Math is an incredibly important subject that can lay the foundation for success in many other areas of study.
When it comes to homeschooling, picking a math curriculum that fits a student’s needs, learning style and pace of learning can be critical in helping them become brave and capable learners.
Two of the most popular approaches to teaching math are mastery and spiral learning.
These methods of teaching are pretty different from one another and each have their own particular advantages and disadvantages that can make them particularly suitable (or unsuitable) to certain students.
To help parents make a more informed decision, we decided to take a closer look at both of these approaches to math, exploring their reasoning, examining their relative pros and cons and discussing how each can impact a student’s learning experience.
What Is A Spiral Math Curriculum?
A spiral math curriculum is an approach to teaching math that emphasizes frequent and regular review of previously learned concepts and topics as new ones are introduced.
In essence, in a spiral math program a larger concept (such as multiplication) is broken up into smaller bite sized chunks (multiplying single digits, double digits, decimals, fractions, etc), which are then taught a little at a time before moving on to another concept.
Students then periodically review and revisit that larger concept, exploring it in more depth and complexity as they do so.
The idea behind spiral math is that students can learn more effectively when they are exposed to a topic or concept in smaller increments a number of times over a given time period, rather than simply learning it once and moving on.
In this way they are given multiple opportunities to engage with and practice what they are learning over time and are given lots of opportunity for practice and review, both of which can help them more effectively encode things in memory.
As an example, let’s explore how a student in a spiral math curriculum might learn some multiplication.
In the first few lessons, students would be first introduced to the concept of multiplication with the multiplication table and some very basic operations, multiplying single-digit numbers with repeated addition, skip counting, arrays and so on.
After a period of practice and review, students might then be introduced to the very basics of, say, division, a concept that would similarly be broken down and introduced incrementally.
At some point, students would return back to multiplication, digging a bit deeper and perhaps learning things like multiplying by 10s, two-digit numbers using the distributive property and so on before again moving on to another topic.
Later in the year, students might review and revisit multiplication again to solidify their learning before applying their knowledge to more complex problems, such as multiplying fractions and decimals or using it with measurements or unit conversion problems.
As can be seen from the example above, with a spiral math curriculum, students are gently and incrementally introduced to new concepts throughout the year, getting a good deal of review and reinforcement along the way.
What Is Spiral Review?
Spiral review is an essential component of spiral math learning and characterizes the learning model to some degree.
Spiral review is a technique that sees a curriculum deliberately incorporate previously learned skills and concepts into new lessons repeatedly over time in order to slowly, but surely, reinforce them and build strong foundational knowledge.
The general idea is that, by periodically reviewing already learned concepts through short review questions, exercises or quizzes, students will stay somewhat connected to previously learned material and help prevent knowledge and skill gaps from forming.
To give a clearer idea of how this works, we’ll look at how a spiral review might be integrated into a lesson.
Let’s say that parents and students are sitting down to tackle a lesson on adding and subtracting fractions with like denominators.
A lesson with a spiral review component might start with a few problems that touch on previously learned concepts, such as simplifying fractions or finding equivalent fractions, in order to prime their prior knowledge and get them ready for a new lesson.
Following this, parents would begin the main lesson with an explanation and maybe some practice work.
Once students have learned and practiced enough, they might be given a set of problems that include questions about the new concept as well as a few that touch on previously learned materials.
In this way, students don’t just learn something new but also work on keeping previously developed skills and knowledge nice and sharp, making sure that they aren’t forgotten over time.
Tight Vs Loose Spiral Math Curricula: What Are We Talking About Here?
One thing that parents might hear when discussing a spiral math curriculum is how “tight” or “loose” its spiral is.
This unusual bit of educational jargon largely refers to how frequently and regularly math concepts are revisited and reinforced over the course of an academic year.
A tight spiral math program is one that touches on a range of math topics but revisits and practices previously learned material pretty regularly over the course of a year.
A looser spiral is simply one that focuses a little more on new material and doesn’t quite repeat and review previous concepts as frequently as others, perhaps revisiting them only later in the year or even in subsequent grades in some cases.
