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Mathematical reasoning is the ability to think logically and make sense of math, to use what you know to figure out what you don’t.
It can be as straightforward as knowing the numbers 4 and 6 represent measurable quantities, and when grouping those known quantities, you can figure out the answer to 4+6.
To put it simply, it’s reasoning your way to the answer.
Mathematical reasoning develops through:
There are two primary types of reasoning used in math:
Both types are crucial to developing mathematical reasoning skills and appear throughout elementary math, pre-algebra, and even advanced courses such as calculus.
Critical thinking in mathematics refers to analyzing, inferring, and evaluating mathematical ideas rather than accepting them at face value. These are sometimes referred to as the three levels of critical thinking.
When critical thinking is applied to a word problem, for example, it might look like this:
I need 10 flowers. I have 5 red ones. How many yellow flowers do I need to make 10?
Analysis is the process of identifying known information. The total needed is 10, the number I have is 5, and the number of flowers needed is the unknown.
Inference involves reasoning from that information to determine a solution strategy. The student infers that the problem is asking for the difference between 10 and 5 and decides to subtract to find the missing amount.
Evaluation is the process of checking whether the solution makes sense in context. After calculating that 10 − 5 = 5, the student mentally verifies that 5 red flowers plus 5 yellow flowers equals 10 total flowers, confirming that the reasoning and answer are correct.
Mathematical reasoning describes how thinking occurs in math. It is bound by formal operations and rules for solving problems with known quantities.
Critical thinking can be applied across many disciplines and allows for ambiguity. It is used to identify assumptions, detect fallacies, and make judgments.
There might be two possible ways to solve a math problem. Evaluating which one to use is critical thinking. Working through that process properly is mathematical reasoning.
Together, the two types of mathematical reasoning (inductive and deductive), along with critical thinking, form a toolkit of math and critical thinking skills, equipping your student for more advanced problem-solving and real-world application.
Strong mathematical reasoning skills are linked to long-term academic success and confidence across subjects, not just math.
Research indicates that students who “understand mathematical relationships and concepts and can make connections between ideas generally achieve more meaningful and transferable learning than those who merely memorize facts and formulas without understanding”
The same research introduced the term productive struggle to describe the kind of critical thinking that leads students to a deeper understanding.
Wrestling through how and why relationships among quantities, operations, and concepts matter enables students to develop an understanding that they can apply flexibly in new problem-solving situations.
Among the many reasons why critical thinking is essential, several are especially relevant:
“When students learn with understanding, they can use their knowledge flexibly, adapt what they know to new situations, and solve problems they have not encountered before.”
— Making Sense: Teaching and Learning Mathematics with Understanding
All About Math equips students to apply critical thinking by focusing on building mathematical understanding with hands-on manipulatives, understanding how math works, and asking students to explain how they arrive at answers.
Reasoning is already happening during everyday learning.
Solid number sense is like the soil that nourishes the roots of sound mathematical reasoning. Refer to this article on number sense to learn how you can start building this all-important foundation for mathematical reasoning in children as young as preschool.
The most important thing to remember when teaching mathematical reasoning in homeschool is that kids learn to reason by reasoning their way through a problem, not by memorizing facts and operations. This makes sense when we consider it, but often, math instruction is limited to memorizing math facts and practicing problems over and over. Developing mathematical reasoning starts with interacting with numbers in a very practical, hands-on way.
As they grow in their understanding of numbers as a representation of quantities, you can teach reasoning in math with these 5 strategies:
Ask:
Manipulatives, diagrams, verbal explanations, and drawings help children connect concrete experiences to abstract ideas. Well-designed multisensory approaches improve comprehension and reduce cognitive load by engaging multiple regions of the brain without causing fatigue in any one area.
Linking fractions to division, multiplication to repeated addition, and geometry to real-world patterns are all ways to connect new concepts to known ones. These connections are essential for mathematical reasoning and modeling.
Mistakes should be approached as a valuable part of education, not as a failure. A child who is taught to accept mistakes as a natural part of learning sees them as a way to evaluate their understanding, identify gaps, and consider new problem-solving strategies. If he learns a healthy mindset toward mistakes at a young age, he will be much more confident about trying new things in the future without being held back by the fear of failure.
Talking his way through steps, such as “make 3 groups of 4,” helps your child internalize logical processes. You can model this using the scaffolding process:
Researchers at Vanderbilt University found “Promoting self-explanation (i.e., generating explanations for oneself in an attempt to make sense of new information) is a recommended study strategy and instructional practice.” They also found that scaffolding of self-explaining when the instructor modeled it first was a crucial, additional step in creating lasting retention of material.
All About Math makes it easy to teach mathematical reasoning and critical thinking by integrating all 5 strategies into the curriculum.
Intentional activities reinforce reasoning without relying solely on drills or worksheets. Games and real-life applications can be woven into your days with minimal prep time, stress, or pressure, so that learning can be fun and relaxing.
These simple, real-world experiences with math prepare students for mathematical reasoning by building understanding.
Right from the first level, All About Math includes hands-on games and critical-thinking activities to build mathematical reasoning. See for yourself in this sample activity book.
A strong mathematical reasoning curriculum embeds reasoning into daily instruction instead of treating it as an add-on.
All About Math supports reasoning by:
In All About Math, reasoning is built into every lesson, not isolated as a skill taught later.
Explore All About Math to see how it turns everyday lessons into powerful opportunities for reasoning and critical thinking.
It’s logical thinking applied to math—using what you know to figure out what you don’t.
Reasoning applies logic within math; critical thinking analyzes, evaluates, and extends those ideas.
They build understanding, confidence, and problem-solving ability.
Ask open-ended questions, encourage explanations, and use hands-on activities.
While no curriculum teaches only mathematical reasoning, it is an integral part of programs such as All About Math. Page 31 of this sample lesson helps children practice critical thinking by asking them to decide if rounding a number up or down makes the most sense for the situation, even if you have to break the normal rules for rounding numbers.
