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Mathematical Reasoning™

Bridging the gap between computation and math reasoning.

Grades: Toddler-11

Mathematics

Full curriculum

•  Multiple Award Winner

Forget boring math lessons and dreaded drill sheets.  These fun, colorful books use engaging lessons with easy-to-follow explanations, examples, and charts to make mathematical concepts easy to understand.  They can be used as textbooks or comprehensive workbooks with your textbooks to teach the math skills and concepts that students are expected to know in each grade—and several concepts normally taught in the next grade. Every lesson is followed with a variety of fun, colorful activities to ensure concept mastery.  The lessons and activities spiral slowly, allowing students to become comfortable with concepts, but also challenging them to continue building their problem-solving skills.  These books teach more than mathematical concepts; they teach mathematical reasoning, so students learn to devise different strategies to solve a wide variety of math problems.  All books are written to the standards of the National Council of Teachers of Mathematics.

Beginning 1 , Beginning 2 , and Level A Contents

Understanding Pre-Algebra This book teaches and develops the math concepts and critical thinking skills necessary for success in Algebra I and future mathematics courses at the high school level.  It was written with the premise that students cannot problem solve or take leaps of reasoning without understanding the concepts and elements that lead to discovery.  The author—with 35 years of experience teaching mathematics—is a firm believer that understanding leads to confidence and confidence gives students the resolve to succeed in higher level mathematics rather than fear it.   It is standards-based, but what makes it different from other pre-algebra books is that it organizes concepts in a logical fashion, stressing practice and critical thinking. It avoids the mistakes—found in many other math books—of trying to teach new concepts before students receive the prerequisite skills and practice necessary for success. The concepts are presented clearly and in connection to other concepts. Math vocabulary is very important to success in higher mathematics, so this book includes easy-to-follow explanations and a user-friendly glossary.

Free Detailed Solutions are available!

Understanding Pre-Algebra Contents

Understanding Geometry The successful completion of this colorful 272-page book will prepare middle schoolers for high school geometry. It covers more than 50% of the concepts taught in high school geometry using a step-by-step approach and teaches the reasoning behind the properties taught in geometry–instead of merely asking them to memorize them. Students are also taught the basics of geometric proofs and coordinate geometry in a way middle school students can understand. Students who struggle with high school geometry usually have lower standardized test scores because it is a fundamental subject in high school standardized testing. A glossary of terms that every student should master is included. This book can be used as a classroom textbook in Grades 7, 8, or 9 (usually over a two-year period) or as a reference for high school students. This book covers more than the National Math Standards for middle school mathematics.

Understanding Geometry Contents

NOTE:  It is our recommendation that students complete Understanding Pre-Algebra (see description above) before attempting Understanding Geometry .

Understanding Algebra I This is a one-year Algebra I course for Grades 7-9. Students who have a solid algebra background will have no trouble with the algebra problems from SAT and even the GRE.  This 384-page book highlights vocabulary and notation, and has examples from the history of math. What makes this book unique and different from other algebra textbooks is that it is built from the experiences of an award-winning algebra teacher with more than 30 years of teaching experience. Many textbooks are written by a committee of authors, and many of those authors have little experience teaching beginning algebra students in middle school or high school. Understanding Algebra I presents the most essential concepts and skills needed to fully understand and gain confidence in algebra in a step-by-step fashion, teaching students that algebra is generalized arithmetic. It helps students see the connection between mathematics that they already know and algebra, so that learning algebra becomes easier and less abstract. This book provides students with real strategies to succeed in solving word problems by using charts and translating strategies that guarantee success.

