As math teachers, we spend much of our time planning for students who need extra time or support to understand mathematics at our grade level. We also worry about how we might create appropriate challenge for students who have a strong foundation in math and are ready to move ahead before their classmates are ready. The last thing we want is for our most able mathematics students to get bored and disengage. Enrichment seems like a good option, but how do we do that?
Why? Goals
Enrichment activies are meant to allow students to
engage in rich tasks and activities at grade level
stay with their peers
replace foundational math tasks that other students require to build understanding
reinforce mathematical thinking skills so that students are even more able to engage in grade-level math
When? Timing
There are at least two opportunities for enrichment within your instructional sequence.
When we pause to help students review and build readiness for grade-level instruction. This might be in the form of a concept review, reteach, or responsive station.
Enrichment tasks can be an option for students who do not need to review or practice foundational skills.
Enrichment tasks that are loosely based on the same concepts as are in the review will help students be even more ready for grade-level instruction.
When we are teaching grade-level concepts and the majority of students require more time to complete tasks.
Enrichment tasks can be an option for students who are done work earlier than their peers.
Enrichment tasks that are loosely based on the grade-level concept, but allow for creative thinking and application can allow students to see mathematics in a creative and contextual way.
What? Characteristics
Enrichment tasks take on all different forms. Some common characteristics might be:
Engaging
Hands-on
Applied to the Real World
Creative
Games
Puzzles
Math Outside of Curriculum
Connecting Math Concepts
Social Justice topics
You can access hundreds of curated resources in this Enrichment Task Google Folder. Feel free to share, download, and use these resources.
Who? Target Students
All students might be able to engage in Enrichment tasks at some point in the year. Different students have different strengths. Enrichment is not exclusive to gifted students.
How? Classroom Structures
A station or enrichment corner are relatively simple to set up. One thing to consider is what tasks to have available at what time. A suggestion is that the tasks in the enrichment station should be loosely related to the math concept being experienced by the rest of the class. This ensures that the students doing enrichment tasks are going to be even more able to complete grade-level math tasks. When enrichment tasks are completely unrelated to the math students will experience next, we are inadvertently creating a time gap in student learning and may actually contribute to lower achievement.
Our school year seems to race by faster and faster every year. A worry that we have is that our students might not be ready for their next concept, next grade, or next step in their education journey. Some of our hopes while year planning would be to:
Prioritize concepts that are foundational to the next school year.
Estimate time so that higher priority outcomes have more time.
Order concepts logically so that math ideas build within a grade.
Cluster outcomes that help kids understand the connections between math ideas.
A prioritized and sequenced math year plan is not a pacing guide. Rather, it is a roadmap that helps you, as teacher, know where you are going next, and provides an estimate for how long to spend on a concept. A prioritized and sequenced year plan should be revisited several times through the year to see where you are at, what might need to change, and what your students need next.
Creating a Prioritized and Sequenced Math Year Plan
Prioritize Outcomes
To prioritize outcomes in our grade level, it is important to know what concepts lead directly into next year’s math curriculum. Some tools that help you do this are Curricular Through Lines. It is most helpful to use the document that shows YOUR grade as well as the grade level after you:
Highlight outcomes YELLOW if they lead directly into next year. For example, multiples and factors in Grade 6 (N6.2) leads directly into adding and subtracting fractions (N7.5) and divisbility rules (N7.1).
Do not highlight an outcome if it does not lead directly into next year. For example, numbers greater than 1000000 in Grade 6 (N6.1) does not lead into any outcome in Grade 7.
Go back to your yellow outcomes. In a given year, you might want to have 6-7 outcomes that are highest priority (GREEN). If you have more than 6-7 outcomes highlighted yellow, which of those would be most important to emphasize? You might want to have a discussion with the next grade teacher to help you determine this.
At the end of this process, you might have:
6-7 HIGHEST priority outcomes.
some MEDIUM priority outcomes.
some LOWEST priority outcomes.
