This blog post is a work in progress! Be sure to come back and visit in a few weeks, as I will be adding to it over time…
It can sometimes feel overwhelming when we look at all of the individual and group needs of our mathematics learners. Building readiness to learn, along with ensuring that we meet the individual needs of students might give us the impression that we need to create an individual lesson plan for each and every person in our classrooms. That sounds exhausting…
But what if we can create structures and use a variety of math instructional strategies within those structures? What if we can create diverse learning experiences that encourage mathematical thinking and growth over key concepts? This is an idea worth investigating!
Our elementary and middle years math curricula in Saskatchewan cover a number of topics, from number to patterns to shape and space and statistics. Ironically, when you look at the skills needed for students to be READY to engage in these grade-level concepts, there are only a handful of pre-skills. These pre-skills are the math concepts that are applied and used in new learning.
For example, when we are learning about adding and subtracting fractions, we need to know about:
addition and subtraction
multiplication and division, multiples and factors
what a fraction is, finding equivalent fractions, improper fractions
So, how do we teach and reteach each of these key concepts in our classrooms? You can find a large number of curated resources in this Google Drive which contains folders of resources
We can also pull out specific concepts and see how they grow. The following concept trajectories were created by a province-wide math leadership group a number of years ago, and show the language, strategies and concepts over time. Each continuum has four instructional strategies listed.
In their book, Reclaiming Youth at Risk, Brendtro, Brokenleg and Van Bockern (1990) proposed a model of youth empowerment and resiliency. This model is called the Circle of Courage, and it is based on research of First Nations philosophy of child rearing. The Circle of Courage is a holistic approach that is described by Dr. Martin Brokenleg in his video.
Dr. Brokenleg’s research shows that every human being has four basic needs: significance, competence, power and virtue. The Circle of Courage makes the following connections:
Dr. Brokenleg has identified what it looks like in someone whose spirit is in a state of weakness, a state of strength, and what might happen if their spirit has been distorted. The following are ideas to support a state of strength by helping to mend a broken spirit.
Belonging: Right now, I belong here.
Belonging is our most basic need. Maslow’s hierarchy (McLeod, 2018) recognizes that a child needs to belong before they can move towards building self-confidence and self-actualization (Peterson & Taylor, 2009). To feel that we belong, we must build relationships with others. Complex kinship systems have existed within First Nations communities for thousands of years (Learn Alberta, 2019). By building a sense of kinship within your classroom, you can ensure that students feel a sense of belonging, culturally socially and physically.
Weak Spirit of Belonging
Distorted Spirit of Belonging
Spirit of Belonging
Distrust
Overly Dependent
Trust
Exclusion
Cult Vulnerable
Inclusion
Detachment
Gang Loyalty
Warmth
Rejection
Craves Acceptance
Friendship
Antagonism
Craves Affection
Cooperation
To build a sense of belonging, create physical spaces for all. The arrangement of the classroom should reflect the need to serve whole group, small group and individual work needs through:
Desk arrangements
How large furniture allow for student movement
If possible, have a quiet space for those who need it
Include all students in time to learn together. Have all students of different abilities, ethnicities and backgrounds working in the same classroom. They can cooperate, communicate and care for one another. By engaging in heterogeneous groupings frequently, students can know one another deeply. Assistive technologies can help to support students who have mild or severe disabilities and help them to contribute the learning of the whole.
Provide opportunities for students to know each other, such as an Interview and Share activity. Provide positive encouragement to try new activities and recognize individuality and creative talents.
Mastery: I may not be perfect at everything, but I will always try to get better.
Mastery is not only of the cognitive domain, but it is also a holistic view of learning that includes physical, social and spiritual competencies. Giving children the opportunity to develop all parts of self can help to grow a strong spirit of mastery.
Using Outcomes-based assessment and tracking progress towards a set standard can help students and teachers know where learning is at and respond accordingly. Dylan Wiliam has identified that clarifying learning targets is key to improving achievement and mastery (2011). When the criteria for success is visible and understood, students can strive for it. Teaching students to set learning goals and solve problems through collaboration can help them to build a spirit of mastery. Communicate teacher expectations clearly and provide specific feedback on student behaviour.
Independence – Earning independence by building trust.
