Give a toddler the option of a plate of two cookies or a plate with one cookie and they will choose the plate with two cookies every time. The toddler is applying their mathematical understanding of comparing amounts.

Fast forward to the same child as a second grader. He must now take his ability to compare amounts using abstract symbols and with much larger amounts to solve a story problem. Each year he progresses he must use more complex reasoning skills and build on more advanced numerical fluency.

There are five fundamentals every elementary age child should master to successfully prepare students for advanced math. Dr. Wilson of John Hopkins University defines the five fundamentals of basic math as:

1. Numbers

2. Place Value

3. Whole Number Operations (addition, subtraction, multiplication, and division)

4. Fractions and Decimals

5. Problem Solving

Children who learn to use mathematical reasoning and have a solid understanding of these 5 elements will excel in math, at even the highest levels. Mastering these concepts requires a systematic and logical sequence of instruction. Young children, given the opportunity to master key math concepts in a self-paced program like Mathnasium, will be able to conquer the rigors of algebra and calculus later.

**Build the Foundation First**

Consider math learning like constructing a building. If you carefully lay a strong foundation with steel beams sunk deep into solid ground and several flat layers of mesh and concrete, you can build a skyscraper on top. The foundation might not be much to look at, but it supports the entire structure. A skyscraper built on a flimsy foundation is doomed to topple.

A strong number sense acts as the steel beams and the other four skills act as layers of mesh and concrete. These five skills support the ability to succeed in algebra, calculus and other advanced math topics. A high school student with a strong foundation can concentrate on complex mathematical analysis and reasoning, instead of getting bogged down in simple arithmetic.

Do the following 3 problems in your head to better understand how foundational skills and mathematical reasoning aid a student.

1. Count backwards by 7s from 28.

Not too hard, right? That’s because you have a solid grasp of numbers and whole number operations.

2. Mentally calculate 4(6.63 + 3.37).

If you understand decimals, place value, and number operations, this problem is still pretty easy.

3. Now** mentally** solve the following word problem:

A teacher told her class of 28 students "divide into 7 equal groups." The teacher then passed out yarn and asked each person in the group to cut the yarn into 2 lengths: 60.63 inches, 39.37 inches. How many inches of yarn did each group have?

Problem 3 requires a person to understand numbers, place value, whole number operations, decimals, and problem solving. If a person lacks any of those foundational skills, solving problem 3 mentally would be difficult. If a person uses mathematical reasoning and connects steps 1 and 2 with step 3, the problem gets much easier.

Take a look at this sample ACT math question.

*Sales for a car dealership were 3 million dollars more the second year than the first. Sales for the third year were double what they were for the second year. If sales in the third year were 38 million dollars, what were the sales for the first year?*

A test taker who can quickly analyze the problem (problem solving), and calculate X = (38÷2)-3 (whole number operations) and then check their answer for reasonableness (mathematical reasoning) can confidently answer 16 million dollars and proceed to the more difficult items on the test.

**Understanding “Mathematical Reasoning”**

You may have heard that Common Core math emphasizes mathematical reasoning. Mathematical reasoning is the logic a student uses to understand a concept or problem using prior knowledge. It is the ability to make sense of a problem, even when there is no routine to do so. In the example of the toddler with the plate of cookies the reasoning would go something like this:

*“I like cookies. I want lots of cookies. Two is more than one. I want the plate with two cookies.”*

Mathematical reasoning requires deep thinking on the part of the student. It uses logic, not a fixed series of steps, to solve a problem. Instructors at Mathnasium of Littleton encourage mathematical reasoning by asking questions like “How do you know…? Will your solution always work? What would change if.…” Open ended questions like these get a child to conjecture or guess, generalize and then justify their thinking. This is the rewarding and hard work of mathematical reasoning. According to the National Council of Teachers of Mathematics, learning to analyze math concepts in this way in early grades is critical to success in higher math.

Mathematical reasoning connects a concrete understanding of the five building blocks. It comes from the child, but competent instructors like those at Mathnasium of Littleton, weave in opportunities for children to explain their thinking daily.

**How Mathnasium Lays the Foundation**

Mathnasium of Littleton finds students’ level of concrete understanding before asking them to memorize a set of steps to get the right answer. Students in elementary school who have a concrete understanding of the relationships between numbers and operations will learn to manipulate them in abstract applications fairly easily. We then progress at the pace of the child to build the foundation. Working at each child's pace is critical to laying the foundation right. The foundational skills of mathematics are big and can take years to master, but the results are amazing.

You may look at the simplicity of foundational math skills and think it’s too easy for your child. But we look at early math as the foundation for a skyscraper.

Invest in the foundation of your child’s math skills. Call or text today (303)-9979-9077 to schedule a no-obligation assessment and consultation.