## 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.

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.

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?

## Low-Risk Formative Assessments – Kahoot

Over the past few weeks I’ve focused in on using low-risk formative assessments in the classroom.  I continue to find that these types of assessments bring out the best in students. I want my students to feel comfortable enough in class to take an educated guess without negative judgement.  Moreover, I want my students to be able to use the formative assessment and teacher feedback to improve their mathematical understanding.

In the past I would give my students a paper exit card.  A typical exit card would have a few questions on a half-sheet of paper.  The questions would relate to the concepts covered in class.  I’d gather up the sheets and write feedback on the pieces for students to read during the next class. I also found that some students weren’t willing to take a risk to showcase their skills.  They might leave a question blank or put a question mark in the blank space.  I wanted to find a way to increase the willingness of the students to take a risk.

I came across the website Kahoot.it after following a Tweet by Matt.  I explored it a bit further and found it to be very similar to Socrative.  I enjoyed using Socrative with my classes and thought that Kahoot had some potential to be used for formative assessment purposes.

After creating a teacher account I decided to browse lessons on the site. I was surprised as there were over 160 thousand quizzes in the lesson bank. Many of the lessons were shorter quizzes, but I found some to use with my math classes. The students used the iPads in the class to go to www.kahoot.it and enter the PIN. Many of the students had no problem with this.  As long as their device had an Internet browser, students could use a tablet, computer or phone to access the quiz.  Once the students all joined the quiz I started it from my computer.  The questions popped up on the whiteboard for students to see.  You can add your own pictures to the quiz.  I found this to be helpful as I took pictures of the classroom and imported them into the quiz.

Students are able to see the whiteboard and read the question.  Students answer questions on their device. Their device looks like the image below.

Students receive a certain amount of “Kahoots” for answering the questions in a certain time period.  I’m a fan of rewarding quality over speed in math so I give students the maximum time allotted.  This can be changed when creating questions.   Students pick an answer and at the end of the countdown the correct answer is revealed.  During this time I can stop the class to check the answer choices that were made.

This can be a great time to clear up student misconceptions as you can see all the responses without names.  I’ve had lengthy math discussions after completing this activity with students. I felt the conversations were rich and gave insight to student understanding.  When finished I opted to download a report for later perusal.  The report gives all the student response and how long each student took to respond to the answers.  Both of these are valuable to me as I can use the student responses to group students and differentiate instruction going forward.

Note:  I’ll still be using general exit cards in class, but I’m finding a variety of tools useful in collecting data and providing feedback to students. I’m finding that diversifying formative assessment measures has its benefits.  It also gives students a variety of options to showcase mathematical understanding.

## QR Codes and Math Stations

Providing feedback to students is important.   I find that the more specific the feedback is, the better.  Teachers use many ways to give feedback, whether that’s verbally or through written form.  Ideally, I’d like to be able to meet with every student in my class and offer them undivided individual feedback to improve understanding and enrich.  That’s not always possible so stations or workshop models become part of the classroom norm.  Math workshop models can improve opportunities to give 1:1 feedback.

During the past two weeks I’ve been using QR code activities (1) (2) for one of my math stations.  One of these activities can last 3-4 math sessions depending on the math concept being covered.  These types of stations involve questions that I’ve found through my PLN.  Some of the QR activities that are used involve scavenger hunts.  Students answer questions in groups or individually and check their answers by scanning the QR Code.  The QR code is unlike the actual teacher’s manual as student’s can’t immediately peek over to see what the answer is.

Instead, students have to scan the code to check their answer.  Students then document and turn in a sheet that indicates whether the students answer was correct or what mistake happened.  I’m looking into creating feedback codes that help students with common errors  with particular problems.  Students are also asked to write in their math journals about problems that were incorrect.  I’m using  this site to create the codes as SMS messages.  If used correctly, QR code activities can increase student reflection opportunities and engagement.  For more information or practical ideas on how to use QR codes in the classroom check out Denise and Edutopia‘s resources.

On a side note, I’m looking forward to using the idea of clickable paper in the classroom at some point.

How do you use QR codes in the classroom?

## Teacher Feedback Tools

This school year I’m emphasizing the importance of offering students meaningful feedback.  By meaningful, I mean that the feedback gives students opportunities to reflect and make better or more informed decisions in the future.  This type of feedback is especially important in math as it allows the teacher to correct misconceptions and help guide students through mathematical processes.   Feedback can come in a variety of forms, such as informal, formal, written, verbal, and even digital.  I’ve found that helping students discover mathematical processes can be accomplished through guidance and timely feedback.

I’ve made a goal this year to give meaningful written feedback to every student more often this year.  In an effort to give more direct feedback, I’ve redesigned my class schedule to include more of the following:

• Increase student-led math conferences
• Increase student collaboration opportunities
• Increase writing in student math journals
• Increase writing opportunities in math class

I believe that these events not only increase student ownership, but also give opportunities to listen and give feedback to individual students.

This week’s #mssunfun post is about “one good thing” this school year.  So for this week I’m going to showcase my newest student feedback tool.  I’m excited to use the Showbie app (free) this year to give student feedback.  Once students complete their digital projects, (e.g. Educreations project) they submit their project through Showbie.  I’m then able to view and give students verbal or written feedback on their project.

Once logged in to their Showbie account, students are then able to hear their teacher’s feedback from an iPad, iPhone, or computer.

It’s been two weeks since I introduced Showbie to my students.  I’ve received mostly positive feedback from my students and their parents.  My students continue to look forward to receiving verbal feedback.  What’s nice is that I can record the feedback anywhere and the students access the recording at a later time.  I’m probably going to grade most of their digital work through Showbie’s voice and camera functions (like taking a picture of a finished rubric).  Students and parents can then access grades and feedback on assignments throughout the year.