Like anything else, tight and loose spirals tend to have their relative advantages and disadvantages.
All things being equal, the tighter the spiral the more reinforcement and repetition of previous facts, something that can be very useful in preventing students from forgetting key skills but also one that can slow down new learning to some degree and even bore some students through its greater emphasis on review.
In contrast, a loose spiral can give a curriculum a quicker pace and give it more time to spend on new material, but may not provide quite as much dedicated repetition or reinforcement of concepts, which can be a bit of a drawback for students who really struggle with math.
Pros and Cons of Spiral Math For Homeschools
It’s incremental by default
It’s important to note that spiral math isn’t just about repeating and reviewing the same concepts over and over, but that the curricula are structured to build on previous learning a little at a time and in greater depth with each spiral.
In this way, these curricula can make math a bit less intimidating as concepts are introduced and covered in smaller, bite-sized chunks, rather than as a single complex concept to master.
Greater opportunity for review and revision
Spiral math programs tend to place a greater emphasis on regularly reviewing previously learned concepts, which can help students (particularly those who struggle with the subject matter) retain information better over time as concepts are revisited throughout the learning process.
Because spiral math tends to take a more incremental approach with its concepts, introducing each a little at a time and shifting between topics regularly, it can feel quick-paced and dynamic, which can help keep kids on their toes and engaged.
Greater coverage in a given time span
Because of the way spiral math curricula move from topic to topic, they can cover quite a bit of ground when it comes to math over the course of a year, introducing students to a broader range of mathematical ideas and allowing them to explore and work on a greater variety of topics and applications.
Can feel jarring for some students
Shifting between topics in math can, at times, make it difficult for students to get their heads around complex concepts, which can lead to knowledge gaps over time.
Wide Range, Less depth
The need to cover a wide range of topics in a given year can limit the amount of in-depth exploration or mastery of concepts that a student can realistically achieve in that time frame.
It can feel repetitive
While some students definitely benefit from the increased review that spiral learning offers, others may not fully understand or appreciate the constant need to revisit and work on previously learned materials, causing them to become bored or frustrated at times.
Who Is A Spiral Math Approach Ideal For?
Although it really does depend on the individual student, we feel that spiral learning can may be particularly well-suited for those who:
- Students who are prone to forgetting skills and knowledge – the more frequent review and reinforcement of previous concepts can help keep them fresh in a students mind, helping prevent them from being forgotten over the course of a year.
- Students who are easily bored – a spiral math progrma’s more frequent changes in topic and activity can make learning seem a bit quicker paced and more dynamic, which can prevent them from disengaging quite as quickly.
- Students who struggle with certain math concepts – because a spiral math moves from topic to topic and introduces them in bite-sized chunks, students who struggle with a concept don’t have to be bogged down by it and can move on to something else and come back to it later with more review and practice.
- Those who like exploring different concepts and topics – spiral math can be a great option for students who enjoy the challenge of tackling new and varied problems, as opposed to mastering one concept at a time.
- Homeschools on a tighter schedule – in a spiral curriculum, students revisit concepts repeatedly over time, which can allow for a faster pace as students build upon their previous knowledge and potentially cover more topics over the course of a year.
Who Is A Spiral Math Approach Not Ideal For?
Although spiral learning can be a good idea for many math students, it’s not a one-size fits all solution and some students may not do quite as well with this approach as others.
These might include:
- Students who struggle with frequent changes – some students do best with a more set and orderly lesson schedule and can be put off by the frequent changes in topics that a spiral math program may involve.
- Students who like to work on one thing at a time – some students prefer to develop a strong and thorough understanding of a topic before moving on and may get frustrating by not diving into its subtopics and mastering it completely
- Students who “get” concepts quickly – The increased review and repetition of a spiral math curriculum can get on the nerves of students who only need brief instruction and practice in order to fully understand and retain information
What Should Parents Look For In A Spiral Math Curriculum
Families considering a homeschool spiral math curriculum should look for a few features to ensure that it stands a good chance at delivering coherent and effective math instruction.
