Student Math Writing

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Last Tuesday I was able to participate in the ICTM chat.  The chat was on the topic of math writing in the classroom.  My teaching journey began as general fourth grade teacher and writing was often placed squarely in the writing/language arts block.  This year I’m trying to find ways encourage students to strengthen their writing in the math classroom.

When asked to explain their writing I find students at the upper elementary level sometimes struggle to find the right words.  Some students have the perspective that writing should be minimal to non-existent in the math classroom.  Mathematical writing at the 3-5th level is often expected on state and district assessments.  Beyond that it’s a bit inconsistent – depending on the teacher.  Generally the math writing prompts ask students to explain why or the steps involved in solving a problem. When students approach these problems I find that they’re sometimes unsure of where to start or are very brief with their submission – often too brief.

Over the past few years I’ve been working for ways to chip away at this issue.  I think part of the concern is due to exposure and practice. I believe students have opportunities on a daily basis to explain their math reasoning.   Quality math tasks can give students opportunities to engage in explaining their math reasoning. I see it more so in classrooms through verbal interactions – not so much in the written realm. Students may present their ideas to the class and/or engage in a dialogue with others about why their answer makes sense to them.  I also see students participate in these types of sessions with partners or a small group.  I find that students can take those meaningful dialogue experiences, but lose some of the substance as teachers attempt to connect the transfer to written form.  Bottom line, I feel like students need more meaningful math experiences with writing and revisions in a math class setting.

How do students get there?  The article discussed in the chat covered four different types of math writing:

  • Exploratory Writing 
  • Informative/Explanatory 
  • Argumentative Writing
  • Mathematically Creative Writing

I find students aren’t expected to write unless they have a prompt to answer – which generally falls into informative or argumentative.  For some students this is the only practice they get in strengthening their mathematical writing skills.

Part of the questions discussed in the ICTM chat revolved around the different types of math writing that educators currently see in the classroom. Organizing this math writing (along with a criteria) into the four categories was fairly new to me and I dug deep into trying to find ways to apply this in my own classroom.  Moving forward I plan on using my student math reflection journals with more frequency. Right now students use them to reflect on their math unit assessments, set goals and progress monitor.  They’re revisited every 1 – 2 months.  I find that’s a valuable use of time, but I’d like to expand and have students more frequently reflect on weekly topics or the skills that are highlighted in class.  I’m also planning on looking at more of the creative math writing component.  I find that interesting and there’s a self-motivation piece that could be helpful.  Planning on using this will take some time, but again, I feel like students will benefit from analyzing the skills discussed in class and applying them to their life.

I think the more practice that students have with writing in math class the better prepared they’ll be to explain their mathematical thinking in written form.  I’m looking forward to seeing how this idea turns out.

 

Takeaways from IAGC 2019

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Last week I had the opportunity to attend the  IAGC conference.  I don’t attend every year, but this year I went with a colleague because we both were presenting.  For those that might not be aware, this conference is really geared towards gifted education in the state of Illinois.  Administrators and teachers primarily attend this conference.  It’s a multi-day conference that’s hosted in a Marriot in Naperville.  I believe last year there was around 500 people (I’m estimating) attend – so it’s more of a smallish conference compared to some of the other education powerhouses in the the state.  The freezing temperatures impacted attendance this year as driving conditions probably kept some away.

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Most of the attendees that I saw last week are teachers that work with students with exceptional needs.  Many of the teachers created sub plans and then school ended up being canceled because of the cold weather.  Go figure.  Whenever I’m asked to be out of the classroom for a meeting, conference, workshop, etc … I always am wondering if it’s truly worth the sub plans – kinda joking here.

What I enjoyed more particularly about this conference was that some of the presenters were teachers.  Certainly not all, but there were a few sessions that I attended that involved teachers presenting to an audience. Some of the themes were related to acceleration, leadership, curriculum, social-emotional and equity with identification.  I attended on Thursday and was able to attend a couple sessions – mostly related to curriculum and differentiation.

