Oct 25, 2018

Making the Most of Math Practice and Reasoning

Girl with handwritten number above her head

The next time you are practicing those basic math facts at home, remember one simple question that can help advance your student’s learning.

Do you remember learning your basic math facts in school? Chances are you’re like many of today’s parents, and your recollections include timed tests and frequent passes through stacks of flashcards. And as a parent, you’re probably noticing that today’s approach to math is a little different from what you experienced.

Today research tell us that to be fluent with basic facts, students need to be more than just fast and accurate. Students also need to be able to take apart the numbers in different ways and be able to choose an appropriate strategy based on the numbers at hand. This research means that when you’re practicing math at home, you will see greater gains in a student’s overall fluency by simply asking your child, “How did you get that answer?”than if you used the old “drill and kill” approach with flash cards.

Student strategies fall into one of three levels of ‘sophistication’, which is a way to demonstrate the depth and complexity of their reasoning. (CCSS Writing Team, 2011). We’ll help break this down for you, and show you how you can help your child regardless of his or her abilities .

Level One – Count All

For addition and subtraction, especially early on in their understanding, you may notice your student having to represent all quantities using their fingers (or a drawing), counting by ones to get their answer.

Take for example, the problem 3 + 5

  • Students using Level One strategies may count 1, 2, 3 on one hand, then count 1, 2, 3, 4, 5 on their other. They will move their hands next to each other and count 1, 2, 3, 4, 5, 6, 7, 8. Or when subtracting 8 – 3, the student might count and draw eight circles, then count to cross out three, then count 1, 2, 3, 4, 5 to see what is left.

Similarly for multiplication, let’s look at 4 x 7

  • Students may represent 4 x 7 by drawing 4 groups of 7 then count 1, 2, 3, 4, ….27, 28 to find how many.

Or for division, let’s look at dividing 21 by 3

  • They may take 21 cubes and deal them out one at a time into three groups, then count to find 7 in each group. Or they may take 21 cubes, and measure out groups of 3 until they realize they have made 7 groups.

As you can see, the key characteristic of a level one thinker is representing and counting all. While the least sophisticated of strategies, these initial understandings about addition and subtraction are how students begin to understand what it means to add and subtract.

Level Two – Counting On (+/-) or Skip Counting (×/÷)

As students become more confident with their early understandings of number, you will notice a shift from Level One strategies towards Level Two strategies. When adding or subtracting, a Level Two thinker will begin to “count on” from a number, and will use skip counting.

Take for example, the problem 7 + 6

  • This student will often start at 7, then count on 8, 9, 10, 11, 12, 13. They realize they do not need to count the first number, 7, and instead can just count up six more. This is called ‘counting on.’

Subtraction often sounds similar. For example, for solving the fact 13 – 7

  • When solving the fact 13 – 7, a Level Two thinker will start at 7 and count up, 8, 9, 10, 11, 12, 13, keeping track of the count on their fingers.

A When multiplying and dividing, a Level Two thinker will “skip count.” When solving the fact 4 x 8 they might say, “8, 16, 24, 32,” keeping track of the skip count in writing or on their fingers. Similarly, when dividing, they will rely on skip-counting. Take for example 36 ÷ 9 – students using Level Two strategies will skip count 9, 18, 27, 36 keeping track of the count on their fingers or by recording on paper.

Transitioning from a Level One strategies to a Level Two strategies is a significant milestone for your student.

Level Three – Using a Known Fact (Putting Together and Taking Apart Numbers in Flexible Ways)

Level Three thinkers are using the most sophisticated strategies. For addition and subtraction, these students are taking apart and putting together numbers in different ways. They are using a known fact to solve a harder fact.

You can see the difference in their strategy for the problem 8 + 7

  • A Level Three thinker may explain their strategy for 8 + 7 by knowing that 2 more gets to 10, with 5 more to get to 15. Or they may know that 7 + 7 = 14, and then add one more.

Again, for the subtraction fact 17 – 9

  • Your student they may explain that they took away 10 to get 7, then added one back on to get 8. Or they may say that they know to take away 7 to get to 10, then take away 2 more to get to 8.

Likewise students are using known facts and taking apart and putting together numbers in different ways to multiply and divide. For example, when multiplying 6 x 7, they may say that 5 x 7 = 35, plus one more 7 is 42. Or when dividing 24 ÷ 4, they may divide 24 by 2 to get 12, then divide 12 by 2 to get 6.

The key characteristic of a Level Three strategy is using a known fact to solve a harder fact and taking apart and putting together numbers in different ways.

Parents can see that students using these Level 3 strategies are really demonstrating the “flexible thinking” component of fact fluency. In fact, these Level 3 strategies will eventually lead to students becoming more efficient with their facts.

But parents, it is important to remember that all levels - Levels 1, 2, and 3 - are valid and mathematically sound. Be careful not to discourage your student by telling them their strategy is “wrong.” Instead, build on what they know! Helping students move through these levels of sophistication is part of the complex stages students pass through towards becoming fluent.

In time, students that arrive at Level 3 strategies for their basic facts will be more prepared to take apart and put together multi-digit numbers, fractions, and decimals in different ways, and eventually algebraic expressions and equations heading in to middle school and high school.

So next time you are practicing those basic facts at home, be sure to take some extra time to ask, “How did you get that answer?” and pay attention to the strategies that your student is using. You will learn a lot about what your student knows, and how you can support them in moving forward in their thinking.



Common Core State Standards Writing Team. Progressions for the Common Core State Standards in Mathematics. K, Counting & Cardinality; K-5 Operations and Algebraic Thinking. (May 2011).