Monday, 17 August 2015

Making Math Relevant: A Holistic Approach to Math Instruction in completion of the Final Assignment for OISE CTL 1799 Summer, 2015


The following blog post contains four 'separate' articles on the topic of Holistic approaches to enhancing Mathematics in the classroom.  Together, they form a series of articles that will be posted sequentially.  

They appear here in their entirety to satisfy the requirements for the final assignment in "Holistic Approaches to Education" CTL1799 at the Ontario Institute for Studies in Education (OISE), Summer, 2015 with Dr. Selia Karsten.

Afterwards, each article will be re-posted separately in a weekly series through the Autumn of 2015 on the website Please feel free to check that website for updates to the article series.  Thanks. ~ D. Ferris

Making Math Relevant Post #1:

Is there a place for ‘Holistic Education’ in the Math Classroom?

"What's in an egg"? Big Ideas help to meet the needs of the whole person with Holistic Education

At heart, all learning, even Math learning, is personal.  Students never really ‘learn’ without that personal connection to the curriculum.  Educators wonder, “How can we help every learner make that deep connection, especially in a conceptually driven area like Math?”
A complex array of factors work together to create deep connection in the math classroom, including ‘Holistic Education,’ an approach that focuses on meeting the needs of the “whole child”.  John Miller explains it this way:  
“Holistic education is an approach aimed at teaching the whole person. Holistic educators reject educational approaches that limit learning solely to the intellect or that train students so that they can compete in a global economy. They believe that we must see the student as a complete human being” (Neves, 2009, p. 8).
Miller’s theory sounds like the heart and soul of passionate teaching, but is it even possible? After all, The Ontario Curriculum seems to tilt in the other direction as it:
“aims to challenge all students by including expectations that require them to use higher-order thinking skills and to make connections between related mathematical concepts and between mathematics, other disciplines, and the real world” (Ministry of Ontario, 2005, p. 3).  
With expectations, achievement levels, and standardized testing, schools appear to focus exclusively on ‘conceptual’ academics, so that students can compete globally, which isn’t really a bag thing, either.  But it isn’t enough. 
Then, the question arises: can global competence and personal relevance co-exist?
Researcher Ana Cristina Neves thinks so.  “In the end,” she writes, “the Ontario curriculum and holistic education share many commonalities and are, therefore, entirely compatible” (Neves, 2009, p. 138).  

There’s the hope.  Now, it falls to the teacher to bring the Ontario Curriculum to life, and to both prepare the student for the world of tomorrow and to make learning relevant to the real-world experience of every child today.  A tall order, to say the least.  The big question is: “How”?  

Key to the process is to create a space for student voice in the learning process.  Over the next few weeks, we’ll consider different 'holistic' approaches to open that space for student voice, including:
  • Constructivist Learning
  • Culturally Responsive Teaching
  • Differentiated Math Instruction
Coming Up Next Week: looks at the role of Constructivist Learning for creating student voice in the Math Classroom.
Please check back in!

Post #1 References:

Neves, A.C. (2009). “A Holistic Approach to the Ontario Curriculum: Moving to a More Coherent Curriculum”.  M.A. Thesis. 

Ministry of Ontario.  (2005). “The Ontario Curriculum Grades 1 – 8: Mathematics”.

Making Math Relevant Series Post #2:

“Building Authentic Learning:” A Constructivist's approach to Teaching Math

Constructing Learning in the Math Classroom:
Constructing and showing their own learning working collaboratively with mentor peers

This is the second post in a series that looks at "holistic" approaches to Math Learning within the Ontario Curriculum. 
Last time, we agreed that all learning, including Math learning, is personal, and needs to make connections to the ‘real world’, and that providing space for student voice helps us understand how the learner is constructing meaning in a holistic way.
“Sounds good”, skeptics might agree, “but what’s the real world?” and, “how do we help students make the connection?” 
The key, according to Dan Meyer, is to focus on math reasoning to look at personally relevant, real-world questions like “which ski hill is steeper?”, or “how long does it take to fill a water tank?”  

To answer these questions, Meyer takes a ‘constructivist’ approach.Constructivists point out that students make their own learning as they go. Learners can’t learn something new unless it ‘fits in’ with what they already know and how they already see the world. 

Ultimately, theorists like Duffy and Jonassen say, “we each construct our own understanding and… learning occurs through the association of previous experience with newly acquired knowledge” (Chao & Stovel, 2002, p. 115).
Dan Meyer meets his students at their level, and let’s student voice set the pace.  Here’s a link to how he hooks his class into learning Math reasoning by asking real questions about the everyday world:

So that’s it.  Constructivists point out that students make their own learning as they go. Learners can’t learn something new unless it ‘fits in’ with what they already know and how they already see the world.