One of the goals of education is for students to own their own learning. Dylan Wiliam (2011) has identified that goal setting and strategies for students to direct their learning path and destination. It is important that if students are pulled out of the classroom for additional support, they understand that it is to help them become independent when they rejoin their classmates. Student empowerment is important to build intrinsic motivation and confidence to persist through learning experiences.
Weak Spirit of Independence
Distorted Spirit of Independence
Spirit of Independence
Impotent
Manipulative
Powerful
Coerced
Dictatorial
Assertive
Unassured
Defies Authority
Confident
Misled
Rebellious
Self-Control
Futility
Reckless
Optimism
Building a democratic classroom can occur through classroom meetings to make decisions, offering choices within curriculum, and co-constructing classroom rules and norms. This can be done through the Circle of Courage itself, with groups brainstorming what it looks like when we work together to build a sense of belonging, mastery, independence and generosity within our community of learners.
By teaching self-regulation, we encourage and build independence and self-control. This can assist students in maintaining focus and personal responsibility.
Generosity – Giving to others makes you feel good.
The virtue of generosity is one of the most valued within Indigenous communities. There is a responsibility to care for others within our community. By being generous, learners can develop a sense of pride and joy. Service projects and creating spaces that allow for generosity to occur can help to strengthen students’ spirit of generosity.
Weak Spirit of Generosity
Distorted Spirit of Generosity
Spirit of Generosity
Emptiness
Driven
Purpose
Rancour
Over Involvement
Empathy
Exploiting
Servitude
Kindness
Vengeance
Co-Dependent
Forgiving
Disrespectful
Plays Martyr
Respectful
Citizenship education can help us to build a caring community within and outside of our classrooms. At the beginning of the school year, create opportunities for students to know each others’ stories. Activities such as collaborative writing, field trips together, assigned student jobs contributing to the well being of the class, and collaborative problem-solving activities can help students experience giving and receiving support. By watching slide shows of classmates completing tasks can all build a sense of care.
Introducing and teaching the Circle of Courage can help your students understand how they are building their own and their classmate’s resilience and spirit.
Peterson, J. M., & Taylor, P. D. (2009). Whole Schooling and the Circle of Courage. Retrieved May 8, 2019, from Whole Schooling Consortium: http://www.wholeschooling.net/
Classroom management is the part of teaching that is invisible when it is working well, and highly visible when we and our students are struggling. Sometimes it may feel like herding cats:
Classroom Management’s purpose is to “provide structure while helping students develop autonomy, awareness and self-regulation skills” (Emerich France, 2018).
Traditional classroom management theories rely on teachers viewing student behaviour as a matter of choices or intentional defiance. As a result, traditional solutions have been a list of consequences and rewards. While consequences may be a part of a classroom management plan, if the goal of classroom management is to have students understand themselves and self-regulate, teachers and students need to first identify the causes of behaviours, attempt to prevent triggers and increase self-awareness.
General principles of Discipline with Dignity (Curwin, Mendler, & Mendler, 2018) are:
This perspective of classroom management allows us to link the effects of trauma and brain development to understanding some of the underlying causes of student behaviours. Classroom management is founded on knowing our students and building relationships with them.
Basic Needs that Drive Behaviour
Curwin, Mendler, and Mendler (2018) and Dr. Martin Brokenleg (Circle of courage, 1990) have identified what our and our students’ basic human needs are in order to be healthy and whole.
Curwin, Mendler, and Mendler
Dr. Martin Brokenleg
Identity – “how we view ourselves and how we feel about ourselves”
Attention – “the need to be acknowledged by others in a way that makes us feel that we matter”
Generosity – to give our knowledge, skills, insight, experience, and resources to others; receiving a sense of purpose from strengthening the community
Connection – “our need to feel that we belong to something that matters to us”
Belonging – to feel valued, affirmed and significant within the group; believing in the importance of a shared purpose; to identify with shared goals
Competence – “feeling that we know how to do something”
Mastery – to develop abilities, skills, and knowledge in order to take risks, to try new things and learn from others; to embrace challenges and learning
Control – “the desire to make decisions that count, to have real choices, to control our environment”
Independence – to feel in control of our learning; to have a sense of competent autonomy and to contribute to responsibility
Saskatchewan teachers and educational assistants used a Circle of Courage graphic organizer to brainstorm what they need to keep in mind to create a learning environment that fosters a sense of belonging, mastery, independence, and generosity in their students:
A school environment that provides opportunities for students to have their needs met is part of a “prevention mindset” (Curwin, Mendler, & Mendler, p. 61). Looking at prevention from the student perspective, appropriate behaviour is generally achieved when students:
Feel connected to the teacher, one another, and the curriculum.
