If you’re looking for a flexible and rigorous concepts-based math program, either as a full curriculum or as a supplement, The Critical Thinking Co.’s Mathematical Reasoning series might be just what you’re looking for.
With its visually impressive workbooks and interesting, yet challenging, problem sets and puzzles, Mathematical Reasoning meets and exceeds standards for pre K to middle school math and can help students learn to more effectively analyze and critically reason their way through just about any math problems they encounter.
What We Like
But watch out for…
What is Mathematical Reasoning?
Published by the Critical Thinking Co., Mathematical reasoning is a homeschool math program designed for preschool through middle school/junior high.
Throughout the series, students learn important skills and gradually hone their logic, critical thinking and problem solving skills by exploring key concepts and completing different activities, exercises and mathematical puzzles.
What Grades Or Ages Is Mathematical Reasoning Intended For?
Mathematical Reasoning is aimed at students in Pre-K through Grade 9, covering everything from the counting, matching and order skills of beginning numeracy to the tessellations, congruences and slopes of middle school geometry and beyond.
To cover this wide range of grades and topics, the series is split into three levels:
- Mathematical Reasoning Beginning
- Mathematical Reasoning Levels A-G
- Understanding Pre-Algebra/Algebra 1/Geometry
| Level | Approximate Grade | Example of Topics |
| Beginning 1 + 2 | Pre K (3+4) | Basic numeracy, counting, essential addition/subtraction, number writing, comparison, measurement, zero, number lines and more |
| Level A | K | Adding and subtracting, odds/evens, patterns, counting/writing to 20, shapes, symmetry, equations, time, money, basic logic and more |
| Level B | 1 | Place value, notation, hundreds, transformation, basic fractions, bar graphs and more |
| Level C | 2 | Carrying, borrowing, 2-digit subtraction, skip counting, intro multiplication, surveys, probability, geometric shapes, placeholders, word problems and more |
| Level D | 3 | Multiplication, division, basic long division, basic operations of fractions, equivalent fractions, order of operations, graphs, perimeter, area and more |
| Level E | 4 | Multiplication and division, remainders, fractions, prime/composite numbers, statistics, inequalities, order of operations, improper fractions, intro functions, negatives, and more |
| Level F | 5 | Fractions and decimals, protractors, angles, volume, complex geometric shapes, graphing, money, reflection, transition, rotation, variables and more |
| Level G | 6 | More complex operations with fractions, decimals, regrouping capacity, coordinates, data analysis, absolute value and more |
| Pre-Algebra | 6-8 | Integers, rational numbers, ratio/proportions/percent, square roots, irrational numbers, 2D geometry, algebraic expressions, functions, graphing, coordinates and more |
| Algebra 1 | 7-9 | Sets, set notation, polynomials, factoring, algebraic function, expressions, ratio problems, percent problems and more |
| Geometry | 7-9 | Geometric notation, lines, planes, angles, pythagorean theorem, tessellations, transformations, 3D geometry, slopes, polygones, quadrilaterals and more |
While the program largely aligns with (and at times exceeds) NCTM (National Council of Teachers of Mathematics) standards, and while each cover also includes its approximate grade level in the top corner, the Mathematical Reasoning series can of course be used at home by students outside of a traditional grade progression.
Its books can, for example, be used by precocious learners of any grade as a core curriculum or as a supplement, should they have the appropriate math background.
That said, due to Mathematical Reasoning’s greater emphasis on strategic and critical thinking, and its frequent use of challenging math problems, we probably wouldn’t consider it an ideal program for older, struggling students who want to brush up on the fundamentals.
Placement tests
Although it is standards-aligned, it should be noted that Mathematical Reasoning can be something of an advanced curriculum, introducing certain concepts ahead of a typical scope and sequence.
As such, it can be a little more challenging for parents switching into it from another program to know where to start with the series.
Unfortunately, at time of writing, there is no real placement test for the series, so parents will have to make a judgment call based on the book’s stated grade level, its topical coverage and their student’s math knowledge and skill level.
What’s Required to Teach Mathematical Reasoning?
Mathematical Reasoning is a pretty compact program, all things considered.
At each level, the combined workbooks and texts are the only real required component for learning.
