Seven Approaches to Learning Mathematics

This blog post is a long excerpt from a forthcoming book that will explain the IOI Process, it is long for a blog post so feel free to skip through and only read the approaches that are of interest to you. Obviously these are my interpretations of the various approaches and how they might be described using the Modern Learning Canvas. The aim of the post is to give you an appreciation of how the Modern Learning Canvas enables educators to develop a sophisticate pedagogical understanding that enables them to understand, discuss, and design innovative pedagogies.

In my previous blog post I shared how the IOI process helps educators move from an everyday folk pedagogical understanding to a more sophisticated understanding of pedagogy.

 

Before I delve into the learning and teaching of mathematics, I do want to make the point that the Modern Learning Canvas is not restricted to building pedagogical intelligence in traditional settings. Learning models visualised using the Modern Learning Canvas are not just restricted to traditional subject and age based classroom organisation. This chapter could be exploring the teaching of any other subjects, or any other pedagogical approaches and the methodology would remain the same. Indeed, the creation of the Modern Learning Canvas was as a direct need for schools and teachers to have tools and processes that allowed them to explore innovative learning and teaching approaches. Representing mathematics as an isolated subject is purely for the purpose of illustrating how the Modern Learning Canvas can be used to visualise and understand a learning model, nothing more and nothing less. Nor are these seven approaches the approaches to the learning and teaching of mathematics, I could’ve also described game-based learning anyone, simulations, … maybe in the next edition!

 

Approach 1: Instructional Teaching

The traditional approach to the learning of mathematics occurs when learners are guided through a set curriculum of mathematical concepts by an expert educator.

What do we see when we enter a classroom using this approach?

We see student sitting either in rows, or in a semicircle configuration, enabling all students a clear view of the teacher at the front of the classroom. We see that the lesson begins with the teacher modeling a new concept, ensuring that she relates the new concept to concepts that have been previously learned. The teacher typically uses a whiteboard or similar to work through a brief series of examples in order to explain the concept to the students. The teacher will most like solicit ideas from students as the problems are demonstrated, to engage them in the mathematical concept and stimulate their thinking.

Once the teacher has satisfactorily modeled the concept, the students are given an opportunity to put their new knowledge into practice. This typically involves solving mathematical problems listed in their textbook, or on photocopied worksheets. Students work individually through these problems while the teacher roams around the classroom monitoring students so as to intervene as required. Students might also seek help from the teacher directly by raising their hand to request help when they encounter a problem that they are unable to solve independently.

To finish the lesson, the teacher will again address the whole class. At this time she will focus on addressing any questions or difficult problems that the students may have encountered, reinforcing the concept and the correct approach to solve them. The teacher will then likely set homework so that the students can further practice this new mathematical concept. Finally, at the end of the delivery of a unit of concepts, the teacher would conduct a test to gauge the level of learning of the students, results of which would most likely inform their final reported grade.

How might we represent the Teacher Instruction model using the Modern Learning Canvas?

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Figure 3.1: Teacher instruction.

In Figure 3.1, you can how the major ideas of this model can be visualised. The first major idea is that learning is an individual cognitive process where repeated practice (both in class and for homework) is needed in order to consolidate knowledge. We see that independent work is valued, with rules that prohibit copying firmly in place. The responsibilities of the learners is limited to monitoring their own understanding, so as to ask for assistance when their understanding of the current topic is unclear.

The second major idea of the Teacher Instructional model is that for learning to be effective it must be structured and sequential. The teacher, with major assistance from the curriculum designer and the textbook author, determines the most appropriate sequence to deliver mathematical concepts. Additionally, when there is more than one way to solve a mathematical problem (and there invariably is) the teacher is the sole determiner of the most appropriate method. The teacher of mathematics must have strong content knowledge, so as to be able to provide clear accurate answers to any student questions.

 

Approach 2: Early Years Numeracy

The early years numeracy classroom shares some key components with teacher instruction classroom. Most notably the priority given to teacher sequencing and selection of mathematical concepts.

What do we see when we enter this classroom?

Instead of rows of tables seen in the instruction based classroom, we’ll see tables grouped together so that rather centred solely on the teacher the students can interact with each other as they learn. This differs greatly from the Instructional Teaching method, where students are more likely seated in rows, to ensure that their focus is on the teacher.

