More Accurate Self-Reflections

Screen Shot 2018-07-17 at 5.07.41 PM

Improving how students reflect on their math progress has been one of my goals during the past few years.  It’s a topic that I’ve been trying to incorporate more in the classroom. That reflection piece in my classroom has changed quite a bit since starting this journey.

Last year students would take an assessment, review their scores and then fill out a reflection sheet.  Students filled out the reflection sheet the best that they could.  The students and I would review the test and reflection sheet to determine the next steps.  Some reflections were spectacular and had a lot of insight, others didn’t. Most of the time the next steps included items like studying more before the test, reviewing a certain concept in more detail, practicing specific skills, or dedicating more time to the subject.  I’ll admit that too many of the nexts steps were vague and wouldn’t match the SMART criteria.  I was glad students were creating goals and following through.  Refinement was needed, but I appreciated that students were lifting up more responsibility for creating their math identities. The students did a fine job following up with the next steps, although this was inconsistently implemented.  I’d check-in on goals during the next reflection time.

While reading Make it Stick (I’m on the second renewal from the library), I found something that I’d like to keep in mind for the new year.  In chapter five the authors discuss the Dunning-Kruger effect.  Research has shown that people (students) sometimes overestimate their own competence.  They “… fail to sense a mismatch between their performance and what is desirable, [and] see no need to improve.”  As I continued reading I found that lower-performing students were the most “out of touch” in gauging how well they were doing compared to the standard.  After reading this I started to think about how students accurately reflect on their math progress.

Students are often asked to compare their work to the criteria for success.  The points/letter on the top of graded work is generally perceived in black and white.  Students either view themselves as doing great or poor.  There’s nothing in the middle.  I rarely have a student that says they had an average test.  This becomes even more evident when students complete the reflection and goal setting sheets.  I’ve had a number of instances where students can’t come up with a goal for themselves.  Through probing questions I’m generally able to help students create a goal that is worthwhile, but this doesn’t always happen.  I believe math confidence and adopted math identities play a role here.  The perception is stuck on the score and it’s challenging to move beyond that number.  Maybe it’s because students aren’t as familiar in gauging how they’re performing compared to the standard? I’ve used different methods to encourage students to look at skills compared to points and this has helped, albeit the success using the table has been inconsistent.

Screen Shot 2018-07-17 at 5.24.03 PM.png

The authors of book discussed an experiment where poor performers improved their judgement over time.  These students received training specifically on the test concepts before the assessment. That time spent improved their self-reflections and they were more in-line with reality.  Basically, the students are better able to show sound judgement during self-reflection if they understand the concepts.  Accurate self-reflection becomes an uphill battle if they don’t.

Moving forward I’d like to spend more time discussing error-analysis and misconceptions with the class.  When students are aware of how these specifically exist then they’re better able to analyze their performance.  Pre-loading that meta-cognition piece is something I want students to keep in mind during the self-reflection process.  I think it will deter students from making statements like “I don’t know what goal to make” or “I need to work on everything.”  These types of statements are disheartening.  I think having exemplars might help instead of just diving in and asking students to reflect.  Having a clearer direction and possibly having a reflection time that occurs more frequently could also help.  Math isn’t always perceived as a subject where students are asked to create some type of narrative and connect to the text/content. I find that students rise to the challenge when I give them an opportunities to do so.  I believe that giving students opportunities to analyze, reflect, and set goals for themselves will empower them to create more accurate math identities.

Advertisements

Embracing Difficulties

Screen Shot 2018-07-10 at 4.35.20 PM.png

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.

Learning and Making it Stick

Screen Shot 2018-06-29 at 4.48.49 PM.png

I’ve been reading Make it Stick during the past few weeks.  It’s been a great summer read and it helps that a few people from the #ICTMChat crew has been reading as well.  Reading as a group adds an accountability piece that I think is needed – especially over the summer.

So far I’ve read the first couple chapters.  I’m finding a few gems and ideas that are great to reflect on before school starts back up in August.  I think the first chapter comes out swinging.  It hooked me from the start. I’m also hoping that writing about these first few chapters will help me remember them by the time school starts back up.  The authors suggest that the ways that people traditionally study aren’t the best methods.  What we’re told about learning is misunderstood. Multiple research studies to show that cramming or re-reading text multiples times is often a prescribed method to study.  It may work well short-term, but not long-term.  We now know that reading content repeatedly in a short amount of time isn’t effective.