Cluster Outcomes
Some of our curricula have several outcomes that would be much more effective if they are taught together. For example, in Grade 7, there are several patterns and relations outcomes and a shape and space outcome that are easier to teach if you put them together:
P7.1 – Relationships between tables of values, graphs, and linear relations
P7.2 – Understanding equations and expressions
P7.3 – Solving one and two step equations with whole numbers
P7.4 – Solving one and two step equations with integers
SS7.4 – Ordered pairs and the Cartesian Plane
When you look at these holistically, it might make sense to cluster these outcomes into:
P7.3 and P7.4 – Solving one and two step equations
P7.1, P7.2, SS7.4 – Representing linear equations as a graph, table of values, equations, words, and pictures/manipulatives.
Once you cluster your outcomes in this way, they act as a single unit within your year plan.
Sequence Outcome Clusters
There are several things to consider when you are creating your outcome sequence:
How might you start the school year? What are they ready for after summer break?
If you follow the order of curriculum, we would begin with place value. While this might be logical at early grades, it can sometimes be daunting for students coming back after summer.
Consider starting with a topic that has lots of hands-on opportunities that can help students understand what numbers and shapes are. This will help get them ready for place value later on.
Example: Consider starting with a graphing and data outcome so that students can use numbers as they create axes for graphs and explore how big those numbers are within a real-life context.
What outcomes are pre-skills for other outcomes in your grade?
There are several examples where one outcome logically comes before another one.
Example: In grade 2, it makes sense that numbers to 100 would be taught before adding and subtracting numbers to 100.
Sometimes we might think something is a preskill that is not. For example, in grade 3 we might think that we need to do addition and subtraction before we do multiplication and division. Because multiplication in grade 3 is limited to 5 x 5, we don’t need to be able to add large numbers before we multiply. In fact, the pre-skills for multiplication are:
knowing numbers to 25.
skip counting by 2, 5.
decomposing numbers.
How might you end the school year? What is best suited for May and June instruction?
Spring is a great time to go outside and extend your classroom. When you consider what math fits into outdoor experiences, you might want to plan for those concepts in the spring.
Example: Grade 3 data and graphing could be based on finding things in the natural environment, creating concrete graphs, pictographs, tallies, and graphs representing what you find.
Create Time Guidelines
The time you spend on each topic is determined mostly by how many highest priority outcomes or outcome clusters you have identified and how many days of mathematics you have in a school year. If you have math every day, you can count on having approximately 150 days of math, as there are always concerts, field trips, and other things that impact instructional time.
Estimate the amount of time per outcome (or outcome cluster):
Highest Priority Outcome – 22 to 25 days
Lowest Priority Outcome – 2 to 5 days
Medium Priority Outcome – 5 to 10 days
Estimate the total number of days you might need for all of your prioritized outcomes.
Is this approximately equal to your instructional time?
Do you need to shorten timelines? Lengthen them?
Plot your units out onto a school year calendar.
Do outcomes begin and end at logical times? What might need to shift for holidays and report cards?
Sample Year Plans
Sample year plans are not meant to show you a ‘right’ answer, but rather to be an example of what a prioritized and sequenced year plan might look like.
It is important to revisit and revise your year plan through the year. Student needs and unexpected disruptions require adjusting along the way. Things to consider when you are reflecting on your year plan are:
Where did I think we would be in our sequence right now? Where are we actually at?
When I look ahead to the end of the year, do I need to adjust the amount of time I am spending on the units I have left?
Are there some of my prioritized outcomes that I need to deprioritize to give us time on what is MOST important?
Might I need to shift some of my lower and medium priority outcomes to stations or centers?
Are there some outcomes that I might connect to other curricula? Examples include:
Some shape and space outcomes moved to Art (transformations, area, 3D objects)
Some statistics outcomes moved to Social Studies (measures of central tendency, graphing)
It is important to keep your year plan current so that can make informed decisions throughout your year. Reflecting regularly can help you build a strong foundation for your students.
Why create a strategic plan for your organization? Whether you are a small business, school, or international company, a strategic plan helps you to
Create a shared vision for where you are heading.
Align actions and impacts from different levels of your organization.
Simplify decision-making.
Develop strategies and actions to create a positive future state.
Outline data and evidence needed to monitor progress.
Allocate resources to initiatives that will create desired changes.
The Unhelpful Side of Strategic Planning
There is a difference between having an ineffective strategic plan and having an ineffective action within a strategic plan. A working strategic plan will have you gather evidence so that you shift your action plans if a strategy is not having the impact you are hoping for. An ineffective strategic plan results in you having no idea if an action is having an impact, causing us to continue doing and dedicating finite resources to ineffective actions. It is possible to have useful strategies within an ineffective plan, making it hard to create systemic and sustainable change.