There’s a practicality component that is involved when a teacher is having a conversation with another educator.  Most of the sessions were small enough that you could engage with the presenter by just raising your voice. Love that.  I find it similar to an edcamp experience as individuals have a similar understanding of what’s expected in schools.  There’s a large difference (at least from what I see) between teachers currently in the classroom and those that aren’t.

Now onto the sessions … I attended a session related to using adapted software in the classroom.  Hearing positive ideas about how to use these in the classroom was different since I’m usually not too fond of this type of learning.  The presenter gave a brief overview of a couple paid/free adaptive programs and spent a good amount of time answering questions about how to use them in the classroom. Another session emphasized the use of Jacob’s Ladder scaffolding techniques.  The questions from the participants spurred additional questions – love when that happens.  Later in the day Cheryl and I presented on using math routines in the classroom.  Participants that attended offered their own ideas on routines that they found helpful.

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It was truly an engaging experience to hear how others use and differentiate math routines.  It was also refreshing to connect with other educators outside of our local area about how to create more meaningful math experiences for students.  I think teachers need to explore outside of their district boundaries from time to time.  Discussing some of the routines afterwards was also encouraging.

All in all, it was decent conference and worth the sub plans.  I left with a better understanding of how certain tools and strategies can help meet the needs of exceptional students.  I’m looking forward to the conference next year.

 

Decimals and Spatial Reasoning

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My second grade students just started a unit on decimals.  Based on the pre-assessment, most students have no problem with identifying the value and place value position of digits in the ones – hundred-thousands place.  It’s a different story for numbers to the left of the decimal point.

Earlier in the week students explored the tenths and hundredths place.  Students connected money concepts to place value and fractions.  They compared 1/2 with 0.50 and $1.50 with 1.50.    They completed similar activities where they needed create benchmarks on number lines and place numbers.  Some were still having trouble and I believe this is partially due to exposure.  Also, I was finding that their were issues with spatial awareness.  Students were looking placing able to approximate benchmarks of half, but placing 0.1 close to the half.  Student practiced using number lines and using benchmarks.  The most tricky piece was looking at the differences between the hundredths and thousandths.  This challenge reminded me of how students develop an understanding of the magnitude of numbers.

Today I grouped students into teams and they used dice to create different decimals.  The decimals ranged between 0 and 3.  Students were given a horizontal and vertical number line on a 11 by 17 paper.  This gave students room to work.  The two number lines were different sizes.  An indicator line was placed at the beginning and end of each line.

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After students created their decimals they started to place benchmarks.  Some students had to get out the erasers as realization set in that the maximum would be three instead of two.  Students also reevaluated their benchmark placement.  Groups noticed that the two number lines were different sizes and had to adjust their benchmarks accordingly.  I found it interesting that some students used the vertical number line top down, while other went bottom to top.

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We’ll be reviewing the number lines on Monday.  I’m looking forward to the discussion and we might even break out the rulers to evaluate the reasonableness between benchmarks.

More on Math Feedback

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Last night I was fortunate to attend ICTM’s chat on feedback.  It was a productive chat and Anne had some great questions cued up for us.  I came away with a few new tools that I need to research.  Chats like these are motivating as the frigid cold of the midwest is ever-present this time of the year and new ideas can spark my planning process.

Teachers know that student feedback is important – it’s everywhere in schools.  It’s on every teacher evaluation tool that I’ve experienced. ASCD describes it as “Basically, feedback is information about how we are doing in our efforts to reach a goal.”  Teachers give feedback all the time – most without even labeling it specifically as feedback.

The chat was still on my mind this morning as my colleagues and I were having a conversation about math units.  After reviewing multiple student papers, I started thinking about feedback in more detail.  Specifically, I started thinking about how feedback takes on different forms and the tools that are used to give that feedback can vary from class to class.  In all cases that I’ve come across, educators want students to actually USE the feedback.

Technology can be used for this although the reliability of the feedback might not match the need.  I’ve also seen cases where the automated feedback is disregarded by students in an effort to score more points.  It depends on what’s needed.  In some cases, a quick verbal prompt might be the feedback that’s needed.  For others, a conversation with a partner can help students identify misconceptions or spur thought.