Clearly outlined learning objectives
A good homeschool curriculum always outlines its learning objectives clearly, but this can be arguably more important in a spiral math curriculum due to its incremental nature and the fact that it tends to revisit and review topics again and again.
Aside from making sure that the program covers the necessary material in some kind of coherent and organized way, having clearly outlined objectives can help make sure that parents (and therefore students) are able to understand and form connections between different math topics.
This can help them see how math concepts fit together as part of a whole and can make the subject feel less like a jumble of random math facts and skills.
Incremental instruction: The curriculum should introduce concepts gradually, building on previously learned material in a logical and organized manner.
Regular spiral review
As we’ve mentioned, spiral review is an important feature of spiral math and any good curriculum should offer regular and intentional review of previously learned concepts to help solidify knowledge.
More than that, a curriculum needs to strike a good balance between old and new material, providing enough practice in the former to help make sure students retain concepts over time but without boring them or slowing down the pace of learning unnecessarily.
Recommended Spiral Math Curriculum
A popular and more traditional spiral K-12 homeschool math curriculum, Saxon places a strong emphasis on repetition and practice and can be highly effective at helping students learn to solve problems quickly and accurately.
Go Math is a Common Core-aligned spiral math curriculum that takes a conceptual approach to teaching math, emphasizing critical thinking and problem solving skills through a variety of activity-rich and multisensory lessons, as well as links to short and fun video lessons.
RightStart Math is a rigorous, Montessori inspired K-8 spiral math curriculum that still keeps learning math concepts approachable and engaging through a variety of highly interactive and fun hands-on and multisensory activities and games.
Life of Fred
It’s rare to find a math program that is actually fun and interesting to read, yet somehow Life of Fred manages to do it pretty effectively.
Life of Fred is a series of living spiral math books that teach K-12+ math concepts through a series of functional and funny stories that make learning both entertaining and educational.
In this way it can be a great potion for students who struggle with or are intimidated by math.
Horizons Math is a highly affordable spiral K-8 Christian math program that can help students develop strong math skills through a particular emphasis on math facts, review and practice.
It also includes a variety of hands-on activities and a particularly notable use of manipulatives throughout its grade ranges, making its lessons quite multisensory and engaging for students.
Simply Good and Beautiful Math
Developed by The Good and The Beautiful, Simply Good and Beautiful Math is a highly interactive, multisensory and overall beautifully designed Christian K-6 spiral math program that uses a combination of books, instructional videos and hands-on learning activities and games to help students learn math concepts in a highly engaging and fun way.
Ideal for busier homeschools, Teaching Textbooks is a self-paced, digital math program that teaches grades 3-12 math through a combination of on-screen video instructions, animations, fun in-program rewards, written notes, automated grading and online assessments help students master key math skills without the need for constant parental oversight and direction.
Abeka math is a strongly Christian and quite rigorous spiral math program that covers the breadth of pre-K to 12 math learning.
The series is notable for its back to basics approach that emphasizes math fact memorization, drill and the development of strong foundational skills in math that ultimately produce students who are able to solve problems very quickly and effectively.
What Is A Mastery Math Curriculum?
A mastery approach to math focuses on having students focus in and develop proficiency in specific math concepts before moving on to new ones.
In contrast to the spiral approach, in a mastery math curriculum students are presented with a series of math concepts and work through each one, starting with its most basic concepts and building on them over the following lessons, moving from skill to skill only when they have personally mastered it.
In effect, students are given ample time to dive deeply into and really study a particular concept, including its various sub topics and skills, so that they can develop a strong understanding of it and be able to apply it to different situations and problems.
Once students reach the end of a particular concept, they are given an assessment and, should they demonstrate their understanding and ability to apply it to a certain level of proficiency, they move on to another concept and won’t really review or come back to that during the year.
So, for example,a mastery math book for grade 4 might have an entire section of lessons on division that might rather comprehensively consist of:
- Division terms
- Division with zero
- Dividing 10s and 100s
- Order of operations with division
- Long division
- Division and money
- Division and average
- Finding fractional parts
- Prime numbers
- And more
Once this section is complete, students then move onto the next section that would cover a completely separate concept, such as geometry or decimals, with division not really appearing again during the year.
