Reflections on #TheMathPod with Cathy Fosnot: The Meaning of Context

 

I recently listened to The Math Pod recording with Cathy Fosnot and Stephen Hurley from VoiceEd Radio.

This podcast really helped me to think about the different lenses that we use to teach math. This is where my thinking went.

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This caused me to connect with ‘lenses’ I have heard, particularly out in the media, include thinking about math as needing to be ‘back to the basics’. I personally assume that this lens implies that math is a ‘pure’ subject. A subject with right and wrong answers, set algorithms that need to be memorized and strategies that are inflexible and rigid.

I also think about the ‘lens’ that I have traditionally used, that includes a) figuring out what needs to be covered in the curriculum; b) finding out where the students are at; c) developing a rich task that allows to enter from their own developmental level; d) providing opportunities for students to build rich math talk and increase the discourse; e) share strategies and learn from one another, assess, consolidate and so on.

But where do you go from there?

This podcast was very enlightening because it was a strong reminder that we need to sequence multiple rich tasks to allow for the progressive development of math strategies and conceptual understanding, along important pathways of learning.

 

It also gave me great pause to think about the significance of ‘context’, and the importance of designing the sequence of rich tasks within meaningful contexts. This blew my mind, because I realised that I was focused on context as a day-to-day construct, not as embedded within sequences of tasks. Providing a meaningful contexts in this sense, helps students to not get lost in the abstraction- a very common occurrence for students still operating at more concrete levels. Context also enables multiple pathways of growth toward becoming efficiency.

I thought about culture, I thought about student identity and decolonizing the curriculum with contexts that include Indigenous Perspectives.

I have realised that it is not about finding the perfect problem, or designing that great problem to be solved. It is about crafting a sequence of problems where students are able to access key strategies, but also able to invent their own strategies – within important contexts that reach across and between problems.

Rather than thinking about this through the lens of moving students along a linear path from the concrete to the abstract, with contexts that change daily, we can use sequencing to enabling multiple paths that foster deep understanding about patterns, relationships and properties about numbers.

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Therefore, I would have to say that using a ‘lens’ of looking at math as a pure subject, would also be to assume that students do not need context, and only learn along linear paths from concrete to abstract.  But we know that students do not learn along a linear path, using the same strategies, and we know that context, if harnessed effectively, can produce very meaningful math learning.

I also think that the deepening understandings that emerge from this sequencing and contexts will lead to greater memorization of basic facts as well, for those of us who do see the importance of freeing up that working memory to do more complex tasks. This understanding is what helps to build more efficient mathematicians.

When it comes to math, there are no easy ways out. The reasoning, skills, procedures, concepts, strategies are all challenging! Does this mean that it is too hard to learn? Never. In fact, I think that it is perhaps the most important things to learn as it helps with thinking in all other areas of life. If we can really get at the heart of helping students to become more efficient in math, then we will have students that can understand relationships, patterns, and ways to figure out what we don’t know in many contexts within our lives.

As for my next step, I am going to really delve in to looking at how to find sequences of rich tasks, and supporting contexts that incorporate Indigenous perspectives.

I will also strive to understand student development and how to help them to become more efficient and develop deeper strategies. This is important to use as a lens for future math work, and moving beyond the lens of ‘this is what I need to cover in the unit, this is where my students are at, and this is the problem they need today’.

What is your next step?

 

Deborah McCallum

 

Language, Culture & Math

I just spent the last 3 days at a Summer Academy for Purposeful Math Planning. I was very intrigued when we were discussing number sense and the need to become more flexible with numbers and how we use them in our world. Only one person brought up the issue of culture and how numbers are perceived. It really gave me pause to deeply consider the impact our culture has on how we perceive math as well. Particularly in the areas of spatial sense.

In the article ‘Does Your Language Shape How You Think’ by Guy Deutscher, I was really drawn in when I read that speakers of geographic languages appear to have almost superhuman senses of orientation, and simply ‘feel’ where the directions are. I couldn’t help but consider how language has deep connections to visual and spatial sense and how we ultimately perform – especially with English when used in our Eurocentric, settler based curriculum.

