How do we design our Math Class?

 

I have been doing a lot of reflecting about math. As a student myself, I have also been doing some research about types of math class situations, and considering what kinds of questions we need to ask to help students develop conceptual knowledge.

 

In our math classes it is important to provide students with cognitively demanding tasks. Rich tasks are great examples of this. Rich tasks provide multiple entry points AND multiple solution paths for students.

 

When we are planning our math classes, we can consider the following situations that we may be providing for our students. Next, we can think about what feedback we need to make the student learning more conceptual.

Situation:

Situation #1 – Students are engaged in lessons that focus on basic knowledge, and procedures. Students need to get to a correct answer vs gaining a conceptual understanding of the strategies used. Students therefore are unable to make connections to deeper math concepts.

 

Key Questions:

  • What do we do to help students develop conceptual knowledge?
  • What supports need to be in place to help make this happen?

 

Situation #2 – students engage in more complex tasks, but then the tasks turn into procedural tasks, thus the students don’t get experience with increasingly difficult and complex tasks

 

Key Questions

  • How do we keep students engaged with a task, and allow them to experience increased cognitive demands and go through the ‘fits and starts’ that learners go through?

 

Situation #3 – Students may engage in rich tasks, with multiple entry points and multiple solution paths, however, they are unable to engage in a whole class discussion about different types of strategies.  

 

Key Questions

  • What steps can we take to conduct a whole class discussion?
  • How do we structure the strategies in a way that helps students to make the best connections?
  • What do we want students to get out of our whole class discussions?

 

Situation #4 – Students are engaged in rich tasks, with multiple entry points, multiple solution paths. Students are choosing from a variety of strategies and honing in new ones. They are able to explain why they chose a certain strategy. They are given time to explore their various strategies, and time to explain their reasoning. Whole class discussions allow for students to explore different strategies, and consider the efficiency of the strategies. Students make deeper connections, learn from peers and apply the new learning to a new task or ‘exit ticket’.

 

Key Questions:

  • How do we push our learning and the learning of our students in this type of a situation?

 

The questions can really help to guide our  thinking into how we are going to design our math class.

I would love to hear your ideas,

 

Deborah McCallum

Copyright, 2017

 

 

Reference

Educational Administration Quarterly

Vol 53, Issue 3, pp. 475 – 516

First Published January 31, 2017

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.

magnifying-glass-29398_1280

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.

fibonacci-1601158_1280

 

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

 

Math & Identity

I just read this great piece by Karin Brodie:

Entitled: Yes, mathematics can be decolonised. Here’s how to begin

 

When we think about math, we often think about the content – but what about the way we think about it, the way it is taught? If think about math in these ways, we are able to consider how identity plays a role in how we teach, understand, and apply math.

What is identity? It is connected to the groups that we affiliate with, the language we use, and who we learned the language from. I believe that we all have different identities depending upon the different groups that we belong to, and that this has implications in terms of the languages and discourses we use.

What is important is that I recognize the intersection of my identity with identities embodied within the Ontario Public School system, my school board, schools, and students that I will be working with next year. In identifying this intersection, can I truly facilitate math learning, and promote higher achievement for students? Especially if my identity is stark compared to the identities that exist within classrooms across Ontario schools?

But this is not comfortable. One of the ways we as educators try to deal with this discomfort is to think of math as a ‘purist’ subject. This is but one way that we can strive to reconcile the dissonance we can feel about dealing with multiple identities in math.

But it is important to hear the identities and cultures of our students, in order to ask better questions about how math can be learned, versus merely finding the ‘right’ answer.

What can we do?

I think that Culturally Responsive Pedagogy (CRP) is one way to begin to address this question and forge a path forward. Inherent in CRP is the idea that I as an educator would continue to use student culture to transcend the negative effects of dominant culture. It becomes a tool to explain the ways in which I will develop deeper cultural knowledge of students, and thus use cultural referents to increase opportunities for student learning.

Here is a great piece to learn more about CRP: Framework for a Culturally Responsive and Relevant Pedagogy:

I do have many questions however.

How do I know that I am actively supporting a safe school environment, and not just thinking that I am because it fits with my own identity and dominant culture in society?

How can teachers situate their own privilege and oppression of themselves, and that of others? It is through this that we can start to understand identity, and understand how diverse our experiences surrounding math can actually be.

