Multi-digit Multiplication Strategies

This past week my third grade class investigated different ways to multiply numbers.  Before diving into this concept I asked the students their thoughts on multiplication.  A few students explained to the class their view on the topic of multiplication.

• double or triple “hopping”
• using arrays
• “timesing”
• Increase the number by “a lot”

Most students were able to showcase examples of the above.  Even though their vocabulary wasn’t exactly spot-on, students were able to come to the whiteboard and show their thinking.

I received different responses from the students when asking them about multi-digit multiplication.  Actually, it was more of a lack of response.  I feel like some of this is due to exposure.  A few students raised their hands and asked to show their process to multiply multi-digit numbers. These students showcased their ability to use the traditional algorithm. The class reviewed this method with a few examples.  Although students were finding the correct product they had trouble explaining the process. Students weren’t able to communicate why it worked or another method to find a solution.

On Tuesday my class started to explore the partial-products algorithm.  Students were able to decompose individual products and find the sum.  This made sense to students.  Students were able to connect an area model with the partial-products method.  They started to write number models right next to each partial-product.

Later in the week students were introduced to the lattice method.  This method seemed “fun” for the students, but didn’t make as much sense as the partial-products method.  Students were able draw the boxes and create diagonals to find the product.  Some students had trouble with laying the boxes out before multiplying.

During the last day of the week students were asked to explain in written form how to multiply multi-digit numbers.  Even though all of the students could use the traditional, partial-products and lattice methods, they were stuck for a bit.  Soon, most students started to lean towards using the partial-products method to explain how and why this method works.  I asked one student in particular why it made sense and she said “I can see it visually and in number form.”  Although most students were able to use the other methods effectively they didn’t seem confident enough to explain why the strategies worked.

Students will be expected to multiply multi-digit numbers on the next unit assessment.  The method to multiply these numbers will be determined by the student, but I’m wondering how many will gravitate towards the strategy (not just the process) that they understand.

Class Math Discussions

A few years ago I remember my school district emphasizing the need to use more of a math workshop approach in the elementary classrooms. The school district even invited a math workshop specialist to present on all the different ways to set up groups and organize guided math.  Some of the teachers gleaned the information and used parts of the model in their own classroom.   The consensus was that some of the guided math approach was better than none at all.

As the years passed the idea of math workshop started to change. Teachers started to change the math instruction block to incorporate small group instruction. Whole group instruction still occurred, just in shorter bursts. The small groups consisted of around 5-6 students and rotated every 10 – 15 minutes.   The groups didn’t meet everyday – that’s almost impossible. I remember barely making it through two rotations 2-3 times per week. The organization involved seemed overwhelming, but doable. This workshop model was modified depending on how the teacher organized their math class. After a couple of years the district changed it’s focus to emphasize reading instruction. One small part of the reading instruction is designed for students to share their understanding with others. After hearing about this type of model I decided to merge this type of model within my math classroom.

As the district changed its initiatives my math model also started to change. Instead of fully devoting time to small group math practice, I decided to incorporate a math discussion within the teacher group for a portion of the time. Half of the time in the small group was used to work on direct problems associated with a standard, while the other time was set aside to discuss the math concept in detail. Over time the conversation started to eat up a larger potion of my small group time. This discussion component ended up becoming more formal after I found the conversations started to impact students’ understanding of math.  The questions that I asked were often related to vocabulary or about a particular strategy that was used to find a solution. Students were given opportunities to answer the question and ask each other questions in the process. For the most part students were on task, but I’d have to reign in or rephrase responses as needed.  I also found myself planning questions to intentionally ask during the small group time. I had to use some type of timer system to rotate groups at the right time. Most of all I felt like students were able to offer their input in a low-risk environment and discuss math while receiving some type of feedback from everyone involved. Also, students were starting to use some of our more formal math conversations in their written explanations. What I’m finding is that I need to be more intentional in creating opportunities for these classroom conversations to happen. They seem to open up additional learning opportunities that were closed off before. I feel as though slowing down the pace and delving deeper into math concepts has brought about this opportunity

Side note: I’ve also used this strategy with a whole-class discussion.  Although it’s benefiting students I need to refine the logistics of using this strategy for the entire class.  Also, I’ve experimented with Math Talks this year – definitely something that I want to explore a bit more in the next few months.

Math Conversations with Tellagami

Having intentional math conversations in the classroom can play in important role in the learning process.  These conversations involve students explaining their mathematical thinking while working with others to complete tasks.  It’s been a beneficial activity and helps students develop confidence while communicating their thinking.  In addition, I’m finding that students are becoming better at explaining their math reasoning in written form.

A few weeks ago I was reading a comment by Mary about possibly using Tellagami in the classroom.  I’ve used Tellagami for an AR scavenger hunt but haven’t yet put the app in the students’ hands.  After researching this a bit and reviewing a few Tweets related to the topic, I thought that the app might have potential in having students explain their mathematical thinking. I made an examples and present it the class earlier this week. I thought that with a few tweaks the project could help students practice having math conversations, while at the same time provide opportunities to create digital content.

The focus of this project was on math vocabulary. The students would be emphasizing math vocabulary for the current unit and use it in a practical situation. The students and I created a rubric for the project. The students added that the background should be related to the math vocabulary word and a minimum time limit be established.  The class came to consensus and decided to use this rubric going forward.

Students were then given about 15 – 20 minutes to create a background for their Tellagami project.  Students were given the opportunity to use the classroom resources to create a background.  Depending on the math vocabulary, students used whiteboards, base ten blocks, student reference books, geometric shapes, coordinate grids, and other math manipulatives in the classroom for their background. The next step in the process was to create their Gami. This didn’t take long as limited clothing and accessories options exist.  Students then wrote out a draft of what their Gami would communicate.  During the next math session students used their draft to record their own voice or used one provided on the app .  Once finished, students then reviewed the rubric, saved the project to the camera roll and uploaded the project to Showbie.  The next step is to move the projects to YouTube or Vimeo.  Overall, I feel this was time well spent and next week the class will be presenting their projects.

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