As the article said:

The convention of communicating with geographic coordiates compels speakers from the youngest age to pay attention to the clues from the physical environment (the position of the sun, wind and so on) every second of their lives, and to develop an accurate memory of their own changing orientations at any given moment”.

The language we use compels our students to pay attention to different cues in the environment. Our language thus shapes our habits in ways that make our spatial understandings feel like second nature.

I was struck by the fact that different languages lend themselves to different languages of space. Some languages explore directions from a more egocentric point of view – ie., directions given in relation to ourselves, whereas others are more geographically oriented. This may not sound like a big deal, until you consider how deeply language shapes our realities and how we perceive and learn about the world around us depending upon the language we have learned.

More questions I have include:

What ‘habits of mind’ form due to the spatial language that we use?

How is our ability to succeed in math class affected by our language?

What if the instructions we give in say a math class is what is preventing a student from understanding instructions?

What about our English Language Learners who may be confused based on instructions that are more egocentric or more geographic?

Do we assume that the student has learning difficulties?

Also, what happens when we are trained via language to ignore directional rotations when we commit information to memory?

 

This is another example in the article that was very powerful to me – basically, if I walked into an adjoining hotel room that is opposite of mine, I might see an exact replica of my own room. However, if my friend who spoke a more geographical language walked into my room, they would not see an exact replica – rather they really would see that everything is reversed, and would have the language to describe that. This has big implications therefore in how we commit events to memory, recall them, solve problems, and critically think about the world around us.

The language we use compels our students to pay attention to different cues in the environment Our language thus shapes our habits in ways that make our spatial understandings feel like second nature.  It therefore will compel our students to think differently about math.

We make so many decisions each and every day about the world around us – so much of this is spatial. We just simply don’t know our language and habits impact our ability to succeed in math.

We really are at the center of our own worlds. If we determine that subjects like math are linear and one-dimensional, with set algorithms and languages to describe, know and understand, then we are absolutely missing the worlds of many of our students. To dismiss language, culture and our identities of our students could very well mean the difference of success and achievement vs failure.

 

Deborah McCallum

D

Spatial Reasoning and Student Success

 

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Spatial Reasoning

This year, I have had the privilege of designing a brand new makerspace for our school. In addition, I have been able to focus on visual-spatial reasoning as the thread that pulls together science, math and technology.

What is spatial reasoning?

According to the Ministry of Education, Spatial reasoning is the ability to engage in reasoning, and understand the location, rotation and movement of ourselves and other objects in space. It involves a number of processes and concepts. More information about this can be found here: http://www.edu.gov.on.ca/eng/literacynumeracy/LNSPayingAttention.pdf

 

Why is Spatial Reasoning important?

There already exists a very strong body of research that spatial thinking correlates with later performance in math. In addition, research consistently demonstrates strong linkages between spatial ability and success in math and science — and those students with strong visual and spatial sense are more likely to succeed in STEAM careers.

It is absolutely clear that early exposure to visual-spatial reasoning is very important.

However, as educators, we traditionally have failed to recognize that our youngest students are actually able to perform way above the expected levels of spatial reasoning. We generally leave these tasks for older students. This has to change.

Not only is this a problem because we are neglecting our youngest students who already come to school with a high level of spatial-reasoning skills, but this also means that our youngest students are not having equal access to spatial reasoning activities that they are able to perform. This is a social justice issue. Especially when we consider that visual-spatial reasoning positively correlates with later performance in math (Mazzocco & Myers, 2003). If we know the research, and have the opportunity to employ high quality spatial reasoning activities for all students in Kindergarten, should we let older curriculum and older beliefs hold us back? Do we recognize when we are teaching in the ways that we used to be taught? What if we had the ability to ensure all of our youngest students engage in spatial reasoning? How would this impact their future?

