Category Archives: collaborative learning

posted on November 26, 2018

I was always concerned in my elementary classrooms when I realized that my students principally saw only finished product in their lives – film, novel, technology, building, business. When are young students these days ever exposed to process other than through concerted efforts on the part of educators to show, for example, how stories are deliberately constructed? Children meeting professional authors offers an essential insight into how ideas are formed, massaged, written down, and ultimately formed into a story through diligent effort and many re-writes. Such is another limited example of students seeing process. In previous times, children were directly involved in family businesses or farming where they were personally involved in the process of production. How does something start, and how do people get it to completion? Not knowing how to connect a topic of study to its practical outcome in the ‘real world’ could lead students to question its relevance to them.

Some of the steps involved in what I will call “the creation of something” are:

– the idea itself
– is it a good idea or not? What are its benefits to me or others?
– is the idea an adaptation? I need to draw information from various sources
– working the idea to a place where it can be explained to others
– seeing if the idea already exists, and if/how this impacts our moving forward
– ascertaining what we need to complete the project
– getting others to cooperate in the execution of the project
– assembling the resources to complete the project
– trying and failing
– trying again and failing
– not getting despondent and keeping resources focused on the project
– trying again, and hopefully succeeding

These are but a few steps in general idea and project conception, process, and completion.

In elementary mathematics, providing opportunities for students to use their hands and minds in a cooperative learning environment are somewhat limited. To that end, we have invented two products for elementary education which help children work physically towards solutions either individually, or in groups.

Aspects of our manipulative products are discussed below:

intooba – intooba HERE

This K-6 construction manipulative offers many lessons in both specific mathematical concepts such as shapes, estimation, and fractions and in engineering challenges. In math, we provide a spiraled program offering teachers instructional ideas in many math concepts. In engineering, we offer students over 25 construction challenges each with varying levels of complexity.
This is hands-on work asking for actionable solutions through construction and mathematics. Resource constraints such as assigning a project budget and input costs makes this learning very relevant.

diskii – diskii HERE

This K-4 manipulative offers students and teachers opportunities to explore mathematical concepts hands-on either individually or in groups. Our ten tokens (essentially representing 0-9) have unique names, faces, and colors offering many layers of complexity in problem construction. Instructional possibilities include logical reasoning, understanding the equals sign, and substituting a token for an unknown algebraic quantity x as in: token + 4 = 10 for lower grades.

Drawing on information to solve a problem is a critical skill. Using diskii, equations can be created where certain token values are given, and students have to use this information to solve for the unknown tokens.

We have created these advanced thinking manipulatives with lesson support to offer easy-to-implement hands-on group problem solving opportunities we feel are unavailable in many physical manipulatives available in classrooms today. We focus on fun, creative visual challenges making math relevant in concrete ways to elementary learners.

In our manipulatives, we are diligently focusing on process as an integrally important part of learning. Students work cooperatively with their hands and minds to solve challenges.
In this environment, drawing on spiraled learning, students have opportunities to produce increasingly complex product through their own efforts.

posted on July 2, 2018

Spiraling through the many years of a K-12 education are various levels of math, language, writing, social studies, the sciences, and the traditional topics we expect in such an environment. The education community is pushing towards STEM/STEAM (Science, Technology, Engineering, Math – with the inclusion of Art) with the realization that the space in which students will eventually be employed has changed dramatically. Not only has the environment changed with advances in technology, globalization, access to information, social and political change, and environmental challenges, but the pace of change is increasing. We are now trying to integrate the disciplines with a view to increased creativity through cross-discipline pollination. Is this enough? Are we fulfilling our essential and critical responsibility in providing opportunities to enable our children to operate successfully in this environment?

Essential features in education looking forward should include:

• learning to correctly and accurately find, describe, and solve complex challenges
• attaining the skills required to work successfully in groups
• the ability to welcome and work through adjustments, modifications, and failures
• demonstrating to students the practical applications of their classroom learning

To these ends, we at Blue Egg Advisory Group LLC have invented and developed a unique manipulative to foster these concepts in young learners (K-6) as they prepare for secondary school and beyond. Our INTOOBA Construction Kit manipulative offers many opportunities for hands-on experimentation through leveled curriculum manuals in math and engineering.

posted on April 24, 2018

BCILD is embarking on collaborative projects to investigate both best practices and innovative ideas in the teaching of mathematics (K-12).

Firstly, we have developed a relationship with Italian high school mathematics teacher, and Global Teacher Prize finalist, Lorella Carimali. Lorella’s biography is listed below, as well as a translation of a recent presentation Lorella gave in Italy.

Secondly, we are collaborating on a math podcast project with the Dawson School in Lafayette, CO. Dawson middle school students are interviewing and producing podcasts with adults in the community on the topic of how they currently use math in their daily lives. This activity helps students understand and appreciate the importance of mathematics in the ‘real world’, and gives them skills in interviewing and producing podcasts. We will provide further details on this project as it unfolds.

BCILD is interested in investigating best practices in the teaching of mathematics and sharing these ideas with teachers internationally. We present Lorella Carimali here as an introduction to an ongoing learning dialogue with her about her teaching philosophy and teaching practice.