Pros and Cons of Mastery Math Learning for Homeschools
In-depth explorations of individual concepts
With mastery math learning, students can really take the time to dive into individual math concepts in their entirety, giving them a more or less complete grade-level understanding of the concept and giving them the time to hone their skills in it.
From the start, mastery math is quite personal and easy to customize around a students needs.
As students really only move on when they can demonstrate proficiency, they move at their own pace and are given the time they need with a subject, which can be beneficial for both struggling and gifted students.
It’s easy to prevent skill gaps from forming in the first place
Mastery math programs usually have students progress skill by skill.
If a student can’t achieve proficiency in a given skill, they don’t move forward, effectively preventing students from moving forward with shaky math ability.
Learning objectives tend to be intuitive
Focusing on specific concepts and progressing through its various skill sets, it’s usually pretty clear in a math mastery program as to what a student is expected to master by the end of a section.
Can be time-consuming
While much depends on the student, there is a risk that mastery math programs can be more time consuming for homeschooling families, as a students’ progression depends on their ability to master a concept before moving on.
Limited exposure to new topics:
In a similar vein, for some students, this approach can also make it harder to cover all grade-level expectations in a 36 weeks, limiting their exposure to new concepts in a given school year.
Students don’t always appreciate spending so much time on one concept
While some students do appreciate the time and depth given to a particular math concept, others (particularly struggling students) can become quite frustrated if stuck on a single concept for several weeks at a time.
Lack of Periodic Review
Pure mastery math programs do tend to offer less periodic review than other approaches.
Once a student has demonstrated proficiency, a concept isn’t really deeply or frequently touched upon again that year, which can be an issue if students begin to forget or if their skills become a bit rusty.
Who Is Mastery Math Ideal For?
Mastery math can be a great program for certain students, such as those who:
- Hate jumping from topic to topic – mastery math programs allow students to spend several weeks really exploring math concepts and their various subtopics and skills, and can provide a good deal of consistency and focus to math learning
- Gifted and struggling students – The more self-paced nature of mastery math, where students move on once they demonstrate proficiency in a concept, can be great for gifted learners who don’t have to spend quite as much time periodically revisiting concepts they are already skilled at.
It can also be a good approach for struggling students who can really take their time with their math learning, ensuring that they master each skill in turn before moving on so as not to develop gaps to begin with
- Homeschools that don’t mind a slower pace – Mastery math learning can be a bit slower paced and may not cover quite as broad a set of topics as spiral, but it does allow students to really dive into specific concepts in an organized and linear manner and get a firm handle on things.
Who Is It Not Ideal For?
- Those prone to forgetting or developing skill gaps in the long run – While mastery math programs can be great at preventing skill and knowledge gaps from forming, as students can only move on when they can demonstrate their understanding and skill, there is no guarantee that students may not forget certain things in the future without a good deal of periodic revision, something that pure mastery programs aren’t really known for.
- Get bored learning one topic for long periods – students who get bored or frustrated studying one specific concept in math for extended periods may not have the best time with the math mastery approach, as each section in a curriculum will likely be based around a single concept and its related skill sets.
- Frequent assessments for mastery – As students will typically be assessed for their proficiency in specific math skills in a given topic and their overall understanding of the concept as a whole, there can be a bit more of an emphasis on formal assessment in a math mastery program than with others, which can stress some students out.
What To Look For In A Mastery Math Program
There are some things we feel that we feel parents should look out for in a mastery math program to really get the most out of them.
Some periodic refreshers or review
For the most part there are very few mastery math programs left that, once a student has shown they understand a concept, simply move on to new ones without ever reviewing previously learned material again.
This is because demonstrated proficiency on a given day does not really guarantee against knowledge or skill gap development as time goes on.
As a result, we do like to see the occasional mixed review or concept revision thrown into a mastery math curriculum, particularly when two concepts interrelate.
As mastery math programs usually encourage homeschooling parents to only move on when a student demonstrates proficiency in a concept, it can be quite easy for struggling students to get stuck.
We do, therefore, like to see a bit more support and tips for differentiated learning with these types of programs.
