Math Explanations

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This year I’ve been working with a group of 20-25 second grade math students.  I see them daily for about 20-30 minutes.  This group has been exploring a number of math concepts related to computation, place value and measurement.  At the beginning of each unit this group is given a pre-assessment and the class investigates a number of concepts for approximately a month.  During that month that class puts together and takes apart numbers, uses math tools, reflects on our math experiences and sharpens  different computation strategies.  The unit concludes with a post-assessment.  The class then reviews the assessment, looks for trends and analyzes possible errors before moving onto the next unit of study.

Every unit assessment has some type of open response which asks students to explain their mathematical thinking.  The performance on these questions has been rough. This isn’t a new experience and I wrote about this a while back here.  Students seem to have trouble creating complete math statements that answer the open response question.  Students have noticed this trend too.  Earlier in the week I ended up having a class discussion about math statements and written responses.  Through our discussion I also realized this is something that I need to learn more about.  This opportunity also had me wondering about past assignments and how often kids are really asked to explain their mathematical writing in written form.

To me, this issue looked like a professional need as well as a student need.  I looked online for additional resources related to helping students identify quality mathematical writing.  I found a few rubrics, but they were very generic and included words like “high-quality” or “fully understands the topic” that I think are valued, but not necessarily quantifiable.  So the class had another brief discussion about what math explanations should look like.  We came up with a list of what should be included:

  • Math vocabulary
  • Restating the question
  • Number models

A draft rubric was built and students completed a pre-assessment using the new expectations.  After writing a mathematical response to an estimation problem earlier in the week, students circled where they felt they were in relation to the expectation.

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I then went through the student responses and added my own thoughts in blue. Next week I’ll be passing these back to my students. This was the first time using this particular rubric and there may be changes to this as the year progresses.  I still need to hone in on helping students recognize what “restating the question” means as I think that’s a bit fuzzy.

The good news is that we’re making progress and students are becoming more aware of their mathematical writing skills.  I’m looking forward to seeing how this evolves over time.  I’ll also be sharing this with my second grade team before winter break.

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Math Visual Model Strategies

There’s power in the picture.  There’s no denying it and I think most people assume this already.  A visual has the potential to bring curiosity and often gives off a certain vibe that brings more questions or leads to someone to swipe to the next picture with disinterest.  Instagram, Pinterest and Snapchat are just a few platforms that have hooked so many people with visuals.  Many people won’t even read an article or blog post if there isn’t a picture included.

Visuals also show up in the classroom.  If you have some time, I highly recommend taking a brief look at YouCubed’s article on visuals in the math classroom. They take the form of classroom posters that are on the walls.  They’re also found in textbooks, on worksheets, on stickers, shirts, shoes, and so many other objects that people see all the time.  I think they also have a role in the math classroom and many teachers and students have used them to make sense of the mathematics that they’re exploring.  Visual models are often used to organize thoughts, make connections, and communicate understanding.  Individual meaning is often associated (although not always) with visual models and educators get to see how how the decontextualizing <–> contextualizing process works with these types of models.

I believe visual models have an important role in math classrooms.  Some students make meaningful connections by creating a map of their mathematical thinking.  Often, these maps are related to what’s introduced in class, other times maybe not.   Students might adopt certain visual models because that’s how they’re told to complete certain problems.  I see this regularly when it comes to fraction models.  The rectangle and pie area models are rampant.  This sin’t a gripe, but just an observation as the visual model might be more emphasized depending on the teacher/grade level.

I believe there’s potential with using visual models, but they’re not always the default. Communicating and having the students bring their own awareness to when/why certain models are used is important.  Rate/ratio tables and tape diagrams lend themselves well in having students organize their thoughts.  When asked to create a visual model I sometimes hear students say that they’re visualizing it in their head and then they write down the solution on paper.  Other times, students thoroughly create large models and use them to find solutions and add more than enough details.

I’m looking forward to seeing how students approach this problem tomorrow.

                                                                            What is 45% more?

Students are familiar with finding 50% or 10% of a number, but this is different. I’m wondering if using a visual model will be part of their problem solving process?   

