Sunday, September 22, 2013

Differentiate in the Math Classroom with Learner Agreements

The readiness levels of our students vary year to year, chapter to chapter, skill to skill. Learner agreements, also known as learning contracts, can help address the various readiness levels throughout a unit of study. Curricular resources and supplemental resources were used to craft this learner agreement. This learner agreement is not the instruction that takes place rather the independent work where students are asked to show proficiency of the standard. Click on the image below and take a look at the learner agreement that was created for fifth grade.

   Graphics by Anchor Me Designs
Learner agreements encourage students to be regulators of their own learning. Students need to be asking themselves...Am I getting this? Do I understand the skill enough to apply it to more complex tasks? Learner agreements provide those students who may be grasping a concept quickly with a more engaging task that they can work through more independently. Rather than wait through repetitious instruction that is at a pace that is not appropriate for their readiness levels, students can be exited out and begin a task that will help address their needs.

So how do learner agreements work? Preassessments can be given prior to a unit of study to get a pulse of where the students are reading to learn. Daily observations and formative assessments also can be used to guide student learning. Learner agreements take some planning and foresight. Standards need to be looked at, student readiness levels need to be considered, materials need to be gathered, and tasks need to be decided upon. It is only then that the learner agreement can be created. However, once created, they are a perfect way to differentiate learning on the spot.

The learner agreement can be viewed as a blueprint for learning. Tiering the tasks allows for multiple entry points for learning. After instruction and depending on the readiness level and the comfort level of each student, students can be assigned a differentiated task to complete. Those students who are more proficient at a particular skill may enter the learner agreement at the progressing level. If they encounter difficulties, they can advocate for themselves and go back to look at/do some of the learning-leveled tasks to ensure foundational understanding is established. Or if a student is demonstrating proficiency at the progressing level, s/he may bump up to the extending-leveled tasks. If students are struggling at the learning-leveled tasks, a small group can be formed with teacher support while other students work at their readiness levels as guided by the learner agreement.

It is important to keep movement through the leveled tasks fluid and based on where students are ready to learn at that moment in time. Keeping in mind the zone of proximal development for students, we do not want tasks to be too easy or too difficult. Some students face tasks that are already in their comfort zone and are not getting the challenge they need. Other students may face tasks where they simply are not developing the understanding necessary.  Tasks need to be "just right." For learning to take place, there needs to be a certain level of challenge. That challenge has to allow for the development of new learnings and skills.

Is this a strategy that might work with your students? What other ways do you differentiate for students in the math classroom?

Monday, September 16, 2013

Concrete, Representational, Abstract (CRA) - Sequence of Instruction

As we work to help students develop deeper conceptual understanding in math, CRA can be helpful. CRA is a system of instruction from the concrete to the abstract where students are first introduced to a concept through concrete experiences. In the area of math that means using concrete objects or manipulatives. Instruction then moves to representational experiences where drawings are used. After the concrete and representational levels are practiced and understood, then students are asked to work with abstract symbols, notation, and numbers in math. The abstract level is what we know as standard algorithms or standard notation. Through the instructional experience of CRA, students can make stronger connections and develop deeper understandings.

When we use concrete experiences with our math students, we help them to develop foundational understandings of a skill or a concept. Think about when students are first introduced to measurement. We do not put a ruler in their hands right away. Students often have experiences measuring with counters, cubes, or links. Using these concrete manipulatives helps to develop an understanding of length. Concrete experiences should be embedded in instruction when new concepts/skills are introduced.

The representational level, sometimes referred to as semi-concrete, is when students draw what would be represented by the concrete objects/manipulatives. For example if students are using counters to make groups when learning about multiplication at the concrete level, students could then draw Xs, dots, or even use mini-stampers (What fun!) to create a visual representation of  counters. The representational level helps to bridge understanding from the concrete to the abstract for students.

Following representational experiences, students can work with a math concept/skill abstractly. Symbols, numbers, and notation are loaded with meaning in math. If students jump to the abstract level before having the concrete and representational experiences there can be a breakdown in understanding. Consider the different standards below and how the activities/strategies focus on the different stages of the CRA framework. Click on the image to grab a copy.


When considering using technology with students, choose programs that are at the appropriate readiness levels for students. View the chart below to see how you can structure technology use at the different stages of understanding for a concept/skill. Click on the image below to download a copy. Visit some of the sites to see the progression of understanding from the concrete to the abstract.


How do you weave these different levels of instruction into your daily routines?