Well-designed assessments of proficiency
A key element in a mastery math program is the ability to tell if a student has developed sufficient understanding and skill in a particular topic or concept so that they can move on.
In order to do so, we feel any curriculum should come with assessments that both align with specific learning objectives (i.e. do they test ability in the specific skills for which they are intended) and provide meaningful feedback for parents that they can use to make a judgment.
What To Watch Out For
Pace of learning needs to make sense
Generally speaking, a mastery math program can’t reasonably cover quite the range of topics and concepts that a spiral curriculum can in the course of a year.
Offering too many sections and concepts in a single grade level book can be a recipe for frustration and disappointment.
Reasonable objectives and definition of mastery
A good mastery math homeschool program should help parents determine what can be reasonably considered proficient for a given grade level.
Setting the bar too low can have students move forward while still lacking certain skills, while setting the bar too high can cause students to get frustratingly stuck on a particular concept or topic, both of which are recipes for disaster.
Popular Mastery Math Curricula We Recommend
Singapore Math is a highly respected and popular mastery K-8 math curriculum based around the Singapore method of teaching math, a curricular approach that helped transform their public schools into global math powerhouses.
Singapore Math offers two math curricula, Dimensions and Primary Mathematics, both of which stress the importance of understanding math concepts and help students better understand complex subjects using their highly multisensory concrete-pictorial-abstract framework.
Math Mammoth is an affordable K-8 mastery math program that clearly and comprehensively explains math concepts using a wide variety of teaching methods, including videos and illustrations.
The program tends to lend itself to self-study a little better than others, which is always good for busier parents, and it tends to include a bit more periodic review and practice than most other mastery math programs out there.
Math U See
Math U See is a K-12 mastery math curriculum that emphasizes conceptual learning and does so through straightforward and down-to-earth instructional videos (aimed at parents) and a variety of hands-on activities and visual aids.
The programs use of visual aids and manipulatives continues far longer than most other programs, well into Algebra 1, which, combined with its videos and approachable teaching method, can make it a great option for students who struggle with math, tactile learners and parents who aren’t really certain about teaching math.
CTCMath is a self-paced digital mastery math program aimed at students in K-12.
Through the use of videos, games, automated practice questions, advanced progress tracking, customizable assignments, diagnostic tests and assessments, students are able dive into math learning on their own, being guided through their lessons without the need for a lot of direct parental involvement.
Thinkwell Math is an engaging and interactive multimedia math curriculum that offers mastery math lessons for students in grades 6-12 with grade-level, honors and AP courses.
The program is self-study and through its use of clear video lessons, exercises, automatic grading and suite of digital manipulatives, it makes studying math a little more approachable even if the content is fairly rigorous.
Learn Math Fast
Learn Math Fast is a fairly gentle mastery math program that covers math learning from elementary through high school.
The program focuses on conceptual math instruction and does so using clear, straightforward and casually written math texts that guide students step-by-step through math concepts in a way they’ll more intuitively grasp and retain, making it a great program for struggling students and those with mathphobia.
Redbird Math is a pretty cool online K-7 mastery math program designed by Stanford University’s Educational Program for Gifted Youth that aims to foster strong, STEM-oriented strategic math thinking though fairly rigorous instruction, high quality digital manipulatives and a variety of interactive project-based exercises and games.
The program is also very adaptable and personalizable, with an adaptive algorithm built into it that adjusts the difficulty of learning and practice to meet a students ability in a particular concept.
Best Academy is a well known, rigorous and challenging K-8 mastery math curriculum that emphasizes high-level thinking and problem solving skills in order to build a deeper and more strategic understanding of math.
The program uses a rather novel approach to teaching, introducing students to math concepts using beautifully illustrated, monster-filled comic books.
At the same time, Beast Academy is highly rigorous and aims to challenge students throughout, making it a fun and engaging choice for advanced or gifted math students.
Art of Problem Solving
From the same company that created Beast Academy, Art of Problem solving is a series of middle and high school math programs created by a former math champion that offers a highly rigorous and thorough approach to math.