The hour long session begins with a modelling activity that lasts about ten minutes and includes questioning from the teacher to gauge how well students understand the concept and would usually reference previous concepts which are related and which the new concept builds upon.

Following the teacher-led modelling activity, the students are divided into their small groups of about five students each. These small groups are grouped according to mathematical ability and therefore the groups don’t change frequently, although they may when a topic ends and new topic starts. In this way, the students are used to working together and confident that their peers can provide them support so they can successfully complete the activity. Unlike the Teacher Led Instructional Approach, this main learning activity is more likely to involve learners using concrete materials, and simulations and games to explore mathematical concepts.

The Early Years Numeracy Model might therefore be visualised using the Modern Learning Canvas as follows:

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Figure 3.2: Early Years Numeracy Model.

How does the Modern Learning Canvas help us to identify the key similarities and key differences from teacher instruction?

The key similarity between instructional teaching and the Early Years Numeracy approach is in the pedagogical belief that the learning of mathematics is most successful when the teacher sequences the concepts and the primacy of the curriculum. In both approaches, the teacher informed by a set curriculum and insight gained through previous lessons, sets the learning agenda and goals for each lesson. In both approaches, repeated practice is valued and assessment is directed by the teacher.

The strategies used by the learners are also similar in the two learning models, in both models the learners use similar strategies to seek help and ask when they identify that they require help. However, the Enablers that support these Strategies differ markedly. While the Instructional Teaching method relies on the teacher to mediate knowledge building conversations and provide help to the learners, the Early Years Numeracy Model allows learners to seek help from and provide help to learning peers. This difference is also identified by the Learner Role, Educator Role and the Culture of the learning model.

Assessment also varies slightly, while both models use testing at the end of of a topic to assess and rate student learning, a pre-topic test is often used by the Early Years Numeracy Model to assist teachers charged with forming the like-ability groups. This also highlights another key difference between the two learning models, in that one believes that all learners will operating at the same level, the other believes that in a classroom their will be students operating a various levels and that the curriculum should be adjusted by the teacher accordingly.

 

Approach 3: Play-Based Learning

Play-based learning models are used to allow learners to experience and explore a simulated learning experiences. For example, young children might explore mathematical concepts around money and transactions by playing shop.

What do we see when we enter this classroom?

We see a space in the classroom set aside for the shop, where the students can buy and sell goods that they create, assigning a price to their items, purchasing items using play money and calculating transactions using a cash register. We might see real world occurrences… too many of the same products which are priced too highly, and therefore do not sell. We might see other products quickly run out stock, as they become the must have item, and are affordably priced.

In such an environment, the children’s play will mimic the observed real life experiences that the children witness. As they play, they take on various roles of shoppers and store keepers, and as such the rules that dictate their play mimic real world rules. The students might bring in items from home to sell in the shop, or create items at school in order to sell them to their peers. The play elements are not competitive, there is no objective to win by amassing the largest amount of money. Testing is usually not used to assess play-based learning, rather documentation is used to identify and record learning as it happens, through photos, artefacts of play and the learners own words. Typically play-based learning such as this would happen over a longer period of time, at least a number of weeks, allowing the rules of the game to be modified as the players discover new insights of the game that they are playing.

A play-based approach to learning mathematics might therefore be visualised as:

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Figure 3.3: Play-based Learning

The visualisation of Play-Based Learning through the Modern Learning Canvas clearly shows that this learning model is almost completely different to our two previous approaches. So much so that it is quite hard to know where to begin! The only similarity, is the with the priority placed on shared language and understanding related to the discovery around mathematics. Both the Early Years Numeracy Model and Play-Based Learning use whole class shared discussions in order for students to share their discoveries. Both these learning models believe that important mathematical concepts will be discovered through their activities, and that deeper learning results from understanding the experiences of other learners in addition to understanding your own experiences.

