Nice to Meet You Area Model

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This week my second grade students have been exploring multiplication strategies.  We started off early in the year looking at arrays and using doubling strategies.  Then we moved to helper facts.  These are still used to this day, but we introduced a new tool this week.  Enter the area model.  Hello!

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Students transitioned from arrays to squares, but didn’t sit at that spot long.  Through the area model, students take apart numbers and partition (yes, we say partition at second grade) the rectangle into parts.  Each part is a partial product.

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I’m fortunate in my position to see this strategy used at multiple grade levels.  The rectangle evolves over time.  As students progress, I find that place value and advanced decomposing strategies become more prevalent.  You can learn quite a bit about a student’s understanding by checking out their math work with an area model.  How they split up the numbers can also tell a story.  Why did they split up the rectangle that way?

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I find quite a bit of value in using this strategy.  For one, it doesn’t immediately move students towards the standard algorithm and it helps build/show conceptual understandings.  My 2-6th grade math students use it in a variety of capacities. My 5th grade crew has recently been using them to multiply fractions. Short story: It makes an appearance at every grade level.  It’s also a a fairly smooth transition to using the partial-products strategy.

Even though it’s a useful resource, I find there are a a couple things that irk me about using this tool.  Sometimes organization skills can hamper the effectiveness of drawing and organizing.  I’ve had more than a handful of students draw boxes that overlap or numbers that might not be decomposed correctly.  Also, it’s not to scale, but that’s not a game changer for me.

As students progress through elementary school they encounter a variety of math tools and strategies.  Manipulatives are generally used to help students build a better understanding of math concepts. The CRA model is often emphasized at this level. Many tools are brought out to help fill gaps and others are continually used.  At some point, I’m assuming the my students will rely on the standard algorithm to quickly multiply numbers (if they don’t have a calculator handy).  They probably won’t understand why the algorithm works, but it just does.  The area model shows multiplication in a concrete way.  Don’t get me started on lattice.

More on Standards-Based Grading

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Last Friday was a teacher institute day.   I spent my time planning, working on report cards, and listening to a speaker in the afternoon.  The speaker spoke to all of the elementary teachers in the district.  The event ended up being in the cafetorium (that’s what we call the auditorium/cafeteria).  It’s a huge wide-open space that usually holds elementary and middle school lunches.  The speaker introduced himself and told everyone that he was there* to chat about standards-based grading/policies.  There’s been talk that the district will be moving towards standards-based grading at some point in the next few years.

The presenter went through questions related to why teachers grade students, why standards are used, and how inadequate a 100 scale is while emphasizing the need to use feedback instead.  I think most educators there were  aware that specific feedback is a more useful tool than points.  The presenter reaffirmed the audience’s beliefs and also  dolled out research by Paul Blake and Dylan Wiliam’s “Inside the Black Box” study.

After about an hour and half the presenter mentioned how he would introduce a  standards-based reporting model.  He also prefaced this saying that there’s not a perfect model.


4 – “Blows the expectations out of the water”

3– “Meets the expectations”

2– “Student needs a little help to meet the expectations”

1– “Student needs a lot of help to meet the expectations”


I’ve never had standards-based grading explained like this and it was refreshing.  I noticed a few teachers nodding and a few commenting about the simplicity behind the reasoning.  The presenter went through a number of submitted questions related to what happens when teachers have different opinions on what “blows the expectations out of the water.”  Questions also came up about how many standards to report for the report card.  There wasn’t exactly a right answer with this, but the presenter mentioned that students have “all of the year” to meet the standard.  There were questions about this.  Consistency with teachers’ expectations was also addressed and many teachers believed this would be a good use of PLC time.

in some schools that use standards-based grading, I’ve seen a number models where teachers use a percent scale and then convert that value to a 1-4.  I’m sure there are plenty of standards-based grading models out there and doubt there’s a fool-proof way to implement this new communication tool.

