Measurement and Reasonable Solutions

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My fourth grade students finished up a project involving area last week.  Students were asked to find the area of different playing areas for certain sports.  They first calculated the areas of the playing field by multiplying fractions and then found the product.

The next step involved creating a visual model on anchor chart paper.  Students worked in groups to put together their athletic park involving the field areas.  They were given the area of the park and then had to place the fields where they wanted according to the team’s decision.  Students also added additional facilities for their athletic field and then presented their projects to the class.

While presenting, students in the audience were required to either 1) ask a question or 2) provide a constructive comment.  Most of the questions that were asked related to why certain fields were placed in specific areas on the field.  One question stood out more than the others … does the distance make sense?

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The locker rooms had to be adjusted (see whiteout) as one student said, “It doesn’t make sense so I changed it to match the 120 yd.”

 

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Students were looking at the length of the fields and observing whether it was reasonable or not compared to the total length.  The class then had a conversation about the terms reasonableness and proportions.  The discussion involved how a double-number line and a grid could’ve helped visualize how the distances match.

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I’m hoping to revisit this idea during the next few weeks as the school year finishes up.

 

Reflections and Math Routines

 

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This year I’ve been using number sense routines* with my 3rd-5th grade classes.  The routines have specifically been put into place to help students strengthen their place value and estimation skills.  The routines last around 5-10 minutes and generally occur during the first part of class  The routines is the first thing on the board as students enter.  Students use a template, complete the routine independently and we discuss the results and process as a class.

Two of the more productive routines this year have been Estimation180 (3rd grade) and Who am I (4th).  Both ask students to use hints or models and then use those visualizations to solve problems.  Students document their thinking on an individual page and then we discuss it as a class through a debrief session.  While working with students this year I noticed that not all students participated to the extend that I’d like.  The conversations were decent and students were engaged, but the reflection piece wasn’t as thorough.  So this year I’ve decided to add an individual reflection component for these specific tasks.  The reasoning actually came from a book that I read back in April that emphasized how sentence stems can be used to help students reflect on their mathematical thinking.

 

I put these sentence stems into practice and added them to a reflection sheet.  I added extra space after the “because” to help encourage students to write more about their own thinking process.

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Students complete each one of these around 2-3 times a month.  Students complete the reflection sheet, discuss the writing with partners and eventually put them in their folders.  The sheets are revisited throughout the year to see the growth over time.


* The images from this post are from a math routines presentation on 5/3. Feel free to check out the entire presentation here.

 

 

 

Spaced Retrieval Practice

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There’s around one month of school left and it feels like the home stretch.  The next month is full of changes.  The weather changes from chilly temps to sunny days (at least in the Chicago burbs), class lists and sections are starting to take form, driving to/from school with the windows down is the norm, and planning for that final month is in full swing.  The majority of my math classes just finished a unit assessment and there’s one unit remaining.  So often I find that students perceive the end of a math unit to “close out” the learning on a particular skill set.  I observe that this idea often gets pushed out as grade deadlines approach.

As my classes start a new unit I’m pausing to reflect on how my practice has changed.  Last year I read How to Make it Stick and I intentionally planned to use more retrieval practices. This year I’ve incorporated more review opportunities through online formative quizzes and by trying to make implicit connections to past learning.  I’ve often asked students how today’s objective connects to this week’s learning.

While digging through my resource materials early this year I found optional mid-year and cumulative assessments. Generally, I find that there’s not enough time to complete all of the assignments/tasks in the resource so these particular tests aren’t used frequently.  This year I decided to use them to help with spaced retrieval practice.  Instead of using a mid-year and cumulative assessment directly following a unit I decided to space out these assignments and take off the grading emphasis.  These types of assignments take multiple days to complete and I often have students work with partners to reflect on their progress.  So far I’ve seen positive progress as students this year are referring back to past skills more quickly and bridging the connections on a frequent basis.  I’m looking forward to using a similar strategy next year.

