Being Mindful of Math Manipulatives

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I’ve been thinking about this blog post for a while, but haven’t had a chance to type it up.  It’s been sitting in the forefront of my mind and I haven’t fully made up my mind about it.  So, what you’ll find below is a free-flow thought of ideas related to math manipulatives and students’ K-5 math experience.

In my current position I get to work with kids in grades K-5.  I’m able to see certain grade levels longer than others, but one my goal is to provide students with meaningful math experiences.  I use district’s adopted-texts and other resources throughout the day. Working with students K-5 is unique and gives me a different perspective as I notice how instructional strategies and resources evolve as students progress from one grade level to the next.

One thing that I notice in my current position is that math manipulatives are used in every grade level.  I believe that’s a good thing.  If you prescribe to the CRA model, then this tends to make sense.  The amount of math manipulative use significantly decreases as students progress through the grades.  There are a lot of factors to consider as in why this happens. At the kindergarten level I find that students use manipulatives (mainly counters) for a decent amount of time.  If I had to pinpoint it, I’d say these types of physical representations are being used in the classroom just bout everyday.  Counters, ten-frames, pattern blocks, dice, and other tools are used in math stations and small groups throughout the day.  What’s great is that students know that these are math tools and they’re given time to explore how they connect to math concepts.  Students are given time to “play” and make connections. Worksheets are also evident, but are generally used as students circle or cross out counter-like representations.  The worksheets are generally not more than one page, front and back.  The staple rarely comes out for worksheets at the kindergarten level.

The first and second grade math manipulatives are still used, but not as frequently as in kindergarten.  There seems to be more of a focus on using workbooks and paper-based assessments.  Part of this is systematic as more academic student data collection is used at this level.  Unifix cubes, clocks, 100 charts, base-ten charts, coins, and fraction pieces are all used with this group.  Counters are still part of this as students group numbers together more fluently.  Students take-away and add-on as needed.  Odd and even are emphasized.  There’s also a larger emphasis on data and charts.  The consumable math journal is used daily at this level.  The rise of the consumable math journal sometimes takes time away from using physical math manipulatives. This is especially evident when grade-level teams are asked to stick close together when completing lessons and assessments.  That means that teachers need to ensure that they’re at the same pace as their colleagues.  The assessments need to also be given around the same time.  Sometimes the assessments that are used are multiple pages, so the stapler definitely comes out for these grade levels – more so at the second grade.

For third and fourth grade, students start to move away from a worksheet counter to more abstract-like representations.   Multiplication and division facts tend to move from arrays with counters, to the horizontal and vertical representations that are often associated with timed-tests. Multi-digit multiplication involves using the standard-agorithm at the fourth grade level. Polygon blocks, card sets, base-ten blocks, place value charts, square counters associated with perimeter/area, and fractions are all used.  There’s a heavier emphasis on transferring students’ understanding of a representation on paper to abstract text.  Similar to first and second, assessments are all paper-based, although students are required show a visual representation to communicate their math reasoning.  Many more word problems are involved at this stage.  Teachers often have manipulatives on hand if students need to use them.  I also find that mini whiteboards are a precocious commodity at this level.  From what I see, students enjoy creating the models on the boards and then transfer their answers/work to a paper-based assignment.  The stapler is definitely used as this level.  Sometimes the assessments are 3-5 pages long and require a heavy dose of time to complete.  Grades are also emphasized at this level, which brings in a heavier focus on assessment points and growth indicators.

The fifth grade level includes a large amount of math manipulatives related to fractions. Fraction computation is heavily emphasized.  Base-ten blocks are also used for decimal concepts.  Counters are brought out to discuss proportional reasoning.  Similar to third and fourth grade, students are expected to explain their mathematical reasoning with visual models and in written form.  Sometimes I find that students work in groups together and  report out answers to open-ended tasks.  These tasks involve multiple answers with an emphasis on explaining their math reasoning.  I find that this level has more problems involving abstract problems more than any other.  Students complete most of their work in a consumable journal. The journals have increased in size since third and fourth. Math manipulates are often readily available, but they tend to be used with students that are struggling with current math concepts.  Assessments are all paper-based and are multiple pages.  As students prepare for middle school, some teachers introduce students to the idea of equations.  Mobiles and Hands-on-Equations manipulatives are sometimes used in those situations.