Believe that success is attainable with reasonable effort.
Feel respected by being heard; feel teachers strongly care about them in personal ways.
Are given responsibility, especially in helping other children. This involves giving appropriate choices throughout the day.
Look forward to sanctioned moments of joy and laughter every day.
Believe that what is being taught is relevant.
Activating Peers as Learning Resources
Dylan Wiliam has identified five strategies for Formative Assessment, which increase student engagement and achievement.
Wiliam’s work guides our instruction in many ways, including how to activate learners as instructional resources. Collaborative learning that encourages group goals and individual accountability is more powerful than simply having students work on tasks in a group. This group work increases student sense of belonging and provides a space for generosity within a community of learners.
Wiliam (2011) identifies why students benefit from helping their peers:
they are working towards a common goal and they benefit from the efforts of all, resulting in increased motivation.
they care about their group members, resulting in social cohesion.
they understand and can address the difficulties their peers are having, resulting in personalization.
thinking together brings clarity, resulting in cognitive elaboration.
Assessment Strategies
Wiliam writes about a number of strategies that can enhance peer support of learning within a classroom.
Peer Evaluation of Homework – The class or teacher create criteria (rubric) for the work. Who is assessing the work changes and is not announced until the homework is complete. Evaluators may be: Self, Other – individual or group or Teacher
Two Stars and a Wish – student feedback on other students’ work – 2 positive qualities and a suggestion for improvement; Teacher needs to instruct students on what quality feedback looks like. Sentence starters can be useful to give to students in advance.
Preflight Checklist – When there are specific requirements or features of a piece of student work, this list is checked by a peer prior to the work being handed in. If there are pieces missing that were on the checklist, the peer is the one held accountable.
Group-Based Test Prep – Within a group, each student is assigned a different part of a unit to review. They prepare and present their review to their group. Peers assess the review using coloured cups (red – not as good as I would have done; yellow – about the same as I would have done; green – better than I would have done).
Teaching and using group work goes beyond learning styles, and can enhance student motivation, engagement and resilience.
Brain research has shown that there are differences between stress behaviours and misbehaviour. If we treat them in the same way, it can be hard on our students and hard on ourselves.
“Trauma has a powerful capacity to shape a child’s physical, emotional, and intellectual development, especially when the trauma is experienced in early life” (JBS International and Georgetown University National Technical Assistance Centre).
There are significant impacts of ACEs on individual health outcomes. Brain science has determined that ACEs disrupt neurodevelopment, which leads to social, emotional and cognitive impairment, adoption of health-risk behaviours, disease, disability, and social problems and eventually can lead to early death.
Continued exposure to trauma creates an environment filled with relentless fear, resulting in the development of a self-defence mechanism in a child’s brain that requires a fight, flight or freeze response. Stress responses are maintained due to:
A persistent fear response that ‘wears out’ neural pathways;
Dissociation from the traumatic event in which the child shuts down emotionally; and
Disruptions in emotional attachment, which can be detrimental to learning.
(JBS International and Georgetown University National Technical Assistance Centre).
Intergenerational and Historical Trauma
The Georgetown University National Technical Assistance Centre for Children’s Mental Health has identified two types of trauma that may have an impact on children in schools:
Intergenerational trauma results when disturbing experiences have not been addressed and their emotional and behavioural legacy is passed down from parents to their children. Parents who experienced persistent trauma in childhood may struggle with their own ability to express empathy, compassion, and self-regulation. Unresolved trauma may make it difficult for parents to build trusting relationships and healthy attachments. This trauma is then transmitted to future generations.
Historical trauma goes beyond a single family to a community caused by historical, systematic abuse and injustice. In additional to family-specific intergenerational trauma, historical trauma may also result in shame and loss of culture and identity. The legacy of historical trauma can result in “repression, dissociation, denial, alcoholism, depression, doubt, helplessness, and devaluation of self and culture”.
