Division and Area Models

 

 

Area Model-01

My fourth grade group just finished up a unit on division.  They spent a great deal of time exploring division and what that means in the context of a variety of situations.  One of the more interesting parts of the unit delved into the use of the partial-quotients method.

You might be a fan of this method if you’ve ever wanted to know why the traditional division algorithm works.  I was in that boat.  During my K-12 math experience I never questioned what was introduced.  I was always encouraged to use the traditional division algorithm.  It didn’t make sense to me why I was dropping zeros or setting up the problem in a certain structure.  I just played school and figured that it wasn’t worth trying to find meaning, but instead just pass the class and move on.

For the past ten years I have been introducing the partial-quotients method to students.  This method brings more meaning to why the division process works.  Students also seems to grasp a better understanding of how partial-quotients can add up to a quotient with a remainder.

This year I introduced something different to the students.  Students were asked to use the partial-quotients method to divide numbers and then create an area model of the process.  I had quite a few confused looks as most students think of multiplication when using area models.  After modeling this a bit I noticed that students continued to have issues with appropriately spacing out their area models.  This was a great opportunity for students to use trial-and-error.

I noticed that students had to first use the partial-quotients method to find the quotient and remainder.  They then had to split up the area model into sections that matched the problem.  This spatial awareness piece is so important, yet I find students struggled with it to a point that they were asking for help.  Maybe it’s a lack of exposure, but estimating where to split up an area model to match the partial-quotients seemed to challenge students on another level.

This activity has me wondering how often students use spatial awareness strategies in the math classroom.  How often are they given these opportunities?  Reasoning and estimating strategies also play a role, but actually spacing out the partitions isn’t what students were expecting last week.

I’m looking forward to seeing how students progress in using area models moving forward.

Advertisements

Second Attempts and Error Analysis

 

Error Analysis-01.png

I’ve been thinking about student math reflections this week.  That reflection can take on many different forms. Giving students a second attempt to complete an assignment can give them an opportunity to reflect on their original performance. This is often (not not always) part of a standards-based-grading approach.  Some teachers allow students to redo particular assignments.  Some teachers have their students complete a paper form of a reflection and/or redo sheet when they didn’t meet the original expectations.  Students fill out the sheet, redo certain problems that need a second look, staple the sheet and finally turn the work back in.  This process has worked well during the past year, but I’m noticing that students are starting to place general statements in the blank lines.  This NY –> M process was starting to become more paperwork than individual reflection.

Screen Shot 2018-02-18 at 1.27.14 PM

Students would avoid writing simple mistake like the plague (since it explicitly says not to do that :)), but they’d write comments that were very general.  I mean VERY general.  Students would write

  • “I didn’t write the answer correctly”
  • “I had trouble with fractions”
  • “I didn’t write the problem right”

Most of the responses were general, and some students wouldn’t even thoroughly review their work before attaching the second attempt sheet.  Don’t get me wrong though.  The sheet was helpful, but I wanted students to delve deeper into their work and become better, or more aware, of where they didn’t meet the expectations moving forward.  Over the summer I was able to attend sessions and workshops related to student goal setting and student error-analysis.  I believe student reflection and error-analysis can be powerful tools for students as well as teachers.  Knowing this, I revamped the second attempt sheet this week.  Here’s the new look.

Screen Shot 2018-10-05 at 6.55.35 PM.png

The blue circles were entered on the sheet based on the most common errors that I found on quizzes.  I made sure to model this with the class before students filled them out.  I gave examples of why someone would check each box.  After a number of questions, students felt more comfortable in deciding which circle to check – some even thought that multiple circles could be checked.  Why not?  I noticed that students would determine which circle to check depending on their perspective.  Check out the three submitted sheets below.  They all are for the same problem, but fit different categories.

MathCoding-01.png

This is interesting because students were starting to analyze their results with a more critical eye.  This is progress, positive progress.  Even with that being said, we have a long way to go.  I need to be more clear on how students should differentiate between a simple mistake and directions.  I also need to clarify and give more examples of what a strategy issue means.  I think some students have been using the updated sheet with integrity, while others might not be using them as well since their perception of the categories isn’t clear.  I believe this is more of a teacher and modeling issue than a student issue.  I’m looking forward to creating a few different activities for next week to help students becoming better at categorizing their errors and misconceptions.  At some point I’d like that awareness to lead to action and eventually goal setting.  One step at a time.