When we consider the multiple identities of teachers and students, we can understand that a standardized test is just one type of outcome for student learning. There are so many additional ways that we can capitalize on to enhance student achievement in math, to help us move beyond the spaces where we simply consume knowledge, into spaces where we can critically examine mathematical knowledge and how it plays out in our lives, and with our own identities.

This is especially important with Indigenous students. Canada has a history of experience with colonizing Indigenous communities. Because Indigenous peoples were on this land first, it stands to reason that the diverse cultures of Indigenous peoples are allowed to be welcomed and understood in our classrooms, as a way to promote and enhance the identities of Indigenous individuals, cultures, and incorporate their diverse experiences with math.

It causes me to ponder the importance and power of language. Language is part of our identity, it forms how we know the world – thus how we understand and know math. We need to learn the languages and narratives of our student identities, and check out our own, in order to co-create the necessary mathematical experiences that will lead toward higher math achievement.

Perhaps it is important to use CRP to help co-create new languages of math in our unique environments of unique identities and cultures – that can help us shape our understandings of different cultures, contexts and sensitive issues. It will be important to have agreed upon norms, and exercise them in ways that help us to foster truth and respect. It will also be important for me to frame this as discourses of education, and not discourses of the individual.

It is also important to facilitate the creation of math opportunities that allow students to discuss their own aspirations for the future.  Noting how students solve problems, and sharing the different ways that problems are solved. I can strive to move away from relying on my own identity and personal experiences to make sense of how math should be solved in the classroom. In this way, I recognize that math is culturally defined, and that I can change the narrative that I learned from dominant culture that math is a pure subject that has the correct answers, and is culturally neutral.

It is time to get really uncomfortable with math.

 

Deborah McCallum

Joseph Boyden

This morning I read the following new article by Joseph Boyden in MacLeans. It gave me a lot to think about.

Here is the article.

http://www.macleans.ca/news/canada/my-name-is-joseph-boyden/

This article really brings into question for me about what being Indigenous really means, and what elements create your own identity. On one hand, it has really struck me that so many ‘others’ are questioning what his identity should or shouldn’t be. However, I have come to know that identity is not all about DNA or blood quantum. It is also about the ‘intimate’ conversations and language you share with the people who shape who you are. It is the language that moves through them and with them that make you who you are – perhaps more than Blood Quantum. I fear that Boyden is benefiting from claiming Indigenous identity, without belonging to an Indigenous community in this way. He states that he identifies most of his life as an anglo-white male, growing up in a mainly anglo-christian household. I am curious about which community or culture is he giving back to? Working to make better?

This is more than just about Joseph Boyden and the fact that he has some Indigenous DNA. What does his story mean for others who define their identity as Indigenous? All of the others who also do not ‘look’ Indigenous’, but are, yet continually asked about what percentage of Indigenous they are – as if there is a magic number that decides.

I am curious as well, did he ever have to live with what it feels like to ‘Live in the hyphen’? Did he experience racism by both white and Indigenous cultures? Was he questioned and rejected by both anglo and his Indigenous cultures – I would infer no, as it appears that he has not been part of such a community his whole life. It feels like he has just decided to ‘live in the hyphen’ now, but only to reap the benefits of both worlds. Not to help heal the traumas, or contribute personally – as one would do with those in our lives who helped to shape us.

After much careful thought, and more reading, I have come to the conclusion that identity is very much about belonging. Whose community do you belong to, and who belongs to you? Blood quantum cannot be looked at in isolation. If you have no community that you can truly claim, and who claims you in return, then how can you truly identify with it? How can you give it a voice? And finally, if you do have a community, how are you using your gift and fame to make your community better, and not just yourself?

This brings up a lot of questions about identity, what community is, and about whose identities deserve privileges, and whose do not. Why do we decide this? We do it without even realising as well.

If you haven’t already heard it, this is a great Podcast to listen to as well: Ep. 73: White Settler Revisionism and Making Métis Everywhere 

I will continue to think about my aporia, and my personal discomfort surrounding this situation with Joseph Boyden.