Better Math Explanations

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One of my primary classes just finished up a math pre-assessment.  One of the questions on this assessment asked students to explain their mathematical reasoning.  Specifically, they were given a prompt, a student example, and then asked to explain in their own words what happened.  Students had a lot of questions about this problem.  Since it was a pre-assessment, I basically kept quiet and asked them to persevere.  Some did, others didn’t.

A few students dropped their faces when they saw their pre-assessment results. Many, and I mean over half of the kids didn’t meet the expectations on the written response.  Instead of putting together sentences, the majority of students created number models and that was that.  Some students even wrote that the character was wrong and didn’t explain anything further.  I was a bit disappointed, but no worries though – this is a pre-assessment.  The actual assessment won’t happen for another couple weeks.

I noticed that I needed to look more closely at how to address the math writing issue.  I also needed to clarify the expectations for written responses.  This was new territory for kids.  Most students are able to tell me (with prodding) their thinking and how it relates to the problem solving process.  It’s a different story when it comes to writing it down.  In the next few week I want to ensure that students are explaining their mathematical thinking clearly and in a way that answers the question.

So this Wednesday students were asked to start looking for specific details in their writing.  We began by having the entire class analyze one math response from another “student” from a couple years ago (ok … maybe I created this).  Students went into teams and analyzed the exemplar and looked for key components in the response.  Students looked for an answer statement, math vocabulary and important numbers.  They then coded the response with circles, rectangles and underlining.  The student teams explained to the class what they thought qualified as an answer statement.  This was a great discussion as students came to a consensus to what qualifies as a statement that answers the question.  Students also discussed the numbers that were important and the math vocabulary that was used.

Later in the day students answered a similar prompt and then switched papers with a peer.  The other student coded the paper and then the pairs discussed what they wrote and why.

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The class will be meeting next week to review more examples.  Afterwards we plan on responding to a different math prompt and code our own writing.  I’m looking forward to seeing how this emphasis on mathematical writing transfers throughout the year.

 

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.

Exploring Fractions – Week One

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My third grade students started a new unit on fractions last week.  They started the unit by learning about part-to-whole fractions and how to identify them.  Student teams explored how fractions are represented in different situations.  One of the first activities asked students to create their own version of a part-to-whole model.  The scissors came out early this week while students cut out fraction area models.

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They compared the pieces within the area models to create equivalent fractions.  This gave students another way to compare and observe equivalencies.  This was time consuming activity, but so worth the time.  Students made connections and played around with the circles/pieces to compare the models.  The only negative was that some students didn’t cut the fractions exactly on the line so the pieces didn’t always line up.  The next day students compared the fraction pieces using <, >, or = signs.

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Students were making progress in comparing fraction area models.  The fraction circles were being used consistently and then the class moved to transitioning to identifying and placing fractions on number lines.  This was a challenge.  We started with a 0-1 line and then identified half.  From there students used benchmarks to compare fractions on line.  Students had some trouble when the number line was stretched from 0-2.  The class also explored how the fractions look on a vertical number line.  A different dynamic was at play there.  Students then practiced a bit more with an Open Middle activity.

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Students were given opportunities to discuss fractions with their peers through a few different fraction math talks earlier in the week.  The time spent today revolved around reviewing different fraction models. On Friday, the class participated in a fraction Desmos Polygraph activity.  Feel free to use the program here.  This was one of the learning highlights of the week.  Students were asked to pick one of the fraction models that they created early in the week, while other students asked questions to help determine the fraction.

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At first students asked questions related to the color of the fraction.  Then they moved to questions involving less than half and more than half.  Students found that clear questions revealed better answers.

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I was excited to see that students were using benchmarks and part-to-whole ideas to help uncover the mystery pick.  Students spent around 20-30 minutes exploring the polygraph with a few different partners.   I even snuck in as a participant.  I’ll be keeping this idea in my back pocket for next year’s plans.

Next week, students will start to add and subtraction fractions.  I’m looking forward to seeing how students will use the experiences this week.  There’s plenty more to this unit and we’re just getting started.

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.

Second Attempts and Error Analysis

 

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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.

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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.

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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.

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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.