7 Questions You Should Ask When Reviewing Your Current Or Future Math Curriculum
Leaders of mathematics are accountable for many things: the foremost of these being student outcomes, staff development, assessment and reporting. Underpinning all of them is a robust math curriculum.
Put plainly, if you don’t have good curriculum foundations it will be very hard to achieve successful outcomes. As mentioned in the article from Education Corner, you need to look at your ‘intent’ first in order to inform your ‘implementation’ and achieve high ‘impact’.
- The principles of a ‘good’ curriculum
- Popular existing curriculum options
- Questions to ask when reviewing your math curriculum
- 1. How will you be developing mastery within your math curriculum?
- 2. Do you know the differences between the various curriculum models?
- 3. What is the context of your staff and student population?
- 4. How does the math curriculum build on what’s come before and lay foundations for the future?
- 5. To what extent does your curriculum reflect developments in pedagogy and cognitive science for learning?
- 6. What are the support materials and resources like for your math curriculum; will you have to create them all from scratch or spend valuable budget on them?
- 7. How easy is it for you to share your curriculum and communicate it to staff, students and parents?
- Checklist for any new curriculum
- Final thoughts on your math curriculum
The principles of a ‘good’ curriculum
It’s actually pretty straightforward.
A good curriculum should have clear aims and objectives and should be communicated clearly and succinctly. Careful thought should be given to sequencing of topics and the rationale for your curriculum model.
A weak curriculum is merely a list or timeline of units of work, independent from each other with limited interleaving.
Here we take a look at a variety of curriculum models, and the questions you should ask yourself before choosing or delivering any effective math curriculum. Look out for the checklist at the end.
Popular existing curriculum options
There are a variety of existing curriculum options available, such as math mastery, Eureka Math, or Dreambox. These curriculums are presented as a full package of sequencing and curriculum maps, resources and assessments, as well as support for professional development.
It is also common for school districts to author and implement their own curriculum to be used across their area. Where there is autonomy to decide on your curriculum, it might be that you embrace a full package or cherry pick from existing curricula to create a tailored offering.
However, regardless of whether you opt for an off the shelf option or opt to develop your own hybrid, it is essential that you have a clear rationale for your choices.
Questions to ask when reviewing your math curriculum
1. How will you be developing mastery within your math curriculum?
The basis of math mastery is teaching for conceptual understanding and it has become extremely popular in recent years. Mastery is a math teaching and student learning approach that focuses on the deep understanding of math concepts rather than rote learning.
An indication that a concept has been mastered is where a child can use their knowledge in problem solving and apply complex reasoning. Influenced by East Asian approaches, a curriculum based on the principles of mastery have become extremely common, especially during the early stages at lower grades in elementary school.
These principles include:
- Representation and structure
- Mathematical thinking
- Variation theory
In mastery math instruction, concepts are ordered in a carefully structured sequence and students don’t progress to the next stage until they have mastered each one.
However, this approach can often come into conflict with grade structure, common core and time pressures of US schools, so it is crucial to consider how you can balance pace with mastery in your curriculum.
You should map out a timeline to indicate how long is expected to be spent on a unit of work while considering the key concepts your students need to grasp. You should also take into consideration if intervention or supplemental learning is required and how this can be done effectively.
This will help to balance between pace and covering necessary content, while ensuring that mathematical concepts are explored in sufficient depth to ensure seamless transition from, for example, Algebra 1 to Algebra 2 or middle school to high school mathematics programs.
2. Do you know the differences between the various curriculum models?
It is important to think carefully about your curriculum model and why this is best for your students.
|Mastery||Progression and sequence of curriculum builds on previous ideas. A topic is mastered before moving to the next stage.||Learners acquire depth of understanding and should move to the next stage with prerequisite skills.||Risk of not covering required curriculum.|
|Big ideas||Focusing on larger areas of the curriculum and drawing on related topics.|
E.g. Multiplicative reasoning (factors, multiplication facts and division facts, area), additive methods and place value (ordering two-digit numbers and three digit-numbers, standard form, decimal places and algebraic manipulation etc).
|Supports students to make links with smaller topic areas. Supports interleaving of topics.||Math is a network of ideas rather than fitting on a linear scale of progression.|
Students may not develop prerequisite skills to enable them to interleave topics early in the curriculum.
|Spiral||Shorter units that alternate between topic areas (e.g. Algebra Unit 2: Inverse Operations, Place Value Unit 1: Whole Numbers, etc) and each time it is revisited, building on prior knowledge. |
You may teach the properties of shapes to draw 2-d shapes in the first unit and then move to 3-d shapes in the second, before revisiting both units in the third when teaching different orientations.
|Frequent revisiting of skills.||Less likely to make links between topic areas. Can increase likelihood that math is viewed as a long list of topics and skills.|
|Topics||Focusing on embedding all of one branch of mathematics, e.g. number skills, before moving on to proportional reasoning, then algebra etc.|
Each branch is studied in its entirety, therefore potentially a whole year on just number skills for example, before studying any algebra.
|Allows more space to ensure skills are mastered.||Limits scope to interleave topics early in the curriculum. Without extensive spaced retrieval practice, skills studied early in the curriculum can be forgotten.|
New students joining the school would miss entire branches of mathematics.