Understanding Algebra I Contents

Essential Algebra for Advanced High School and SAT

Discover Essential Algebra for Advanced High School and SAT , a 241-page math book in the esteemed Mathematical Reasoning series written by award-winning author and teacher with 30 years of expertise in secondary mathematics. This powerful resource teaches the ‘essential’ connection of arithmetic and geometric concepts with algebraic concepts. Without this understanding, students tend to memorize Algebra I problem-solving steps—which is sufficient to pass Algebra I—but leaves them unprepared for math courses beyond Algebra I and the SATs. Algebra, the essential language of all advanced mathematics, lies at the core of this book's teachings. By delving into the generalized arithmetic that underpins algebra, students develop a solid foundation in the rules governing number and fraction operations, including factors and multiples. This vital knowledge empowers students to move beyond mere memorization of Algebra I problem-solving steps and confidently tackle the complexities of math courses beyond Algebra I. Without the knowledge and skills taught in this book, students often struggle or even fail in advanced mathematics courses and on the SATs. Imagine a good high school student who sees a problem like 3•x•y•4 and hesitates to write 12xy due to uncertainty about the rules governing multiplication. Or not understanding how to add 2x to 1/4y to combine it into a single fraction. Or why –6 2 is different than (–6) 2 . It is easy to see that not having a strong understanding of the foundational rules of algebra can stop even the smartest students from succeeding in advanced high school math courses. Essential Algebra for Advanced High School and SAT serves as a companion to an Algebra I course or aids in post-Algebra I readiness. To ensure students’ long-term success in advanced math beyond Algebra I, this book teaches the following 'essential' mathematics skills and concepts:

• Understanding Terms and Order of Operations
• Understanding the Family of Real Numbers
• Rationals and Irrationals
• Working with Terms and Polynomials
• Polynomial Division, Factoring, and Rational Expressions
• Solving Equations and Inequalities
• Ratio, Proportion, and Percent
• How Algebra is Used in Geometry
• Understanding Functions
• Working With Quadratic Equations and Functions

Mathematical Reasoning™ Supplements These supplemental books reinforce grade math concepts and skills by asking students to apply these skills and concepts to non-routine problems. Applying mathematical knowledge to new problems is the ultimate test of concept mastery and mathematical reasoning. These user-friendly, engaging books are made up of 50 theme-based collections of problems, conveniently grouped in self-contained, double-sided activity sheets that provide space for student work. Each collection contains relevant math facts at the end of the worksheet in case students need hints to solve the problems. Calculators are allowed on activity sets that have a calculator icon at the top of the front side of the set. Each activity set is accompanied by a single-sided answer sheet containing strategy tips and detailed solutions. Teachers and parents will appreciate the easy-to-understand, comprehensive solutions. These books are a wonderful enrichment tool, but also can be used to assess how well students have learned their grade level's math concepts.

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All products in this series.

    • Our eBooks digital, electronic versions of the book pages that you may print to any paper printer.     • You can open the PDF eBook from any device or computer that has a PDF reader such as Adobe® Reader®.     • Licensee can legally keep a copy of this eBook on three different devices. View our eBook license agreement details here .     • You can immediately download your eBook from "My Account" under the "My Downloadable Product" section after you place your order.

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Mathematical Reasoning Review

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

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. 

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

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.

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.

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-Algebr a – $42.99

Understanding Algebra  1 – $39.99

Understanding Geometr y – $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.

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.  

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Mathematical Reasoning Beginning 2

By the critical thinking co..

Mathematical Reasoning™ helps children devise strategies to solve a wide variety of math problems. This book emphasizes problem solving and computation to build the math reasoning skills necessary for success in higher level math and math assessments. This book is written to the standards of the National Council of Teachers of Mathematics. The Beginning 2 book is a complete curriculum for children of age 4.

These highly effective activities take children far beyond drill-and-practice by using step-by-step, discussion-based problem solving to develop a conceptual bridge between computation and the reasoning required for upper-level math. Activities and units spiral slowly, allowing children to become comfortable with concepts but also challenging them to continue building their math skills.

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Mathematical Reasoning Level D Workbook, Bridging the Gap Between Computation and Math Reasoning (Grade 3) Paperback – January 1, 2009

Purchase options and add-ons, additional details.

• Reading age 8 - 12 years
• Print length 384 pages
• Language English
• Publisher The Critical Thinking Co.
• Publication date January 1, 2009
• ISBN-10 1601441614
• ISBN-13 978-1601441614
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• Publisher ‏ : ‎ The Critical Thinking Co. (January 1, 2009)
• Language ‏ : ‎ English
• Paperback ‏ : ‎ 384 pages
• ISBN-10 ‏ : ‎ 1601441614
• ISBN-13 ‏ : ‎ 978-1601441614
• Reading age ‏ : ‎ 8 - 12 years
• Item Weight ‏ : ‎ 2.3 pounds
• #449 in Children's Arithmetic Books

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Carolyn anderson.