The students in Meyers’ math class make their own learning meaningful by focusing on real-world, complex problems where math becomes the pathway for answering real questions.
Meyer’s approach of posing ‘real world problems’ to groups of students and asking authentic questions leads all the way back to thinkers like Dewey and Vygotsky. 
Dewey saw that learning occurs from personal “experiences which have meaning and importance” to the learners themselves (Chao & Stovel, 2002, p. 116), and Vygotsky said that ‘authentic’ learning occurs in a “social context”—like a classroom—where students can learn alongside others who share similar interests (Chao & Stovel, 2002, p. 116).
A quick glance at the Ontario Curriculum reveals many chances to put Meyer’s constructivist approach to work.  The Curriculum itself encourages teachers to adopt a ‘constructivist’ approach in the way it frames curriculum expectations and ‘sample’ problems, such as this example from the Number Sense and Numeration strand of the Grade 3 curriculum:

"solve problems that arise from real-life situations and that relate to the magnitude of whole numbers up to 1000 (sample problem: Do you know anyone who has lived for close to 1000 days? Explain your reasoning)" (Ministry of Ontario, 2005, p. 55). 

In Meyer’s example, just asking something as simple as, “How long does it take to fill a water tank?” creates the opportunity for ‘constructivist’ learning to take place.  It’s a great way to focus on ‘math reasoning’ when considering the ‘real world’ challenges of the Ontario Curriculum. 
Coming Up Next Week: looks at the role of Culturally Responsive Pedagogy for creating student voice in the Math Classroom.
Please check back in!
Post #2 References:

Chao, T., & Stovel, B. (2002). Nothing but the Blues: A Case Study in the Use of Technology to Enrich a University Course. In P. Rogers (Ed.), Designing Instruction for Technology-Enhanced Learning (pp. 114-133). Hershey, PA: Idea Group Publishing. doi:10.4018/978-1-930708-28-0.ch007

Meyer, D. (2010). TED Talks. Math Class Needs a Makeover.” Online Video. 

Ministry of Ontario. (2005). “The Ontario Curriculum Grades 1 – 8: Mathematics”.


Making Math Relevant, Series Post #3: 

“The Student's Voice:" Holistic Math Learning and Culturally Responsive Pedagogy

Deep knowledge of the students' lives allows students to share their lived experiences  

This is the third post in a series that looks at "holistic" approaches to Math Learning within the Ontario Curriculum.  
Last time, we agreed that all learning, including Math learning, is constructed by making connections with what students already know about to the ‘real world’.  

Building up from there is the need to recognize that the 'whole' child does not come to school without prior experiences, and that these experiences shape the context of the student's ability to interact with the curriculum.  Teachers need to make the Ontario Curriculum "responsive" to the different cultural contexts in the classroom.

Culturally Responsive Pedagogy 
describes a way of teaching that integrates a student's background knowledge and lived community experiences into the classroom environment (Ministry of Ontario, 2013, p. 2).  
Culturally Responsive Pedagogy

As we saw in two previous posts, both 'holistic' and 'constructivist' points of view recognize that students understand the classroom out of the context of their actual lives and lived experience.  It makes sense, then, to design instructional activities that align with a child's lived experiences.

Students need to see themselves and their lived realities reflected in the life of the Math classroom, and it's through student voice that learners have the opportunity express their lived realities in response to the curriculum.

A skeptic might wonder: "that all sounds well and good in theory, but what does it look like in the classroom?" 

Three important criteria that teachers demonstrate in the classroom include:
  • High expectations for all students
  • A deep knowledge of their students' backgrounds
  • Teaching strategies and practices that are culturally responsive

It also means honouring the lived experiences of children who grow up in a a culture that differs from the dominant social culture, such as: 
  • race and ethnicity
  • faith
  • family structure
  • English Language Learners and New to Canada
  • sexual orientation
  • ability
  • special education
  • mental health
  • Socio-economic status (SES)
Teachers can and identify ways to remove discriminatory biases and barrier to student achievement, and plan to integrate, honour, and represent students' lived experience in the classroom activities.

