You’ve been teaching your students their multiplication facts. It’s been a couple weeks but they seem to be getting it by the end of math class on Monday, so you’re feeling pretty good. Your students come back to class on Tuesday and you throw out some easy warm-up problems: 4 x 2, 3 x 5.

And...crickets.

It’s as if you never taught the concept! In my classroom, I sometimes felt like I was teaching and reteaching the same math facts over and over again. 

So what’s your next move?

My answer might surprise you!

Any of these teaching strategies could work, depending on the grade level of your students and their stage in the learning process.

In this blog post, you’ll learn the right approach to help your special education students achieve fact fluency, even if they’ve been struggling with their math facts for years.

Ages and stages influence our math fact fluency instruction

Let me explain why grade level and stage matter:

When it comes to fact fluency instruction, we have two categories of students:

• Group #1: Those students who are learning their math facts for the first time. For example, this might be your 3rd graders who are learning about multiplication and division, having never formally been taught multiplication and division before.
• Group #2: Those students who have learned their math facts before, but haven’t become fluent yet. For example, this might be your 5th or 6th graders who learned multiplication and division years ago, but they’re still struggling with their times tables.

Each of these groups needs different teaching strategies to help them learn- and master- their math facts to become fluent. 

You also need to use different teaching strategies depending on what stage of the learning process students are in.

If you’re first introducing addition, subtraction, multiplication, or division, you’ll want your instruction to focus on conceptual understanding.

If you’ve already introduced computation and students can show you what it means to add, subtract, multiply, or divide, you’ll want to focus your instruction on developing efficient strategies for solving computation problems and building recall.

As you can see, choosing the right teaching strategy isn’t always straightforward! 

As we know, our students with learning disabilities will often make it to 3rd grade without mastering their addition and subtraction facts. Those same students then make it to middle school without mastering their multiplication and division facts.

Some teachers rightly think they need to spend more time shoring up these basic math skills. Andrea, your struggling 3rd grader, needs to learn those addition and subtraction facts soon or she’ll really struggle with later math concepts.

But the solution isn’t to take out those flashcards and spend 10 minutes of your math class every day doing addition drills with Andrea.

That’s what Andrea’s 1st-grade teacher did. And her 2nd-grade teacher. Andrea’s practiced and practiced her addition facts and seen those flashcards countless times before. They didn’t work then and they won’t work now.

So if math fact fluency drills don’t work with our special education students who are already behind, what should we do, instead?

First, we need to take a step back and find out what stage of learning your students are currently in.

The 4 stages of fact mastery

Dr. Arthur Baroody, a math professor, describes different stages that students progress through on their journey towards math fact fluency. If we use multiplication as an example, students progress from understanding that 3 x 6 is three equal groups of six objects to “just knowing” that 3 x 6 equals 18.

Stage 1: Conceptual understanding

In the conceptual understanding stage, students learn the meaning of the operation. They learn that addition means putting groups of things together, subtraction means separating or taking away, multiplication is making equal groups, and division is like repeated subtraction.

Stage 2: Counting strategies

In the counting strategies stage, students count to find the answer to the computation problem. They use their fingers, manipulatives, drawings, or verbal counting to solve problems.

For example, Andrea might draw 3 groups of 6 tally marks and count all the tally marks to figure out that 3 x 6 equals 18.

Stage 3: Reasoning strategies

In the reasoning strategies stage, students rely on “known facts” to solve for unknown problems. They apply reasoning skills to flexibly use related facts.

For example, Andrea might not know what 3 x 6 is right away, but she does know that 2 x 6 equals 12. She knows that 3 x 6 is the same as 2 x 6 plus 6 more, so she’s able to use this related fact to figure out that 3 x 6 equals 18.

Stage 4: Mastery

In the mastery stage, students have mastered addition, subtraction, multiplication, or division.

Mastery here means that they can efficiently and accurately provide an answer to a computation problem. They know the math facts from memory and don’t need to use their fingers, draw pictures, or use repeated counting to help them figure out the answer.