Carolyn is the President of Anderson Associates, a real estate firm specializing in Greenwich residential properties. Prior to opening Anderson Associates she designed and renovated many restaurants and residences in Greenwich. She has written a number of cookbooks as well as the Anderson Guide to Greenwich Connecticut, now in its Eighth Edition.

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Critical Thinking Company

Building thinking skills series.

The Building Thinking Skills series is probably the most comprehensive resource for thinking skills for younger students. Each reproducible student book (except for Primary ) includes an answer key. While students will be able to complete some lessons independently, others will require teacher interaction—more so with younger students. Each lesson should take about ten to twenty minutes to complete and requires minimal preparation.

The first four books in the series are written for preK through sixth grade. The books are quite large, ranging from 224 to 416 pages, increasing in size from the youngest level. (There are two additional books in the series, both for grades seven through twelve, but I have not reviewed them.)

The first book, Building Thinking Skills Beginning is suggested for ages three to four. This colorful book helps children develop pre-reading and math skills, auditory processing, and logic and spatial concepts. Attribute blocks might be used along with some of the lessons, but they are not required.

In Building Thinking Skills Primary for kindergarten through first grade, children deal with similarities and differences, sequences, classifications, and analogies. Visual-figural skills get a workout in these lessons, too. Attribute blocks and interlocking cubes are required for some lessons... Primary is the only book in this series that has a separate teacher’s manual which you need as an answer key as well as for instructional information.

Building Thinking Skills Level 1 , suggested for grades two and three, begins with visual-figural skills then shifts to more abstract verbal work. For example, children work with figural analogies then with verbal analogies. Many of the lessons require children to analyze relationships between objects and words. Among other topics and skills addressed are deductive reasoning, classifying, describing, figural sequences, parts of a whole, spelling, vocabulary building, Venn diagrams, mental manipulation of two-dimensional objects, and sorting words into classes. This level is also available on CD-ROM.

Level 2 , suggested for grades four through six, does all of the above, expands to additional types of analogies, and adds map skills and directionality, branching diagrams, “if-then” statements, overlapping classes, and more. Some activities require students to write out their answers or explanations. Activities vary in difficulty, so select those that seem most appropriate for each child. This level is also available on CD-ROM.

Mind Benders series

These are books for kindergarten through twelfth grade that work on logic "grids." See my review here.

Daily Mind Builders™ series

Three Daily Mind Builders books are available, each with a different subject area emphasis: language arts, social studies, and science. Books are geared for older students in grades 5 through 12 according to the publisher, although I'd probably recommend them for grades 7 through 12. The format is the same for each book. While subject matter for stories and exercises in the science and social studies books is drawn from those areas, the subject matter of the language arts book is very broad. All three books really fit best as critical thinking resources rather than tools for improving skills in any of the three subject areas.

There are 101 lessons in each book, and each lesson should take about 5 to 10 minutes to complete. You can use these in whatever order you please, whenever you want. Each lesson is presented on one page and is reproducible, and each student needs his or her own page on which to write.

Each lesson has two exercises. The first exercise has a very short story, usually based on an actual historical event. Students need to read it very carefully to be able to answer the question at the end of the story. Stories are actually somewhat like riddles, offering up clues as to the answer. While careful reading and use of clues within the story helps, some of these "riddles" are going to be impossible for your students to figure out on their own. There is simply not enough information, or students might come up with plausible but incorrect answers. However, they can guess at possible answers, and that exercise is useful in itself, especially if two or more people do it together as a discussion.

Each book includes a reproducible "Daily Mind Builders Thinking Map," a visual organizer that might help students sort out information that might help them arrive at a conclusion. However, using it is optional.

The second exercise has five different variations, but all of these are generally activities students can do independently. They might be logic puzzles such as figuring out the order of five people based on information from a short story. Others involve word associations, vocabulary, identifying English translations of foreign words, and working with rhyming words. All of these exercises require critical thinking in one form or another. Students should be able to arrive at correct answers for these exercises.

A complete answer key with explanations is at the back of each book. All three books are available in either print or digital editions, and sample pages can be viewed on the publisher's website.

Mathematical Reasoning series

One of my Top Picks! See my review here.

James Madison Critical Thinking Course

Students in high school and beyond—even adults—might want to tackle the James Madison Critical Thinking Course . This huge, 534-page worktext is more challenging than The Fallacy Detective but it seems more approachable than most other formal logic courses. This is probably because most lessons are presented as cases being investigated by a police detective. Other situations and subject matter are also used, but almost everything is taught within a practical context. Instruction on each new topic is relatively brief and includes examples.

