Collaborative Math – Page 2 – Educational Aspirations

Math Podcast Project

For the past few years my students have completed a classroom evaluation form near the end of the school year. The purpose of the evaluation is to reflect on the progress made throughout the year and to highlight beneficial learning experiences. Based on past survey results, students seem to rate learning experiences related to technology and collaboration higher than independent projects. Not a big surprise here, but my elementary students seem to thrive when given a choice on how to present content with a technology component.

One of the more highly rated projects this year was the inquiry based math podcast project. The math podcasts gave students an opportunity to use technology and incorporate creativity into their projects. Notice I said their. I think what helps make this project so beneficial is that students take ownership of the project. That ownership isn’t often related to any extrinsic reward (maybe peer pressure?), but a self-conscious effort to communicate what they’ve learned. Many students noted the topic of student ownership and this project on the end-of-year evaluation.

As with any student project, there were guidelines and a lot of planning. I used a rubric to help guide the projects along and had specific check-in points to give feedback. Time can really get away from the teacher if guidelines aren’t established and enforced. Click on the images below to find Gdoc templates.

The students were split up into groups of two to complete the project. The teams were randomly chosen. Each team received the directions and a rubric page. Each team then created a script that was eventually approved by the teacher.

The students were given approximately 30 – 60 minutes once a week for approximately two months to work on this project. The free program Audacity was used to create and mix the recorded sounds. Creative Commons sound effects were used and can be found with a quick Google search. This can also be an opportunity to have a conversation about attributing credit to sources. Some students needed extra time and it was given. Student groups then presented their podcast to the class and answered questions from the audience. The projects were then shared with the community. Overall, I thought that the skills reinforced/learned through this activity justify the amount to time that was dedicated to the project. I’m hoping to incorporate more of these projects into my classroom next year.

How do you use math projects in class?

Student Groups and Debates

Teachers often have students work in groups to solve problems. Educators may recite that “two heads are better than one” or something of that sort when talking about the power of effective collaboration. I’ve seen firsthand how student grouping can impact decision making and student learning. How a group interacts will often influence outcomes. Positive interactions between group members often spurs a team to meet their goals. I believe most teachers encourage positive talk during group activities and many set up a norm/expectation list for behavior. Learning is often stretched when students are encouraged to explain their answers to others.

What happens when a student explains an answer and the other party isn’t receptive? Or, what happens when students disagree on an answer or how to solve a problem? This is bound to happen from time to time, but I don’t think this is necessarily a negative. Students should be able to stay on topic and analyze their own argument without expressing frustration towards the idea (not people) that they disagree with. Disagreement may conjure anger if not carefully managed. This requires clear expectations and modeling by the teacher. Easier said than done? Yes. Often “I agree” statements can overshadow academic misunderstandings, while students just follow what the leader is saying in the group. I’m aware that some classrooms encourage debate and I think that in some cases that benefits the classroom. I should also note that having a classroom/group debate depends on the problem and is purely situational.

Students, no matter what their age, need to be able to communicate their ideas in order to meet goals. It’s perfectly fine for students to disagree with the group. How that disagreement is communicated and received charts the course for the group. Individual insights hold value and each contribute to the overall goal of the group. Students need to be able to disagree respectfully, but understand that the team is working towards the same goal. Students that have this mindset are able to offer differing opinions, but innovate as a team.

Having a balance is key. Groups should work together but also be open to differing ideas. Disagreement often forces other students to justify their positions. Justifying provides opportunities for students to analyze their own argument, which gives the teacher a better understanding of a student’s understanding of a particular topic/concept.

I think this also plays a role in how adult teams operate as well (see Ringelmann). I’m going to end this post with a quote from James Surowiecki, the author of The Wisdom of Crowds.

“The wisdom of crowds comes not from the consensus decision of the group, but from the aggregation of the ideas/thoughts/decisions of each individual in the group.”

Picture Credit: S. Miles

Math Debates in Elementary Classrooms

Over the past few months I’ve dedicated a good amount of time to to having math conversations. These math conversations occur when the class is unsure of how to solve a problem or when disagreement ensues over what particular strategy should be used to tackle a problem. The math conversations (or debates) allow students the freedom to openly discuss logical reasoning when solving particular problems. These conversations can be sparked by the daily math objective or follow another student’s response to a question. It’s not necessarily planned in my teacher planner as “math conversation” in yellow highlighter, but I do make time for these talks as I feel that they bring value and encourage student ownership. The conversations also give insight to whether students grasp concepts and are able to articulate their responses accordingly. Mathematical misconceptions can also be identified during this time.