An ineffective strategic plan might have the following features:
It does not reflect current state.
A working strategic plan reflects current trends, research, and actions being taken within our organizations. It is a living document that shifts over time.
It does not have buy in.
A working strategic plan is co-constructed by key stakeholders and is known to all employees and managers.
It is not used.
A working strategic plan is used to determine resource allocation, professional development plans, actions of staff, and evidence gathering.
It is too broad.
A working strategic plan is simple and focuses on what is most important within your organization. Too many actions and outcomes can spread time and resources out too much and reduce impact of what is most needed.
It is too vague.
A working strategic plan has clear actions, impacts, timelines, and responsibilities. Using specific terms can help everyone in an organization know expectations and roles.
It includes ineffective strategies.
A working strategic plan is a form of action research. There is no guarantee that an action is going to produce a desired outcome. What is important is that data collection is timely and specific so that your organization can shift strategies if needed. Be aware, however, that often time is needed for implementation and understanding in order to have a desired impact.
Traditional Strategic Planning: A Deficit View
Traditional strategic planning methods go through a cycle of problem identification and solution generation and implementation. Traditional strategic planning methods can lead to a deficit view.
Rather than a deficit view of what your organization is not doing, or not doing well, a strategic planning process should identify where you want to go, what you are already doing that might help, and how you might leverage the strengths of your organization to get there. By empowering your employees with a positively intentional view of your organization’s impact, it is possible to design a future state and the steps to get there. When an organization works with others to achieve its goals, identifying the influence you have on others and the influence they have on you, along with the steps you might take to achieve the influence you are seeking should be a part of your strategic action plan.
Hopeful and Helpful Strategic Planning Processes
There are three key theories of strategic planning that work well when viewed as part of a holistic process:
Appreciative Inquiry
Outcome Mapping
Logic Modelling
Appreciative Inquiry
There are a series of stages built into the Appreciative Inquiry process:
1. Discovery: What are we currently doing well?
Ideas are themed and patterns emerge related to organizational strengths. This process allows organizations to focus on positive capacity.
2. Dream: What is the world calling us to become?
What are the things about our organization that no matter how much we change, we want to continue into our new and different future?
3. Design: What should be our ideal state?
Co-constructed ideas are grounded in what we are currently doing well, opportunities that are apparent, and organizational capacity.
4. Destiny/Delivery:
How do we empower, learn and plan for actions to reach our ideal state?
Along with steps in the Appreciative Inquiry cycle, there are a number of foundational principles that guide conversations and planning:
You can learn more about Appreciative Inquiry with the following resources:
Where does outcome mapping fit into Appreciative Inquiry? Within the Design Phase, those organizations that work collaboratively with other groups in a mutually influential role can utilize facets of outcome mapping to identify actions, outcomes, and desirable observable behaviours. Here is a helpful introduction to Outcome Mapping from Sara Earl.
Strategic action planning and program evaluation involve the following steps:
In order to have an impact, people and organizations need to change behaviours. This occurs within the Outcomes stage. When we see ourselves and our organization as a part of a large web of interdependent entities in relationship with one another, we can view change as
Continuous
Complex
Non-linear
Multidirectional
Not controllable
People and organizations contribute to the goals of others through influence rather than control.
Program Evaluation as Formative Assessment
Program evaluation is built into strategic planning so that progress can be monitored. When viewed as formative assessment, program evaluation can provide real-time information to inform decisions regarding action plans, inputs, and innovations within your organization. Are the actions you are taking resulting in the outcomes you have identified?
Traditional Evaluations
Evaluation as Formative Assessment
Purpose
Supports improvement, summative tests, and accountability. Renders definitive judgments of success or failure.
Supports development of innovation and adaptation in dynamic environments. Provides feedback, generate learnings, support direction or affirm changes in direction.
Roles & Relationships
Positioned as an outsider to assure independence and objectivity.
Positioned as an internal team function and ongoing interpretive processes.
Accountability
Focused on and directed to external authorities and funders.
Centered on the innovators’ deep sense of fundamental values and commitments.
Design
Design the evaluation based on linear cause-effect logic models.