The book begins with an introduction to critical thinking and continues with topics such as distinguishing between fact and opinion and hazards such as ambiguity or vagueness. At first, students spend most of their time working through exercises which are mostly multiple-choice questions—but these questions are not easy!

In the fourth chapter, students begin to analyze and construct arguments and conclusions using letters to stand in the place of statements. In the next two chapters, students frame arguments and conclusions using symbols.

From there it takes a plunge into propositional logic and categorical syllogisms, devoting about 250 pages to these topics in two very long chapters. It wraps up with a brief investigation of inductive arguments and informal fallacies.

A quiz concludes each chapter. The instruction guide is vital since it has all the answers plus explanations which will help when both student and teacher are stumped.

See the Critical Thinking Company's site for many other excellent resources.

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James Madison Critical Thinking Course, Teacher

• $10.99 at Christianbook.com
• $10.99 at Rainbowresource.com

Mind Benders Verbal

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• $9.99 at Rainbowresource.com

James Madison Critical Thinking Course Studnt

• $39.99 at Christianbook.com
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Building Thinking Skills Book 1 with Answers

• $19.99 at Rainbowresource.com

Building Thinking Skills Book 3 Verbal with Answers

• $36.99 at Christianbook.com
• $36.99 at Rainbowresource.com

Building Thinking Skills Beginning

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Building Thinking Skills Book 3 Figural with Answers

• $29.99 at Christianbook.com
• $29.99 at Rainbowresource.com

Daily Mind Builders - Science

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Daily Mind Builders - Social Studies

Daily mind builders - language arts, homeschool extras.

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Disclosure of Material Connection: Some of the links in the post above are "affiliate links." This means if you click on the link and purchase the item, I will receive an affiliate commission. Regardless, I only recommend products or services that I believe will add value to my readers. I am disclosing this in accordance with the Federal Trade Commission's 16 CFR, Part 255 "Guidelines Concerning the Use of Endorsements and Testimonials in Advertising."

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The Critical Thinking Co: Mathematical Reasoning E and Language Smarts Level E!

Complimentary Product Received!

We have been long-term users and reviewers of The Critical Thinking Co. My son’s first math book was Mathematical Reasoning Beginning 1 and his first logic book was Building Thinking Skills Beginning 1 we developed a love for this company’s products over the years. I was really excited when granted an opportunity to review the Mathematical Reasoning Level E and Language Smarts Level E .

About Critical Thinking Co.

Just to give you a little background on The Critical Thinking Co. it was established in 1958 and its mission has been to supply learning resources that are fun and have students work to their highest abilities. All of their resources integrate logic and critical thinking skills.

“If we teach children everything we know, their knowledge is limited to ours. If we teach children to think, their knowledge is limitless.” – Michael Baker, President

Mathematical Reasoning Level E

Mathematical Reasoning Level E arrived just in time when we were doing a complete overhaul of our math curriculum choices. My son was not struggling with math concepts he was disliking his current math curriculum layout and busywork approach. I looked over the Mathematical Reasoning Level E with him and the first thing he said was “It has color!”. Then he noticed how the book didn’t look like your typical math text. Mathematical Reasoning Level E uses vibrant colors, and festive illustrations to teach math. There isn’t a lot of wording it’s straight to the point then kids practice the skills. The text is 384 pages of slowly spiraled concepts that include easy-to-follow explanations, charts, and examples.

“This book emphasizes problem-solving and computation to build for success in higher-level math and math assessments.” The Critical Thinking Co.

Each page is an adventure in learning 4th-grade math. Mathematical Reasoning Level E is a comprehensive full math curriculum that can be used as a supplement or a full curriculum. Mathematical Reasoning Level E teaches all the concepts for 4th-grade math and also includes concepts that aren’t presented until 5th-grade math. What I like about Mathematical Reasoning Level E is that it provides my son with just the right amount of rigor while promoting mathematical reasoning.

Mathematical Reasoning Level E is available on hardcover or ebook. The Mathematical Reasoning series has books for Toddler- 6th grade. Keep in mind from my experience with the grade-level books they will cover those grade-level concepts as well as introduce the next grade. Mathematical Reasoning Level E is a full curriculum this isn’t a workbook. We are using the book as a full curriculum alongside our RightStart Level E this is a perfect pairing. We love Mathematical Reasoning Level E so much that I went ahead and picked up Mathematical Reasoning Level F .