Unlike other math programs out there, there are no teacher’s manuals or kits to purchase, although the company also sells a line of supplemental books that can be used to strengthen learning.
Mathematical Reasoning Workbooks
For the most part, students and parents largely work out of the Mathematical Reasoning workbooks.
These are ~200-400 page, consumable, full-color softcover (or non-consumable digital) workbooks that include all the materials needed to work with the program. The books include basic math instruction and teaching tips, various exercises and puzzles, as well as the occasional link to some digital resources.
The Mathematical Reasoning series is highly visual and its books are filled with page after page of big, colorful drawings, math puzzles and puzzles.
Each page is color coded by math topic, so finding a particular concept to work on is quite easy to do, which is a definite plus when using the program as a supplement.
Similarly, each math strand or category is referenced at the top of the page, so students (and/or parents) can easily keep track of what they are doing.
In terms of instruction, the books start out mainly offering simple exercise instructions, perhaps reflecting the relatively fundamental math involved and the greater need for parental supervision with younger students.
As the series progresses, however, there is a marked increase in math instruction, which is largely aimed at the student.
Although generally pretty brief, the instruction offered in lessons is quite clear and should be easy for students to understand (being written at around the right grade level in our opinion).
Yet, despite this approachable tone, these books do consistently use the proper mathematical terminology for concepts, which is something we appreciate.
By staying approachable but not dumbing things down, the Mathematical Reasoning books can make it a lot easier for students to work more independently while still getting a thorough and professional exposure to math theory.
At the end of each book there is an answer key and glossary, something that can be very helpful for parents guiding the learning, especially at later levels.
For more advanced concepts, the company also offers free detailed solutions manuals that offer step-by-step explanations and solutions for questions, which is very useful for both students interested in learning more and parents who might be a little rustier in their own middle school math skills.
One thing that parents should be aware of with the Mathematical Reasoning workbooks is that, due to the way the program is structured, lessons and pages in the books can look quite different from one another on any given day.
Some lessons may have pages with a lot of problems and formal instructions on them, while others may contain fewer problem sets.
Students who really do best with routine and consistency between lessons may therefore find the series a little more challenging to get through, although it can be a good option for students who get bored with repetitive lessons.
Optional Supplements
Manipulatives
Although Mathematical Reasoning isn’t quite as hands-on as some other programs out there, at times (especially earlier in the series) it can recommend parents use manipulatives to help students grasp some of math’s more abstract concepts, and does offer some exercises and ideas for instruction around their use.
While the program doesn’t come with any physical manipulatives per se, these can be picked up fairly inexpensively on the internet.
Further, for those who don’t mind working a little more digitally, the workbooks do offer links to the National Library of Virtual Manipulatives, which offers a wealth of simulated math manipulatives, as well as related activities and ideas for using them to explore different math concepts.
Mathematical Reasoning Supplemental Books
In addition to the main Mathematical Reasoning workbooks, the Critical Thinking Co. also offers a series of supplemental books to go with the program.
Each supplement covers a range of grades, such as grades 2-4, 4-6, 7-9 and Essential Algebra, so they don’t necessarily correlate 1:1 with the main series and are best used after completing a couple of levels.
By and large, these supplemental books dive a little deeper into the concepts taught in the main program, offering additional work to help build skill fluency and challenging students with more complex and/or unusual takes on traditional math problems that can stretch their critical thinking skills a bit further.
Printed in black and white and having far fewer illustrations than the main series, these supplemental books aren’t quite as visually appealing, but do contain quite a few more interesting puzzles, exercises and challenges with (thankfully) full solutions printed in the back.
Approach To Teaching Math
Conceptual Math
Mathematical Reasoning is a conceptual math program.
In other words, during the course of their studies with the program students will dive more deeply into understanding the why behind math, that is why math operations work the way they do, why and how certain formulas and procedures work the way they do and how to analyze a problem in order to understand its essential logic.
As a result, and perhaps unsurprisingly given its name, Mathematical Reasoning has a stronger focus than many other math programs on getting students to critically analyze and reason their way through math.
In practical terms, rather than having students do a lot of drill and rote memorization, the lessons place more of an emphasis on problem solving, math puzzles and even (at upper levels, particularly) exploring multiple approaches, strategies and solutions to problems.