Design the evaluation to capture system dynamics, interdependencies, and emergent interconnections.
Measurement
Measure performance and success against pre-determined goals and SMART outcomes.
Develops measures and tracking mechanisms quickly as outcomes emerge and evolve.
Evaluation results
Aim to produce generalizable findings across time and space.
Evaluation engenders fear of failure.
Aim to produce context-specific understandings that inform ongoing innovation.
Evaluation supports hunger for learning.
Complexity and uncertainty
To control and locate blame for failures.
Learning to respond to lack of control and stay in touch with what’s unfolding and thereby respond strategically.
(Quinn Patton, 2006)
Boundary Partners
Boundary partners are a term used within Outcome Mapping to describe those groups or organizations that you work with directly and anticipate opportunities to be mutually influential. Outcome Mapping identifies behavioural changes in your boundary partners as being a key measurable towards your goals. This is due to the fact that “development is done by and for people. Although a program can influence the achievement of outcomes, it cannot control them. This is because ultimately the responsibility rests with the people affected” (Earl, 2008).
Progress Markers
Progress markers (Earl, 2008) are a set of statements describing a progression of changed behaviours in a boundary partner. These describe:
Actions
Activities
Relationships
Leading to the ideal outcome. These markers show the complexity of the change process and have the following characteristics:
Can be monitored and observed
Permit on-going assessment of boundary partner progress, including unintended results
The ladder of change can be applied to any progress marker:
Beginning: Expect to see.
Mid-Term: Like to see.
Final: Love to see.
To understand more about Boundary Partners and Progress Markers, watch Sara Earl’s Explanation.
You can learn more about Outcome Mapping through the following resources:
A logic model (Taylor-Powell & Henert, 2008) has many different forms and is a structured way to note specific inputs and activities of an organization and how they lead to outcomes over short, medium and long-term timelines. A logic model can be used in conjunction with other strategic planning processes and is a useful organizer.
You can learn more about creating a logic model through the following resources:
The following Seven Steps for Strategic Planning encompass all three of these foundational processes.
Area of Focus
What from your state or provincial priorities resonate with you as an area of need for your students and staff? What are you seeing that indicates this is something your organization should focus on?
What does your data say? What is your present state?
What are you currently doing that is particularly effective in this area? How do you know?
What do your students need you to focus your creativity and energy on?
What changes would benefit your students and staff?
Who are you trying to influence?
For schools, students are the most obvious group you are trying to influence.
Who else’s behaviour are you hoping to influence through this strategic plan? What individuals, groups, or organizations?
What might happen if your plan is successful?
In general terms, what impact are you hoping to have?
What are the desired behaviours, relationships, beliefs and actions of those you are hoping to influence?
What would you expect to see? like to see? love to see?
What might you as an organization DO to contribute to these changes?
What are you currently doing that works towards these impacts?
What might you do in the future?
Who might lead this work? Who might participate in this work?
What resources might you need?
What systemic structures might need to be in place?
What time, money, and other resources are required?
Where might these resources come from? be reallocated from?
What are the outcomes you expect to see? There are two ways of viewing outcomes:
Logic Chain:
What are the short term, or direct behavior changes that might result from your actions? These are often LEAD indicators (behaviours that might predict a future state or success). These short term outcomes are often teacher behaviours.
If those happen, then what medium term then long term, or indirect behaviour changes and outcomes might be influenced by your actions? Some of these are LEAD indicators, while others are LAG indicators (behaviours in current state that are based on past actions and performance).
Behaviour Progression:
Consider the group whose behaviour you would like to influence. What might you EXPECT to see? LIKE to see? LOVE to see?
How might you monitor your progress?
What evidence might tell you about the implementation of your actions?
What evidence might tell you about the efficacy of shifts in organizational practices and structures?
By working through an integrated process for strategic planning, you can empower your organization to look beyond a deficit view of present state and work towards a desired future. By having an action plan for the steps to getting to your future state, it is possible to measure progress as formative assessment, providing a continuous feedback loop for your organization.