Student Self-Reflection and Common Math Errors

Screen Shot 2018-09-22 at 2.31.17 PM

My fourth grade students took their first unit assessment of the year last Wednesday.  This is the first class to take an assessment this school year.  The unit took around four weeks and explored topics such as area, volume, number sentences, and a few different pre-algebra skills.  This year I’ve been approaching student reflection and unit assessments differently.

Students were given their study guide during the first couple days of the first unit.  The study guide included questions that covered topics that would be taught throughout the unit.  At first students were confused about how to complete items that we haven’t covered yet.  Eventually students became more comfortable with the new study guide procedure as we explored topics and they completed the study guide as the unit progressed.  There were a couple of students that lost their study guides, but they were able to print it off from my school website.  I reviewed the study guide with the class the day before the test.  It took around 10-15 minutes to review, instead of around 40-50, which has been the norm in the past.

After students finished the study guide the class reviewed the skills that were going to be assessed.  Students informally rated where they were at in relation to the skill.  I decided to move in this direction as I’m finding that reflection on achievement or perceived achievement doesn’t always have to happen after the assessment.

Students took the test and I passed back the results the next day.  Like in past years, I have my students fill out a test reflection and goal setting page.  This page is placed in their math journals and I review it with each student.  I decided to use Pam’s idea on lagging homework/coding and add this to my student reflections.  Last year my students used a reflection sheet that indicated problems that were correct or incorrect and they developed goals based on what they perceived as strengths and improvement areas.  This year I’m attempting to go deeper and have students look at not only correct/incorrect, but also at error analysis.

So I handed back the tests and displayed an image on the whiteboard.

mathcoding -01.png

I told the students that we’d be using coding in math today.  I reviewed the different symbols and what they represented with a test that was already graded.  Each question would be given a code of correct, label / calculation error, misconception, or math explanation. I gave multiples examples of what these might look like on an assessment.   I spent the bulk of my time introducing this tool to the misconception symbol (or as some students say the “X-Men” symbol) to the students.  After a decent amount of time discussing what that looks like, students had a good feel for why they might use the math explanation symbol.

I then passed out the sheet to the students.

Screen Shot 2018-09-22 at 2.07.02 PM.png

Students went through their individual test and coded each question based on the key.  At first many students wanted to use the label/ calculation error code for wrong answers, but then they stopped and really looked at why their answer didn’t meet the expectation. In some cases, yes, it was a label issue.  Other times it was an insufficient math explanation.  Most of the students were actually looking at their test through  different lens.  Some were still fixated on the grade and points, but I could see a shift in perception for others.  That’s an #eduwin in my book.

This slideshow requires JavaScript.

After students filled out the top portion of the reflection sheet they moved to the rest of the sheet.

Screen Shot 2018-09-22 at 2.11.02 PM.png

Students filled out the remaining part of the reflection sheet.  They then brought up their test and math journal to review the entry.  At this time I discussed the students’ reflection and perception of their math journey and I made a few suggestions in preparation for the next unit.

At some point I’d like create an “If This Than That” type of process for students as they code their results.  For example, If a student is finding that their math explanations need improvement then they can ________________ .  This type of growth focus might also help students see themselves as more owners of their learning.  I’m looking forward to using this same process with my third and fifth grade classes next week.

Third Grade Math Confidence

Screen Shot 2018-09-15 at 3.06.58 PM

My third grade students have been working on rounding and estimating this week.  It’s been a challenge as these concepts are fairly new to the entire class.  We’ve only been in school for only three weeks but I feel like we’re in stride now.  Kids and teachers both are in a routines and tests are already on the schedule.

