Why 'new' math isn't new and why you shouldn't be scared of it

When you look at math curriculum or at explanations of math methods, it may seem like math has changed from what you learned as a kid.

Despite what it may look like, math hasn't really changed, but kids are now tasked with gaining a deeper understanding of why math works. 

For those who want to homeschool math or help their children with math homework, these 'new' approaches can be confusing, but it doesn't have to be this way. Keep reading if you're wondering how you can help your kids learn math.

This article explains this 'new' approach to teaching math, including why it isn't as scary as it may look and how you can embrace it when homeschooling math.

Conceptual vs procedural math

'New' math focuses on children gaining a conceptual understanding of math. Today children must understand why math works and how different areas of math are related.

'Old' math focused on procedural understanding, which means understanding how to follow a set of rules or steps, but not knowing why they exist. 

Procedural math is the math most parents learned.

Are you wondering if you learned math procedurally? You were if you try to solve the problem 416 x 36, you might set the problem vertically and start multiplying the digits from right to left, 'carrying' a 3 from the first column to the second column before moving to the next line and placing a 0 on the right. 

However, most people who were taught procedurally wouldn't be able to explain why you needed to place the 0 on the second line or why you 'carried' the 3.

The problem with procedural math

You could think of procedural math as creating math robots. 

Imagine you successfully 'programmed' (aka taught) a robot to follow the steps to carry out a specific math operation or solve a particular problem. The robot would be great at carrying out what you programmed it to do, but it wouldn't be able to:

• Recognize when they have made a mistake or the answer doesn't make sense.

• Develop their understanding and become more efficient.

• Look for connections between different areas of math.

• Apply their knowledge to multiple problems or situations.

• Say they understand the math - after all, they are just carrying out a series of instructions.

Similarly, if we teach children math procedurally, they may be able to replicate the steps, but they won't really understand why they are carrying them out.

Unfortunately, this means they won't be able to spot mistakes or apply their knowledge to other problems and areas of math.

The benefits of conceptual math

When children learn math conceptually, they understand why math works. They are aware of more than just isolated facts and methods. This conceptual understanding allows them to grasp why a math concept is essential and how they can apply it in different situations. 

For example, suppose a child has a conceptual understanding of multiplication and can multiply a two-digit number by a one-digit number (e.g., 23 x 6 =?). With practice, they are more likely to apply this to multiplying a three-digit number (e.g., 345 x 7 = ?) without being taught new methods or steps.

They would then also be able to apply their understanding to 'long' multiplication, like the example we looked at earlier, which a fourth grader may solve using an area model.

Are you confused by the area model? Don’t worry, we'll explore some of the different models used in 'new math' later in this blog. 

Once kids have this conceptual understanding, they can develop their learning. In later grades, children will often build on this understanding and use methods that appear similar to traditional calculation methods. However, they will know why the methodology works and be able to use it fluently.

Conceptual math builds the foundations for success

Teaching math conceptually creates the conditions for all kids to succeed and become confident mathematicians who use and understand number flexibility.

Research has shown that the level of conceptual knowledge of math a child has in elementary school strongly predicts their achievement in high school. 

Children are now being exposed to these concepts so they have a solid foundational understanding of math and can flexibly apply their skills to various problems and scenarios.

Jo Bolar, a leading math educator, has explored many research studies around math achievement and the importance of conceptual understanding. She has concluded that:

"Low achievers are often low achievers not because they know less but because they don't use numbers flexibly – they have been set on the wrong path, often from an early age, of trying to memorize methods instead of interacting with numbers flexibly."

'New' math isn't really new at all

While many parents call teaching math with a focus on conceptual understanding 'new' math, there isn't anything new about it. 

The concepts you're encouraged to teach when you homeschool math have existed for as long as math has existed, and they are the reason why the procedural methods, tricks, and 'shortcuts' that most parents learned at school work. 

If teaching math conceptually baffles you, then Outschool can help!

As well as offering online math tutoring, Outschool has hundreds of different math classes that allow your kids to develop a conceptual understanding of math alongside others their age.

When was 'new math' introduced?

The importance of conceptual understanding, fluency, problem-solving, and reasoning was central to the Common Core math standards, which were initially adopted by 46 states.