During these conversations I have manipulatives, chart paper, whiteboards, iPads and computers nearby to assist in the discovery process. I emphasize that there’s a certain protocol that’s used when we have these discussions. Students are expected to be respectful and listen to the comments of their classmates. To make sure the class is on task I decide to have a specific time limit dedicated to these math conversations. Some days the conversation lasts 5 minutes, other days they may take upwards to 15-20 minutes. When applicable, I might use an anchor chart to display the progress that we’ve made in answering the questions. I should also mention that sometimes we don’t find an answer to the question. Here are a few questions (from students) that have started math conversations this year:

Why is regrouping necessary? (2nd grade)
What can’t we divide by zero? (3rd grade)
Why are parentheses used in math? (3rd grade)
Why do we need a decimal point? (1st grade)
When do we need to round numbers? (2nd grade)
Why is a number to the negative exponent have 1 as the numerator? (5th grade)
Why do you have to balance an equation? (5th grade)
How does the partial products multiplication strategy work? (3rd grade)
Why do you inverse the second fraction when dividing fractions? (5th grade)
Why is area squared and volume cubed? (4th grade)

Above is just a sampling of a few of the math conversations that we’ve had. Afterwards, students write in their journals about their experience finding the solution to the problem.

Of course this takes additional time in class, but I believe it’s time well spent. The Common Core Standards focus on depth of mathematical understanding, rather than breadth. This allows opportunities to have these conversations that I feel are beneficial. They also emphasize the standards of practice below.

CCSS.Math.Practice.MP1 – Making sense of problems and persevere in solving them.
CCSS.Math.Practice.MP3 – Construct viable arguments and critique the reasoning of others

Photo Credit: Basketman

Do you have math conversations in your class?

Equivalent Fractions Tweak

A few days ago I started gathering resources to supplement a math unit on fractions. The classroom was studying equivalent fractions and I thought there might be a variety of resources available on a few of the blogs that I regularly visit. I generally follow the #mathchat hashtag and find/share ideas that relate to mathematics. While reading a few math blogs on fractions, I came across John Golden’s site that has some amazing ideas that can be used in math classroom. His triangle pattern template sparked my interest.

John provided a template that’s available on his site. I printed out the template and began filling out each triangle with fractions. I ended up with a sheet that looked like this.

So what happened?

First a lot of brainstorming and error checking. Then I decided to have students cut out the triangles and compile equivalent fractions. This is what happened …

Students in fourth grade cut out each triangle and combined them to make equivalent fraction squares. Students worked in collaborative pairs during the project. I observed students using math vocabulary and having constructive conversations with each other to finish the assignment.

Before giving the assignment to a fifth grade class I decided to eliminate two triangles on the sheet above. It was the job of the student to find what triangles were missing and create equivalent fractions to complete the squares. The students were engaged in this activity from start to finish. Some students even wrote the equivalent decimal next to each square.

Overall this project took approximately 45 minutes to complete and it was worth every minute. Students used the terms fraction, improper fraction, mixed number, numerator, denominator, multiplication, division, and pattern throughout the project.

Just as I did, feel free to tweak this project to best meet the needs of your students.

Dice and Math Computation

Since the beginning of the school year I’ve been searching for different ways to incorporate guided math in my classroom. Guided math has many benefits although organizing the groupings can bring a few challenges. Guided math looks different depending on how the teacher implements the structure. For example, one math group might be working with the teacher while two other groups are using math games or participating in problem based learning activities. The groups will rotate according to a specific time schedule. I’m finding that groups that are not with the teacher need specific instructions and expectations.

For the past few months I’ve been using dice games to emphasize number sense skills. These dice games have peaked student interest and work well in increasing computation fluency. I decided to collect multiple formative data pieces to validate whether the dice games were contributing to student success. By analyzing student data and observing over a period of time, I found that students were becoming more fluent in adding, subtracting, and multiplying small/large numbers.

The games have worked for me, so I’m passing it along to others that might find it useful. Needed materials and pdf files are below.