Chapter two discussed retrieval practices.  One example that I thought was interesting involved using testing as a tool for learning.  I was brought up in an education system where a test signified the end – the end to a unit or end to a bunch of concepts that were studied together.  A grade was plopped on the top of the test and that was that. Once the test was given it was up-and-onward to the next unit.  Researchers found that giving multiple tests (low-risk) with a high cognitive demand helped students perform better on a final exam.  I put the emphasis on low-risk and high cognitive demand.  If the assessments have consequences attached than students might not be as willing to perform.  If tests are too easy or hard, then students might not take the tasks as serious. I see this as a balance.  The words formative assessments come to mind when thinking about low-risk tests with high cognitive demand. Most educators give these types of assessments throughout their courses.  This isn’t new and is often brought up during during teacher evaluation process. Repeated retrieval through the use of these formative assessments or self-quizzing techniques can produce better outcomes.  I see this repeated retrieval taking the form of study guides, formative assessments, exit cards, and even group tasks. While planning out the new school year, I’m thinking of being more intentional in picking spots within a unit to insert self-quizzing and formative assessment opportunities.

The third chapter brought some head-nodding.  Variety is the spice of life – or so they say.  It seems it’s the same in the classroom.  Massed practice of repeating or re-reading the same thing over and over again produces results that don’t last.  Cramming is often the go-to before a big test.  I’ve used it before and I’m assuming you have as well. You might retain something, but it’s generally gone before too long.  I’m into the learning that lasts.  I want what’s being introduced in September to still be swirling in students’ heads in March.  One of the practices that the author highlights revolves around the notion of spaced and interleaved practice.  Spacing out practice sessions gives students time to process new learning while making connections. Interleaved practice is similar to spiraling assignments where students need to recall different processes and find (often informally) the relevancy in how they’re connected.  Varied practice is something I find often at the elementary level. Concepts are introduced and practiced, but not necessarily mastered before moving on to the next sequence. The example of Coach Dooley and his practice regiment was interesting.  Coaches often have their players working on a variety of skills throughout the week. They need to review playbooks, look at tape, work on fundamentals, practice individual skills and practice team skills. Having a schedule in place to address all of these is helpful and the improvements are often slow and steady. According to the research, this type of learning is better for long-term acquisition.  The same can be applied to the education world.

Practice and repetition is an important part of the learning process.  I think this book has me thinking more critically about why practice is emphasized so much.  The research involved makes compelling points.  Students should be given time to practice and process their math experiences. Teachers want students to move beyond memorization and have them apply their learning in different contexts.  Students need to discriminate what the problem is asking and delve into their toolbox to pick the right way to approach the task. Being more aware of interleaving, spacing, and variability practices can help teachers be more intentional in how students practice skills.  Adding a student reflection component to a practice session also helps bridge connections.

I’m actually enjoying the approach of taking my time and reading this over the summer. This post was designed to as a medium to reflect on what I’ve been reading.  I’m looking forward to checking out chapter four over the next week or so.

Moving Away from the Gifted Math Label

Screen Shot 2018-06-18 at 8.46.22 AM.png

I’ve been paging through Jo Boaler’s Mathematical Mindsets lately.  I’m finding a few takeaways as I’ve been reading sections over the past few days.  My school has embraced some of the ideas in the book and we’ve been taking small steps each year.  One idea I found interesting relates to gifted students.  Page 94 discusses the “myth of mathematically gifted child.”  I feel like that statement is ripe with controversy as many are for or against the idea.  Most parents, teachers, and students have at some point in their life been told or shown their math identity.  Then that math identity may or may not be adopted and confirmed by the student. That communication can come from a teacher, parent, or somebody else.  Sometimes it comes from a single teacher or constant grades on assessments/assignments. Usually it’s developed early in life and continues with that individual.  People often can’t shake the generalization that they’re “good” or “terrible” at math.  I hear this at parent/teacher conferences, at school meetings and on EduTwitter.  Of course this is a generalization, but I find that this math stigma has lasting consequences.

I believe that the same stigma has the potential to occur with the “gifted” label.   I find that this can happen as early as the elementary level or even before then.  In an effort to address the needs of all students sometimes elementary schools group students by perceived math ability – emphasis on perceived. This often takes students and places them in different math classes during instructional blocks.  Students are moved to these different levels based on standardized test scores, classroom tests, teacher recommendations, or some other data that the team feels is necessary.  The groups can be fluid and change every unit, but sometimes they don’t. In some instances, school also have advanced math classes for students in upper elementary.  These classes might have a gifted label associated with them.   Although they’re labeled “gifted math” the roster doesn’t match the label.  The classroom rosters are often based on a criteria.  Sometimes the criteria is heavily weighted towards one single test-score cutoff, accounting for 40 or more percent.