Let’s jump back to the choices I posed at the beginning of this blog post:

Many of our students with learning disabilities are stuck in stage 1 or stage 2, even years after they’ve first been introduced to addition, subtraction, multiplication, or division.

We see these students still using inefficient strategies and coming up with unreasonable answers (4 + 9 = 2) without applying reasoning skills to realize that answer couldn’t be correct.

If this describes any of your students, you’ll want to assess their knowledge of the math facts to find out which stage of fact mastery they’re in.

Assess their understanding and prescribe a solution

First, you need to assess students’ level of conceptual understanding.

In other words, do they understand what it means to add? (or subtract, multiply, divide?) Can they explain it in words or in a picture, or use manipulatives to model the procedure?

Here’s an assessment protocol for you to try.

Start with one student that has been struggling with his or her math facts.

Sit one-on-one with this student in a quiet space apart from his or her peers.

Take out some manipulatives (can be unifix cubes, bears, circular counters, etc.) and say to your student:

I’m going to give you a math problem. I want you to solve it any way you want. You can use these counters if you want. What is 7 + 3?

Depending on what math facts you’re working on, you can give an addition, subtraction, multiplication, or division problem.

*Note that you’re not giving your student a word problem, just the equation. Our students with disabilities may have expressive or receptive language delays that make interpreting word problems tricky.

If you embed a math equation as part of a word problem, you’re assessing both their ability to compute AND their ability to interpret the words in the problem. Let’s just focus on their computation skills, for now.

As the student is solving the problem, take note of how they’re solving it.

Are they using their fingers, drawing a picture, or counting aloud?

Do they seem to be solving it in their heads or using the manipulatives you provided?

Also, note approximately how long it takes for your student to solve the problem.

When your student gives you an answer, note to yourself whether it’s correct or incorrect, but don’t say anything about the accuracy of their answer to your student. Instead, say this:

I’m really curious about how you solved the problem. What you were thinking or doing to figure out what 7 plus 4 equals?

Give your student some time to answer and explain how they solved the problem. Their answer will reveal whether they understand what it means to add, subtract, multiply, or divide.

Watch to see if their actions or words show that they know to combine groups to add, take away objects to subtract, make equal groups or do repeated addition to multiply, or do repeated subtraction or make smaller groups to divide.

If they don’t say much, prompt them with: Can you tell me more?

If they say “I don’t know,” say: Let’s pretend you’re teaching a younger kid how to add. How would you explain to the younger kid how to add 7 plus 4?

After you go through this procedure with one math fact, you should try a few more. Aim to ask students to solve 3 to 5 computation problems and listen to their responses.

Remember that this isn’t a teaching opportunity (although our instinct is always to teach!)

You’re not trying to correct their mistakes or tell them better strategies to use. In fact, you should barely be talking, but genuinely listen to your students’ answers and let them do most of the talking.

Teaching will come later. Use this as an opportunity to assess their understanding.

Decide what stage to start in

The results of this math facts assessment will tell you whether you want to start your remediation instruction in stage 1, stage 2, or stage 3.

Begin in stage 1...

if your students cannot accurately explain or demonstrate the meaning of the operation. This means you need to start from the very beginning and use concrete materials to build their conceptual understanding.  

Begin in stage 2...

if your students demonstrate strong conceptual understanding but still get the wrong answer. This means they understand the meaning of the operation, but they’re making procedural errors.

In this case, you can start your remediation in stage 2. Your students likely have a glitch in their counting skills (which is very common in our special education students), so they’ll need to shore up their counting abilities before progressing in the stages of fact mastery.

Begin in stage 3...

if it’s clear that students have a strong conceptual understanding AND can answer each computation question correctly, but they rely on inefficient strategies that slow them down.

For example, they might draw 7 circles and draw 4 more circles then count all the circles to figure out what 7 + 4 equals, instead of counting on from the larger number or realizing that they already know that 7 + 3 equals 10, so 7 + 4 is just one more, 11.

Now that you have an idea where to start in, you need to use a teaching strategy that makes sense for the stage that your students are currently in.