Let’s take this problem:

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This particular students was able to identify the rule and complete everything but the bottom problem.  Being able to anticipate misconceptions can lead to better student feedback. There are a few questions that I might have before approaching the student and giving feedback.

  • How can this student divide 14 by 7, but still have trouble with the bottom problem?
  • Does this student think of “divide by two” as half of the in?
  • Was this a simple mistake?

Or here’s another one:

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  • I notice that there isn’t any work or model here
  • Did the student notice that the denominators weren’t equal?
  • What strategy was used here?
  • Was this a simple mistake?

Last one:

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  • Did the student miscount the boxes?
  • Is the students missing pieces? (yes, this has happened before)
  • How did the student get 6 as the numerator?

In all of these cases a simple mistake is probable.  I’m working with K-6th grade math this year and sometimes rushing leads to simple mistakes.  I try (as much as I can) to limit that option when deciding to give feedback.  In all three of the cases I could ask the student to recheck their work.  Some students will, while others won’t.  I could also write on their paper a statement or question about wondering what strategy they used.  I could also have the students meet with a peer and discuss the problem in more detail.

There are so many ways to communicate feedback and it’s not a simple issue.  Some students are more responsive to written feedback, while other students want to have a conversation with another peer to discuss their strategies.  As students get older the type of feedback also changes.  Many of my upper elementary students prefer a brief comment on a paper or a quick underline, question mark, or specific arrow to help them move towards a goal.  Having a 1:1 feedback conversation with a student is my number one option because then I can see how receptive they are and answer any follow-up questions.  If you don’t have time to do that with every kid (who does?) then you use other options.

There are a ton (I mean a TON) of apps out there that “help” students along their math learning journey. I tend to be a bit caution when deciding to use them in the classroom. Is the feedback appropriate for their needs?  Is the feedback helping them in their efforts to reach a goal?  In some cases it may, but I think it’s worthwhile consider the ways in which feedback is given.

We’re Back!

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School is offically back in session and 2019 is here.  Teachers and students are getting used to the routine again as my school approaches our third day back tomorrow.  It’s been quite a week already and the adjustment continues. Yesterday, my classes spent a decent amount of time discussing what happened over break and previewing what we’ll be learning about over the next few months.  I taught six classes on Monday and each class took some time to debrief, briefly review the expectations/routines and lay out a plan on what needs to be accomplished before the end of January.

There’s definitely a layover impact when students (and teachers) have long breaks away from school.  Students came in on Monday and were excited to see their friends, but also need time to chat about their experiences and reconnect withe the community.  That time is so important.  This isn’t news to any educator, but relevant – especially when students get back from a long break.  Similar to the first day of school, that first day back from winter break takes on a community building component.  That takes on a different form depending on the class.  I’m sure this looks a bit different and the middle and high school levels.  This year I had the students tell the class “one positive experience” that they had over break.   I made sure to put a disclaimer out that this shouldn’t be an opportunity to brag and that they could pass if they want.   It took around 10-15 minutes to hear about the experiences for each class.  I was silently proud as I listened to some of the students talk about seeing relatives, reading books, going to movies, eating holiday meals, and waking up without an alarm (that was mine too) types of experiences.

I know some teachers want to dive right into content and feel as though students will acclimate accordingly.  I’m minimally there when it comes to that perspective, but slowly moving outside of that zone.  It’s only taken a decade or so of teaching.  Haha.  I appreciate having a certain amount of structure in place.  There’s part of me that really wants to get back into the routine as soon as possible.  State testing, curriculum guides, formal evaluations, and other factors all play a role and can limit this time to reconnect.  I’m glad that I spent the extra minutes to reconnect.  I won’t be able to get that time back, but students will remember it.  I think it’s worthwhile to continue to share and discuss our experiences and look forward to our shared learning moving forward.  I believe it’s worth considering taking a look at how teachers structure their lessons after long breaks.


Side note:  One of my “experiences” over break involved organizing a site related to percent images.  I’m hoping to add to the site this year and give possible prompt options.

