November 2014 – Educational Aspirations

Exploring Fractions

Fourth grade students explored fraction computation last week. Since the beginning of the year they’ve been periodically reviewing how to add and subtract simple fractions. About a month ago this same group of students used fraction pieces of a pie to show a visual model of adding/subtracting different fraction less than one. Last week students identified and compared fractions and mixed numbers. They started to convert mixed numbers into fractions and vice versa. I’m finding that as the students became more comfortable with converting fractions they’re becoming better at fraction computation. Not all the students are at this level, but many are ready to add/subtract mixed numbers.

Over the past few years I’ve used a fraction computation activity that I often refer back to throughout the year. Every year I tweak it a bit more to fit better with my students. This year I felt my students were ready for the challenge. The students cut out the fraction pieces below. Students are then given time to explore how different fraction pieces are equivalent.

Fraction Blocks

Photo Nov 20, 11 10 56 AM

I asked the students to model different types of fractions with their pieces. The class came to a few different conclusions on how fraction sums were calculated. I didn’t really hear students talk about finding common denominators; instead I heard students saying the words “equivalent” “matches” “is the same as” throughout the conversation.

Students were then asked to combine their fraction pieces to find certain sums. For example, students were asked to show 1 1/2 using 7 pieces. Students wrote the number model below their visual representation. I was encouraged to see that some of the students showed fraction multiplication in their number model eg. (5 x 1/6) + 1/3 + 1/3 = 1 1/2 .

$fraction pieces$

Through trial and error students started gaining traction in finding the sums. Students had to place all the questions out on their desk and match the fraction pieces to find the sum. After all the fractions were found students taped/glued them to their paper. The class then discussed how this activity could be completed in a variety of ways. Next week students will reflect on this activity in their math journals. The activity described in this post can be found here.

Addressing Misconceptions

misconceptions-01

Students in third grade are exploring measurement this week. As students progress through the unit I feel as though they are becoming more efficient in converting Metric units. Near the end of the class today students started debating the differences between US Customary and Metric. The class than started completing an activity where they had to measure different insect lengths. Students worked in groups to accomplish this task.

During this time I traveled to each group and intentionally eavesdropped on the conversations. Students asked me questions and I listened and asked questions back. I then moved on to the next group. I wanted the students to work together and persevere. Some students started to talk about the measurement of different objects around the room. I especially paid close attention to the questions that students were asking each other. This was a great opportunity to check-in on some of the misconceptions that were flying around the room. I jotted down some of the conversations as the students came back to the large group.

We had around five minutes left in class to review the questions that I noted. I wrote the questions that I heard on the whiteboard. I was able to clarify some of the responses and answer other questions. This time was definitely worthwhile. The students seemed to appreciate the time as well. During our next group activity I’d like to do something similar, but not completely rely on my less-than-stellar eavesdropping skills. Instead, I’m thinking of having the students periodically use Post-it notes to ask questions. This could turn into a “wonder wall” type of activity. The students could then place the questions in a bin and we can review them throughout the unit. I think this type activity is one way to proactively address misconceptions and answer questions as students grow in their mathematical understanding.

Using Multiple Strategies in Math Class

howandwhy-01

Last week a few upper elementary classes started to explore different methods to divide multi-digit numbers. Many were familiar with repeated subtraction for numbers more than one and some even had experience with the long division algorithm. I asked students to explain their reasoning for the steps needed to divide numbers using repeated subtraction and long division. All of the students were able to explain the reasons for using repeated subtraction. Students gave quality answers and were able to communicate why each step was performed. Some students even related repeated subtraction to a number line. I then asked the students the reasoning for using the long division algorithm. I heard responses like “it’s quicker” or “that’s what I was told to use” or “you just do this and this.” I could tell that there was a disconnect between the shortcut and having a conceptual understanding. Students understood the steps but couldn’t provide solid reasoning to why you would bring down the next number.

partialquotientswhy-01

The class then had a conversation about the importance of being able to clearly explain their mathematical thinking. The students that knew how to use the long division algorithm were getting correct answers, but couldn’t tell me why. Blindly following procedures can lead to holes in understanding. Explaining the reasoning behind completing a problem is important. Honestly, I don’t mind if the students use algorithms like the above picture if they already have a descent understanding. The problem I have is that sometimes this is the only way students are taught how to divide large numbers. The problem becomes steps –> answer without understanding.