Lorella Carimali HERE

Lorella Carimali – Mathematics Presentation – 2017

To develop an all-inclusive classroom setting which encourages the understanding of mathematics in a holistic way, we need to take account of both the process of being involved with a mathematical problem from start to finish and develop a comprehensive understanding of the discipline. Becoming a competent mathematician involves not merely the memorization of facts, but entails a deeper development and enjoyment of confidently utilizing intuitive skills and creativity in the process of problem solving. ‘Creative enthusiasm’ must become part of the learning process in mathematics, and it can be used across other learning disciplines as well.

A community of learners engages high-achieving students in the process of teaching their colleagues. In this way, the high-achievers refine their own thought processes through improved articulation while the students who find the topic more challenging feel included in the learning process.

Student mistakes in approach or calculation are not seen as negatives in the inclusive classroom but are instead seen as opportunities to learn. Mistakes open doors rather than close them. Use mistakes to motivate students. No student should ever feel inferior because they make mistakes. Students should see the usefulness of mistakes in their process of academic growth.

The methodology used in the teaching of mathematics should be neither mechanical nor automatic. Teachers need to use creative materials and approach students in a way that reinforces the idea that student intelligence and ability is not fixed, but rather something malleable and changeable with practice, concentration, and support. Teachers need to put tools in the hands of students, and support those tools by providing the student with an awareness of their own cognitive processes and abilities. Success in this endeavor is achieved when everyone in the classroom is equally involved, leaving nobody behind.

Points to consider:

(1) Teach the process before you present the problem. Use imagination and creativity in your practice. Each individual has an area of personal growth. Potential does not always lead to actual output.

(2) The teacher should act as a change agent for the student, uncovering collaboratively the student’s personal cognitive processes. Autonomous thinking depends on understanding your own capacity and process.

(3) The teacher and student should always have the same mindset and be available for growth opportunities.

(4) Teachers need to work with students in understanding the entire process of mathematics from problem understanding, to working through the process of solving the problem, to being able to articulate their thinking and problem solving processes to others. High-achievers need to have the ability to explain how they process their understanding.

Integral to the learning success of every student is having their progress carefully and consistently monitored each and every day, thereby fostering continuous engagement between the student and the teacher.

In teacher planning and preparation, plan your approach around who you have in the classroom as learners. Leverage the skills and abilities of each student to both their own, and the group’s, benefit.

An interesting approach with students would be to create a theatrical play around mathematics learning because it is important to develop skills such as intuition, imagination, projecting, theorizing, deducing, controlling and evaluation to be able, in a second moment, to ordinate, quantify and measure sustainability of facts and phenomenon belonging to reality.

I experimented with this model in different contexts, applying it following the scientific method and I could verify that it works and that it is replicable.

Students could express their discomfort with mathematics in a theatrical fashion, and they could safely explore ways to explain concepts and understandings to their peers. Play creation is similar to mathematics in that concentration, considering each detail, curiosity, routine, and repetition are all involved. In plays, as in mathematics, you recognize your own limitations, and can view them as opportunities for growth.

Studying math means essentially to learn to think mathematically and to become conscious of one’s own reasoning method, mastering procedures and not simply applying them, and therefore being able to transfer to other contexts the mathematical skills assimilated. Another approach would be to create a game in which students create a company where they have to use mathematics in building a business and making appropriate decisions regarding marketing, management, and the financing of a business.

A student wrote: “she made me re-evaluate math by teaching me to see it not as a set of formulas but as a way to see life and be able to simplify it by reasoning”.

Confidence is competence. Confidence is built through each student knowing their own mental processing capacity. What did you do? How did you do it? Why did you do it? The sharing of these insights between students is vital to a student’s overall competency. The students develop into a team working together where mistakes are not ostracized, but instead act as learning opportunities. Sharing is a process that involves everyone, with all participants feeling included and having the ability to refine and define their own thinking. In this way, we can make a comparison between students and the individual players in an orchestra: they sound good playing on their own, but sound much better as a symphony.

posted on September 1, 2016

In integrating collaborative work in our approach to K-12 education, we are teaching students the benefit of listening, learning from others, coming to consensus on ideas, and other group dynamics. These are all essential skills in group problem solving exercises in school, and in adult project based work. What we should not lose sight of is both valuing the individual as a contributor, and building personal communication skills thereby promoting effective group dialogue and collaboration. It takes a very skilled teacher to nurture the individual as a person of abilities, aptitudes, and evolving capacity while at the same time teaching effective group dynamics. Productive group work is predicated upon individual skills in communication, and group skills in collaboration. We want children to know that their opinions and observations are highly valued, we want them to have the skills to communicate them effectively, and we develop collaborative skills to make project based learning effective.

In early development of these skills, it may well be the case that using physical manipulatives in the classroom facilitates the development of communication skills across curriculum topics. As personal skills in, for example, vocabulary, persuasion, reasoning, and advanced thinking develop, children could use manipulatives to assist them in communicating their ideas with peers. This is evidenced in the example of Kim Haines, 4th grade teacher at Dawson School in Lafayette, CO who used the INTOOBA Construction Kit in developing communication skills in listening, giving directions, providing clarification, and in either being a giver or receiver of information in her math class:


Essentially, teachers can observe individual thinking and development of these essential skills through the use of manipulatives while also noting the child’s functioning within a collaborative setting. Children here are supported in the learning of specific collaborative language through the use of their hands.