Back to rounding and estimating.   So students have been struggling a bit with these two concepts as we head towards using the standard algorithm. With that struggle comes a shake in math confidence.  Students needed to be reminded of our class expectation of “lean into the struggle” many times during the past week.  It’s interesting how a student’s math confidence changes throughout a unit, or even throughout the year.  This third grade class in particular is working on becoming more aware of their math performance compared to what’s expected.  In order to reach that goal, I dug back into my files and found a simple, yet powerful tool that might help students on this awareness math journey.

Screen Shot 2018-09-15 at 2.50.55 PM.png

Basically, students first read the top row goal. They were then given a die to create an example of the goal.  During this process students circled one of the emoji symbols to indicate their confidence level.  The extremely giddy emoji indicates that they could teach another student how to complete the goal.  The OK smiley means that you’re fairly confident, but feel like you might not be able to answer a similar question in a different context.  The straight line emoji means that you’re confidence is lacking and you might need some extra help.  This paper wasn’t graded and that was communicated to the students.

Regardless of the emoji that is circled, students are required to attempt each goal.  Some students were very elaborate with explaining their thinking, while others tried to make their answer as concise as possible.  After completing this students submitted their work to an online portfolio system so parents can also observe progress that’s been made.  So far it’s been a success.  I’d like to use this simple tool for the rest of the first unit and possibly the next.  It takes time, but as usual in education, the teacher has to decide whether it’s worth that time or not.  In my case, the student reflection has meaning and it’s directly tied to the goals of the class.  I’m looking forward to seeing how these responses change over time. Feel free to click here for a copy of the sheet if you’d like one.

Volume and Group Tasks

2examples-01.png

My fourth grade students have been exploring measurement and geometry during the past week.  They started out by learning about perimeter, area, and are now in the midst of discussing volume.  Students are working on a project where they’re building rectangular prism models and constructing cities.  They document the dimensions, cut out the rectangular prism that matches the measurements and places it on a map.

The groups have been working diligently over the past days.  What I found interesting on Friday were the student conversations.  As I peered over each group I eavesdropped on what was being said.  Students are in groups of 2-3 and there’s plenty of conversation happening.  Students are recognizing that the length, width and height all impact the volume of a rectangular prism.  Mistakes are also happening.  That’s a good thing.  Students have had to use multiple grid sheets because they either cut out the faces too large or too small.  They just grab another grid sheet and start over again.  Their perseverance and being able to “lean into the struggle” is evident and I made sure to remind them of that.  Students were even getting creative in adding sunroofs and open decks with their prisms.

Projects like this take time, but they’re often worthwhile in helping students build conceptual understanding.  I’m looking forward to adding a question component for the groups on Monday.  The questions will relate to cubic units and how the volume is found when combining multiple rectangular prisms.

I believe that this activity helps students apply volume formulas.  I want students to come away with a better understanding of why the formula V = b*h is used and have them feel more confident in being able to see geometry/measurement relationships.

After this activity students will take a brief formative checkpoint where they’ll be answering questions similar to the below image.  This is also how students will be assessed in a couple weeks.

3dcomboshape

The projects should be finished by the end of the next week.  I’m looking forward to seeing how the cities finish up and the student reflections that follow.

Math Exposure Isn’t Enough

A few months ago I remember sitting in a meeting where teachers were discussing students and their math placements.  The conversation revolved around the topic of whether students should change math placements for the next school year.  For example, should a student stay in a homeroom math class or be part of an accelerated class?  How will we provide additional math support for particular students?  These types of questions tend to occur throughout the school year, but action for the next year often takes place near the end of the school year.

The decision to change placements is based on a variety of factors, but many schools/districts narrow down their criteria using standardized assessment data.  That data is often in the form of achievement, cognitive, and/or even aptitude tests.  Each district that I’ve been in has had a different process to determine subject placements.  This placement process becomes even more apparent as students travel from elementary to middle or middle to high school. Students’ birthdates, norm-referenced test scores, and percentages often take center stage during these decisions.  Sometimes the conversation evolves into whether students would be able to transfer the skills to a more rigorous math program than the one that they’re currently attending.  The conversations are usually productive and emphasize how to best meet the needs of students.