Here are different math teaching strategies that work for each stage of fact fluency:

Strategies for each stage

Stage 1: Conceptual understanding

Storytime.

Everyone loves a good storybook. What makes stories special is that we can often relate stories to our own lives, they’re engaging, and they bring abstract concepts to life.

Studies show that when students relate math facts to their own lives, they find them more meaningful and develop stronger conceptual understanding.

Use storybooks, scenarios, or relevant word problems to help students develop their own understanding of computation concepts.

You can also use math storybooks that illustrate the computation concept. Here are some of my favorite storybooks for teaching math:

[Disclaimer: The links below are affiliate links, which means I may get paid a commission if you purchase through those links, which comes at no cost to you.]

Math storybooks for teaching addition and subtraction: 

• Quack and Count by Keith Baker
• Elevator Magic by Stuart J. Murphy
• Ten Red Apples by Pat Hutchins
• Rooster's Off to See the World by Eric Carle

Math storybooks for teaching multiplication and division:

• One Hundred Hungry Ants by Elinor J. Pinczes
• A Remainder of One by Elinor J. Pinczes
• The Great Divide: A Mathematical Marathon by Dayle Ann Dodds
• Amanda Bean's Amazing Dream by Cindy Neuschwander

Concrete, explicit instruction

Using concrete, explicit math instruction in the special education classroom is one of the strongest findings in research.

If you have students who’ve struggled to master their math facts for years and lack conceptual understanding, you’ll need to take out those manipulatives and go back to the basics.

As you use stories to demonstrate computation, model the computation with physical materials. Talk aloud as you combine or separate unifix cubes and explain: If I eat 5 more pretzels, I’m going to add these 5 cubes to the stack. Now I’ll count all the cubes to figure out how many I have, altogether.

Stage 2: Counting strategies

Verbal counting practice

It’s not unusual for students with learning disabilities to struggle to recite the counting sequence, even in middle school grades and beyond.

The foundation of computation skills is counting. If their counting is weak, their computation will be weak!

You can help your students improve their counting skills by doing 5-minute verbal counting warm-ups. If only a few of your students need counting practice, pull a small group and only do the warm-up with them.

Start by counting from 1: All together: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12...

Stop them in the sequence and count backwards: 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1.

Then practice counting on from different numbers: Let’s count on from 39: 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51.

STOP! Now let’s count backwards from 74: 74, 73, 72, 71, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61.

It’s important to match verbal counting with physical numbers, especially in the beginning. When I do verbal counting practice, I keep a number line visible either on students’ desks or on the smartboard. As we count, I point to the numbers (or have a student point to the numbers.)

It also helps to emphasize a rhythm when counting. I have students clap or tap their fingers or feet as we count to keep a rhythm.

Teach counting on

Weak counters will often count all instead of counting on.

For example, if given the problem 5 + 2, they’ll start by laying out 5 counters, then 2 more, then count them all.

A more efficient counting strategy is realizing that you can start from the bigger number and count on. So start from 5, lay out two more counters, and count on: 6, 7.

You can teach counting on by having the student circle the larger number and continue the counting sequence from that number.

Stage 3: Reasoning strategies

Explicit, strategy-based instruction

Many of our special education students won’t develop efficient reasoning strategies on their own. They need to be taught these strategies or guided towards coming up with their own strategies.

We talked earlier about the importance of explicit instruction. If your students are struggling to derive unknown facts based on known facts, you’ll need to explicitly teach them how.

One way to do this is by modeling while thinking aloud.

I don’t remember what 7 + 4 equals. Do I know any facts that can help me solve this problem? I do know 7 + 3 equals 10. 7 + 4 is just one more, so it must equal 11.

Combine modeling with using visuals or manipulatives.

Give students problems to solve, but don’t tell them to find the answer. Tell them to name a fact that can help them solve the problem, instead.

Have students explain aloud how a related fact helped them solve a problem.

For example, 5 + 6 = 5 + 5 + 1 (5 doubled is 10 and one more is 11)

Teach the commutative property

As adults, we know that 2 x 7 is the same as 7 x 2. However, this isn’t as obvious to students.

