Estimating in the Elementary Classroom

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Using Estimation180

My school finished its ninth day of school yesterday. It’s been a journey as students are understanding class routines better.  At this point in the year, students and teachers are starting to become more solid in their processes.  Many of my students arrive to class at different times. Some students are at an elective or leave class a bit earlier/later than the rest of their peers. Regardless of the arrival time, when students enter the room they follow a flow chart. Students have their own folder and materials inside that are ready to go. I usually have some type of visual brainteaser for the week and a grade specific Scholastic math magazine. In the past I’ve used different types of math warm-up activities to start class.

This year I adapted my warm-up strategy. I wanted to individualize the type of responses within that warm-up time slot. After researching a few different tools, I decided to try Andrew Stadel’s Estimation180 this year. I think of Estimation180 as an opportunity for students to develop a stronger sense of numbers and practice estimation skills in the process. Initially, I thought that the site would be great for middle or high school students. I then found the below sheet and site that seemed helpful. This is one way in which student can document their thinking.  The template also includes lessons that could link to Fawn’s Visual Patterns site.

Click to download template

This template inspired me to adapt the sheet to fit an elementary classroom. I changed the template a bit to work with a third grade math class.  A few colleagues and I will be using this sheet early next week.

So now, students enter the classroom, pick up their folder and begin to work on their daily estimation challenge warm-up  sheet.  The estimation is displayed on the whiteboard. Students pick a high, low and exact estimate. I ask the students to prepare to tell me about the reasoning that they used to come to the concluding estimate. The class then completes the online portion of the site and submits a response. We then look at other responses and reasoning.

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After a brief discussion the result is revealed. Students write in the correct numbers and find the + / – . The entire activity takes about 5 – 10 minutes.

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I’m planning on using Estimation180 a few days a week and incorporate Visual Patterns for the rest of the days. The template also includes a few different reflection pieces.  I feel like these activities provide students opportunities to produce a product and reflect on the results. At some point I’d like to add a journaling component to encourage more reflection and possible goal setting.

Thoughts on Questioning Techniques in the Classroom

photo credit: mag3737 via photopin cc

photo credit: mag3737  cc

Every year I find that pairing the right math activity while asking specific questions can yield some amazing student learning experiences. I would assume that most math teachers would agree that only giving a specific solution to a student doesn’t necessarily help them understanding concepts. Offering solutions without feedback or questions can encourage students to care only about finding the answer. The act of “answer finding” limits understanding and diminishes curiosity.

When I started teaching I spoke constantly. I would give examples and statements that I thought would help all my students. Looking back, I spoke more than I should. As I progressed in my career I found that constructing a mathematical understanding doesn’t always ignite from just listening to the speaker.  There’s a time and place for listening, but being engaged in the learning process is vital.  I soon found that a balanced instructional approach was needed so I decreased the amount of talking and started to ask math related questions instead.

Although statements are beneficial, effective questioning techniques can provoke a response from the student. Offering guiding questions, or questions that encourage students to delve deeper in their explanation benefits the student. I feel like part of my job is to create an environment where students are able construct mathematical understanding. When students struggle with that understanding, questioning techniques can be another tool that teachers utilize. Questioning also helps students think more independently and explain their mathematical reasoning in a verbal or written form. Students need to be able to explain why and how they find solutions.  This type of communication is an important skill to develop.  Before planning on using questioning techniques in the classroom there are some important points to consider.

The environment

Students have to be open to answering the questions that are posed. In order for questioning techniques to work, students need to feel comfortable enough in the classroom to offer their ideas and explain their mathematical thinking. This environment is often intentionally built by creating a positive classroom learning community early in the school year.  Students will often participate less if they feel as though their input isn’t valued.

The timing 

Teachers can spend extensive time planning, but I find the best times to use effective questioning techniques are in the moment. Learning can be messy and teachers need to be able to have questions available depending on where students are in their mathematical understanding.   I’ve seen great question techniques used in whole class and small group settings.