Materials

A variety of dice (6, 10, 20, 30, etc. dice) Here are some examples:

Templates (in pdf form)

Roll to 150 (multiplication)

Roll to 125 (addition)

Roll to 100 (addition)

Roll to 45 (addition)

Roll from 50 (subtraction)

Roll from 95 (subtraction)

Roll from 35 (subtraction)

The Marshmallow Challenge in the Elementary Classroom

Using Food to Learn — Creating Structures with Collaboration

Approximately two months ago I noticed a Twitter post about something called the Marshmallow Challenge. The tweet led me to this TED video. Many of the examples indicated that the challenge could be used with adults as well as students. The official Marshmallow Challenge website offers many useful instructions and tips for facilitators. I decided to use the challenge with a fourth grade classroom. The session, from start to finish, took approximately 45 minutes. The standard 18 minute time limit to work on the project was perfect for my classroom. Of course the focus of this project emphasizes teamwork, but I decided to add a few measurement standards. For example, the students were required to measure the length of each pasta stick used and find the volume of the marshmallow (as a cylinder). The total height of the structure was also measured. Here are a few pictures from the event:

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The class had a debriefing session after the event. During this discussion, students revealed their strategy. Here were some of the questions that were discussed.

What will the base of the structure look like?
Will we use all of the materials?
What are our roles?
How will we work as a team?
How does working as a team help us succeed?
Will we wait to put the marshmallow on top at the very end or test it throughout the project?
Should we write out a plan in advance?
How should we work together?
What are other groups doing?

Overall, this learning experience gave students an opportunity to use critical thinking in a collaborative setting. I’m planning on having students complete a plus/delta chart and complete an entry in their math journals next week.

*Picture credit: Stoon

Measurement and Mini Golf

Approximately a week ago I was paging through my math curriculum. Through a pre-assessment I found that students were in need of a review on angle classification and measuring skills. The curriculum lessons offered a number of worksheets and angle measuring drills. Although these lessons seemed beneficial, I felt the need to create a more memorable learning experience for my math students. At this point, I decided to search for measurement projects. While following #mathchat, I came across this Edgalaxy site. The project seemed to match many of the objectives that needed strengthening in my class. I changed up the directions and modified some specifics in order to best meet the needs of my students.

So … a week has passed and almost all of the projects are complete. I listed the project steps below. Feel free to use any of the ideas below in your own classroom.

1. Had out the direction sheet. Here is a Word template (via Google Docs) for your use.

2. Review many of the different vocabulary words associated with the project: acute, obtuse, right, parallel, perpendicular, trapezoid, etc.

3. Show possible examples. I tend to show just a few examples as I don’t want to give them a mini golf course to copy.

4. Group the students into pairs. If you prefer, this project could be implemented as a collaborative group activity.

5. Students choose their construction paper color (11″ x 20″)

6. Students draft their course in pencil (on grid paper). The draft gets approved by the teacher and then is transfered to scale on construction paper.

7. Students present their final projects to the class.

Estimation Challenge

Image by: Akeeris

Over the past few years, I’ve been working on ways to utilize technology to improve student learning. Understanding what objectives are being assessed helps me plan on what technology will be used and in what capacity it will be used. One of the second units in my class emphasizes the importance of estimation. The fifth grade Chicago Everyday Math curriculum asks the students to do the following:

Notice that the question says “location given by your teacher”. Instead of giving all the students a specific destination, I decided to have the students pick an establishment (Culvers, Kohls, local park, school, etc.) in the town that they reside. In the past, I’ve found that student choice can be a motivator for students. The destination had to be within 15 miles of the school. The students were grouped in triads and were given a computer to complete this task. Students were asked the following:

What is your destination?
About how many miles is it from your school to your destination?
About how many steps will it take to reach your destination? (they used the conversion in the journal above)
How long would it take you to reach your destination?
What breaks would you take during your walk to your destination?
If you left at 8:00 AM on Monday, when would you arrive at your destination?

The students were also given the Google Maps website to start their estimation challenge. Most students were able to navigate Google Maps and find the “get directions” tab and enter in the school address. The groups were able to find the establishment address fairly quickly, although some groups needed prompting. The student groups needed to find out what route to take to their destination. Some routes were quicker than others, but involved a lot of stopping at cross walks. Other routes were scenic, but took longer. Each group decided which route to take and found the Google Maps distance to the destination. Here is a sample of what the students were looking at:

Some of the groups extracted the “Google Maps time” to answer the questions. Other groups thought that it seemed odd that it would take a specific amount of time for everyone to reach the destination at the same time. One of my students remarked that not all people will take 2 hours and 56 minutes to reach the destination. I thought this was a prime opportunity to bring in the topic of ratios and proportions. One of the groups decided to time themselves walking 10 feet and then find out how long it would take to walk an entire mile at that pace. I was impressed with the groups that went beyond the artificial time given by Google Maps. Even more, the topic of ratios and proportions is typically introduced next school year.