Of course the major difference between Play-Based Learning and our other two approaches is in terms of curriculum. Rather than a belief that curriculum being set by the Educator and sequential in form, this model views curriculum as emergent as it is discovered by the learner. Obviously when playing shop, it is expected that learners will encounter basic mathematical operations such addition, subtraction and multiplication, and this would be the major reason that the teacher chose the game. Yet the teacher does not expect that the curriculum needs to be encountered in a strict sequence as to understood by the students. As such it is very likely that these learners will encounter mathematical concepts, such as supply and demand, that traditional curriculums deem far outside their current ability.

A major responsibility of learners then is to question and explore, and to take creative approaches to playing the game. While there are rules, refer to the Policies component in Figure 3.3, these rules are flexible to the extent they can be changed to improve the game. To this end, a supportive culture is needed where a freedom to explore using a process of trial and error, as opposed to the repeated successful practice of our previous two approaches. To this end, the Educator helps guide the play, providing assistance and enabling students to experience success and solve their own problems.

 

Approach 4: Papert’s Mathland

Papert’s “Mathland,” which we briefly explored in Chapter One, has a lot in common with our Play-Based Learning approach. In Mathland students would learn mathematics as naturally as children who live in France learn French. Papert’s best example of what Mathland might look like is realised in the children’s programming language LOGO.

What do we see when we enter this classroom?

We’d see students programming LOGO on their laptops. They might be working individually or in pairs. We would see children trying to create with the LOGO turtle, we’d see failures and frustrations, and success and celebrations. Children would excitedly share their discoveries with their peers, and we would see them seek help from each other when they were stuck. The teacher would occasionally stop the class so a student could share what they had discovered was possible, so that the whole class could benefit from the discovery.

With LOGO children are encouraged to take on the role of the LOGO turtle, a programmable object, from which they can they explore a mathematical environment as the create and solve problems. Initially learners control their turtles using the programming commands, such as move forward and turn right, to simply explore what the turtle can do. In LOGO these explorations largely involve creating images and artworks based around geometric shapes, created by writing simple iterative programs. Empowered by this ability to write powerful iterative programs, define algorithms and develop theorems, these learners can then use LOGO to design experiments to solve problems and answer questions of the world outside the LOGO environment.

Immersive environments such as LOGO are seen as useful for learning in areas where it is otherwise dangerous or difficult to do so. Battlefield simulation are used by defence force personal where the potential for injury is high. Aircraft simulators are used by pilots to train for much lower costs. It has been suggested that World of Warcraft is a similar natural learning environment, but unlike LOGO which is designed for the exploration of mathematics, they point to how players who play in teams can explore and master leadership skills.

Papert’s Mathland might be visualised using the Modern Learning Canvas as follows:

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Figure 3.4: Mathland

As you can see from the visualised learning model, Papert’s Mathland has many parallels with play-based learning. In both models, learner exploration and trial and error, and learner from and with peers feature heavily. These learning models are favoured by those that believe learning is not necessarily linear, and that outcomes should emerge as needed as being predefined. The sharing and celebration of discoveries and learning is also crucial to both models, so that learners can not only build upon their own knowledge but also the shared knowledge of their peers.

The Educator Role is similar to their role in the Play-Based Learning model, in that their role is to support and guide the exploration of Mathland. The Educator helps facilitate whole class sharing and language as it emerges, as opposed to directing it in the Teacher Instruction model. They follow what their students are doing, looking for breakthrough opportunities to celebrate. As such the Learner’s Role requires them to playfully explore, taking on the role of the LOGO turtle. Rather than look to the teacher or their peers for direction, these students need to be self motivated and self directed in their explorations of what the LOGO turtle can do.

The nature of the LOGO environment, with its fixed rules based on mathematical concepts, means that the environment is not able to be modified and the rules and policies are not able to be as they can be in the play-based learning environment where rules are negotiated and governed by imagination. In this way, Mathland has narrower learning outcomes than play-based learning, with its focus on computational thinking and mathematics theorems, and less focus on social and other interpersonal skills.

 

Approach 5: Authentic Projects

If Papert’s Mathland is an attempt to allow learners to immerse themselves in an authentic mathematical world in which to explore mathematics, then the Authentic Projects learning model (for want of a better label) suggests that the real world is also useful for this.

What do we see when we enter this classroom?