Many questions come up regarding the actual percentage for students that are identified as gifted.  Most gifted specialists tend to agree that the amount is less than 10%.  Yet, these classes that are labeled as gifted tend to have 1/4 or even 1/3 of the total grade level population.  These classes may be accelerated, but not necessarily be meeting the needs of all the students that are identified.  Moreover, moving students to and from these classes can prove difficult as social/emotional consequences play out.  Often these classes aren’t as fluid and the roster doesn’t change as much since the students are accelerated from day one.

When shopping for school districts I sometimes find that parents are looking for whether schools have gifted classes for their child.  Schools might communicate on their website or through brochures that they have “gifted” classes, but in reality they’re accelerated subject-oriented classes.  Gifted students have academic and social/emotional needs and funding isn’t always available for this need.  It’s up to the local districts to create a system to meet the needs for these students.  I’m assuming positive intentions for the schools and districts in this scenario. In an effort to please the community and potential registrations, districts might used the term gifted to mean that the needs for high-achieveing students will be met.  Also, students that participate in these classes are artificially given the gifted label and they adopt the identity.  For some students they thrive in the class and it’s just what they need.  For others, it’s the opposite. Students struggle and feel contempt for math as they attempt to live up to the label of the class.  Having this happen at the elementary level sets the stage for a student’s math identity into middle school and beyond.

Labeling a class as gifted has consequences.  I want students to be able to create and maintain their own math identities.  Creating engaging math experiences for students with a heavy does of individual reflection can help students decide for themselves how they feel about math.  Regardless of their assigned math identity, I’m hoping my math class provides an appreciation, curiosity, and enthusiasm for mathematics.

Class Survey Results – 2018

Screen Shot 2018-06-13 at 2.24.51 PM.png

School officially ended around a week ago and I’m starting to look back at the year.  During the last few days of school I gave students a survey.  I’ve used end-of-the-year surveys before, but this year I wanted to get the students’ perspectives on lesson structure and math interactions.

The reason I went in that direction because I noticed some trends while recording myself teaching. These recording were taking place because video recording were required for a certification that I’m pursuing. I went with the volume approach and decided to record three days a week for the first couple months of school so students would feel more comfortable around the camera.  At first, students would wave, make faces, dab and do all sorts of unrelated math actions around the camera.  That died down once students started to see that the camera wasn’t leaving.

After reviewing many different recordings (this took what seemed like forever), I started to notice trends related to how I was designing and implementing lessons and tasks.  This was a humbling experience.

I noticed that students were doing more independent work that I’d like to admit. There’s nothing wrong with independent work, but I wanted students to engage in math conversations with one another more frequently.  My lessons weren’t generally designed to have these math conversations occur regularly.  I used quite a bit of whole-group math conversations to spur mathematical thinking.  Although that seemed to be a good use of time, I noticed that not everyone was engaged.  There were students that hung out in the background and didn’t engage unless called upon.  I found through this experience that I was spending too much time on certain instructional elements and not enough on others.  I want the time that students spend in my classroom to be valuable and useful.  So I decided to start varying my strategies more often.  Short story: lesson design and pacing is undervalued.

So I made a few changes related to how often students work together.  The class created norms associated with how students should be working with one another.  I decided to increase that amount of “collaboration” so that each class had more of the time dedicated to working with other students.  I used visible-grouping strategies so students could see that groups were randomly chosen and all ideas are valued.  During this partner/group time students were completing math tasks, short-term assignments, long-term projects, Scholastic magazines, Desmos, Nearpod, and Kahoot! activities.   I noticed that students were having better math conversations at a more frequent pace.  While students were in groups I walked around and asked questions to help ignite or guide the discussion.  This slight shift seemed to play dividends as the year progressed.  Students became more confident with their math communication skills and the quality of those conversations increased over time.

Here are the survey results that came out during the first week of June.

Screen Shot 2018-06-13 at 1.05.41 PM.png

Not quite sure about the 2s and 3s.  I reviewed what the term “appropriate” meant and how it applied to their personal learning before students answered the question.   I think that some students prefer to work independently, while others thrive during group work.  I have to take this into consideration while reviewing these results.  Also, some students might be rating the actual work that is being done during these two scenarios.