 

Deborah McCallum

Teaching and Assessment with Math Processes

geometry-1044090_1920

Teaching and assessment in math go hand in hand. What ties them together are the mathematical processes. Our job as teachers is to help students build mathematical knowledge and skills of the curriculum through the 7 mathematical processes. They  include:

  • problem solving
  • reasoning and proving
  • reflecting
  • selecting tools and computational strategies
  • connecting
  • representing
  • communicating

For instance, here are the math processes for Grade 4 from the Ontario Curriculum:

math_processes_II.png

In order to begin to assess what our students have learned through the expectations and processes, we must set some learning goals to help our students learn.

A Learning Goal must:

  • Have a sense of purpose
  • Build on student ideas about math
  • Engage students
  • Help students develop mathematical ideas
  • Help teachers to assess student progress
  • Connect with the classroom activities
  • Connect with math processes

 

The kinds of activities that we engage in during math class that embody math processes may include the following:

* something to ponder: can you think about what math processes can be embodied in each of the following? Are there any more we can add?

Questioning:

It is important to ask the right questions. The questions help us to facilitate the discussion that will follow. Questions are also used to raise issues and problems

 

Inquiry Based Learning.

As students solve problems, they will develop their ability to ask questions and plan investigations to answer those questions and solve related problems. The goal is to invite student entry into the math problem, and facilitate their exploration of the math.

 

Gallery Walk

The focus of a Gallery Walk is on the student work and interactive discussion shared around the classroom. Students have the ability to read different solutions and provide written and verbal feedback to each other, communicate, and solve problems together.

 

Bansho

Here, the Chalkboard becomes a record of the entire lesson. This really helps us to model effective organization to our students. It also includes cooperative learning strategies including Think-Pair-Share, Think-Talk-Write & Placemat.

 

Math Congress

Here, the purpose is to support development of mathematicians in classroom learning community vs fixing mistakes in student work. We focus the whole-class discussion on 2-3 student solutions that are selected strategically by myself, the teacher. Students also share work with one another, check answers and strategies, ask questions to provoke clarification & elaboration, and defend and support mathematical thinking.

 

Assessments we use:

Assessments will include rubrics, performance tasks, formative and summative tasks, observations, portfolios, journals, interviews and products. Assessment will be based on Learning Goals, expectations, processes and the following Achievement Categories:

Knowledge and Understanding. Subject-specific content acquired in each grade (knowledge), and the comprehension of its meaning and significance (understanding).

Thinking. The use of critical and creative thinking skills and/or processes,3 as follows: – planning skills (e.g., understanding the problem, making a plan for solving the problem) – processing skills (e.g., carrying out a plan, looking back at the solution) – critical/creative thinking processes (e.g., inquiry, problem solving)

Communication. communicating mathematical ideas and solutions in writing, using numbers and algebraic symbols, and visually, using pictures, diagrams, charts, tables, graphs, and concrete materials).

Application. The use of knowledge and skills to make connections within and between various contexts.

 

All of the instructional and assessment practices can be interconnected with the Math Processes as defined in the Ontario Math Curriculum:

math_processes

 

Makerspaces & Math Links

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

STEAM Job descriptions for Curriculum Planning

 

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Using job descriptions can facilitate program planning and student learning. A job description provides us with rich opportunities to extract content areas, learning goals, success criteria, and rich tasks for learning. It just doesn’t matter if the position is paid or not, volunteer or mandatory. The point is that you will often find key information about skills that are important in our world today, and perhaps discover more relevant ways to teach those skills.

In my quest to make learning relevant for students, I have begun to look at job postings for S.T.E.A.M. related work, and think about ways that I can apply them to the curriculum. There are a great number of possibilities that crop up when we consider how our curriculum can be interpreted through the lens of a real job.

Consider the following job description in blue. As you review it, consider the cross-curricular, and integrated learning opportunities that may present themselves. Consider the project-based learning opportunities you can use to help students gain the necessary skills to apply for this job. Where do various technologies fit into this picture?

Check it out: 

_______________________________________________________________________________

BRIDGE DESIGN TECHNICIAN

Organization: Ministry of Transportation
Division: Provincial Highways Management
City: London
Job Term: 1 Permanent
Job Code: 12682 – Engineering Services Officer 3
Salary: 
$1,122.02 – $1,410.37 Per Week*
*Indicates the salary listed as per the OPSEU Collective Agreement.
Understanding the job ad – definitions

Posting Status:

Open
Job ID:
99401
Apply Online
View Job Description
Are you looking for a new challenge? Would you like to apply your knowledge of civil engineering technology and computer abilities in a new way?
Consider this opportunity in structural design while contributing to the safety of Ontario’s transportation system.