The good news is that I believe teachers are already using standard-based practices.  Some teachers are eliminating points and percentages on some of the assignments. They’re also moving towards a “Not Yet” or “Met” policy with tasks.  Report card grades tend to reflect unit assessments. I know of some classrooms that are already using classroom policies that reflect a standards-based model, while others don’t. Moving forward, I believe there’ll need to be support in developing consistency as districts move towards new reporting models.  Some Illinois districts have moved towards or have already started using standards-based policies and some have encountered turbulence.  I believe there’s consensus that averaging grades isn’t always the best option.  Moving away from that will cause some to squirm and ensuring that there’s a smooth transition won’t be easy.  Communication and consistency will play a major role in how it’s received by all stakeholders.

*Bonus – the presenter introduced the think, ink, share process.  I wasn’t aware of this and am planning on trying it out in a couple days.

Division and Area Models

 

 

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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.

Study Guide Issues

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This past week I started working on my school website.  It’s a journey every year, but this year is a bit different since my district adopted a new math text and quite a few of my links need to change.  In the past I’ve posted homework, a few math strand practice sections, and class newsletters to my website.  I’ve decided to change a few things up for the 18-19 school year.  I’m nixing the homework section and adding a study guide piece.  The reason I’m adding a study guide section is because I’m not thrilled with how it’s currently being used.

Study guides have been a sticky issue for me over the years.  There are so many different ways that they’re used.  My class tends to give and review a study guide a day before the unit assessment.  Each teacher in my school uses them slightly differently, but the process usually follows this sequence:

1.) Students use class time to complete the study guide

2.) Study guide is reviewed by the class and teacher

3.) Students correct their answers and feedback is provided

3.) Students use the study guide to prepare for the test

It seems that most teachers use some version of a study guide or review before an assessment.  Some teachers use games, while others go the paper and pencil route.  I think it truly depends on the teacher and their students.  This study guide process works well for many students, but I think it needs some tweaks and to a certain extent, improvements.  This summer I’ve been reading Make it Stick and it affirms some of what I’m seeing when it comes to memory retrieval.  Teachers want students to be able to retain what’s experienced in the class and giving a study guide with only that night to prepare isn’t as helpful as other strategies.  I’m starting to become more critical when it comes to questioning study guide practices. I sent out a Tweet indicating my concerns.

Here are my issues:

  • Students aren’t being given enough time to process what’s being discussed on the study guide
  • Students aren’t benefiting from enough retrieval practices
  • Students solely rely on the study guide to review for the test
  • Students might not be engaged or they decide to copy the answers from their partner
  • Students aren’t aware of how to study (this could be a whole different blog post)
  • Students aren’t experiencing enough reviews throughout the unit
  • The questions on the study guide are very similar to the actual test

I’m aware that some of these issues will occur regardless of the policies or procedures that are put in place.  I’d like to specifically address the blue issues in this post.

  • Students aren’t being given enough time to process what’s being discussed on the study guide 

In order to give students more time to process the study guide I’ve decided to give the packet in advance.  This requires more planning on my part (let the uploading and copying process begin!).  I’m planning on posting the study guides on my school website and giving students a paper copy at the beginning of the unit.  Students will have 4-5 weeks to finish up the study guide before the assessment.  In addition, this will help students preview the learning, as Mary pointed out.  It’s likely that some students will lose the sheet as they’ll need to hold on to it for about a month.  That’s why I’m deciding to post the study guides.  I’m also planning on having students code their work with a few self-monitoring strategies. I really like the completed, mistake, misconception, and correct coding.  Occasionally the class will review concepts discussed on the study guide so that the class won’t have to wait until the last day before receiving feedback.

Giving students more opportunities to experience math has its benefits.  Being more  intentional in how retrieval practices look is important.  I currently have specific exit cards and review checkpoints that are used for particular units.  I’m planning on creating more and placing them strategically throughout the units.  I’d like to give students multiple opportunities to address standards and receive feedback.

  • Students aren’t experiencing enough reviews throughout the unit

Moving forward, I think students need to have the opportunity to review topics as the unit progress.  The text my district uses has reviews, but the students need more opportunities to address skills that are taught at the very beginning, middle, and end of the unit.  Like David said, I’m planning on adding deliberate interleaving of concepts to the study guide.  That may add additional questions to the packet, but I think it’s worthwhile.  I also need to keep in mind that students will have around a month to work on the packet.