I’ve seen my students solve a problem like 2 x 7 then go to the next problem (7 x 2) and start from scratch to figure it out, instead of realizing they just calculated the answer a moment ago!

If you explicitly teach the commutative property of multiplication and addition, your students won’t need to learn strategies to figure out the entire addition and multiplication table. The commutative property cuts the table in half!

Stage 4: Mastery

Make practice fun by playing games.

Those flashcards and timed worksheets are not only anxiety-inducing, but they make addition practice and multiplication practice dull and mindless.

Make math fact practice motivating by using games. My struggling math students are the most engaged when I turn math practice into a game.

Some of my favorite commercial math games are:

[Disclaimer: The links below are affiliate links, which means I may get paid a commission if you purchase through those links, which comes at no cost to you.]

• Sum Swamp game for addition and subtraction practice
• Sums in Space for addition and subtraction practice
• Multiplication Bingo (a classic!)
• Math for Love Prime Club for addition, subtraction, multiplication, and division practice

Spaced practice

Spaced practice, also called distributive practice, means spacing practice out in smaller sessions over a longer period of time. This strategy has been shown to work powerfully over and over again in research studies. 

What DOESN'T work is cramming a lot of math facts into a few weeks and trying to memorize all those facts.

What DOES work is choosing a few math facts to work on at a time and spreading out practice for just a few minutes per day.

What does this mean? Students aren’t going to become fluent with their facts in just a few weeks. In fact, it will probably take months, if not the entire school year.

And that’s OK! You’re helping your students gain math fact fluency now so that know their facts forever. Invest the time now and students will reap the benefits for years.

Additional strategies that work in each stage:

Build confidence

Students who have been struggling to master their math facts for years KNOW that they’ve been struggling to master their math facts for years.

They’re probably not very motivated to keep trying, especially if they see their peers rattling off answers to computation problems with ease. They might have been called “babyish” for needing to use their fingers or draw pictures to help them with their addition facts or multiplication facts.

All of this makes them reluctant to give it another go.

Here’s where you come in. You’ll need to build up their confidence again the let them know that they CAN learn their facts.

Set and track goals

It’s rare if our students with learning disabilities don’t know ANY of the facts. They usually know some of their facts better than others and don’t need to learn the entire addition or multiplication table from scratch.

We can do a quick diagnostic to find out which facts students have mastered, which facts students “sorta” know, and which facts students don’t know at all. You can color-code each of these facts on the addition or multiplication table.

Focus on the “sorta” know and set those facts as a goal. As students master those facts, have them change each fact to “master.”

Then make your way through the “don’t know at all” facts. In this way, students are self-monitoring and keeping track of their own progress. As students see themselves making progress, they’ll increase their confidence in themselves and their ability to learn math.

Wrapping it up...

In this post, we learned about the 4 stages of math fact fluency and strategies to help your students progress through each stage until they master their math facts.

There are long-term benefits when you give students plenty of opportunities and experience building their conceptual understanding with physical materials and pictures and exploring derived facts before expecting them to recall the facts automatically. 

But let’s get real for a moment: There’s only so much time in a day to teach math and there’s a lot in your math curriculum.

However, students with disabilities often need more time to engage with a new concept before they understand it. They’re not going to learn their addition and multiplication facts in a few short lessons.

One problem I encounter with teachers is that they don’t want to invest the entire school year practicing math facts. They want their students to master the facts in just a few weeks, and when our students with learning disabilities inevitably don’t, teachers resort to drills or just give up entirely.

What students need is practice and experience problem solving spread out over the course of the year. That’s right, the entire school year.

You’ll see true, transformative change when you invest the time to find out which stage your students are in and target your intervention to helping them, where they are. You need to match your intervention strategies to their learning needs.

Remember to frequently highlight their growth and remind students how far they’ve come during the school year and how many new facts they know and how their strategies have developed. Give them hope that they CAN learn their facts and become fluent.

What are some of your favorite strategies and games to help your students learn their math facts?