Category Archives: Teaching and learning theory

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 May 9, 2018

Students should bring as many of their senses to bear on a topic in order to enhance their learning. All information entering the brain for processing, understanding, manipulation and output comes through the use of one, or more, of the five senses. As an interesting and engaging add-on to the curriculum, I used smell to teach Math in my classroom. Virtually any topic can be covered. I usually used an exercise of multiplying two digits by two digits, by assigning a smell to a number. The smells are cooking extracts from the supermarket.

In this way, children are able to place a math problem in their brains through their noses as opposed to the more traditional auditory or visual approaches. Students are highly engaged and interested in this activity.

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 March 22, 2018

Logical thinking, the process of moving sequentially from one thought to another in order to reach a conclusion or solve a problem, is an essential skill. Teaching this complicated process to young students can be challenging. However, using the INTOOBA Construction Kit, students can construct physical models and see an explanation of how logical thinking works.

Exercise: teacher assigns values to the rods and connectors. Students are asked to logically establish what is missing in the figure and what they need to complete the construction. They are asked to give values to what they see, what is missing, and what a completed piece would be worth. Students are asked to complete the model, and share their thinking process with the group.

Students use logic to establish what is missing, and then use their hands to solve the challenge.


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.

posted on March 27, 2015

When rote learning education models closely mirrored early industrial production based economies, outputs in the classroom followed the teacher driven input- predetermined output style. Modern theories of education include a far greater degree of student involvement in their learning. Additionally, in an attempt to mirror rapidly changing political, environmental, and economic factors, problems presented to children often involve a necessity for self-directed inquiry where outcomes are far from preset. In such an environment, it is vital to provide an opportunity for children from an early age to interact with information inputs in the process stage of learning.

As such, developing ways for children to understand what process is includes opportunities to work mindfully in this space. Examples of process learning might include working with mathematical manipulatives, reading a recipe and making a cake, investigating the scientific method in a collaboration on a specific topic with a university professor/researcher while in high school, or identifying and investigating a social need and developing a product to meet that need. Processes already exist in many areas of education, but seldom are they identified as a specific learning goal in and of themselves. Design thinking, critical thinking, art appreciation and discussion, the scientific method, negotiation, and conflict resolution are just some of the topics which may fall under this category. Applying these techniques to multidisciplinary subject areas enhances students’ capacity to solve problems creatively.

Students need to delve into, and work through, process in order to grapple with the complexities and demands of modern society. Process inquiry meets the curiosity of students who are always asking why and how in their daily learning.

posted on October 22, 2014

I was very interested to read the attached Mindshift article about how we view student academic struggle in schools. Eastern cultures appear to see struggle as an integral part of the learning process, whereas the article states that western cultures see struggle as a weakness, or something needing immediate correction. It would be interesting to have a discussion linking these perspectives with Angela Duckworth’s idea on developing grit.

Mindshift article HERE

posted on October 2, 2014

How much more engaging would our teaching of young people be if we paid careful attention to how we as humans process sensory inputs, delivered an understanding to our students of how our brains work, and had a healthy grasp of the connection between the brain, emotions, and feelings?

I was fascinated to read about the work of Antonio Damasio in the MIT Technology Review magazine (June 17, 2104) recently. We all realize the vital importance of grit, tenacity and dogged determination in learning. However, being able to connect our understanding of the fundamentals of brain mechanics and human perception with the curriculum is very compelling.

Click HERE for MIT Technology Review:

Click HERE for TED talk


posted on February 9, 2014

Central to our success as humans is learning, both by ourselves, and from others. Inextricably tied to our learning is our capacity to do so, and the skillfulness with which we are able to transmit understanding. Hattie and Yates have expertly crafted and condensed copious amounts of research data and analysis into succinct and user-friendly chapters on topics of vital current interest to teachers and administrators. Their layout provides actionable summaries supported by clear units of study. They realize that teachers are not merely sterile conduits of information, nor are students generic recipients of information. Central to their argument is that a keen understanding of the process through which we teach is vital to content transmission, accessibility, understandability, and eventual student ownership. This includes building trusting relationships, giving cognitive load-appropriate lessons, and providing cogent feedback.

The authors additionally pay careful attention to the role and experience of the individual learner. Topics covered include the need for deliberate goal-oriented practice, understanding how to effectively engage memory for information storage and later retrieval, and giving credence to student learning styles.

posted on October 21, 2013

I have found it extremely useful to assist children to think of themselves as independent people. Quite obviously, pre-adolescent children are not independent in many respects, but creating an environment where they think of themselves as independent opens doors to many positive outcomes in the classroom. An independent young person is responsible for personal thoughts and actions, assignment planners, and homework. A vital and shared component of all these, and other, activities is the ability and responsibility to make decisions. Establishing oneself as a decision maker sets the stage for life as an independent person.

A very useful activity to undertake on the first day of school is to provide students with a blank sheet of paper approximately 18 inches long by 12 inches wide. On the board in front of the class write the word BORN with a period under it. Proceed to explain to the children that you want them to make a DECISION TREE outlining all the important decisions they have made in their lives to date. A decision tree will have branches for paths both taken and not taken. Paths not taken will end there, and paths taken lead to a continuation down the decision tree of life to the present day. It is imperative not to give too many ideas to the students at this point as to what specifically constitutes an important decisions. Of necessity for the purposes of this exercise, students should be thinking carefully through what decisions they think they have made over the course of their lives to date. Leaving it very open-ended without much guidance tends to produce varied and interesting results. Such results might include what school they ‘chose’ to go to from amongst alternatives they have heard of, what instruments or after-school activities they chose to do, or possibly where they have lived if they have moved apartments or cities. Whether they themselves actually made these decisions is immaterial. What is  important is for them to think about making decisions, and to have some defined course they followed over their young lives. They should see that a large part of our human experience is essentially selecting from amongst a series of options. Once you establish yourself as a decision maker, then you are an independent person with regard to thought and responsibility for actions.

At times, one could give a few hints to those individuals who are struggling with the project in order to get them into a productive mode of thinking through this exercise. Once all students have finished this project, have students share their thoughts and ideas with each other. Studying the Robert Frost poem, The Road Not Taken at this juncture would be a very useful exercise.

After completing and discussing the poem, ask the students to turn over the sheet of paper upon which they have outlined their major life decisions to date. Again go the classroom board and write the word FUTURE on it with a period under it. Explain again the concept of a DECISION TREE to the students. This time, however, ask them to put together a set of possible scenarios of major decisions they think they will make for the remainder of their lives. Leave this discussion as vague as possible in order to foster creativity. Interesting outcomes here have been college, marriage, career(s), vehicles, sports, number of children, retirement, travel and in some instances even death. Here again, we are establishing the concept of independence through the ability to make decisions.