After the class had a discussion about long division we explored the partial quotients algorithm. I explained to the students that this was another method to divide larger numbers. As we progressed through and explained the steps, students became more aware of how partial quotients seemed to make sense to them. For many, this was the first time exploring the partial quotients algorithm. The students were able to explain why each step was taken in the process.

partialquotients-01

I believe some of the students were also relieved that they didn’t have to get the partial quotient “correct” the first time. By breaking apart the problem students were starting to see the correlation between repeated subtraction and the partial quotients algorithm. More importantly, students were able to explain their reasoning for completing the problem with this method.

As we move forward, I feel as though students are becoming better at explaining their mathematical thinking. It doesn’t matter to me what method the student decides to use in the future as long as they are able to justify their reasoning. This thinking could also lend itself to just about any type of computation skill. Last week reminded me of the need to expose students to multiple strategies to complete problems. Providing these strategies can assist students in becoming better at explaining their mathematical thinking.

4/9/15

As my students progress through their fraction multiplication unit I came across another example of why using multiple strategies matters. In the past students learned how to multiply mixed numbers by 1) Convert the the mixed numbers into improper fractions 2) Multiply the numerators and denominators 3) Covert the improper fraction back into a mixed number. This is how I was taught to multiply mixed numbers. Although this method seems to work, students had trouble explaining why they completed each step. So early this week I decided to use a different strategy in class.

$muliplyingfractions-01$

Students were able to visually represent this multiplication problem and the steps to solve seemed more logical.

Math Genius Hour – Research and Presentations

topofpost

This is my first year using a math genius hour model. My third, fourth and fifth grade classes all started their genius hours at the beginning of September. The beginning of our journey can be found here. Students narrowed down their question and have conducted research over the past month. The research process has been an eye-opening experience. Before beginning, I was able to set aside some time to have a conversation with students about finding appropriate resources for their project. Even though the classes were during the math block I thought discussing this was important, especially if we’re having more than one genius hour per year. I thought that having the conversation would pay a few dividends later in the year.

The majority of the research will be conducted online. The class discussed the importance of reviewing the ending of website addresses. We reviewed the url ending (.gov .edu .org .com .net) and how to conduct research in an effective and meaningful way.

We analyzed different red herring websites (1, 2) and I believe students are getting better at identifying sites that seem legitimate. This took a large amount of time and many questions were asked. I feel like an entire course could be dedicated to this topic. After a while, the class and I created a sheet that the students would fill out to organize their sources.

Although my district provides a facts database for students (www.facts4me.com), the majority of the research that needed to be conducted was beyond the site. So students began to explore research outside of the box. I soon found out that many students were under the impression that they could Google their question and use the first link that appeared. Students also found that Yahoo Answers wasn’t necessarily the best source either. Through a good amount of exploration, students found sites that were adequate and provided legitimate information that they could use in their project. The students became much more independent once they understood the research parameters.

At this point students are starting to explore how they will present their project. Last week the classes took time to review different presentation tools. Many of the students used a variety of presentation tools last year so they were fairly comfortable in picking a tool. Eventually the class decided to use the sheet below to help make an organized decision.

Screen Shot 2014-11-01 at 8.00.29 AM — click for file

After students pick a tool they will start creating their presentation. Some students are at this point, others are not. My fifth grade classes helped create a rubric that students could follow. I wanted the rubric to be flexible in allowing students to present in a way that they wanted yet a minimum criterion was established. I wanted to also make sure that students’ creativity and voice were part of the presentation. A self-reflection piece is also incorporated into the rubric.

Screen Shot 2014-11-01 at 7.47.41 AM — click for file

I’d like to thank Denise Krebs and the genius hour Wiki contributors for the self-reflection sheet. Addition resources on genius hour can be found in Joy’s genius hour Livebinder site.

Overall, the math genius hour is a work in progress and I’m assuming the students will present at some point in December. I continue to look forward to how this project progresses throughout the year.