Classifying Polygons

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One of my classes is in the middle of a unit on geometry and measurement.  They’ve identified shapes before, such as rectangles, squares, triangles and hexagons.  Earlier in the year they found the area and volume of shapes involving rectangles, squares and triangles.  The current unit investigates how polygons (specifically triangles and quadrilaterals) are similar and the study of shapes progress as students create hierarchies.

  • CCSS.MATH.CONTENT.5.G.B.3
    Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.

 

In order to dig deeper into the above standards the students starts the classification process.  This was fairly new for most of the students.  I explained what classification meant and gave a few examples related to the characteristics of triangles and quadrilaterals.  Students were given a sheet of quadrilaterals to cut out and classify.  The next question I was asked was related to how each shape should be categorized.  The class reviewed different vocabulary words associated with polygons and then I left the students create their own categories.

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This student decided to split up the shapes into three categories.  3-sides, 4-sides and 4+.

After discussing equal side lengths and parallel sides two of my students created the classifications related to those terms.

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Almost every student had a different way to organize their shapes.  Students went to different tables and observed how their peers classified the shapes and then the class discussed similarities.  Next week students will classify the shapes with a hierarchy chart.  I’m looking forward to seeing what they create.

Stretching in Math Class

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This post has been marinating for a while and I’ve been waiting to write it up.   State testing is just around the corner and I feel like this is a good time to press send.

One huge emphasis that I see in schools is related to the idea of student growth.  This is communicated in schools, during teacher pd meetings, when talking about Hattie’s next best effect-size list and can even be part of teacher evaluation criteria.  I see this when school districts use MAP, state testing, or a similar type of tool that measures growth over time.

Schools, teachers, and parents want students to grow.  In schools the focus of growth primarily related to academic content.  How is that measured?  Well that depends on the teacher/district/organization.  Some teachers go the route of using a pre vs. post-test.  Others give multiple formative checkpoints and then use them along with student reflection components to show growth on the summative.  Case in point – there are multiple methods to show student learning and growth.

Here’s my not so small gripe.  In an effort to show growth some educators may feel rushed to “get through” as much content as possible.  I hear this a lot more in math classes than other content areas at the elementary level.  Math has a subjective linear vibe that I think some teachers hold onto. This idea is often reinforced through the structure of some of the adaptive standardized tests that communicate math growth to teachers and school administrators.  This can be a bit troublesome if these types of tests are used for evaluation purposes as it brings along additional pressure.

I find students make meaningful math connections when they’re given time to process and apply information.  I believe rushing through concepts or stretching to just expose students to higher-level concepts that aren’t part of the lesson isn’t as beneficial as it seems.  Moving off the pacing guide or lesson to stretch to other concepts might not be the best idea. When I first started teaching I remember having a teacher talk about exposure all the time.  The teacher would say, “If they’re just exposed… then they’ll complete those problems correct and that’ll push them to the next concept.”  This teacher was truly amazing (and made some great coffee in the mornings), but I questioned this then and still do now.  If we’re exposing students to math concepts so they’ll score well on an adaptive test then that’s another issue altogether.

If a lesson is looked through a linear math lens a teacher might feel as though they should introduce fraction multiplication if students are doing really well with multiplying whole numbers.  Is that the right move?  Should a teacher stray from the lesson plan to possibly reach a few kids that seem like they’re ready?  I’m not saying yes or no because it depends on the station and students, but I’m more in the no camp.  Stretching math concepts in a lesson/task for exposure sake doesn’t last.

Last summer I was able to reading Making It Stick and came away with some applicable ideas related to changing my study guides and how retrieval practice benefits those wanting to learn.  I find that there’s sometimes pressure to stretch to another concept for exposure sake.  Instead of stretching concepts, there should be opportunities for students to enrich their understanding through connections.  This looks different and is more challenging in my opinion than just pressing the accelerate button temporarily.  I believe that taking time for students to process, reflect and engage in meaningful math tasks will last more than a glimpse optimistic exposure that may soon be forgotten.

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.