Evaluating Summative Assessments

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This past week has been a great opportunity to reflect and plan out the new year.  I’m writing this as I sip my coffee on a chilly day in the midwest.  As I contemplate my plans I’m reminded of always to “keep the end in mind” as I think of how instruction eventually leads to a summative assessment.  I usually see the following procedure used for math instruction:

Introduction–> Formative Assessment –> Reflection –> Summative –> Reporting

Summative assessments have changed over the years.  One major factor that determines how these look at the K-5 level is based on the district resource that’s used.  There are so many math publishing companies that push their resources to districts and the unit tests generally accompany them.   There are formative assessments that teachers use, but they’re often used depending on teacher preference. Grade level teams often plan out the formative assessments, although it’s up to the individual teacher to use them.  The summative assessments are another story.  These are set in stone and given at specific intervals throughout the school year.

The summative assessments in some K-5 classes take a long time to complete.  The upper elementary classes take a lot more time than the lower.  The accelerated classes can take 1-3 instructional days to complete one summative assessment. Usually there’s another day reviewing the assessment afterwards. The unit assessments often include written component and an open response section.  Each math class is allocated around 60 minutes for math and students are given (not always though) as much time as they need to complete the assessment.  These summative assessments are solely used to communicate how well students are performing compared to the standards and that’s put on the report cards.  As schools move more towards standards-based grading the reliance of the summative assessments may increase. Middle and high school might manage this time differently as final exams and tests become more of a priority.  I wonder how this summative assessment time could be better managed moving forward.

As far as I can see, there a couple options to consider.

  • Prioritize the standards within a math unit and adjust the summative
  • Rely more on class quizzes and emphasize the summative less
  • Create new summative assessments that better reflect the standards
  • Continue without changing anything

There are consequences depending on what’s picked and I’m sure other options exist that I can’t think of right now.  Prioritizing standards will inevitably leave out some standards and students won’t be exposed to certain skills/concepts as frequently.  Relying on quizzes for grades has me wondering how many attempts are given before students are actually assessed.  This can be a challenge as some students might be exposed to a concept three times, while another might have seven.  Is one student better prepared for the summative than the other?  Creating new summative assessments might be a good option if time is available to do so.  I’ve heard multiple times that teachers aren’t trained to write assessments, but I find that there input to be very valuable in the process.

I don’t believe there’s one right solution to this issue, but it’s worth contemplating a few different ideas.  I’m interested in hearing how others manage the summative assessment process.

Continuing the Math Writing Process

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As some of you might already know, one of my goals this year is to find strategies to help my students write better mathematical explanations. My students have been making progress towards that goal.  I see glimpses here and there where students are putting together more concrete statements and transitioning them into coherent explanations.  It’s good news – and making progress helps the students see that growth is happening.  We all need that boost every once in a while – or even more than that. Last week, students were asked to fill out a rubric and evaluate their own mathematical writing.  I then went over and highlighted my responses in a different color.

I passed those papers back early last week and the class had a conversation about the difference between my scores and theirs.  It was a productive conversation and I believe the kids left with a better understanding of what the categories in the rubric mean.  For the next couple days the class reviewed measurement concepts and place value.

The next stage of this mathematical writing process was for students to evaluate the writing of another student.  I’m finding myself using more retrieval practice strategies for this particular process and critiquing others writing multiple times has helped (at least I believe) them become better at recognizing rubric elements in their own writing.  I also want to give students multiples opportunities that are spread out with assessing and self-assessing strategies.  I tried this before earlier with some success, but this was also before students created the writing rubric that we’re using now.  Students read the prompt on magnitude estimates, read over what the writer created and filled out the rubric.

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The students carefully read through the prompt and then throughly read through the response.  During this time the classroom was so quiet.  Some students used highlighters while others were very critical with the pencil.  I even had a few students ask the writer about their response on the paper – future teacher maybe?

Near the end of the class I reviewed the responses and the students voted on where they thought this writing would fit on the rubric.  I’d say around 50-60% of the students were on target with all the rubric selections.  This improvement is telling and I’m excited to see growth in this area.

Next week I’m planning on introducing a different math writing prompt.  This will be their second attempt and will be used as a formative checkpoint.  The class will then continue this journey after winter break.

  • Shoutout to the MAA site and people for helping me thinking of additional ideas for this math writing process