work well was to have students go to stations and then I passed out the assessments.  I realized later that they just compared the results when they left the classroom.  I want to shift the paradigm to more of an individual growth model.  It’s a challenge.  Through the years, I believe progress has been made in this, but more needs to be done.

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The student math goals are interesting.  I had to have a brief mini lesson on the topic of math goal setting as many students wanted to initially make a goal of “getting everything right on the next test.”  I think many students were more interested in thinking of what their parents wanted and not necessarily a specific goal for themselves. Keep in mind these are 3-5th graders.  After a few different attempts, students started to make goals that were more skill focused.  Some students are now writing goals about “becoming better a dividing fractions”, “divide decimals accurately”, “become better at solving for x with one-step equations.”  While conferring with the kids I’m reminding them that the goals need to be measurable.

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After the assessment students review their math journals and monitor whether they’ve met their goal or not.  If not, they write down why or possibly change their goal.  I’ll then meet with the student and sign-off on the goal.  My next step is to involve parents in the goal and have a more frequent monitoring process.

Addressing Local Math Misconceptions

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Over the past week I had time to disconnect a bit and spend time with family. I was able to stay with relatives in another state and spent most of the time catching up with people I haven’t seen in a while. It was a great time to refresh and reflect on the past year.  While relaxing one relative in particular asked me about this “new” math that’s in the schools now. I was asked why schools are changing how they teach math and why it needed to change. Specifically she spoke of the different strategies used to compute numbers.  I’m assuming she meant the extensive use of the number line and compatible numbers.  I defended the reasons for a more conceptual understanding of mathematics, especially at the elementary level. Many of the “new” strategies help build that understanding and enable students in developing a foundational understanding of numbers. The relative was receptive and asked more questions related to this topic. I felt like her understanding of the topic became clearer as we discussed the use of  multiple strategies utilized to teach computation. This was a small part of our longer conversation, but the topic had me thinking about how to provide opportunities to address misconceptions. In particular, I thought how my conversation could apply to addressing math misconceptions in schools.

I feel like one of the more important issues with student misconceptions stems from a lack of addressing them. They tend pile up and build over time. I vaguely remember having a math teacher that asked if his students had any questions. I remember looking around and wondering if I was the only one in the class that had multiple questions. Unfortunately I kept my hand down even though I was lost. The teacher then quickly surveyed the room and seeing that no one had their hands up, moved onto the next topic. I found that the less I asked questions that less comfortable I was with the current concept and the process continued until I finished the class. Looking back, I’m sure there were other opportunities to address my misconceptions and questions; I just don’t remember any of them at the time of this writing. I didn’t learn as much as I should from the class, but I started to understand that I needed additional opportunities to ask clarifying questions.

In addition to including many opportunities to address misconceptions, the classroom environment plays a pivotal role in having students feel comfortable in offering input.  The strategies below can be used in a variety of settings. I’ve had success with the strategies, although some have been more successful than others.

Classroom Math Conversations

Classroom conversations can be a powerful strategy in gaining a better understanding of students’ viewpoints. Using open-ended math question and having groups respond to the class can offer opportunities for a healthy math debate. For example, I’ve seen some teachers use Always, Sometimes, Never with great success. Math Talks can also be an avenue in which classroom conversations can develop. Through these conversations teachers can glean important information and possibly misconceptions that can be addressed later or at that time. These types of math conversations, accompanied with anchor charts can document the classroom’s learning journey.  The anchor charts can then be revisited as students construct their understanding.

Formative Assessments

These types of assessments can take different forms. Some teachers prefer to use exit cards, while others use a quiz model. Formative assessments can be used via technology means and some may take the form of a paper/pencil quiz. Regardless of the form, the student’s response can give teachers an indication of understanding. In order for the teacher to give feedback the question needs to be appropriate. Students need to be given the opportunity to explain their reasoning or steps involved in solving problems. If not, the problem is wrong/right, and the teacher is unaware of where the mishap is occurring. Using written or verbal feedback to address the misconception can lead to a more in-depth conversation at a later time. Some students may need the reinforced conversations while others may not. I believe most teachers understand their students and at what level to scaffold feedback.

Journaling

Similar to answering a question in a classroom student group, journaling can provide students a low-risk venue to showcase their understanding. Through a prompt, math journaling can allow students to explain their mathematical thinking and processes in a written form.  Students often become more aware of their growth as the year progresses.  I find that students might not know that they have a misconception until it’s brought to light.  It’s the you don’t know what you don’t know dilemma. The concept of math journaling can be used for teachers to write feedback to individual students and ask questions that give students opportunities to reflect on their writing and math process. Allowing a bit of extra time to confer with a student after their math journaling process can be beneficial as teachers may want to review specific concepts with students.


All of these strategies above seem to go well with a heavy dose of teacher feedback and student self-reflection.  Through reflection, students can help internalize and address the misconceptions.

How do you address misconceptions in the math classroom?

 

Reflection before report cards

photo credit: woodleywonderworks via photopin cc
photo credit: woodleywonderworks via photopin cc

Last Thursday marked the end of the first trimester grading period.  After a few unit assessments, quizzes and special projects, my students are given a report card.  The report card splits into two categories: academic grades and behavior skills.  I tend to give my students their report card a few days before it’s actually sent home.  Once the reports cards are passed out I find that students focus only on the letter grade. Not the personal teacher comments, learning strands, checked boxes, but the letter grade is what gets the focus. Over the past few years I’ve challenged this type of thinking and laser-like focus on grades.  I’m slowly but surely moving my class towards a standards-based grading model, although the district requires teachers to use a traditional A-F model.

Before passing out the report cards this year, I gave my students an opportunity to journal about their math journey so far.  Math journaling has been a larger part of my teaching this school year. Students use math journals in my class to complete different types of math problems and for self-reflection.  I try to have the students journal approximately once every two weeks. During the journal time I turn off the lights in the classroom, turn on some music in the background and allow the students to go anywhere in the room to write up their response to the journal prompt. Some students stay at their desk while others find a hide-out in the corner of the room, on a comfy chair, or underneath a table.  As the year has progressed students are beginning to ask to have additional time to journal.

This year I gave each student their academic file before journaling. Enclosed in the file were all the past unit assessments and quizzes that took place during the first grading period.  Students were asked to analyze their own file and answer the questions below in their math journal.

  • What learning experiences stand out in your mind?
  • What do you feel are your strengths?
  • What would you consider a “growth” area for the next grading period?
  • What is one SMART goal that targets one growth area?
  • Create an illustration that matches any of the prompts above

After the students respond, I’ll review the responses and write short comments back to each student.  This does take some time, but definitely worthwhile.  I generally comment on their strength and ask questions that encourage students to reflect on their progress and growth areas.  This process also gives me an insight into what a particular student thinks and values.  By analyzing their own data, reflecting on progress made, and creating an action plan, I feel students are better prepared to take ownership of their own learning.