Consequently, students can get a very strong understanding of what they are doing with Mathematical Reasoning, becoming more comfortable trying out different strategies to solve problems, which can ultimately translate towards greater confidence when faced with new or unusual math problems.
This more conceptual approach to teaching math stands in contrast to more traditional programs, such as Saxon, where the focus is more on how to do math.
In such programs, students tend to spend more time learning math facts, doing computational drills and practicing very particular steps and procedures in order to solve math problems quickly and accurately.
On the downside, with a stronger emphasis on understanding and critical thinking, Mathematical Reasoning tends to have fewer practice problems per page and per topic, particularly when it comes to fluency-building drill and computational problems, than a traditional, procedural program might have.
Students who tend to need more practice to “get” a concept may therefore need to supplement the workbooks with further problems (either with the ready supplements or with those from another program).
Spiral Curriculum
Mathematical Reasoning is also a spiral curriculum.
This means that it tends to break math concepts down into smaller, bite-sized pieces rather than approaching them as a whole.
It will then introduce one of these topic pieces, work on it for a while and then move onto another topic, revisiting the first in greater depth later on in the book and the series.
This is in contrast to a mastery approach to teaching math, which approaches each math topic as a whole, exploring them completely and only moving on to another one when a student can demonstrate a level of proficiency (“mastery”) with it.
As a result of its spiral approach, with Mathematical Reasoning students get something of a break after working with each math topic, moving on to new topics pretty regularly, which can keep learning fresh and lessons a little more dynamic.
In addition, this spiral of learning tends to revisit topics pretty frequently across the program, which gives students more opportunity to practice and review concepts over time, which can help strengthen their understanding and skill level in the long run.
That said, it should be noted that while many students enjoy and benefit from a spiral approach, others can perceive it as a bit jarring or rushed and may prefer to take their time and work on one thing at a time until they really understand it thoroughly.
Visual Learning
Particularly at the Pre-K through elementary grades, Mathematical Reasoning takes a very visual approach to teaching math.
The workbooks frequently use colorful graphics and work with visual representations of numerical concepts.
During their lessons, for example, students might examine pies to develop an early sense of fractions, work with drawn number cubes, use various drawings of people or things to represent numbers in an equation, and much much more.
Not only can this visual approach make the books look more appealing to look at and make them more useful for visual learners, but they can help younger students come to terms with many of math’s more abstract or intimidating concepts by providing them with a more readily understandable and approachable representation.
Additionally, by frequently using symbols and drawings in the place of numbers, Mathematical Reasoning can also help students get more comfortable with algebraic reasoning from a younger age by helping them understand that symbols can often represent unknown numbers in equations or problems, which is kind of cool.
How It Works
Mathematical Reasoning is fairly straightforward to use as far as math curricula go.
Each book in the series is divided into a number of skills adhering to (and often going beyond) NCTM (National Council of Teachers of Mathematics) standards for that grade.
Because it is a spiral curriculum, as we mentioned above, each skill is broken up into smaller bits and introduced a little at a time over the course of each book.
As a result, at least until about the middle school level, books in the series tend to have a large chart at the front of the book rather than a traditional table of contents.
This chart provides the different locations in the book for each skill, as well as readily showing their connection to broader stands or categories in math, as can be seen in the example below.
There are two ways that Mathematical Reasoning can be used by homeschoolers – as a standalone math curriculum or as a supplement to another curriculum.
If it is used as a main curriculum, students can simply start at the beginning of a book and work forward.
The books tend to begin with a brief section aimed at parents that serves to both introduce the math topics at hand and provides some important teaching advice for more effectively introducing concepts to students, as well as some troubleshooting tips that can help when things go wrong.
After this, lessons then begin.
Where necessary (mostly starting in the elementary grades), each lesson in the book begins with a brief introduction of a math concept, including step-by-step demonstrations.
At upper levels, because Mathematical Reasoning is a strongly conceptual math program, these introductions will often discuss alternative strategies and approaches that students can try out to solve problems in different ways.
Following this concept introduction, students are then presented with a variety of related exercises to complete.
