This week I introduced function machines to one of my primary classes. The activity yesterday revolved around the concept of number patterns and perimeter. Student groups were given a pile of square geometry blocks. The groups were asked to find the perimeter of one square. The perimeter was quickly found, which ended up being four inches. Students then found the perimeter of two squares connected.
and then three squares …
Students started to recognize a pattern as they filled out their in/out table.
Students were then asked to explain a rule for finding the perimeter of the square shapes. Many of the student groups attempted to find a rule, but found a single digit addition or multiplication rule that didn’t work for all the numbers. The groups started to struggle in an attempt to find the rule. One group finally came up with a rule indicating (in x 2 )+ 1 = out. Students were excited that they were on the right track. After a few minutes another group came up with a different answer (in + 1) x 2 = out. A couple of the groups asked how can there be more than one rule? This allowed an opportunity to have a conversation about equivalent rules.
After students found the rule(s), they were asked to find the perimeter of 423 blocks. I told the student groups that I didn’t have 423 blocks, so they will need to use a rule to figure this out. Students began to understand the usefulness of math rules. Even more, I was glad that they were able to explore the advantages of having math rules on their own.
During the past few days my class has had elaborate discussions regarding the importance of student ownership and the learning process. The class agreed that being able to explain our mathematical thinking is important. Many of the math projects that I assign focus in on the eight mathematical practices and highlight the ability to articulate the “how” and “why” certain steps are/were taken. That process can enable students to understand math concepts more clearly and apply their learning. The class then discussed how applying our learning can lead to innovation. That conversation then cascaded into the importance of being an innovator today and how modern-day technology often evolves through the refinement of ideas. These ideas may come from inventors or entrepreneurs that take a risk and create a new product/idea.
One of my goals this year revolves around the concept of enabling students to be digital content creators. Digital content creation happens all the time and there are many examples (positive/negative) of this. I want to encourage my students to create ideas, projects and connections this year. So earlier this week I noticed that @mwhitedg posted a tweet with the #dg58learns tag about how his class is now using the Showbie app to turn in digital content. This idea intrigued me as one of my focuses this year is to have students create digital work for their eportfolios. One of the main problems that often exist at the elementary level is that students aren’t allowed/don’t have email addresses, so emailing work to the teacher can be problematic. It seemed that this particular app might help solve this problem. I discovered the Showbie app that night and started to look at what student content creation apps exist. I found plenty and many content creation apps that I haven’t heard about. Click the below image to access the site.
I’m familiar with the app Educreations and used it as a primary whiteboard in the past; not delving too deep into it’s capabilities. After finding Showbie, I begin to upload my student roster into the Educreations site and started to find that the app has potential. The next day I modeled the app with the class and showed students how to login and send assignments via Showbie. My students were asked to compare prime and composite numbers (4th grade) and compare a kite and rhombus (3rd grade) with Educreations. I allowed the students to explore the different functions of the app. Most students found a comfy and quiet place in the room to record their lesson. Without even telling the students, some started taking pictures of objects in the classroom and importing them into their lesson. After 20 minutes, all students were done and submitted their videos. As a class, we reviewed the videos and made a plus/delta chart on the quality of the projects. Below you will find some sample screen shots of the projects.
We learned a lot about the Educreations app and how to position the iPad while speaking (hint: don’t cover the microphone or move the iPad with force). The class also had a conversation about the importance of having guidelines during the creation process. In the future we’ll be using a form of this rubric and possibly work in groups to create a number of projects this year. This may change though, depending on the quality of the projects and learning experiences that are in our future. The students were extremely excited to view their content and thrived on the idea of being able to create their own work. Moreover, I found that five students actually went online and viewed their creation video over the weekend. Becoming a responsible digital creator is an important skill to have and I believe we are starting to make headway.
How do you encourage student creation at the elementary level?
Today was unofficially factor day in my fourth grade math classroom. The lesson focused on factors, prime/composite numbers and prime factorization. For some students, the lesson reinforced preconceived notions, while others were introduced to a fairly new concept. The goal of the lesson was for students to develop a deeper understanding of factors and the role that they play in mathematics. I decided to use a variety of math games to review the concept, as well as to extend the concept of factors. One of my favorite methods to review and enrich the learning experience is to use math games in the classroom. Math games often encourage students to take risks and use strategies in an attempt to win. In the process they often have to work together to ask questions and clarify their mathematical understanding.
Today began with a brief mini lesson on factors. I then split up the class into stations. Each station was designed to reinforce and provide enrichment opportunities. Students worked in partners at every station. In some stations they worked together, while in others they were competitors. Here are the stations:
Factor Captor – This is a staple game in my classroom. There are three different levels and students progress to the next level when they feel ready. To play this game students need to be able to identify prime and composite numbers. Here’s a short video of the game in process. A template with sheets can be found here.
Divisibility Dash – This iPad app is designed (at least for me) for students to work in groups to identify various factors. I found this app for free about a year ago and took advantage. Many McGraw Hill apps are free during special times of the year. Students record their scores/factors on a separate sheet of paper.
Sliding Factors is a computer game that encourages students to find factors of composite numbers. There’s a two player function, which definitely comes in handy. While browsing the #mathchat tag I saw @Richard_wade post a link to this game.
I should also mention that one section of the classroom was designed to be an “exit card” station. Students completed a quick three question formative assessment and I discussed the answers with them. This is another opportunity to give direct feedback that may help the student clear up misconceptions and help them make mathematical connections.
When used correctly, math games can truly benefit students. When the students are in stations I like to sneak by the groups and listen in on the math talk that’s happening. The math talk often gives students an opportunity to defend their mathematical thinking. Students often correct each other, but are generally respectful in the process. Tomorrow the class will be writing up a quick reflection of the station/factor experience in their math journal.
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.
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.”
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
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.