  • The questions on the study guide are very similar to the actual test

I’m conflicted with this.  I think students should be aware of what skills are on the test and the format shouldn’t be a surprise.  Being unfamiliar with the questions or format can cause anxiety.  There’s already enough anxiety surrounding testing.  I think sometimes giving questions that are too similar can cause students to be overconfident.  I think there’s a balance, I just haven’t found it yet.  I’m placing this bullet point in the ‘to be continued’ section.

From here, I’m currently updating my school site to include study guide materials.  It’ll take a shift in expectations as I loop with many of my students and they’re not used to that process.  Change is inevitable and I believe being aware and making a shift will benefit students. I’m looking forward to seeing how this process plays out and will write a post about it at some point.


 

 

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A simple mistake or something more?

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I’m grading tests this weekend.  My third grade group just finished up an assessment on fractions and multiplication.  It’s been about a 1-2 month journey full of investigations on this particular topic.  Many formative checkpoints were planned along the way and the unit assessment was scheduled for last week.

While reviewing the student work, I’ll sometimes write questions or direct students to a part of the question that would’ve made their answer more complete.  There are moments of pride and moments where simple mistakes drive me a bit crazy.  You see, there are only eight units with my new resource, so the assessments influence grade reporting quite significantly.

After grading the assessments, I have student analyze their results.  They comb through the test and look at how each question aligns with certain skills.  They also determine if a missed question was a fixable mistake.  I want students to be able to recognize when this occurs and fix them when they can.

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In my experience, 9/10 times students believe that the reason they missed a question was because it was a fixable mistake.  That’s not always the case.  There’s a certain amount of self-reflection and humility that’s involved in this process. Being able to be a bit more honest and communicating what a simple mistake is and what it isn’t might be in order before the next assessment.  So, what steps do students take if a missed questions isn’t a fixable mistake?  It’s one step in the right direction to admit that it isn’t fixable, but then what happens next?  Do students and teachers have plan for this, or do we move on to the next unit?

So, was it a simple mistake or something more?  This question comes up more often than not while I’m grading student work or reflecting back on a class conversation.  Some of the answers are more positive than others.  A simple calculation error can vastly impact an answer, but it may be a simple mistake and the student has a solid conceptual understanding of that skill.  But, a number model that doesn’t match the problem tells me that the student might not be certain about the operation that needs to be completed. Was the simple mistake putting the wrong operation sign in the number model?  I guess you could go down many different paths here.

The question type can influence how well and thorough a student responds. Some questions are quite poor in giving educators quality feedback to help inform instruction.  Right off the top of my head, multiple-choice and true/false questions fit that bill.  They sure are easy to grade by human or a machine.  Hooray!  But, they don’t give me quality feedback that I can use immediately.

This also has me wondering about the quality of the assessments that are given.  Measuring how proficient a student is on a particular concept doesn’t always have to come from standardized or unit assessment results. Classroom observations and formative checkpoints are beneficial and give teachers insights to what students are thinking. I want to make students’ math thinking visible.  Whether that’s using technology or not, making that thinking visible puts the teacher in a  better position. From what I observe, some of the best math task questions are open-ended and tend to have a written component where students are asked to explain their thinking.  The quality of the question and openness of the answer helps educators dig deeper into how and what students are thinking.  I think that’s why teachers are always looking out for better math tasks that help students demonstrate their understanding more accurately.

How do you help students determine if a mistake was simple or something more?

Meaning and Algorithms

My third and fourth grade classes are exploring computation algorithms this week. Taking apart and putting together numbers is one of the skills needed to tackle higher level math concepts. I find it interesting when this topic is brought up as I sometimes hear students commenting that they already know how to add/subtract. Many of my students have been instructed to use the traditional algorithm to add large numbers.  After digging a bit deeper I started to see students falter with their explanation of the traditional method.  I asked them to explain their thinking. I found that their reasoning started to turn into comments about the process. Students explained the steps involved to find the product.  One student told me about the example below.