In Math, like all subjects, it means focusing on the "Big Ideas" in the curriculum to guide lesson planning, rather than the General and Specific Grade Expectations.  Consider the implications of this Grade 3 'sample problem' in the Ontario Curriculum: 

Sample problem:  You spend 5 dollars and 75 cents on one item and 10 cents on another item.  How much did you spend in total? (Ministry of Education, 2005, p. 56).

Seems simple enough: $5.85

But how does it appear to: 
  • a student from abroad who has never used Canadian currency? 
  • a student from a family where the parents receive living assistance and do not handle money themselves? 
  • a student from a lower SES family whose parent struggles with low income daily?
  • a student who has grown up in a technologically advanced household that only uses electronic commerce, and who never sees physical money exchanged?
With a deep knowledge of their students' lived realities, teachers can look for ways to express these lived realities through student voice.

Coming Up Next Week: looks at the role of Differentiated Instruction for creating student voice in the Math Classroom.
Please check back in!
Post #3 References:

Ministry of Education. (2013).  "Culturally Responsive Pedagogy: Towards Equity and Inclusivity in Ontario Schools." Capacity Building Series. Secretariat Special Edition #35.  

Ministry of Ontario. (2005). “The Ontario Curriculum Grades 1 – 8: Mathematics”.


Making Math Relevant Series Post #4: 

“Open and Closed Tasks:" Differentiated Instruction In the Math Class

With Differentiated Instruction, Parallel Tasks give every student the chance to demonstrate Big Ideas.
Here, a student demonstrates time concepts using paper, computer technology, and a manipulative
This is the fourth post in a series that looks at "holistic" approaches to Math Learning within the Ontario Curriculum.  
Last time, we agreed that all learning, including Math learning, needs to be Culturally Responsive, as it recognizes and honours each students' lived experience and reflects their lived realities in the life of the Math classroom.  

To ensure success in the Math classroom, teachers need to begin to do things differently.  As holistic, or constructivist, or culturally responsive pedagogy suggests, students come first, not the curriculum.  After years of trying to bend the student to fit the curriculum, holistic educators agree that we need to bend the curriculum to fit the student, because: 

Sameness is not Equality

The key to ensuring equity then, is to 'differentiate' the curriculum so that all students can succeed.  

Teachers need to be aware of unspoken biases that limit learning: "when teachers are aware of their students' prior knowledge and experiences, they can consider the different ways that students learn without pre-defining their capacity" (Ministry of Education, 2008, p. 1). 

Starting Point: Gather prior assessment data, otherwise known as "Assessment for Learning" or "Diagnostic Assessments"

Differentiating, which is "an organized yet flexible way of proactively adjusting teaching and learning to meet kids where they are" (Ministry of Education, 2008, p. 1), is as simple as assessing a child ahead of time to find out what they already know, and developing suitable instruction to move them forward from that place of prior learning.

By 'diagnosing' a student's current level of knowledge and skill, a teacher can avoid re-visiting learned material--which risks boring the student into disengagement--or confusing the student with difficult material, which risks alienating the student through frustration.  

By aiming neither too high nor too low, teachers can find a target zone--what theorist Vygotsky called the 'Zone of Proximal Development' (Ministry of Ontario, 2008, p. 1)--and meed the need of every child in the classroom.

Every child learns. It's what we want, right? But it's a lot of work.

Managing the Workload: Lesson Planning and Differentiation

So, how does a teacher manage the planning required to 'differentiate' for every student in a classroom? 
  • Focus on key concepts, otherwise known as the "Big Ideas" in the curriculum.
  • Develop "Open Tasks" - design problem solving activities that can be solved using a broad selection of strategies
  • Develop "Parallel Tasks" - design two--or more--similar tasks that meet a variety of needs and that both focus on learning the same key concept, but address the concept at different levels of sophistication 
  • Develop multiple ways of responding, so that students can demonstrate and display their learning in personally relevant ways
  • Use a Problem Solving Model to guide all students as they approach a task from multiple perspectives
Over the past four weeks, we have looked at how a 'holistic' approach to education can support students in the Math classroom.

Next we'll turn our attention to less 'theoretical' concepts, and talk about instructional strategies that can help kids succeed in Math.  All of the strategies coming up, though, will reflect back on these approaches: 

  • Holistic Education
  • Constructivist Learning
  • Culturally Responsive Pedagogy
  • Differentiated Instruction
I hope you'll continue to join us!

Coming Up Next Week: looks at the role of Big Ideas for creating student voice in the Math Classroom.
Please check back in!
Post #4 References

Ministry of Education. (2008). "Differentiating Mathematics Instruction". The Literacy and Numeracy Secretariat Capacity Building Series.

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