We see students working individually or in groups, on hands on activities. We see them working on different projects and using different tools, some might be using multimedia, others writing, and yet others using concrete materials. Students interact with each other, asking for help and giving advice

Typically an Authentic Project would take the form of students, identifying a problem that needs to be solved, designing a solution for the problem, creating the solution and then finally presenting their finished solution. Usually, the teacher would define the scope of the problem and by doing so define a certain set of parameters that might be addressed as part of the problem and solution. Yet ultimately, the student or group of students make all of the decisions about the project

The teacher might offer structured learning opportunities to provide instruction on a topic or teach a skill. Non-negotiable instruction would cover content that the teacher decided their students should know about, while negotiated instruction would cover content students identified as being necessary for them to solve their problem. These negotiated and non-negotiated learning opportunities mimic professional learning in the workforce, where workers access structured learning opportunities, some at the direction of their employer and some based on their own assessment of their needs.

These non-negotiable learning opportunities would most likely contain mathematic concepts, where the teacher would introduce processes and tools which the students could then use in their projects. In this way, students would use mathematics in authentication situations to solve real problems. During the creation stage, the teacher supports and mentors their students, providing feedback and assistance, and identifying potential pitfalls.

An Authentic Projects learning model might be visualised using the Modern Learning Canvas as follows:

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Fig 3.5 Learning Model depicting an Authentic Project

The Authentic Projects learning model is also similar to Play-Based Learning and Mathland, in its Strategies of discovery and trial and error. The Pedagogical Beliefs with a focus on experiential learning, and rejection of a strictly sequenced curriculum in favour of an emergent authentic curriculum is also similar.

We do however see some differences in the Learner Role and the Educator Role largely due to the removal of the artificial construct of LOGO and the rules of play. As such the student now has an expanded role and makes all major decisions about their learning, what problems they focus on, how they tackle the problem, and what is produced as the solution. As a consequence there is a very strong likelihood that the content of the problem may be outside of the teacher’s knowledge domain. Rather than providing scaffolding to understand the mathematical rules of the LOGO world, or the rules of the game of shop, the teacher now acts as a mentor and provides methodological guidance. The key is to help their students make better decisions through all facets of the project journey.

The Authentic Project model is unique in that it produces, a finite product that illuminates the success of the student’s learning. However, like Mathland and Play-Based Learning models the problem solving and project management competencies that the students display while solving problem, make up the bulk of the evidence of successful learning.

 

Approach 6: Massive Open Online Courses

Massive Open Online Courses are an attempt to cheaply bring scalable education to the masses. In chapter one, we had a brief look at Sebastian Thurn’s Remedial Math MOOC, and now we’ll return to explore how MOOCs might be used to learn mathematics.

MOOCs are designed to mimic the typical university course model but to leverage the scaling opportunities provided by being totally online. They have a prescribed curriculum that learners must work through, which is sequenced by the course lecturer and released to course participants week by week. Course participants are required to watch these videos during the week they are released, resulting in all participants engaging with the material at roughly the same time. Video lectures are generally shorter than would occur in an on campus course, with lectures generally under 30 minutes in length.

Lectures may be interrupted periodically by multiple choice questions which are used to assist the learner to self-assess their understanding. With the learner given immediate feedback on whether their answer was correct. Learners are also able to use the MOOC discussion forums to ask questions of other students and to generally engage in meaning making conversations. Assignments might be featured throughout the course, which form part of the final assessment. In the case of Udacity’s Remedial Math MOOC a final exam was used to assess student learning of the course content.

The MOOC learning model might be visualised using the Modern Learning Canvas as follows:

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Fig 3.6 Learning Model depicting a Mathematics MOOC

The MOOC learning model shares some similarities with the first two learning models that we examined, Teacher Instruction and the Early Years Numeracy approach, in the priority of a set, sequenced curriculum. The lecturer is solely responsible for determining what their students need to know, and learning is seen solely as cognitive activity. Like the Teacher Instruction model, all students work on the same mathematical concepts at the same time. The weekly release of lectures means that students are not able to skip ahead in the course, nor are the able to slow the release of the weekly sessions without getting too far behind. Repeated practice is also valued, but this practice is performed in isolation from the teacher with sole responsibility on the learner to recognise and rectify any misunderstandings.