Screen Shot 2018-06-13 at 1.06.01 PM.png

I was pleasantly surprised to see these results.  I tried to focus on giving students multiple ways to show their mathematical thinking.  Looking back, I used whole-group and independent math routines, projects, journal pages, individual tasks, math reflection journals, and  math class discussions throughout the year.  I was hoping to give students multiple opportunities to learn about math.  I believe students enjoyed some of the structures more than others, but having a variety of them gave them opportunities to see math from different perspectives.

The last survey question is below.

Screen Shot 2018-06-13 at 2.10.10 PM.png

This has been an issue that I’ve been tackling for years.  There are not simple answers for this. Students often view math as a speed game.  The quicker they are, the better.  I believe students see and are experiencing mixed messages when it comes to their math journey.  Fortunately, I get to loop with students and they get the message that math is a journey and there’ll be challenges and wins along the way.  Now what they do with that message differs.  When students perceive math as a journey, they develop a deeper understanding of the concepts that are introduced in class.  There’s also less anxiety related to speed and algorithms.  I’m going to keep this in mind while planning out what messages are being sent to students and the community next year.

An Elementary Desmos Journey

Screen Shot 2018-06-06 at 8.31.26 AM.png

During the past few months I’ve had a chance to explore Desmos.  I started seeing the platform on Twitter a while back through different educational chats.  It was prevalent in #msmathchat as well as in the EduTwitter math world.  I saw it across my screen, but didn’t really dig into how it could be used until NCTM.  During that time I attended a session titled “Which Comes First:  The Equation or the Functions?  Come Stack Cups and Use Demos to Find Out!” The session was presented by the University of California staff.

The session started out by having the participants sit in groups with a stack of styrofoam cups.  We were asked the question below.

Screen Shot 2018-06-06 at 7.32.11 AM.png

My table took the cups and started stacking them. They found the measurement of one of the cups and decided to use millimeters to be more accurate.  We came to a consensus and decided to extend the pattern and graph our pattern.  The presenters then asked us to head to a Desmos url where we could plug in our numbers.  The audience filled out this step-by-step sheet for the next 20-30 minutes.  Being a newbie to Desmos, I appreciated how the sheet guided us through the different components of Desmos and how it graphs the line as the numbers change.  The sheet also includes teacher directions that I found useful as I replicated this exact lesson a couple weeks after NCTM.

The second part of the session went in a different direction.

Screen Shot 2018-06-06 at 7.40.51 AM.png

Basically, we were asked to find out the option that would give us the most money during the next 14 months.  My table filled out this sheet and used this Desmos template to find a solution.  There was a lot of trial-and-error during this time and some perseverance, but it was a worthwhile journey.  Listening to others at my table was enlightening.  Hearing how other teachers, mostly at the middle and high school levels, gave me a different perspective.  Being able to find patterns and develop algebraic reasoning was at the forefront.  Using Desmos to create or check our predictions was helpful.  Moreover, I was able to learn more about the Desmos platform and how it could be used in my classroom.

Near the end of the session I started gathering all the links to save for later.  I started to think about how this could be used in my own classroom.  I began by using a few different tasks with my sixth grade standards.  The inequality activities were very helpful as I was becoming more familiar with the logistics of using Desmos in the classroom.  Desmos isn’t a math tool that’s used frequently in elementary schools.  The calculator included can be intimidating for elementary students as well the teachers.  Although, when my class first started using the platform my students were totally interested in wondering what all the symbols meant on the calculator.

During NCTM I was fortunate enough to come across Annie’s post on how to use Desmos in the elementary classroom.  I reviewed all of Annie’s examples and decided to graduate and move on to creating my own.  I started with copying a couple card sorts for a kindergarten and first grade group that I see four times a week.  The sort included having students put together base-ten blocks, written names, standard form, and expanded form statements.  The lesson went well and I had to intensely model how to login and ended up using a QR code for students to access the url on an iPad.

I then started getting used to how to create better questions using “what do you notice” or “what do you wonder” types of prompts.  About three weeks ago my third grade students needed to review polygons and relationships.  I decided to put together an original activity.

Screen Shot 2018-06-06 at 8.08.57 AM.png
Link

I ended up making the shapes in Illustrator and inserted them into the activity.   I used this task with a third grade group of students that were working on fourth grade standards.  Students worked in pairs as I didn’t have enough Chromebooks and iPads for everyone that day.  I used the pacing function and made sure all the students were on the first slide.  I love that Desmos includes that function as many of my students want to speed through an activity.  Stopping with “pause” to review all the different responses was also helpful.  This helped encourage a class discussion.