What can I expect to do in this role?

In this role you will:
• Prepare scale drawings depicting bridge details and materials for review and approval;
• Prepare associated contract documentation according to Ministry standards using required software;
Review bridge site plans and preliminary geometry information supplied by consultants;
• Carry out quantity calculations and cost estimates;
• Provide and assist in the training of regional staff in bridge inspections, in the use of computerized bridge detailing systems and bridge management systems;
• Provide interpretation of standards, specifications and policies as required;
• Assist in bridge inspections by carrying out inspection of simple structures, and updating and maintaining related databases;
• Provide technical guidance, training and advice to junior staff on bridge drafting and contract preparations, durability and construction issues with complex structural details and innovative techniques ensuring safety and economy;
• Answer queries on technical issues from other jurisdictions as required.

How do I qualify?

(aka learning goals and success criteria, criteria for rubrics and other assessment methods)

Knowledge of Bridge Design

• You have knowledge and skills in the design, detailing and contract preparation of provincial bridge contracts.
• You have knowledge and skills to be able to inspect bridges.
• You have knowledge in bridge design and detailing principles, and ability to consider various constraints such as materials, fabrication and production techniques.
• You have practical working knowledge of the varied and complex safety issues related to the design of bridges.

Communication Skills

• You have well-developed oral and written communication and presentation skills.
• You can use consultation skills to identify needs and maintain effective working relationships with regions and other functional teams
• You are committed to customer service.

Research and Project Planning Skills

• You can understand and interpret engineering plans and profiles, technical reports and relevant codes of practice.
• You have knowledge of project planning in order to design, detail, implement, lead and manage a number of concurrent projects of varying degrees complexity, individually or within a team environment.
• You have demonstrated analytical, planning, scheduling, project management and work coordination skills.

Computer Skills

• You can use computer systems and their applications, including Computer Aided Design (CAD) systems and database systems.

_________________________________________________________________________________
Now that you have had a chance to look at this, tell me you are not inspired by the sheer opportunities to connect science, math, technology and literacy? How many skills can be extracted and channeled into balanced literacy and math activities? How many rich tasks can be created? What projects and inquiries can be facilitated? How will they culminate into an end of unit(s) assessment task that includes applying for this job?
How can we help students figure out what they need to do next in order to ‘prove’ that they have the skills to apply?
What if my students were given a small bank of job descriptions, and they need to choose one that looks interesting that they will apply for.
Here are a few steps to consider:
1. Conduct your hypothetical job search
3. Teach the feedback skills that enable all students to engage in higher quality feedback and assessment as learning processes.
4. Find the Big Ideas
5. Plan your projects, centers, and assessment protocol.
6. Reflect
7. Share
Job searching can provide key information into the skills and knowledge that are important in our world. They can even help inform our curriculum planning and instructional design. Next time you are wondering how to infuse math, science, literacy and more into your short and long range plans, consider starting with a job search.
Deborah McCallum
c 2016

Helping kids to find their Writing Superpower: by Allison Tait

The following is a guest post by Allison Tait, Author of The Mapmaker Chronicles

 

 

As an author who’s regularly asked to visit schools for talks and workshops, I have one main question for educators: Who am I talking to?

I find that the candidates for small group workshops tend to be made up of what I like to call the Keen Beans – kids who LOVE writing and simply can’t get enough of it. Most of the Keen Beans are writing their own novels by the time they’re eight.

Large group workshops, however, are a different matter. There’ll be one or two Keen Beans – answering all the questions for me – and 28 kids who simply stare at me as though I have two heads if I start talking about plots or characters or, heaven forbid, paragraphs.

This crew perks up immeasurably, however, when I mention superpowers. In particular, writing superpowers – and the fact that everyone has one.

They get even more excited when I tell them that I’m going to help them find their own writing superpower.

 

What is a writing superpower?

A writing superpower is a special strength that you bring to your writing. Everyone’s got one, but they’re not always what you might imagine. It’s not necessarily about the way that you use words, though this, of course, is part of it. It’s more about where you get your ideas from and what you do with those ideas. It’s about whether or not you can get to The End of your story, pushing through when it gets hard. Sometimes it’s about the ability to plan your story out, taking it logical step by logical step, and sometimes, for other people, it’s more about huge leaps, pushing an idea as far as it will go.