Brain Malleability
While trauma affects brain development, causing structural and hormonal changes, the malleability of the human brain brings us hope. Because the brain can heal, and new neuropathways can be developed, teachers and schools can impact the long-term social, emotional and physical impacts of trauma. Trauma-informed schools are those where the school climate, instructional designs, positive behavioural supports, and policies are created so that traumatized students have what they need to achieve academic and social competence (Craig, 2016).
The goal of PBIS is to create a positive school climate, in which students learn and grow.
A process through which children and adults acquire and effectively apply the knowledge, attitudes, and skills necessary to understand and manage emotions, set and achieve positive goals, feel and show empathy for others, establish and maintain positive relationships, and make responsible decisions.
Dr. Bruce Perry has done significant research in the area of mitigating the impact of trauma on children.
Dr. Perry contributes to the Child Trauma Academy, which makes a number of free resources available.
Solutions in classrooms directed at brain stem activity can help create new pathways in the brain’s cortical areas. These include
patterned rhythmic activities
walking
dancing
singing
meditative breathing
create a calmer cognitive state so higher order thinking can occur
scaffold new information on prior knowledge
use classroom discourse purposefully
peer collaboration
provide direct instruction
Instruction and school environments that provide calming opportunities can help children learn to be more self-aware and foster their own emotion regulation, which then impacts the behaviour-consequence cycle.
As teachers, when we understand the underlying causes of student behaviour, we can not only respond, we can avoid triggers. Classroom Management is more than just the strategies we use, it is the relationship and understanding of the children we teach in order to help children build new neural pathways, understand their own behaviour and learn to self-regulate.
When and Where to Refer Students?
Sometimes, we might recognize that one of our students requires additional supports. Central Office personnel or support teachers may be able to help you assess where your students are at. For parents, Amy Morin writes about some of the indicators that a child might require targetted services in her blog post “When Should Parents Seek Help for a Child’s Behavior Problems?”. Supports from outside of your school system may be available through your regional health authority, which is listed on the Saskatchewan Health Authority’s Website. In the Prairie North Health Region, for instance, there are a number of youth-focussed supports in North Battleford.
Craig, S. E. (2016). Trauma-Sensitive Schools: Learning Communities Transforming Student Lives K-5. New York: Teachers College Press.
Curwin, R. L., Mendler, A. N., & Mendler, B. D. (2018). Discipline with Dignity: How to Build Responsibility, Relationships, and Respect in Your Classroom. Alexandria: ASCD.
Comprehension strategies are ways in which we make meaning of the information we take in through text, media, and sound. Often considered the domain of the English Language Arts teacher, we now realize that it is important that teachers across subject areas and grades encourage meaning making and use tactics that cause learners to use strategies that they may or may not be fluent in.
Depending on which researcher you follow, you may categorize comprehension strategies into 6, 7, or even 13 different strategies. Following the framework of Ellin Keene (1997) in her book Mosaic of Thought: Teaching Comprehension in a Readers Workshop, this post identifies seven strategies that help learners make meaning of complex information. A detailed comprehension strategies summary describes the actions of readers, writers, mathematicians, and researchers.
While categorizing strategies and tactics is helpful, we need to keep in mind that the comprehension strategies are all interconnected. Visualization is grounded on activating and connecting knowledge. Determining importance and summarizing are closely related. Monitoring comprehension often creates self-questions. Each person has different comprehension strategies that their mind tends to use. Our challenge as teachers is to create a pause and opportunity for students to try stretching to new mental pathways.
Activating and Connecting Prior Knowledge
This comprehension strategy involves students connecting their learning to past experiences, events in the world, and to other learning they may have in and out of school. We simply can’t understand new information that we hear, read or view without thinking about what we already know. It is important that teachers teach learners to:
Relate to prior experience.
Make connections between text, media, and personal observation.
Connect the new to the known – use what learners know to understand new information.
When students are using this comprehension strategy, you may hear them say…
This reminds me of…
I noticed…
It made me think of…
I never knew…
That changed my mind…
This is different from…
Some tactics that teachers might use to promote activating and connecting prior knowledge include:
This comprehension strategy involves students recognizing and acting on their own confusion, and their self-questioning to determine understanding. We monitor our comprehension and keep track of our thinking in a number of ways. We notice when text makes sense and when it doesn’t. We ask questions, infer, activate background knowledge, and make connections, all in the effort to promote understanding. It is important that teachers teach learners to:
Become aware of their thinking as they read.