Screen Shot 2018-06-06 at 8.11.36 AM

At first I used the “anonymize” option, but soon found that students wanted to be identified.  Students used a similar prompt with a trapezoid and triangle.  I then went over a few different vocabulary terms before heading to the next slide.

Screen Shot 2018-06-06 at 8.15.18 AM.png

Students checked off the criteria that matched the shape.  I then displayed the correct solution and this led to additional questions and conversations.  When creating this I decided to look back at some of the most engaging Demos activities that I’ve used.  Most of them ended with some type of card sort.  I decided to do the same.

Screen Shot 2018-06-06 at 8.18.13 AM.png

I decided to have a “doesn’t have a match” column to increase students thinking.  It took a while for students to complete this as certain criteria worked with only one shape while other statements didn’t match any.  When I first created this I noticed that some of the statements that were created matched multiple shapes.  Ooops.  It was fixed before I gave it to my third graders, but that could’ve caused a few issues.  After around 10 minutes we review the answers as a class.

Screen Shot 2018-06-06 at 8.21.45 AM.pngStudents self-checked their work to see how accurate they were.  Some students didn’t actually match the statements completely.  They brought the statement to the column, but didn’t attach them.  I should’ve explained how to do this before completing the activity.  This happened with around three students.   The next time we completed an Desmos activity those students were fine and attached the statements.  There’s a small learning curve with these types of activities and this was one of those moments.

My next step is to learn how about the computation layer and possibly how to use it more effectively with my K-3 students.  I think it’d be great to be able to rotate or drag a shape on a coordinate grid.  Students could then use a digital protractor to measure the internal and external angles.   I’m also looking at how to use polygraphs more effectively next year.  My students had a blast using them during the last week of school. Maybe I’ll learn about that during the summer.  I’m looking forward to using this more with my elementary students next school year.

Exploring Scale Factor

Screen Shot 2018-06-02 at 10.54.08 AM.png

My fifth grade students explored scale factor and dilations this week.  This was their last week of school and it was a great topic to study as they head off to middle school next year.  Some of the geometry that they’ll see in middle school involves this specific topic. The class started early in the week with a brief Kahoot! on similar figures and enlargements as well as reductions.  This was a bit challenging as some of the shapes were rotating and students had to identify a particular side.  After discussing words like dilation, scale, and factor, I gave students a multi-day project.  This project actually stems from one of Allison’s Tweets that I read earlier in the week.

At first glance the image in her Tweet made me laugh.  My second thought was … wait… this might work in my classroom. You know how a Tweet can spark additional ideas?  Yes.  That happened here.  I asked Allison about the project and she provided additional details about the pre-image.  So I put together a direction sheet and rubric.  I decided to have students use a 5 cm x 7 cm grid.  The class then discussed how enlarging the image would change the horizontal and vertical dimensions.  I decided to not have them reduce images since the pre-image was already so small to begin with.  We did discuss how me might need a magnifying class though.

Students went through their Math Magazine.  They looked for an image inside the magazine that they found interesting.  Some students found something immediately, while others took some time.  Students then traced over the 5 cm by 7cm card, used a straightedge to create grid lines, and then finally cut out the sheet.  Students then pick their scale factor page.  Just about everyone picked the 15 cm by 21 cm grid.

Screen Shot 2018-06-02 at 10.54.02 AM.png
Note to self:  In the future I need to have students randomly  (maybe with dice) pick a different scale factor page.  

Students were then off to work in creating the image.  Some students asked questions about whether they needed to enlarge everything on the image.  Yes.  Interesting … some students thought it was just the picture that was enlarged and not the text.  Everything in the gird was increased by the same factor.  Once that was covered students made steady gains for the next 20 minutes.  They started with pencil and then shaded in the rest with color.  Students were proud of what they accomplished during the first session and many wanted to finish them up at home.

Screen Shot 2018-06-02 at 10.53.48 AM.png

Students finished up their scale factor projects on Friday.  Since it was the last day of school I sent them home.  I gave each student a survey (this might be another post) before leaving and it was interesting that many commented about how they enjoyed learning about scale factor and ending the unit with a project like this.  Maybe it’s because it was the first thing in their mind, but I thought this type of project was worth the time and will help students moving forward as they discuss similar figures next year.

I’d like to find more time over the summer to create more memorable math activities similar to this. Kids can then hold onto these experiences past the last day of school and look back on them as they make connections next school year.