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Traditional

The process, although important, doesn’t reflect mathematical understanding. Students didn’t mention place value at all.  Place value within the traditional method is evident, it’s just that the students weren’t able to explain it with confidence.  It reflects more of an understanding of the procedures required to find a solution. Most of the students that I see are used to adding/subtracting numbers on a number line. They can hop up and down the number line with ease.  Once they feel comfortable with that, parents or others start to show students the traditional algorithm. Students become familiar with the terms carry, borrow, cross-out, add a zero, and others without necessarily knowing the reasoning behind the process.

This past week I introduced the partial-sums algorithm. This is one of my favorites as students start to see the reasoning behind the process that they’ve been following for years.  Although at first it can seem clunky, students started to see how place value can be a key indicator in determining the sum. That’s an #eduwin in my book.

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Partial-sums

Later that hour students were given an assignment where they had to find the sum using the partial-sums method. Some students struggled a bit to move out of the traditional algorithm process and move to a different algorithm. Afterwards I gave student the option to use whatever method best meets their needs. It’ll be interesting to see the progression that students make as they decide on an algorithm. When asked about how and why the algorithm works I’m hoping they’re able to confidentially create their own proof.

We’re making progress!

Learning Number Concepts

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This summer is moving by quickly and thoughts of the upcoming school year are in view.  I’m in the process of preparing materials and the first few lessons.  This year my math sections are larger than before and it changes some of the ways in which I prepare. While researching a few ideas I came across a Tweet from Jamie.

 

I’ve heard of Kathy Richardson and her role in the math education world, but haven’t delved too deep into what she’s written.  This book interested me mainly because of the use of Kathy’s critical learning phases.  As students progress in school they visit different stages of mathematical understanding.  It’s not always linear, but these stages tend to follow each other. I wanted to learn more about this process.

So the book arrived yesterday and I was able reading through the first section. Kathy states that children learn number concepts in different learning phrases.  She coined the phrase critical learning phases to describe the stages that kids pass on their way to making meaning of the math they encounter.  The first section of the book focuses on understanding counting.  On first glance I thought this would be very basic.  Advertised and delivered.  It’s basic, but also intriguing and gave me a few takeaways.  After reading this section I started to draw parallels to how my own students make sense of numbers.  I then wanted to reach out to my colleagues to direct them to this section. Understanding counting, although basic, is a foundational skill that is built upon.  My paraphrased version of Kathy’s learning phases are below.


Counting objects

  • Counts with 1:1 correspondence
  • Knows “how many” after counting
  • Counts out a specific amount
  • Spontaneously adjusts estimates while counting to make a better estimate

Knowing one more/one less

  • Knows one more and one less in a sequence without relying on counting out
  • Notices if counting pattern doesn’t make sense

Counting objects by groups

  • Counts by groups
  • Knows quantity stays same when counted by different sized-groups

Using symbols

  • Uses numerals to describe amount counted.  Connects symbols to amount counted.

 

As I read through this I started looking through my school’s teaching materials for grades K-3.  Some of the materials follow a linear progression while others tend to favor spiraling.  I have a few takeaways after reviewing this first section of the book and looking at my district’s materials.

  • If counting is important than students will start to see why keeping track/organizing numbers is important.  If it doesn’t matter students won’t care whether they come up with a different amount when a specific amount exists.
  • Being able to “spontaneously adjust estimates while counting to make a closer estimate” is a skill that’s found as students progress through the elementary grades. This plays a major role in measurement as students are asked to estimate using non-customary  tools.  I also see a connection to Estimation180 as students might use proportional reasoning after initially counting to find an estimate.
  • Moving from 1:1 correspondence counting to group counting can be confusing for students. Understanding that counting by 2s, 5s, or 10s is a strategy that’s more efficient and that may be learned in time.  Students might not initially come to this conclusion and replace 1:1 with 2:1, 5:1, or 10:1 if they don’t understand the reasoning and if they’re taught in the same period of time. For example, I find this evident when there’s a disconnect between counting by 10s and grouping numbers into 10s and then counting the groups.