While some of these exercises are familiar computational and word problems, Mathematical Reasoning workbooks also include a wide variety of mathematical puzzles and challenges that are designed to get kids analyzing and thinking more deeply/critically about how to apply a math concept, reasoning their way through math so to speak.
Depending on the level, for example, students might have to figure out the area of an oddly-shaped house floor plan, use their math skills to navigate grids and solve a riddle, find the wrong answer in a list of numbers, or determine whether certain math logic statements or rules are true or false.
At higher levels, such as in Pre-Algebra, Algebra 1 and Geometry, students are often challenged to prove a particular math statement, explaining why something may or may not be true based on their own logic and math skills and demonstrating their reasoning.
For those using Mathematical Reasoning as a supplement, the Critical Thinking Co. has actually made it pretty easy to do.
In every book, skills are color-coded, so their relevant workbook pages are easy to find, being visible from the side of the book when closed, and each particular concept in math is printed atop the page for easy reference.
As a result, students can work with the curriculum of their choice and then test their skills by finding and working with some challenging, conceptual problems in their Mathematical Reasoning workbook and even learning alternative strategies and approaches that they can add to their math toolbox, should they so choose.
Our Thoughts
Overall, we feel that Mathematical Reasoning can be a very effective and useful math program.
The workbooks do dive fairly deeply into math concepts, even at the younger age levels, offering lots of challenging puzzles that we feel can encourage students to think flexibly and strategically about the problems they come across.
The books also offer students a wide variety of math puzzles, exercises and challenges to work through, including computational sets, word problems, multi-step problems, riddles, surveys, crosswords and more.
As a result, students aren’t as likely to get bored or tune out during lessons as they might with more repetitive programs.
Yet, despite the challenge of its curriculum, Mathematical reasoning introduces its concepts in an approachable manner that’s pretty easy for students to understand and that uses a variety of visuals and diagrams that can help them better grasp more complex and less intuitive topics.
Further, Mathematical Reasoning keeps its lessons pretty short.
Usually only a couple pages long and with exercises that often take a few minutes each, the lessons are easy to fit into even busy homeschool schedules and aren’t likely to cause a lot of frustration with students whose attention tends to wander after a while.
Finally, by including clearly written and straightforward instructions and explanations in each lesson (often aimed directly at the students), Mathematical Reasoning is a program that can be used more independently by students once they are comfortable with their reading skills.
This can be of great use to parents looking to encourage their student’s capacity for self-study, as well as for busier homeschools, as students can be set to work on each lesson with parents only needing to pop in and out to oversee the work and provide essential guidance here and there.
On the downside, there isn’t a huge amount of in-depth instruction with Mathematical Reasoning.
Although each lesson contains a topic introduction with step-by-step demonstrations, these tend to be relatively brief and to the point, and overall the workbooks tend to move students into the practice component pretty quickly.
By and large, it isn’t a program that will provide parents and students with long and detailed lessons about a given concept or one that has a more detailed teacher’s manual to accompany it.
As a result, some students who are using it as a main curriculum may have a hard time developing a full understanding of a math topic before starting practice, and may need to look elsewhere for the more in-depth explanations they need.
That said, this may not may not be an issue for more talented math students who pick things up quickly and is certainly not a problem for those intending to use Mathematical Reasoning as a supplement to another curriculum.
Similarly, while certainly interesting and challenging, there aren’t always a lot of practice problems in each lesson as there are in other programs, with some pages containing as few as one puzzle or exercise to complete.
As a result, some students may find themselves needing to supplement the program to get the practice they need in certain math topics.
How Does Mathematical Reasoning Compare To Other Math Programs
Conceptual Math Learning
Mathematical Reasoning is a conceptual math program that is designed to help students better understand why they are doing what they are doing and, more importantly, to use their critical thinking and logical skills to solve tricky or unfamiliar math problems.
In this way, it is similar to programs such as Singapore Math, Math U See, Math in Focus and more.
This focus on math concepts does set the program apart from more traditional, procedural curricula and supplements, such as Saxon or even Math With Confidence, that tend to have students memorize and apply math rules, formulas and procedures and do a lot of computational drill exercises in order to help them learn how to solve problems quickly and accurately.