Whichever model you opt for, the sequencing and interleaving of topics needs careful consideration. Timing and scheduling units of work is a theme across any curriculum model.
3. What is the context of your staff and student population?
Choosing a curriculum model will depend on your context. If you have a strong and stable team of specialist math teachers with good subject knowledge, then models such as mastery or big ideas provide an opportunity to explore mathematical concepts in depth.
If your student population is transient then you would want to opt for a spiral curriculum rather than topics, as this would provide opportunity to frequently revisit with less reliance on prior knowledge.
To deliver mastery and big ideas you will need to ensure there is regular time for collaborative lesson planning to ensure teachers are confident with the required depth of conceptual understanding.
If you’re a middle or high school, you may also want to liaise with your feeder elementary schools to build on the foundations they have established, especially where schools have embedded a mastery curriculum.
If you opt for a mastery curriculum you are likely going to require various manipulatives. This may provide an opportunity to utilize your existing resources to greater effect, otherwise significant investment in apparatus is required.
4. How does the math curriculum build on what’s come before and lay foundations for the future?
As you would with a lesson, plan with the end in mind. Children’s knowledge and understanding of math starts in elementary school and will go well beyond their high school diploma, so it is important to factor in progression from elementary to high school to real-world math!
Think of it as layers of planning, each stage zooming in with more detail. Lessons should not be thought about independently from one another but as part of a bigger picture. This is the same when considering the sequencing of units of work and how it relates to the wider grade group.
When planning your high school math program, look back at the state standards for elementary school. This will give insight into which topics you can expect students to have an understanding of and which topics may need more attention.
For example, geometry is not typically taught in high school until 10th grade. Students typically go through two years of pre-algebra and algebra, in 8th and 9th grade respectively, and may need a fresher when it comes to geometry in 10th grade.
Read more: Teaching Elementary Math: A Guide
Reflecting on previous grade levels will help to inform your planning and scheduling as you think about what information might need consolidating and what may need more time to develop conceptual understanding. This will help to ensure that progress occurs at a suitable pace and that students remain engaged and challenged.
We also need to consider what we are preparing students for, aside from achieving great test results. You will need to consider the next pathways for students, be that employment, further or higher education.
Not only may we need to prepare students to progress to higher study of math, but also subjects with significant statistical focus such as geography and psychology.
When considering curriculum progression, think about your context. Do you want to view the curriculum as separate entities for each grade level, or will you view it as a grade level progression plan?
Whichever you decide, ensure you have a clear rationale for your choices and have accounted for enough development in earlier grade levels to equip students with skills for life rather than focusing on passing end of year tests.
5. To what extent does your curriculum reflect developments in pedagogy and cognitive science for learning?
There have been many recent developments in pedagogy and theory which are features and principles often incorporated into a curriculum and its resources.
Many of these are covered in Third Space Learning’s series of blog posts around How I Wish I’d Taught Maths by Craig Barton, starting with this one on cognitive load theory.
Things to consider are:
- Variation theory: Variation theory covers a rigorous approach to question setting. By carefully selecting the questions and progression of the questions you ask, you are encouraging ‘intelligent practice’, reducing cognitive overload and supporting reasoning and making connections.
- Hinge questions: Hinge questions are carefully considered questions which assist a teacher in diagnosing if a student or a class is ready to move on to the next stage.
- Multiple choice questions: These should be diagnostic questions. The incorrect solutions need to be carefully considered to enable teachers to identify and address students’ common misconceptions. The correct answer should not be obvious from elimination.
- Spaced retrieval practice: Just because students grasped a concept at the time of teaching does not mean that this will remain the case. Continually revisiting skills and concepts keeps them fresh in students’ minds so they can draw on them when required and reduce the need for intensive revision.
- Interleaving: Students need to apply learning to unfamiliar contexts and by drawing on a range of skills to demonstrate mathematical ability. Interleaving means that topics studied earlier in the curriculum are linked in to current learning to make links and revisit prior learning. Thinking about how topics can be revisited and interleaved. For example, when studying angle properties, forming and solving equations can then be incorporated, as well as problems involving 2D and 3D shapes (knowing perpendicular lines bisect at right angles).
- Low stakes quizzing: Supporting the retrieval and providing effective formative assessment for learning, enabling teachers to be responsive. This can range from a short topic test completed in class to support feedback and future planning to multiple choice questions or starter questions.
- Metacognition: Metacognition is one of the most effective and least expensive interventions. Providing opportunities for students to reflect on their learning, their progress, and steps to improve can have dramatic results on their outcomes in math.
6. What are the support materials and resources like for your math curriculum; will you have to create them all from scratch or spend valuable budget on them?
A curriculum needs to be clearly articulated to support teachers with their planning and implementation of the curriculum, as well as ensuring consistent, high-quality delivery.