In fact, students who experience issues with math, often have difficulties with geometry and visual spatial sense (Zhang, et al., 2012). This to me sounds like an amazing opportunity to understand mathematical achievement via spatial reasoning. The earlier we recognize this, the earlier we can respond.

Wouldn’t it be great if we gave all students the ability to access higher level learning associated with visual-spatial sense right from the get-go? Imagine the impact this could have in overall math achievement throughout our students entire school career, and beyond, in their STEAM based careers.

To me, I think this behooves us to ensure we have access to makerspaces – regardless of where they are located in our schools – to promote visual spatial reasoning skills.

What do you think?

 

Deborah McCallum

c 2016

References:
http://www.edu.gov.on.ca/eng/literacynumeracy/LNSPayingAttention.pdf
http://tmerc.ca/research/
http://www.pme38.com/wp-content/uploads/2014/05/RF-Sinclair-et-al.pdf
Mazzocco, M. M. M., & Thompson, R. E. (2005). Kindergarten predictors of math learning disability. Learning Disablilities Research & Practice, 20(3), 142-155. doi:10.1111/j.1540-5826.2005.00129.x
Mazzocco, M. M. M., & Myers, G. F. (2003). Complexities in identifying and defining mathematics learning disability in the primary school age years. Annals of Dyslexia, 53, 218–253
Zhang, D., Ding, Y., Stegall, J., & Mo, L. (2012). The effect of Visual‐Chunking‐Representation accommodation on geometry testing for students with math disabilities. Learning Disabilities Research & Practice, 27(4), 167-177. doi:10.1111/j.1540-5826.2012.00364.x

Makerspaces & Math Links

https://www.tes.com/lessons/sGvLjtLFbRRUZA/math-and-makerspaces?feature=embed

Math Assessment, Home Connections

Ontario Math Resources

 

Here is a list of some of the best resources to support math in Ontario. Please feel free to add more to the list in the comment section. You can also check out the following blog:

Ontario Math Resources

 

Teaching Math for FNMI Students

 

The dominant ways in which math has always been taught in our Western society includes drill, rote learning, and a focus on math ‘authorities’ including the teacher.

This poses very serious problems for many of our mathematical learners, particularly for our First Nations, Metis & Inuit (FNMI) learners, whose perspectives and ways of knowing may not be included in the traditional curricular frameworks. Therefore, we are faced with very serious issues when it comes to considering who gets to learn math, and who will be included.

Math that is inclusive of different cultures and ways of knowing the world, is built on the awareness that math itself is about knowing the world. It is my view that we as teachers can do many wonderful things in the classroom to integrate basic skills with constructivist and culturally responsive ways of teaching math that will support multiple ways of knowing – particularly for students who are FNMI.

How do we use strategies and approaches that both facilitate learning in math, AND infuse FNMI ways of knowing? We start by recognizing the importance of connections, communication and contextualization of the learning of FNMI students.

What strategies help to infuse FNMI ways of knowing, perspectives and content?

Strategies

The following strategies can be designed to infuse FNMI ways of knowing, perspectives and content into the Math Curriuclum.

First, recognize that students learn by attaching meaning to what they do. Students need to construct their own meaning of mathematics.

2. Integrate Inquiry Based Learning into math. Check out the following website from OISE on Inquiry in Math. 

3. Provide holistic learning experiences that include cultural and social interactions through dialogue, language and negotiations of meaning.  This would include allowing other students, community leaders, Elders, Senators and other diverse resources to teach, facilitate, share and learn in our classroom.

4. It is impossible to isolate math from culture. It is important to strive to help change mindsets about what ‘real’ math is. Ask ourselves questions including is math about making financial transactions? Is it about complex beading, knitting, or making intricate porcupine quill boxes? Are our cultural routines linked to math? Become aware of how math is linked with culture.

5. Aim to create equal opportunities for Math learning for Aboriginal students. However, exercising caution not to merely integrate holidays, artifacts, stories and more merely as a form of ‘tokenism’. Also, exercising caution not to make FNMI students solely responsible for adding culture and learning to the math classroom.