There is also a recognition that knowledge building conversation amongst peers is important for clarification and testing understanding, with the discussion forums provided for students to engage in these kinds of learning conversations. The educator doesn’t play any role in these discussion, which is in stark contrast to the Teacher Instruction model where almost all conversations go through the educator. Also, unlike the Early Years Numeracy model, there isn’t any importance placed on developing shared language and understanding. Individual understanding is all that matters, as they attempt to align their understanding with that of the course lecturer.

This learning model has the most learner autonomy of the seven approaches to learning mathematics that we’re exploring. Learners are responsible for auditing their own understanding, to re-watch video lectures if any concepts are unclear, and to identify when they need to engage in the discussion forums. Some MOOCs do employ tutors to monitor discussion forums and provide official answers as the need arises, but these are the exception and is only possible when the number of course participants is small.

 

Approach 7: Khan Academy

Khan Academy is an online video tuition website with over 6,000 short video lectures and over 100,000 practice problems. Although Khan Academy has been primarily designed for learners to learn mathematics independently, research into the use of Khan Academy in schools published in March 2014, shows that this is not how schools use it. For our visualising of how Khan Academy proposes that students learn mathematics we will focus on how the site intends people to use, rather than how schools are actually using it. For the Educator Role, Policies, and Pedagogical Beliefs, I will be considering Khan Academy the educator.

When a learner logs in to Khan Academy they are faced with a list of subjects with the question “What do you want to learn?” Learners can choose to learn Math, Science, Humanities, Economics and Finance, and Computer Science. Each of these is further divided into sub-categories. There are slightly different approaches that Khan Academy uses according to the different subjects. As we are investigating different approaches to learning mathematics, we’ll explore the subject of mathematics.

After choosing a sub-category the learner is presented with a series of pretest questions so that Khan Academy can identify what you already know. After completing the test your results are displayed reporting the various skills and their levels that the learner has mastered. From here the learner is presented with a list of skills which the learner can practice, alternatively the learner can search from the 482 mathematical skills that Khan Academy has identified.

When learning a mathematical concept or skill, learners are presented with questions which are then automatically assessed as right or wrong. A learner is said to have mastered a skill when they are able to correctly answer five questions in a row. Video lessons, which describe the process of the mathematical skill, are available for the learner to reference at any time.

The Khan Academy learning model might be visualised using the Modern Learning Canvas as follows:

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Fig 3.7 Learning Model depicting a Khan Academy’s approach to learning mathematics

Like almost all of our learning models, except for Play-Based Learning and Mathland, the Khan Academy believes that mathematical knowledge needs to be sequenced by an expert in order for successful learning. It also views learning as a purely cognitive exercise, learners watch video lessons and then use their new knowledge to correctly answer questions. Mathematical concepts are seen as discrete units, which need to be repetitively practiced until their are mastered. All Khan Academy students learn the same concepts in the same way.

Unlike the MOOC learning model, learners do not have to audit their understanding, instead the Khan Academy questions give immediate feedback, notifying the student if they are right or if they are wrong. The dashboard displays their complete learning history, what they know and can do, and what they don’t know and can’t do. That said, learners can work at their own pace and re-watch video lessons for clarity in the same way that MOOC participants can re-watch the video lectures.

Khan Academy, in its role of the educator, is responsible for providing everything that the student needs. There is no scope for knowledge building conversations between other learners as shared knowledge is not valued at all. Learners do have some choice in what they learn, in that they can leave a mathematical skill without mastering it, and then return to it at a later stage.

 

Teams at the IOI Weekend will use the Modern Learning Canvas to understand, discuss and design innovation learning and teaching practice.

A free fast paced three-hour taster on Friday night will provide you with a shorter experience of the IOI Weekend. This is a free event and will be held at:

May 15th 6PM – 9PM at New Era Melbourne
Level 2 141 Capel Street North Melbourne VIC 3051

Over three hours we will give you a taste the IOI Process highlighting:

  • IOI Pedagogical Quality Framework,
  • IOI Learner Development Profile,
  • the Modern Learning Canvas,
  • how pedagogical quality, effectiveness and capacity can be measured,
  • and get you on your way to develop an Innovation Thesis.

Please RSVP to if you intended to join us to help us with catering (light finger food and drinks.)

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