 The questions 

The questions that are posed truly matter. When I started teaching my questioning techniques were less than stellar. Through time I’ve learned to expect more from my students. When given a chance, students are fully capable of expressing their thinking. Teachers need to allow students opportunities to do just that. The questions should prompt a response from the student beyond yes or no.  I want to get the students talking about their math process and learning.

 

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Other classroom questioning resources are below.

Effective Questioning Techiques
Asking Questions
Using Questioning to Stimulate Mathematical Thinking
Leveled Math Questions

 

 

 

 

 

 

 

 

 

 

 

Bridging Procedural and Conceptual Understanding

Yesterday I was putting together a few math projects when a Tweet caught my eye. The Tweet below started a short conversation that I thought was interesting.

David’s Tweet had many responses.  Most responses revealed that educators tend to side with solving one problem ten different ways rather than having students solve ten similar problems.  I started to reflect on how teachers give assignments that ask students to complete repetitive problems that often reinforce procedural mathematical thinking.  I also started to think how in an effort to provide practice, teachers may focus on procedural aspects first and then move towards practical application.  I find this happens frequently with math concepts at the elementary level.  What I don’t find often is the viewpoint that practicing procedural aspects can be embedded in solving specific problems multiple ways.  This type of thinking reminds me of number collection boxes.

Regardless of the assignment I want to be able to give specific feedback.  A larger problem that involves multiple steps can provide opportunities for teachers to pinpoint where misconceptions are and give direct feedback.  This isn’t always possible with ten similar shorter problems.  Below is an example of a few problems that you may find in a fifth grade classroom.  I don’t condone using these types of problems as they are definitely utlized, but I think we need to ask what’s being assessed when students complete this type of problem?  Students are simply asked to find the volume and show a number model.  I appreciate how the problems ask students to show their number model, but these types of problems seem to measure procedural understanding.  Do students know the formula?  Yes, well then they can answer many of these problems, even 10 in a row.

Procedural

 

I think the above problems have a place in the classroom, but shouldn’t necessarily be the norm.  Usually these types of problems are found on homework sheets.  The problem below which was adapted from a recent fifth grade test is more challenging, but gives students opportunities to showcase their own mathematical understanding and persevere.  Some would say that these two problems are completely different.  I would agree, but similar concepts are being assessed.  They do look different and the second requires more skills to complete.  Students need to be able to use their procedural understanding and apply it to the situation.  Also, one key element that’s missing from the first problem is the student explanation.  Students are required to show their mathematical thinking in the second problem.  This is big shift and can reveal student misconceptions more clearly than the first problem.  I struggled with the decision, but eventually had students work in groups to complete the problem below.  Students were allowed to use any of the tools in the classroom to find a solution.

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At first, all groups struggled with this problem.  Near the end of class all the groups presented their findings.  What’s interesting is that all the groups had different answers and ways in which they came to their conclusions.  I was able to offer opportunities for students to see and ask questions about different math strategies.  During the next class I was able to pull each group and give feedback.  This activity took a good amount of time to complete, but I feel like it was worth the commitment.

Through this experience and others I’m continuing to find that it takes a “bridge” to connect the procedural and application pieces.  At times I feel like there’s an assumption that if students are able to answer 10 similar procedural problems that they will be able to simply apply that knowledge in a multi-step problem.  This isn’t always the case and sometimes the bridge doesn’t fully form immediately.  Performance tasks, similar to the problem above can be one way in which teachers can help the transition from procedural understanding to practical application.  Being able to apply that knowledge to a math performance task can be a challenge for some students.  When teachers focus so much on the procedural, that’s the only context that students see and practice.  A blend between procedural and application needs to be established within the classroom.  I feel like activities like this help bridge this gap.


How do you bridge mechanical and conceptual understanding?

Math and M.C. Escher

 

Math and M.C. Escher

Math and M.C. Escher


During the last week of school my students started to explore topography concepts. Topography usually isn’t the first thing that is thought of when someone mentions the word math. That’s why I find it so interesting.  I truly enjoy teaching this topic because it often brings out the best from my students.  I find that most upper elementary students tend to thrive when given geometric shapes and asked to explore, rotate, translate or even turn them inside out.