There are 10 writing superpowers

  1. X-Ray Vision: These kids are great at describing what they see. They think in pictures, and are often good at drawing as well. Encourage them to imagine a scene in their heads and simply write down what they see.

2. Supersonic hearing: This is one of my superpowers, and is a great source of not only story ideas, but natural-sounding dialogue. Lots of writers I know are eavesdroppers, and I encourage kids to look for story ideas in the daily conversations around them. Mum telling stories about the ‘olden days’ might be a story starter, as might two younger kids in the playground talking about how cool it would be to fly to the moon.

  1. The ability to leap: While it’s important that kids learn to plan a story, those Keen Beans who can start with an idea and a sentence and then follow the story to the end have a superpower. It’s a crazy way to write (I know because I do it) and can go horribly wrong, but if you have a Keen Bean who works this way, encourage them to push their idea as far as they can – as long as they finish the story.
  1. Endurance: If there’s one thing I’ve learnt about writing in the many years that I’ve been doing it, it’s this: most people are really good starters. But the ones who get really good at writing have a very special superpower – they keep going until they finish the story. Kids who finish are superheroes and should be treated as such.
  1. Analytical thinking: Kids who are good at maths often think they’re not good at writing, but that problem solving ability they have can be a writing superpower. When I talk about this superpower, I use Ironman as an example. People who are plotters and planners make up a huge proportion of published authors for one simple reason: they finish their novels. When you have a logical blueprint, you never end up with your hero stuck in a hole with no way to get out (as once happened to me).
  1. Memory: In The Mapmaker Chronicles, my hero Quinn has a photographic memory, which I think is a writing superpower. Kids who have good memories are able to recall not just the things that happened to them, but how they felt about those things. This is indispensable not only for coming up with story ideas, but for using small details to make the stories feel real. I encourage all kids to keep a journal or diary to help develop this superpower.
  1. The ability to shrink and expand at will: While Ant Man is not often associated with writing, I use him as an example of the value of editing your work. He can shrink himself when he feels like it, or be larger than life. Writers who can do that to their work have a superpower – being able to go through your words and remove the stuff that’s not necessary, or add in details that are, is a rare skill. Kids who understand the importance of editing – and are good at it – are miles ahead.
  1. Spidey senses: By the time they get to grade four, most kids have heard that they need to use all five senses when they’re writing a story. But it’s a rare kid who actually does it. If you have a kid in your class who describes the salty taste of the air at the beach, or shows you humidity by describing the sweat rolling down a character’s arms and the damp stickiness of his clothes, you have a superhero right there.
  1. Batman’s voice: this is perhaps the greatest writing superpower of all. One of the questions I’m often asked in high school workshops is this: how do I stop myself from writing like John Green/ Suzanne Collins/ Rainbow Rowell? The only way to do it is to tap into your own writing voice, which is basically the way that you put things together – the words you choose, the sentences you use, the little jokes you put in. The best writers write like they talk – only better.

What does this have to do with Batman? Everyone has their own Batman Impersonation (mine is particularly impressive now that I’ve had to do it at countless workshops). We’re all trying to sound like Christian Bale or Michael Keaton or Adam West – and yet we all still sound different.

Writing is the same. We’re all writing a story, but the thing that makes the story special is our writing voice.

A kid who has developed his or her own writing voice is a superstar.

  1. Bravery: Writers who write what they think and feel, and are willing to let other people read it, are really, really brave. The best thing about this writing superpower is that it can be developed with time and practise.

Why do writing superpowers matter?

Every kid, even the ones who don’t think that writing is for them, can find something on this list that they’re good at – or can become good at (in the case of bravery, for instance).

I encourage kids to identify one writing superpower and use it to give them the confidence to keep writing. Because when you’re confident that you’ve got at least one thing going really well, then it’s much easier to take risks with writing and to try different things.

And, as we all know, the best way to improve writing is to keep writing.

 

Allison Tait (aka A.L. Tait) is an Australian author, who has been working professionally as a writer for 20+ years. The Mapmaker Chronicles, her bestselling middle-grade trilogy, will be available in the US and Canada from 1 June 2017 through Kane/Miller Books.

Find out more about Allison at allisontait.com and more about The Mapmaker Chronicles at themapmakerchronicles.com

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