Notice when meaning breaks down and their mind wanders.
Employ ‘fix up’ strategies – reread for clarification, read ahead to construct meaning, use context to break down unfamiliar words, skip difficult parts and continue to see if meaning becomes clear, check and recheck answers and thinking, and examine the evidence.
When students are using this comprehension strategy, you may hear them say…
I think…
I understand…
This doesn’t make sense…
Oh, now I get it…
A part I had trouble with…
Some tactics that teachers might use to promote monitoring comprehension include:
This strategy involves students actively wondering about topics and questioning facts and information. Questioning is the strategy that propels learners on and is at the heart of inquiry-based learning. Humans are driven to make sense of the world, and questions open the doors to understanding. It is important that teachers teach learners to:
Wonder about the content, concepts, outcomes, and genre of text.
Question the author.
Read to discover answers and gain information.
When students use this comprehension strategy, you may hear them say…
I wonder…
I’m curious…
My big question is…
Why…
Do you know anything about…
Some tactics that teachers might use to promote questioning include:
This comprehension strategy involves students predicting, hypothesizing, interpreting, and making conclusions. Inferential thinking allows learners to grasp the deeper essence of text and information. Readers infer by taking their background knowledge and merge it with clues int he text to draw a conclusion or arrive at a big idea that is not explicitly stated in the text. It is important that teachers teach learners to:
Use context clues to figure out meaning of unfamiliar words.
Draw conclusions from evidence.
Predict outcomes, events and observations.
When students use this comprehension strategy, you may hear them say…
I’m thinking…
It seems to me…
I’m guessing that…
I predict…
Probably…
Some tactics that teachers might use to promote inferring include:
This comprehension strategy involves students evaluating information, making judgments about information, and identifying key ideas and concepts. We read nonfiction to learn, understand, and remember information. It is important that teachers teach learners to:
Target key information.
Choose what to remember.
Construct big ideas from smaller ideas.
When students use this comprehension strategy, you may hear them say…
This is really important…
The most important ideas here are…
So far, I have learned that…
I think this part means…
Some tactics that teachers might use to promote determining importance include:
This comprehension strategy involves students making mental pictures and/or mind maps of ideas and how they interconnect. Visualization builds complex connections and involves more than just how something looks. It extends to other senses such as smell, touch, sound, and taste. It is important that teachers teach learners to:
Gain information from the images they construct and view.
Create mental images drawn from background knowledge and observations.
When students use this comprehension strategy, you may hear them say…
I am getting a picture…
I can see (smell, hear, taste)…
I have a picture in my head…
Some tactics that teachers might use to promote visualization include:
This comprehension strategy involves students recalling, retelling, creating new meaning, and remembering information. Synthesizing information allows us to see the bigger picture as we read or observe. Thoughtful readers merge their thinking with information to come to a more complete understanding. It is important that teachers teach learners to:
Add to their knowledge base.
Paraphrase information
Move from facts to ideas.
Summarizing and synthesizing are often used as synonyms. While they are related, they are different strategies that readers use. Sarah Eaton, a professor at the University of Calgary (2010, Summarizing vs Synthesizing), identifies these differences:
Summarizing
Synthesizing
A basic reading technique.
An advanced reading technique.
Pulls together information in order to highlight the important points.
Pulls together information not only to highlight the important points, but also to draw your own conclusions.
Re-iterates the information.
Combines and contrasts information from different sources.
Shows what the original authors wrote.
Not only reflects your knowledge about what the original authors wrote, but also creates something new out of two or more pieces of writing.
Addresses one set of information (e.g. article, chapter, document) at a time. Each source remains distinct.
Combines parts and elements from a variety of sources into one unified entity.
Presents a cursory overview.
Focuses on both main ideas and details.
Demonstrates an understanding of the overall meaning.
Achieves new insight.
Summarizing is taking the details of information apart while synthesizing is putting those details back together into a new and unique whole.