Common Core aims to provide consistency in standards between states. It sets out the math skills and knowledge that students in each grade should develop, alongside eight overarching mathematical practices that math educators should develop in the children they teach: 

1. Make sense of problems and persevere in solving them

2. Reason abstractly and quantitatively

3. Construct viable arguments and critique the reasoning of others

4. Model with mathematics

5. Use appropriate tools strategically

6. Attend to precision

7. Look for and make use of structure

8. Look for and express regularity in repeated reasoning

In the list above, we've color coded the practices that specifically relate to conceptual understanding, fluency, reasoning, and problem-solving.

Some states have now moved away from following the Common Core in favor of their state standards. However, state standards often incorporate large parts of the Common Core.

The core aims and objectives remain the same or very similar, meaning teaching math conceptually remains at the center of most modern math curriculums 

Conceptual understanding, problem-solving, and reasoning are also key in the states that never adopted Common Core. For example, the math curriculum in Texas has similar aims to the Common Core, and teaching methods used in states that follow Common Core are also used in Texas. 

If you’re homeschooling, consult your state’s laws before you select your curriculum, as requirements will vary from state to state. Some states expect the state standards to be followed by homeschoolers, which are often copies of or very close to the Common Core. 

The importance of fluency, problem-solving, and reasoning

Alongside developing a conceptual understanding of math, children should also be taught math in a way that develops their fluency, problem-solving, and reasoning skills.

Fluency

The National Council of Teachers of Mathematics (NCTM) describes fluency as:

"...having efficient and accurate methods for computing. Students exhibit computational fluency when they demonstrate flexibility in the computational methods they choose, understand, and can explain these methods, and produce accurate answers efficiently."

Fluency doesn't mean just 'knowing' math facts.

Instead, it means selecting the most appropriate method, or math knowledge, for the task they are faced with and being able to carry out the methodology efficiently with an understanding of why it works.

Of course, as part of developing their fluency, children should also be encouraged to learn to recall key math facts automatically, such as multiplication facts up to 12 x 12. But, it's important that they do this by having a conceptual understanding of multiplication.

The cognitive scientist Daniel Willingham explains that this automatic recall is essential as it helps free our minds to think about and develop an understanding of the underlying concepts behind math. 

In other words, if you always have to calculate 6 x 3 = 18, you've got less mental capacity to calculate and understand why 60 x 30 = 1,800.  

However, kids should develop this automatic recall through understanding the concepts and relationship between the math they are learning. You can find hundreds of online math classes on Outschool that will help your child develop fluency in math facts in an engaging way that supports their conceptual understanding. 

Problem-solving

Problem-solving in math means finding a way to apply existing knowledge and skills to answer unfamiliar problems.

For example, a 6th grader should be able to use their understanding of ratios to work out how to make 18 cupcakes from a recipe that makes 12. 

When homeschooling math, if you want to expose your children to varied and engaging problems, consider enlisting the support of an expert Outschool teacher through an exciting problem-solving online math class.

Reasoning

Reasoning means children can make logical links and connections within and between areas of math. It is the glue that bonds kids' mathematical skills together.

Reasoning allows kids to make the connection between fluency and problem-solving. It also allows them to use their fluency to extend their math knowledge and solve problems.

When you homeschool math, you can help your child develop their reasoning skills by encouraging them to make an informed guess (which mathematicians like to call a conjecture) about a math concept and find proof that proves or disproves it. 

For example, a third grader might reason that if it doesn't matter what way round you complete a multiplication, the answer is always the same. They would be able to look at different examples (for example, 4 x 6=24 and 6 x 4=24) and use various math models, such as arrays and number lines, to determine why the answers might always be the same. 

We explore more about what reasoning is and how to develop it when homeschooling math in this blog post.

Confused by new math? Outsource to Outschool

As seen above, there are many good reasons to homeschool math in the 'new' way to help kids understand math conceptually.

If you want an expert teacher to help, why not outsource to Outschool? Outschool offers a wide range of online math support. From private math tutoring, logic courses, calculus, and math camps to classes that cover the full math curriculum. Or check out a class on percentages, math facts, long division, and math word problems. 

You have an option for whatever you need to support your kid on their journey to becoming a confident and flexible mathematician. 

Tim HandleyTim is an experienced educator and author who has written over 70 textbooks, workbooks and guides to help support teaching and learning. When he's not writing, he loves spending time at the coast or getting lost in a good book.