# Wendy Goodman & Maurice McFadden, Tyendinaga Public School, Hasting and Prince Edward DSB

## Engaging students through VNPS with rich probems

Utilizing low-floor, high-ceiling rich tasks in math class is a great way to engage the diverse learners in the classroom. One approach is to implement problem solving through Vertical Non-Permanent Surfaces (VNPS), or as staff and students of Tyendinaga Public School fondly call them... “Wall Problems”!

The strategy of using VNPS is one aspect of Peter Liljedahl's work on Building Thinking Classrooms. WipeBook Flipchart and the use of dry erase markers are great for supporting this strategy.

They allow students to record their thinking and problem solving in a risk-free manner where they can adjust strategies as they tackle the problem. Since the work is done vertically it is easy for the teacher to monitor student thinking, as well as making thinking visible to the other student groups in the classroom.

## The tax collector: Making thinking visible

The Tax Collector problem is an example of a low-floor, high-ceiling rich task. This is a problem modified from the New York Times “NumberPlay” column:

Start with a collection of paychecks, from \$1 to \$12. You can choose any paycheck to keep. Once you choose, the tax collector gets all paychecks remaining that are factors of the number you chose. The tax collector must receive payment after every move. If you have no moves that give the tax collector a paycheck, then the game is over and the tax collector gets all the remaining paychecks. The goal is to beat the tax collector.

The students need to think critically about how the Tax Collector collects the 12 paychecks (based on factors) - in fact it is best if you play a round or two against the students so they can discover on their own that the factors are the key aspect of the problem. The challenge is then increased as students are tasked with determining how they can beat the Tax Collector.

Further challenge can be provided to those who solve the initial problem by increasing the number of paychecks presented (eg. 18, 24, etc.). In the end, the lesson can be consolidated with the students to highlight the strategy for beating the Tax Collector, and the best first move - taking the highest prime number. The vertical nature of such tasks make it easy for the teacher to facilitate the consolidation of the task and connect elements of the solutions to support students in developing new strategies as they learn from each other.

## Wipebook flipcharts provide students with lots of vertical surfaces for VNPS and thinking classroom

During these types of rich tasks the WipeBook Flipcharts are pivotal in creating flexible vertical surfaces to provide students space to organize and record their collective thinking. The WipeBook Flipcharts are easy to put up in the classroom and instantly create “thinking space” for students, where previously a space didn’t exist. Due to the vertical nature of these tasks, the teacher, at a glance, can see the thinking of the students.

This visually displayed information, in conjunction with the observations and conversations that take place as the teacher monitors the task, is used to inform the assessment for learning process so that the teacher can formulate next instructional steps to support student learning.

## Collaboration and communication: Teachers' thoughts on using VNPS and thinkingclassroom

Grade 6/7 teacher, Moe McFadden reports that “The WipeBook Flipcharts allow students to orally communicate their thoughts and strategies with each other in a small setting. It also allows for misconceptions to be cleared up, discourse to occur and for students to actively engage with the math and feel comfortable taking risks.”

Principal, Jen Hawkins shared, “Using VNPS allows for the transfer of knowledge around the room. The students get to see the work of other students and they learn from their peers. It allows the students to pave the path of numeracy learning as they engage in mathematical discourse with peers and share new strategies for solving rich problems. Their voices are heard, they are engaged, and teachers can be more responsive based on hearing their students’ thinking and thought processes shared openly.”