When students use this comprehension strategy, you may hear them say…
I never knew… now I know…
I think the big idea is…
I have learned that…
Now I understand that…
Some tactics that teachers might use to promote summarizing and synthesizing include:
When do we explicitly use Comprehension Strategy Tactics?
If the information that we are having students interact with is complex, it is important to use instructional methods that help them make meaning. As well, if we notice that students are struggling with a specific skill or content area, we can view comprehension strategy instruction a possible solution to those learning barriers. Teaching with comprehension in mind will lead to increased cognitive engagement and deeper meaning-making.
Eaton, S. E. (2010, September 26). Reading Strategy: The difference between summarizing and synthesizing. Retrieved from http://www.drsaraheaton.wordpress.com
Johanson, T., & Broughton, D. (2014). Exploring Comprehension in Physics. Saskatoon: McDowell Foundation.
Keene, E., & Zimmermann, S. (1997). Mosaic of Thought: Teaching Comprehension in a Readers Workshop. Portsmouth: Heinemann.
Public Education & Business Coalition. (n.d.). Thinking Strategies for Learners: A guide to PEBC’s professional development in reading, writing, mathematics and information literacy. Retrieved December 15, 2018, from Public Education & Business Coalition: https://www.pebc.org/wp-content/uploads/publications/thinking-strategies.pdf
Math fluency is the ability to perform mathematical operations quickly and accurately. Math automaticity with basic facts is part of fluency. John Munroe (2011) indicates that we as learners have a finite amount of working memory. It is important that student working memory is available to learn new math concepts, solve complex problems and think creatively in mathematics, rather than being used to recall basic math facts.
So, how do we promote automaticity of basic math facts without endless worksheets, mad minutes and text-book assignments? Math games, puzzles and routines related to grade level concepts allow for flexibility in thinking, practice and student engagement and fun. The following documents can help us connect practice with curricula:
There has been a significant amount of research in the area of mathematics routines to enhance learning, building automaticity and fluency. There are a number of key resources that are helpful to teachers from grades 1 to 10.
Productive Mathematics Discussion
Margaret Smith (2011) has created a structure for planning for and implementing classroom discussion in a mathematics classroom. Discussion and sharing mathematical thinking is the key to most mathematics routines.
This sequence of thinking can be applied to a teaching strategy, Learner Generated Examples (Crawley, 2010), to create a powerful way of sharing student thinking in mathematics.
Learner Generated Examples:
Brian Crawley, a teacher from Saskatchewan, did extensive research on learner-generated examples. Overlaying the Five Practices, you create a cycle that can be applied to many number routines:
Move through this cycle three times. This allows students to push beyond the knowledge that they find easy to access and move to more and more complex ideas.
Examples of Math Routines
There are several powerful math routines gathered from many Math Routine Resources that can be adapted to different concepts over grade levels. The following is a short list:
The key to math routines influencing math fluency in your classroom is to choose appropriate concepts and numbers related to your curriculum. For example, in the routine “Today’s Number”, if you are in early primary, you might choose a number between 1 and 10. If you are teaching ideas around skip counting, perhaps choosing the number 6. If you are working on perfect squares, perhaps choosing a number like 36.
Math Puzzles
Math puzzles allow learners to explore mathematical ideas and practice thinking flexibly. Well-designed puzzles are engaging and logical, with most of the time focussing on solving the puzzle mathematically. As with routines, puzzles can be adapted to the range of numbers and concepts appropriate for a given grade level.
Extensions can include puzzles that go above the 1-10 line and below the 91-100 line on a hundreds chart. What numbers exist below 0? What numbers exist above 100?
Place value puzzles can help students overcome misconceptions. They were originally created by Naoki Inaba from Japan, and then adapted to decimal numbers on the “Embrace the Challenge” blog.
When choosing and setting up games in your classroom, along with considering the math concepts you are emphasizing, it is also important to consider student grouping and classroom norms.
By creating homogeneous groupings, it is possible that some groups will use tools to help them calculate or identify numbers, such as:
In heterogeneous math game groupings, it potentially frustrating for both the stronger student and student who needs more time to compute or recall math facts.
There are many excellent math games sites that include Math online games. As well, there are many games that require few materials and are engaging for students.
The key to finding and using math games in your classroom is to know the math that you would like to build fluency in, and then search for those concepts. There are literally thousands of great sites that you can access both online and paper copy games.