6. Engage in Culturally Responsive Teaching of mathematics. When we don’t include culture in math, we are essentially positioning people ‘outside’ of math. Serious implications thus arise as FNMI students are at a greater risk of being forced into negative math mindsets and math deficiencies. Culturally responsive teaching is about understanding surrounding communities, and making the program ‘Student-Centered’.

7. Step outside of traditional curriculum frameworks. Not Big ideas and high expectations, but the pedagogical frameworks. When we try to add culture, content, perspectives and ideas to math, we can change the traditional curriculum frameworks. Mathematical learning that incorporates FNMI perspectives, content and ways of knowing, should not be an add-on. We need to make sure that we change our traditional frameworks lest we inadvertently continue to promote the ‘othering’ and exclusion from math.

Math is about knowing the world around us. FNMI students deserve to be included in our curriculum. How will you strive to equally include FNMI students in your curriculum?

 

Deborah McCallum

2016

 

Teaching Math for FNMI Students

 

The dominant ways in which math has always been taught in our Western society includes drill, rote learning, and a focus on math ‘authorities’ including the teacher.

This poses very serious problems for many of our mathematical learners, particularly for our First Nations, Metis & Inuit (FNMI) learners, whose perspectives and ways of knowing may not be included in the traditional curricular frameworks. Therefore, we are faced with very serious issues when it comes to considering who gets to learn math, and who will be included.

Math that is inclusive of different cultures and ways of knowing the world, is built on the awareness that math itself is about knowing the world. It is my view that we as teachers can do many wonderful things in the classroom to integrate basic skills with constructivist and culturally responsive ways of teaching math that will support multiple ways of knowing – particularly for students who are FNMI.

How do we use strategies and approaches that both facilitate learning in math, AND infuse FNMI ways of knowing? We start by recognizing the importance of connections, communication and contextualization of the learning of FNMI students.

What strategies help to infuse FNMI ways of knowing, perspectives and content?

Strategies

The following strategies can be designed to infuse FNMI ways of knowing, perspectives and content into the Math Curriuclum.

First, recognize that students learn by attaching meaning to what they do. Students need to construct their own meaning of mathematics.

2. Integrate Inquiry Based Learning into math. Check out the following website from OISE on Inquiry in Math. 

3. Provide holistic learning experiences that include cultural and social interactions through dialogue, language and negotiations of meaning.  This would include allowing other students, community leaders, Elders, Senators and other diverse resources to teach, facilitate, share and learn in our classroom.

4. It is impossible to isolate math from culture. It is important to strive to help change mindsets about what ‘real’ math is. Ask ourselves questions including is math about making financial transactions? Is it about complex beading, knitting, or making intricate porcupine quill boxes? Are our cultural routines linked to math? Become aware of how math is linked with culture.

5. Aim to create equal opportunities for Math learning for Aboriginal students. However, exercising caution not to merely integrate holidays, artifacts, stories and more merely as a form of ‘tokenism’. Also, exercising caution not to make FNMI students solely responsible for adding culture and learning to the math classroom.

6. Engage in Culturally Responsive Teaching of mathematics. When we don’t include culture in math, we are essentially positioning people ‘outside’ of math. Serious implications thus arise as FNMI students are at a greater risk of being forced into negative math mindsets and math deficiencies. Culturally responsive teaching is about understanding surrounding communities, and making the program ‘Student-Centered’.

7. Step outside of traditional curriculum frameworks. Not Big ideas and high expectations, but the pedagogical frameworks. When we try to add culture, content, perspectives and ideas to math, we can change the traditional curriculum frameworks. Mathematical learning that incorporates FNMI perspectives, content and ways of knowing, should not be an add-on. We need to make sure that we change our traditional frameworks lest we inadvertently continue to promote the ‘othering’ and exclusion from math.

Math is about knowing the world around us. FNMI students deserve to be included in our curriculum. How will you strive to equally include FNMI students in your curriculum?

 

Deborah McCallum

2016