Near the end of the project, the students presented their answers to the class. Each group chose a different location and I could tell that their answers were well thought out. Overall, the skills utilized in this project are applicable outside of the classroom and I feel that the students were fully engaged and met the objectives for the lesson. This is one of the projects that I’m planning on using next year. This is my second Google Maps activity, my first lesson can be found here.

Math and Multiple Solutions

Image by: Krishnan

For the past few days I’ve been reviewing a math unit and have found that the lessons included have very few problems with multiple solutions. I have nothing against one correct answer scenarios, although I feel as though students should be exposed to problems with multiple solutions. There are cases where having one solution in math is mandatory, but there are other cases where multiple solutions are possible. I believe the project in this blog post isn’t completely “open-ended”, although it does have multiple solutions. The concept of open-ended math is important because I believe that this idea is relevant in and outside of the classroom. Students often seem more intrinsically motivated to complete open-ended problems, as it’s different than the norm.

Recently, I came across a math activity designed for the upper elementary level, (although it could work at middle school) that offers multiple solutions. Since I didn’t personally create this activity, I’d like to give credit to NRICH Project for the original idea. Multiple math concepts are found in this project. The concepts covered in this project include a great amount of number sense concepts: factors, multiples, square numbers, even, odd, prime, composite, and triangular numbers. This assignment covers many concepts and a teacher could informally assess students in the classroom as they facilitate the learning process.

Here’s the process that I used:

1. Download the Word documents (you can easily edit them to meet your needs). Here is the Word file.

2. Review the concepts of multiples, factors, square numbers, even/odd, prime/composite, and triangular numbers.

3. Pass out the sheets to the students. (I had one of the pages a different color than the other – better for organization)

4. Students cut out and glue the project together. (my class took approximately 30 – 40 minutes)

5. Review the project with the students

6. Have the students journal about their math problem solving experience. Extension opportunities can be found here. The reflection and assignment could be used to show growth over time and might even be useful in a student portfolio.

Here are a few possible solutions:

Additional answers may be found here.

If you use this, please let me know how this project works in your classroom.

Reflection Journals in Math Class?

Image by: Samana

In the past, I’ve used reflection journals for language arts assignments. Allowing students to reflect via journaling was one way that I could informally assess whether students were making connections to the literature. After utilizing the idea of journaling for my language arts class, I thought that it might be useful to integrate this strategy with math. Before starting this adventure I decided to complete some homework on the idea of math journaling. In the past I’ve used standard reflection sheets. While collecting ideas, I also looked for math journal writing prompts and rubrics 1 2 3 . I found many ideas and strategies for math journaling here and at Monica’s website. If you’re unsure of how to introduce the topic of math journaling, this Word example may help. If you’re curious of where to start, I’ve found that this site provides terrific examples. So, after researching a few options I decided to label all of my journals and prepare for uncharted territory.

After giving a unit assessment, I gave my first math writing prompt:

How do you feel about your performance on the last unit assessment?
What type of math concepts do you find interesting? Why?

Students were also asked to include a picture with their response. Why a picture? I thought that allowing students to draw a picture may portray how they feel regarding their performance. Some students decided to draw more of a picture, while others decided to write more with words. Allowing this type of flexibility gave students an opportunity to communicate their response to the writing prompts differently. The students then turned in their journals and I wrote a short response to each individual response. I feel as though the students really enjoy the fact that I personalize my response to each student. I also feel as though this builds a positive classroom environment, as each student is shown that their opinion is valued. The journals can also be used during parent teacher conferences, although it might be a good idea to disclose this to the students before they write.

What happend?

After completing a plus/delta chart, students thoroughly agreed that the math journals enabled them to reflect on how they are doing in the class. Some students even communicated that the journals were a way to set specific math goals. Currently, I give students an opportunity to complete a journal entry approximately every two weeks. A byproduct of using the journals may also lead to personal goal setting and more academic involvement from the student.

What’s next?

I would like to incorporate the idea of utilizing specific math vocabulary in the journals. Not only should the math journals be used for reflection, but they can also be used as another opportunity to practice mathematical concepts. As an elementary school teacher, I think it’s important for students to have a solid understanding of math vocabulary at a young age. Having consistent definitions is also important. Certain math vocabulary words that are utilized in first grade will accompany a student throughout their entire life. For example: multiply, divide, sum, fraction, etc. Overall, I feel that students will become better at understanding math vocabulary and reflect on their learning through the math journals. The journals will be used consistenly, so students will observe the progress that they have personally achieved throughout the year.