Rigor
Mathematical Reasoning is also something of a rigorous and advanced math program that can introduce concepts a little ahead of other math programs and tends to eschew rote memorization and application in favor of more complex exercises, word problems, multi-steps and even puzzles.
In this way, and much like programs such as Beast Academy and Art of Problem Solving, Mathematical Reasoning can challenge students to analyze and think more deeply about what they are doing, rather than simply follow a set procedure, and can even encourage them to use alternative strategies and approaches to do so.
A Spiral Learning and Conceptual Approach
Generally speaking, conceptual math programs tend to favor a mastery learning format, where students spend multiple lessons exploring a single topic.
In contrast, Mathematical Reasoning favors more of a spiral approach, spacing out math topics and re-introducing them periodically throughout each book (and the series).
As a result, it can provide students with a greater opportunity for review than more mastery-based programs, as well as having more dynamic lessons.
Pros And Cons
Pros
Highly flexible as a math program
In general, parents and students have some choice in how they can use their Mathematical Reasoning books.
Due to its clear, step by step introduction of math concepts, the program can be used by students as a complete math curriculum.
At the same time, the books are color coded and organized in a way that makes them easily used as a solid, concepts-based supplement to another math program.
Rigorous and advanced approach to math
Mathematical Reasoning is a fairly advanced and rigorous math curriculum that dives fairly deeply into the concepts and theory of math and tends to introduce topics ahead of their normal grade level.
The books also contain a variety of very interesting, and often very challenging word problems, multi-step problems, math puzzles and more.
Highly visual and fun to go through
The books in the Mathematical Reasoning series, particularly at the elementary school level, are very visual and colorful.
Not only are they enjoyable to look at but their use of visual representations can get students more comfortable with some of the more abstract ideas of math and can start them thinking more algebraically from a younger age.
Clearly and understandably written
Mathematical Reasoning introduces and explains math concepts in a clear, straightforward and easy to understand way, often using step-by-step demonstrations and even visuals to help things along.
At the same time, at no point did we feel that the program dumbed down the math or used anything less than accurate, thorough and professional terminology.
Varied and interesting exercises and math puzzles
Not only does Mathematical Reasoning offer a lot of thought-provoking puzzles and exercises, but there is also quite a bit of variety that can keep students from getting too bored or zoning out..
At any given time in a book, students might be asked to do computations, solve riddles, do crosswords, solve word problems, work with diagrams and charts, complete puzzles and much, much more.
Can help students develop strong analysis and problem solving skills
Mathematical Reasoning really emphasizes the use of critical thinking skills and reasoning when approaching complex math problems.
Rather than just being given a set of procedures to follow, the program challenges students to think through why they are doing what they are doing, to explain their reasoning and even to use different approaches or strategies to solve a problem.
Relatively short lessons
Although it varies, generally speaking lessons in Mathematical Reasoning aren’t that long, usually taking up only a couple pages or so.
As a result, the lessons shouldn’t feel too intimidating or strenuous for students to go through and can fit a lot easier into a busy homeschool schedule.
Can be worked on independently
Due to its clear instruction and use of visuals, unlike some other programs out there, once a student is capable of reading fluently on their own they can start to use Mathematical Reasoning on their own, allowing parents to shift into a guidance role and freeing up their time for other tasks.
Cons
Not the cheapest curriculum around
Although compact and unlikely to break the bank, with each individual workbook costing around $30-40, Mathematical Reasoning isn’t exactly the cheapest math program around, either.
Not always a huge amount of instruction in each lesson
Although they do include a brief topic introduction, often with a step-by-step demonstration, lessons in Mathematical Reasoning don’t usually have a great amount of in-depth, thoroughly written explanations and text for each topic and may be a little sparse for some students to learn from as a sole curriculum.
Those using as a main curriculum may need to add in extra problems
While Mathematical Reasoning does use a variety of interesting puzzles and exercises, oftentimes there aren’t all that many per lesson.
As a result, some students may need to add more of their own to build skill fluency or work on particular skill gaps.
Who Is Mathematical Reasoning Ideal For?