Therefore, information about a unit or series of lessons detailing the math skills and concepts, ideas for differentiation and misconceptions provides clarity and purpose to teachers’ planning rather than merely a series of presentation slides.
Think about incorporating the following materials:
- An overview of the year so that teachers understand the context, progression and timing of the concepts they are teaching.
- Unit overviews which contain information about the key objectives, representations, methods, misconceptions, suggestions for deep questions and resources.
- Assessments which range from low stakes quizzes in class to summative assessments which are likely to form the basis of reporting to parents. You may opt for a topic diagnostic test and a topic test to assess the understanding to inform planning prior to the unit and assess the effectiveness of learning after it.
- Feedback and reflection are key parts of the assessment cycle. Following an assessment (high or low stakes), how are students supported to reflect on, improve and address misconceptions? In preparation for high stakes assessments, you could incorporate exam questions or a problem in context to solve deep questions later in a lesson.
- Starter activities which could be, for example, interleaved review questions written by the class teacher tailored to their needs, rich exploratory questions to set the scene for the lesson or exam questions.
- Lesson resources which include presentation slides, worksheets and activities, one-step and two-step questions, as well as multiple-choice questions. Think about what will ensure quality and support teachers with workload. For example, providing quality deep questions, as these are resources that can be onerous to source or require more time, creativity and good subject knowledge to create.
- Knowledge organizers are required by some schools for all subjects, to support students with self-study. Math requires practice rather than the recall of any given number fact but you may decide to incorporate knowledge organizers where it would support students with recall of facts where appropriate.
7. How easy is it for you to share your curriculum and communicate it to staff, students and parents?
Developing a robust curriculum is very complex. It requires a balance of providing depth, guidance and consistent high standards across a team of math teachers while embracing simplicity by way of structure and consistency so that the essence of the concepts to be taught are not lost.
Communicating and sharing your curriculum
A good curriculum is only going to be effective if it is implemented well by the team, therefore communication is key. You need to communicate the following:
- Key objectives that need to be understood by students.
- Time scales as a guideline so that appropriate depth is given while maintaining pace to ensure the curriculum is delivered.
- Common teaching approaches. You may have specific approaches that should be common across the team such as use of bar modeling when teaching problems involving addition and subtraction, multiplication and division, for example.
- Representation and methods: which topics require mental methods and which formal written methods? Are statistics lessons going to include bar charts, pictograms, etc?
- Outline of the resources available.
- Ideas for differentiation.
- Summary of potential misconceptions.
- Prior learning and next steps.
- Homework links – either suggested resources or clip numbers and links if you use an online homework platform such as Zearn.
Teachers are busy so it is essential that this information is concise and accessible and an integral part of daily planning practice. I have found that a unit overview, a one paged document per unit, which summarizes the key information is valuable.
These can be brought to planning meetings, encouraging staff to annotate and reflect on them, promoting engagement with time to discuss and amend for future use.
Although resources and presentation slides for lessons are powerful for consistent delivery of a curriculum, supporting documents to ensure clarity are essential too.
Checklist for any new curriculum
Here is a checklist of things you may want to consider and talk about with your colleagues regarding any curriculum change. Documenting agreed responses is always a good idea.
- Storage and presentation of your curriculum: in an increasingly digital world, this is likely to be stored online but how can this be efficiently organized?
- Statement of intent: How will you communicate the aims and vision of your curriculum?
- Are you all able to articulate the rationale for content, choices and sequencing for your mathematics curriculum and present the long- and medium-term thinking? Concise unit overviews and road maps for the year can be very useful.
- What evidence and opportunities will there be in books for students to be able to build schema and recall learning?
- Buy in from staff: How will you communicate your rationale for your choice of curriculum model?
- Would your context benefit from reflecting on your state’s standards to increase awareness of prior learning?
- What are you preparing students for? E.g. further study or employment.
- Implementation: How is each layer of your curriculum presented to show progression? E.g. using a curriculum map or flight path? This is where you think about the sequencing and timing of topics to prepare students to achieve top grades and equip them with the necessary skills for their next stages.
- Communication and sharing of content: How will it be communicated clearly to teachers?
- Math resources to support your curriculum: as above.
- Professional learning program to support effective delivery.
- Assessment: how will you assess students on the curriculum? This could be by including high order thinking questions, low stakes quizzing and summative assessments.
- Impact: How will you know if your curriculum is successful? Look beyond student assessment data and consider how you can capture staff and student voice and review effectiveness from learning walks.
Final thoughts on your math curriculum
Life is always full of new challenges in education. Priorities shift, the state standards are audited and modified, and pandemics occur…
It is impossible to account for every eventuality but as recent events have shown, whether delivered online, at home, or at school, a strong and effective curriculum which is understood by everyone from leaders to staff, students and parents, can prove to be an essential uniting focus when everything else seems so uncertain!
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The content in this article was originally written by secondary math teacher Lindsey Ford and has since been revised and adapted for US schools by elementary math teacher Christi Kulesza.