Stavros, J., Godwin, L., & Cooperrider, D. (2016). Appreciateive Inquiry: Organization Development and the Strengths Revolution. In W. J. Rothwell, J. Stavros, & R. L. Sullivan, Practicing Organizaiton Development: Leading Transformation and Change (4th Ed) (pp. 96-116). Hoboken: John Wiley & Sons Inc. doi:https://onlinelibrary.wiley.com/doi/book/10.1002/9781119176626
What were many of us MOST looking forward to for Christmas? Food, Fun and Family… Fun in our family over the years has been board games, cards and dice games. And then new health orders came to Saskatchewan and that hope seemed to be dashed… BUT… there was a huge ‘aha’ when I was asked to design and facilitate a “Let’s Play: Math Games Online workshop” this past week. I put my research hat on and found some online platforms for play. There are SO many cool fun platforms that are FREE to everyone. There will be Cribbage for Christmas after all!!
I have always believed that my confidence and competence in math came from the games I played as a child. Cribbage with my Grandpa and Great Grandpa, Monopoly with my friend Tommy, Go Fish with my sister, 31 with my Mom and Dad… So, how might we play these games online with Jori and Michael in Martensville, Erin in Comox, and Adam in Saskatoon? And better yet, how do we play them with the kids’ cousins Paige in Halifax, Amanda and Christian in Winnipeg, and then there is Auntie Sandra in Melfort and Grandma Carol down in Mesa this Christmas? My family is FAR too large to list everyone here, but you get the idea. Here are some games and platforms that might bring some cheer to your homes and families over the holidays.
Each of the following does NOT require an account or sign in. For each, you:
Start a game.
Copy the link and send it out by text or email to your game friends.
Set up a phone call or video call so that you can talk to everyone in the game. (Did you know that your iPhone can create a group call up to 5 people??) Or use Zoom, Google Meet, Facebook Rooms, What’s App… SO many possibilities. If two of you are in the same house on your own devices, consider either using your phone on speaker, or each of you use headphones/ear buds so you don’t get sound feedback.
Each of these is designed for each player to have their own device – they seem to work on computers or tablet/iPad or smart phone.
We have rich Saskatchewan curricula, and an important step in planning is to make sense of what curriculum is asking students to know, do, and understand, and to connect to our local context. I just recently had a chance to work with the CTEP students in Cumberland House under the guidance of their instructor Lily McKay-Carrier. She calls this Nistota Curriculum (Understand Curriculum). Place matters, our students matter, and our own professional and personal knowledge matters when planning for instruction and assessment. Our professional judgment and expertise are what helps us design units of study that honour who are where we are teaching.
We know that outcomes are what students need to know, do and understand. They are the destination of instruction, while indicators are the ways that students might show us that they know.
So what is a process that we might use to make sense of curriculum? Mind mapping is a visual way to see connections between curricular ideas and link to our teaching context.
Steps for Mind Mapping:
Determine what course(s) and outcome(s) you are going to cluster into one unit of study.
If you are creating a cross-curricular unit, then you might want to start with one course and then link to others.
Some curricula cluster outcomes into strands that make sense to teach as one unit (i.e. science and social studies), while others make sense to cluster outcomes from different strands or teach them alone (i.e. mathematics), as a strand is too large.
Identify what concepts students would have interacted with BEFORE this unit that would have provided a foundation, and what concepts this unit feeds into next.
Read over the outcomes to get a general sense of what the unit of study will be about. If you were to describe this unit to someone who is not a teacher, what might you say in a sentence or two?
Highlight the main concepts identified in the outcomes and indicators. These are often the NOUNS.
Mind map the main concepts to see how they connect. Often, there is repetition between outcomes, so this helps to streamline the unit.
Ask yourself what activities based on your community or your personal and professional knowledge might connect to curricular ideas. Add these to your mind map.
Sample Unit: Mind Map of Diversity of Living Things generated in collaboration with CTEP students fall, 2020.
Once you have generated teaching ideas, ask yourself if these honour the intent of the indicators in your curriculum. If a student did these things, would they be able to show that they know, do or understand this outcome?
Developing your Mind Map into a Unit of Study
Once you have created a mind map of key concepts and teaching activities, you can
develop essential questions that pull together the unit.
Develop an instructional sequence that includes:
Learning Activities
Assessment – both formative and summative
Materials/resources required
You can see the beginning of a draft unit of study created in collaboration with CTEP students in fall, 2020 focusing on Diversity of Life in Science 6.
Terry Johanson Consulting 