Those looking for a strong conceptual math supplement
Mathematical Reasoning is a conceptual math program that dives into the why behind math concepts, helping students better understand why they are doing what they are doing, encouraging them to develop stronger reasoning skills and exploring multiple approaches to problem solving.
Its puzzles and exercises can be quite varied, thought-provoking and challenging, and can make it an interesting supplement option, particularly for those looking to add a bit of conceptual work to a more memorization and drill-heavy traditional or procedural math curricula.
Stronger math students looking for a rigorous math curriculum
Mathematical Reasoning adheres to and often exceeds NCTM standards, exploring math concepts fairly deeply and can be a good option as a full curriculum for talented math students, particularly those who don’t need a lot of in-depth explanation.
Visual learners
Mathematical Reasoning books, particularly those at its lower levels, include a lot of illustrations and graphical representations of math concepts, which can make learning math a lot easier for those who are more visual learners.
Homeschools looking for a more self-directed math workbook
With its clear instructions, well-structured lessons and step-by-step demonstrations, many Mathematical Reasoning books can be used by students on their own without a lot of parental supervision.
Those looking for a compact math option
Unlike many other math curricula, theren’t aren’t any teacher’s guides, test books or other learning materials to buy and keep track of with Mathematical Reasoning./
Most of the learning and practice is contained within its workbooks, making it a very compact and efficient program.
Who Is It Not Ideal For?
Those looking for a procedural math program
Mathematical Reasoning is a conceptual math program that spends a good deal of time exploring the why’s of math and getting students to think critically about solving math problems and puzzles.
As a result, it can be quite different from a traditional, procedural approach to math, where students memorize math rules, do a lot of computational drills and are taught particular ways of solving math problems.
Those looking for a full curriculum with lots of detailed math instruction
While Mathematical Reasoning does include math instruction, and while it is pretty clearly written and systematic, it can be pretty brief and some students may need a bit more detail and formal instruction in math concepts before feeling comfortable enough to start practicing.
Price
Note: Prices correct as of writing. All prices in USD.
As we’ve mentioned, there aren’t a lot of moving parts to Mathematical Reasoning, with most of the learning in each grade contained in the workbooks.
Students and parents can, however, choose to deepen their learning or get more practice with the associated supplemental books.
Mathematical Reasoning Beginning 1 – $33.99
Mathematical Reasoning Beginning 2 – $35.99
Mathematical Reasoning Levels A-G – $37.99-42.99
Understanding Pre-Algebra – $42.99
Understanding Algebra 1 – $39.99
Understanding Geometry – $36.99
Supplemental books – $24.99-26.99
Essential Algebra for Advanced High School Math and SAT – $36.99
As always, it is important that parents check out the latest prices for Mathematical Reasoning, as well as any current deals or offers that may apply.
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Is It Worth The Price?
Although the Mathematical Reasoning series isn’t necessarily the cheapest math curriculum out there, we believe it can provide a lot of value for homeschooling families.
The curriculum itself is fairly rigorous, meeting and exceeding NCTM standards for education and is capable of providing students with a very strong background in math and key math concepts.
To do so each book provides brief but very understandable instruction and is filled with a wide variety of interesting and colorful exercises, problem sets and puzzles that are designed not only to get students to practice their math skills but also to get them to apply their own logical reasoning and critical thinking skills when doing so.
In this way, students are challenged to engage with the math they are learning in a deeper and more analytical way that can really help them better understand what they are doing and why, which in turn can help them better cope with any complex or unusual problems they may encounter in the future.
Finally, Mathematical Reasoning is a very flexible math program.
It can be used by stronger math students as its own, rigorous curriculum, and it can be used by those studying with another curriculum as a solid, concepts-based supplement.
And, importantly for busier families, it can be used by students more independently, freeing up parents to handle the myriad of other tasks that homeschooling can involve.
Bottom Line
If you’re looking for a flexible and rigorous concepts-based math program, either as a full curriculum or as a supplement, The Critical Thinking Co.’s Mathematical Reasoning series might be just what you’re looking for.
With its visually impressive workbooks and interesting, yet challenging, problem sets and puzzles, Mathematical Reasoning meets and exceeds standards for pre K to middle school math and can help students learn to more effectively analyze and critically reason their way through just about any math problems they encounter.
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.