Language Smarts Level E

I was pleasantly surprised by the Language Smarts Level E , I had no idea this textbook was so comprehensive. We have been using the Word Roots Beginning and the Reading Detective Grades 3- 4 this year we love these 2 texts and now we are enjoying Language Smarts Level E . The Language Smarts Level E is a fantastic language arts textbook it provides 400 pages of rigorous, vibrant, vocabulary, and critical thinking skills. Students will also learn skills meant for 5th grade which means you can stretch this book into your student’s 5th-grade year. Language Smarts Level E can be used as a supplement or a full curriculum.

The textbook also has easy explanations and provides students a fair amount of practice. What I really enjoy is the simplicity of the lessons, my son didn’t feel overwhelmed and he could do lessons independently.

I highly recommend The Critical Thinking Co. Mathematical Reasoning Level E and Language Smarts Level E comprehensive full curriculum textbooks. They are both available as physical and ebook. I also highly recommend any resources from The Critical Thinking Co. This is one company that will be a permanent fixture in our homeschool. The website provides excellent sample pages and details of all of their books. You will find your student loving these vibrant funny pages that engage them to think critically and logically.

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Published by Little Learner and Mom

De-Shaun Jordan is the founder of Little Learners Mom Consulting and Little Learner and Mom Homeschool Blog. For the last 8 years, she has exclusively homeschooled her son. DeShaun holds a Bachelor of Arts in Healthcare Management and is Montessori Certified. She is married to her high school sweetheart and mother to their 10-year-old son. DeShaun enjoys being a mom, cooking, organizing, and helping others looking to homeschool or in need of guidance with education intervention. View all posts by Little Learner and Mom

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Artificial intelligence is taking on some of the hardest problems in pure maths, arguably demonstrating sophisticated reasoning and creativity – and a big step forward for AI

By Alex Wilkins

10 April 2024

Simon Danaher

In pure mathematics, very occasionally, breakthroughs arrive like bolts from the blue – the result of such inspired feats of reasoning and creativity that they seem to push the very bounds of intelligence . In 2016, for instance, mathematician Timothy Gowers marvelled at a solution to the cap set problem , which has to do with finding the largest pattern of points in space where no three points form a straight line. The proof “has a magic quality that leaves one wondering how on Earth anybody thought of it”, he wrote.

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All of which has prompted mathematicians to ask if their field is entering a new era. But it has also emboldened some computer scientists to suggest we are pushing the bounds of machine intelligence, edging ever closer to AI capable of genuinely human-like reasoning – and maybe even artificial general intelligence, AI that can perform as well as or better than humans on a wide range of tasks. “Mathematics is the language of reasoning,” says Alex Davies at DeepMind. “If models can…

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Quantitative Biology > Populations and Evolution

Important: e-prints posted on arXiv are not peer-reviewed by arXiv; they should not be relied upon without context to guide clinical practice or health-related behavior and should not be reported in news media as established information without consulting multiple experts in the field.

Title: Mathematical modeling and prediction of COVID-19 in Moscow city and Novosibirsk region

Abstract: The paper formulates and solves the problem of identification of unknown parameters of mathematical models of the spread of COVID-19 coronavirus infection, based on SEIR type models, based on additional information about the number of detected cases, mortality, self-isolation coefficient and tests performed for the Moscow city and the Novosibirsk Region from 03.23.2020. Within the framework of the models used, the population is divided into seven (SEIR-HCD) and five (SEIR-D) groups with similar characteristics with transition probabilities between groups depending on a specific region. Identifiability analysis of the SEIR-HCD mathematical model was carried out, which revealed the least sensitive unknown parameters to additional measurements. The tasks of refining the parameters are reduced to minimizing the corresponding target functionals, which were solved using stochastic methods (simulating annealing, differential evolution, genetic algorithm, etc.). For a different amount of tested data, a prognostic scenario for the development of the disease in the city of Moscow and the Novosibirsk region was developed, the peak is predicted the development of the epidemic in Moscow with an error of 2 days and 174 detected cases, and an analysis of the applicability of the developed models was carried out.

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