I generally introduce the unit with M.C. Escher.  The class learns a bit about the life of Escher and his contributions to the world of art.  Moreover, we discuss how art and math are related. This is often a deeper conversations as students start to expand on the notion that mathematics can be found throughout our world.  Topics like the golden ratio and Pi often get brought up during this time.

After learning about Escher’s life and his influencers, the class looked at his different artistic creations. Usually my students recognize at least a few different creations.  Students seem to gravitate towards his optical illusion pieces or the famous Waterfall work.  As each work of art was discussed the more students found mathematics as an integral part of Escher’s work. After reviewing the different pieces of lithograph art, the class watched a short video on how Escher’s design and math are connected.

After the video the students were asked to have a conversation about how math can be found in most art.  The words symmetry, rotations, slides, translations, reversals, surfaces, and perspective were all brought up during the discussion.  What’s nice is that the vocabulary was brought up naturally as students spoke to one another.   I was able to highlight the words and facilitate the discussion as needed.

Eventually the discussion ended and the class moved to the next activity.  I planned to have the students create their own Escher-like artwork.  The students reviewed how to have “Escher-like eyes” when creating their own pieces.  I was proud of the student responses and the imagination that came forth during this discussion.  The class then reviewed the directions to create their own Escher-like creations.

The students went through the directions and asked questions.  Once the expectations were clear I passed out a 8 inch by 8 inch square to each student.  Students created their own tessellation template.  In the future I’m probably going to cut the square dimensions in half so the patterns become more evident.

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Eventually the students used the template to create an Escher-like creation.  Students showcased their work to the class using the vocabulary mentioned above.  The students were able to bring their work home on the last day of school.  All in all, this is a lesson I’m intending on using next year and a definite #eduwin in my book.

 


How do you incorporate art and math?

 

 

 

 

Exploring Rates in the Classroom

 

Exploring Rates in the Classroom

Exploring Rates in the Classroom

The topic of mathematical rates was introduced earlier this week.  Personally, I tend to find this unit enjoyable as there are many opportunities to connect the topic outside of the classroom.  To introduce the topic my classes go home and find examples of rates in their kitchen’s pantry.  The next day the class shares out what they found. This usually leads to an in-depth conversation about rates and patterns.  After our conversation I felt as though more examples and experiences were needed.

That evening I found some masking tape in my desk.  I decided to create a race path around the classroom.  The path varied in width and it purposely had a few sharp turns.  A roll of masking tape was used as well as a few proactive comments to the janitorial staff to not pick up the tape overnight.  When students came into the room the next day they saw this:

The raceway

The raceway

When the students walked into the room they were surprised.  A few started to jog around the track and ask questions about the room.  Already I was fairly excited as the students were pumped to see what I was up to.   I explained to the students that we were going to use the track to discuss rates, patterns and measurement.  The class then measured out the track and found that it was 66 feet long.  We had a conversation about how this track could be used to emphasize rates. I then introduced the students to the sheet below.

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Rates Sheet

Students were starting to see the big picture of this activity.  Students then took turns and quickly walked the course.  While they walked I had a few students become referees to make sure that no one stepped outside of the path.  I used an online counter and displayed the results as students quickly walked.  Once all the students completed their route and wrote down their results the class reviewed how patterns can be developed with rates.  Students were able to find the amount of feet traveled per second and then used that information to find how fast they walk one foot.  I was finding that students were trying out different mathematical strategies to find a solution. I gave them opportunities to work with each other to find solutions.  I asked clarifying questions when needed, but for the most part the students were on track. When the class finished this part of the sheet I gave them the second part.

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Converting feet to miles

This portion of the activity was more challenging.  Students were able to find the total amount of seconds, but converting the seconds to minutes was a struggle.  Many students asked how they could convert 12.9 minutes to minutes and seconds.  I was proud to see that students understood that 0.9 doesn’t mean 9 or 90 seconds.   This was a great opportunity to explore the concept of converting decimals to actual minutes.  The class used different calculations and found that 12.5 would actually be 12 minutes and 30 seconds.  As progress was made students started to find a conversion strategy to correctly convert the decimal to seconds.