 

The above is not an all-exhausting list and include my observations.  As I write this, I’m also remembering that I forgot to include the use of number lines.  Number lines are heavily emphasized throughout K-5.  They are found in all of the consumable math journals.  Students are also expected to include number models at every grade level.

I forgot to include the role of technology with math manipulatives.  I’ve seen and used technology versions of math manipuatlives at all of the levels indicated in this post.  A digital math representation can be used as a powerful tool.

As I finish up this post, I’d like to bring up one issue that I’m continuing to observe. Across the board, I’m a bit concerned with the reluctancy to move out of the consumable math journal from time to time.  An over-reliance on using a consumable math journal isn’t the only options when it comes to engaging students in powerful math learning experiences.  I’ve always thought that math manipulates are put away too quickly.  I think they have a role at every grade level, but in an attempt to appease systematic policies, they’re occasionally sidelined and consumable journals take their place.  In my ideal world, I’d have every elementary teacher observe how math manipulatives are used in kindergarten and first grade classrooms.   I think it would give teachers a different perspective on the use and purpose of math manipulatives.

That’s just my two or three cents.

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Fraction Blocks and Strategies

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My second and third graders started a unit on fractions last week.  Students are used to identifying typical pie fraction pieces.  Generally, I find students are introduced to fractions using this type of visual representation.  Students then count the amount of pieces and place that number as the numerator.  I find moving towards mixed-numbers has some students changing their strategy as they can’t just count the pieces, but they have to recognize that a certain amount of equal parts are one whole.  Based on their pre-assessment results, it seems as though my second grade and some of my third grade students are at this point.

Using a number line has helped.  Placing the fractions on the line has brought a better understanding of the placement of fractions in relation to a whole number.  Currently, students can identify certain benchmark fractions on a number line.  We’re working on bolstering this skill and connecting it to fraction computation in the near future.  Before that happens I want to ensure that they have a decent understanding of mixed numbers and where they fall on a number line.

On Thursday and Friday I introduced students to a fraction block activity.  Students were given a sheet with fraction parts.  Each block was split into a certain amount of equal square parts.

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Each student was given an envelope to put their pieces in once they were finished with the activity.  Students cut out each block and were asked to put them in order from least to greatest value.  Students were able to complete the task.

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We then had a conversation about quarters, halves and wholes.  I then gave each student the card below.

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Students placed the A block near the top of their desk and started comparing the different blocks.  The class completed the first question together.

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I then gave students time to work on the rest of the problems.  Students were then given time to use trial-and-error to find which blocks worked for each problem.  I went around to the different table groups and asked students questions about their strategies. Students ended up matching the squares with other shapes to determine what was a quarter, half, almost a half, and what happens when you combine shapes.  After about 10 minutes the class reviewed the sheet and found that some problems could be answered with multiple solutions.  Students put the sheets in their envelopes since we ran out of time.

The next day students completed some more challenging half-sheets involving their blocks.

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Students struggled a bit with this as they had to look at A as half instead of one whole.  This changed the value of all of the other blocks.  I allowed students to work in groups for about five minutes and then independently for another five.  This gave them an opportunity to gain another perspective and a different strategy.  Afterwards, I reviewed the possible solutions with the class.

Next week I’m taking this activity one step further and using the blocks without markings.  I’m borrowing this idea from Graham’s post on defacing manipulatives.

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Students will complete similar half-sheets, but without the evident markings. I’m looking forward to seeing how students’ strategies change and the math conversations that follow next week.   Click here to download the activity that I used.