The topic of exposure is often brought up when making these types of math placement decisions.  A quick Google search will bring up one of the definitions – “introduce someone to a subject or area of knowledge.” I have heard on more than one occasion the following paraphrased statements/ideas:

  • If students haven’t been exposed to the content then they won’t be prepared
  • Those students weren’t exposed to above grade level work so they won’t be ready for that class
  •  The reason the student scored at the ___%ile was because he/she was exposed to that skill before the test
  • If they’re not exposed to this class then they won’t take higher-level classes in high school

I feel like these types of phrases are thrown around lightly and in a way that doesn’t hit at a bigger issue  Being exposed to content doesn’t necessarily equate to applying it in different situations.  Showing a students how to complete a specific skill/process doesn’t mean that they fully understand a particular concept.  Students might understand a process, but are limited during the application stage.  Also, educators need to keep in mind whether an above grade level curriculum is developmentally appropriate for students.


I believe the bigger issue here is equity.

  • Are all students receiving high-quality math instruction?
  •  Do the tasks and math routines allow students opportunities to explore mathematics and build solid understandings?  
  • Do students need enrichment opportunities instead of acceleration?
  • Will being exposed to a new curriculum/topic/grade-level be the panacea to move students to a higher math placement?  Is that even a goal?  

So many questions are above and I’ll admit that I don’t have a solid solution for them.  I think we have to go back to what a school/district values. I do know that I want students to be curious about math and dive into its complexities.  Classrooms should develop a culture where taking mathematical risks is the norm.   High-quality math instruction takes investment from a school and district.  Ensuring that this instruction is occurring and support is provided is also important.  Mathematical tasks that encourage students to observe, create, and apply their understanding beats limited exposure any day. Exposure is the first step and it doesn’t end there.

Community Building and Content

I think it’s safe to say that I’m slowly transitioning into school mode.  It’s inevitable and happens every year, but the month of August seems to fly by as a new school year approaches.  Over the past few weeks I’ve bought items for my classroom and have started some planning here and there.  Next week I’m planning on dropping by my room and start the unpacking process (I changed classrooms).  That is unless HGTV decides to makeover my classroom over the weekend.  So right now I’m drinking coffee and being a bit reflective.  I’ve opened up my planbook and am starting to ink in the first couple days.  While doing this a few questions have crossed my mind.

Will students be receptive to the beginning of the year tasks/activities? Are the activities related to my content area and does that matter?  Will the activities be remembered one day, five days, or even five months from now?  How will the activities impact the rest of the year and how will students remember them?

Many students get excited about new tasks or activities.  I find this happens quite frequently at the elementary level. The beginning of the year often yields plenty of classroom community building activities.  These may or may not be associated with the content that’s taught.  The emphasis is on building a positive classroom environment and often helps set the stage for the rest of the school year.  During this time students often work in groups and there’s generally a reflective piece near the end where a consensus is made.  Sometimes the classes develop norms and touch on the idea of growth mindset.  Usually these activities end after the first few days of school.  As the community building time ends students know what’s going to happen next.

Screen Shot 2018-08-10 at 9.58.50 AM
Should this be the process?

A shift is approaching and then it comes.  Kids know this and so do the adults.  All of a sudden homework starts being assigned and lesson sequences arrive.  It’s no longer “community building time” and we’re now in (insert your content area) time.  It’s often expected that the norms that were established and community building will last throughout the year.  It’s been established, right?

Not so much.  I find that teachers have to revisit the community building, norms and other themes periodically – not just after a long break. Otherwise those themes become like the posters on classroom walls – ignored after a certain amount of time. Students are used to playing the game of school.  Having novel beginning of the year activities and building a classroom community aren’t mutually exclusive.  Students and teachers are often reminded that the culture of the classroom is always evolving.

There’s often a perception that teachers need to dive into curriculum as fast as possible.  This is often perpetuated with scope-and-sequence guidelines and expectations.  Why not blend the community building activities and your content area?  That’s why I’m a fan of having math as part of the community building process.  Blending in content and community building can happen and I think it helps the sudden transition that sometimes becomes apparent.  I think also revisiting some of these community building activities throughout the year can give perspective and remind everyone of the importance.