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Click for Word template

 

As a class we shared our results and found patterns and the mean.  This activity worked so well that I used it with a few other classes this week.  I’m finding that students are developing a better conceptual understanding of rates while participating in a learning experience that I hope they don’t forget too quickly.

How do you introduce rates in the classroom?

 

 

Understanding Volume

Constructing Conceptual Understanding

Constructing Conceptual Understanding


This past week second and third grade students at my school are learning about measurement. The students are making progress. The classes have become more fluent with understanding perimeter and area, and are now starting to explore the concept of volume. Throughout the process students have used various manipulatives, such as prisms and nets to deepen mathematical understanding.   Even with all the activities  some students that are still struggling with the concept of volume. In about three weeks or so students will be assessed on this particular topic. Providing extra sessions for students to develop a conceptual understanding of volume is important. I wanted to find or create a math task that gave students intentional time to review geometry and measurement terms, while at the same time allow opportunities for students to create different products. After reviewing different options I decided on having students use the project detailed below.

Students were given a full sheet of colored centimeter graph paper.  They were then asked to read through the directions.

Directions: Create a net for a rectangular prism using the graph paper provided. The rectangular prism you build should have a volume of 20 cm3. Cut out your net and build a rectangular prism using glue or tape. Write the dimensions of the prism you built in the charts below.

Looking back, it seems like there were more than enough questions about what was expected.   Students always seem to have questions when there are multiple solutions/products.  After I answered their questions I took about 10 minutes to model the activity with the students.  This was important as it cleared up expectations for the activity.  I then passed out the assignment.

Click for sheet

Click for sheet

Students then used the centimeter grid paper to create a rectangular prism net.  They then filled out the top portion of the sheet.

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Some students had to use multiple attempts to create a net with a volume of 20 cubic centimeters. This was great opportunity for students to show perseverance and find a solution that worked.  I went around the classroom and asked students questions to help them think of a solution.  The students then cut out the nets and constructed their prisms.  The students then presented their rectangular prisms to the class.

 

 

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How do you construct meaning in geometry?


Representing Fractions with Thinking Blocks

Many classrooms in my school are in the midst of reviewing fraction concepts. Throughout the school students are finding fractional pieces, converting fractions to decimals, and identifying fractions on number lines.  For the past week students in second grade have been identifying fractional parts.  Earlier in the week students completed the page below during a math station.  Students did well on the first two pages, but struggled a bit when identifying fractions on a number line.

Representing 3/4 in many different ways

Representing 3/4 in different ways

This was a challenge for some students as many are more familiar with identifying fractions within objects (in a circle/rectangle).  Moving from identifying fraction to placing them on a number line can be a stretch.  Many students have already started to decompose numbers and have completed “fraction-of” problems.  These types of activities have helped reinforce the number line and fraction connection.  Next week students will be assessed on the fraction unit and many classrooms move into geometry concepts.  Before focusing in on geometry, I wanted to give student an opportunity to visualize fractions and use them with more complex word problems.

As I was looking for supplemental material I came across a Tweet by Paula (@plnaugle). She referenced Thinking Blocks  as a resource that she uses with an interactive whiteboard. I looked into the site and thought that it might be useful for my grades 2-3 classes since the app allows students the opportunity to solve fraction problems visually.  Specifically, I downloaded the fraction app on the school iPads.   Yesterday a second and third grade class used this app in their classroom as a guided activity.  The app was introduced to the class and I modeled the different steps involved in solving the problems.

Modeling

Modeling

The students were then asked to find a comfy place in the room and complete a minimum of three exercises.  What’s nice is that the problems are picked at random, so students aren’t on the same problem at the same time.  There’s also a feedback box that assists in guiding students towards labeling the correct parts of the fractions.

Click to enlarge

Click to enlarge

I helped the students as needed, but many were able to use the virtual manipulatives and generated feedback to stay on track.  Some students completed three problems, while some went beyond and tried out five.  After about 12 minutes the class gathered and we reflected on the perseverance that was needed and celebrated successes. This activity gave students an opportunity to make mistakes and persevere.  I’ll be keeping this app in my repertoire for the future.