As well as aiming to build a strong foundation in math concepts, the books place a particular emphasis on developing strong problem-solving, flexible and critical thinking skills, making them particularly popular with gifted students and those interested in math competitions.
Are There Blended Mastery/Spiral Math Programs?
Not every homeschool math curriculum out there fits neatly into the mastery and spiral dichotomy.
Recognizing that both approaches have their downsides, a few programs have tried to blend elements of both.
These nearly always involve teaching specific concepts in depth and periodically testing for proficiency, as in mastery, while including some spiral review, i.e. periodically giving students mixed practice (to assess previous learning) and more focused concept reviews throughout the year.
As with anything else, there can be downsides to this hybrid approach.
For one thing, it can be hard to effectively strike a balance between mastery and spiral.
If a hybrid program leans too much towards mastery there may not be enough periodic review, while if the program has too tight a spiral there may not be enough time to dive into the main learning.
Similarly, while aiming for the best of both words is always nice in theory, students may not always appreciate a blended approach.
For example, some students may still feel overwhelmed with the increased review and revision of previous concepts, while others may become irritated at the time spent on specific math concepts.
Hybrid Mastery / Spiral Math Programs To Consider
Math With Confidence
Math With Confidence is a K-6 math program that makes it easy for parents to teach critical math skills to younger students through the use of highly interactive and multisensory lessons, real life demonstrations and clear, open-and-go lesson plans.
The program largely teaches for mastery but also incorporates spiral review , with lessons frequently revisiting and refreshing previous concepts and skills during lesson warm ups.
Math Lessons For A Living Education
Math Lessons For A Living Education is a rather interesting, Charlotte Mason-inspired Christian homeschool math curriculum that leans on fiction stories, centering around two twins and their adopted siblings, to provide helpful context when introducing and explaining math concepts.
The program is largely mastery based, although it incorporates far more dedicated, periodic review of previous concepts than most, and interestingly enough also includes things like copywriting and crafts to keep things interesting and to (at least partially) take the place of standard drill and quizzes.
Summary Table: Comparing Mastery Math Vs Spiral Math
|Focus||Mastering one skill before moving on||Revisiting multiple skills in each lesson|
|Approach||Learn one topic until students demonstrate understanding and skill||Building knowledge through incremental instruction and practice|
|Review||Review until mastery, then move onto another topic||Review of previously learned skills periodically throughout year|
|Pace||Self-paced learning and fixed pace, teacher-led learning||Self-paced learning and fixed pace, teacher-led learning|
|Assessment||Assessment after mastery of a skill||Ongoing assessment, tracking progress over time|
|Ideal for||Students who require a deep understanding of concepts||Students who need regular reinforcement and practice|
|Pros||Emphasizes foundational skills and mastery||Builds and strengthens prior knowledge and skills through repetition|
|Provides clear goals and measurable outcomes||Helps to prevent gaps in learning and keep concepts fresh in memory|
|Allows for individualized instruction and pacing||Provides ongoing opportunities for review and reinforcement|
|Can be helpful for students who struggle with retention or recall||Can be engaging and motivating for students who enjoy variety|
|Cons||May not cover all topics or skills in a given year||May require more planning and organization for homeschooling parents|
|Students can get bored||Students may not like jumping around between topics|
|Not always as much review||Constant review can be a bit much for some students|
|Look Out for||Make sure mastery and objectives are defined||Make sure curriculum organization seems logical|
|Make sure there is enough review so students don’t forget concepts after mastering them||Make sure the spiral review is tight enough that students don’t develop gaps between incremental study|
While many homeschoolers and even educational specialists tend to have quite strong opinions about them, both mastery and spiral math approaches have their advantages and disadvantages.
More than that, each approach can have a significant impact on topical coverage, a student’s overall pace of learning and the amount of review and revision they receive over the course of a year.
Ultimately, which is best really depends on the needs and preferences of the parent and student and homeschooling parents should really explore these approaches in some detail before deciding on a curriculum that is (at least largely) based on one or the other.
About the Author
David Belenky is a freelance writer, former science and math tutor and a tech enthusiast. When he’s not writing about educational tech, he likes to chill out with his family and dog at home.