Assessing Math Fact Fluency And Reasoning Strategies
If done well, the assessment of basic math fact fluency can provide a teacher with invaluable information about what a student knows, doesnt know, as well as any misconceptions the student may possess. These assessments do not need to be lengthy or complicated its actually the use of a few well-constructed questions that can yield an abundance of information about student progress and next steps for instruction.
According to the article Assessing Basic Fact Fluency by Gina Kling and Jennifer Bay-Williams, there are three important ways to glean important data about students understanding.
- Engaging in a brief discussion with a student provides a window into their thinking and reasoning. Also, as students talk through how they are solving problems, there is an increased likelihood that they begin to check the reasonableness of their answers and self- correct when it is necessary. Listed below are some sample questions that can be used when interviewing individual students:
- Write a problem on an index card. Ask the student the answer to the question, as well as how they found the answer and if they know of another way to solve the problem.
- Ask students how they can use the answers to facts they know to solve facts they dont know. For example, as questions like How can 5 + 5 help you to solve 7 + 5? or How can 3 x 3 help you to solve 4 x 3?
What If Students Do Not Have A Parent Available To Play The Fluency Games
While most of the games were designed as partner games, many can be played alone, or be easily modified for students to play alone. Rather than competing to see who might be the first to get 4 in a row, students might just play until they get 4 in a row. Rather than competing to see who can fill a grid first, students might see if they can fill it in 12 rolls or fewer. Each online game is in Word format, so the directions are customizable. Changing a few words provides students with directions that work at home or when they are playing alone.
Happy International Day Of Math
We hope you and your students enjoy celebrating the wonder of mathematics. If youd like to know more about how math impacts our daily lives and the planet we live on, check out the International Day of Mathematics mini-site.
Do you have a favorite math fact we didnt include? Share it in the comments!
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Why Are Math Facts So Important For Children To Learn
Even though it might seem old school, knowing your math facts fluently and thoroughly is still a crucial skill. So, before delving into the importance of math facts, lets first define the term.
What are Math Facts?
Basic number combinations for addition, subtraction, multiplication, and division are known as math facts. Children should be able to recall these problems within a few seconds. They are addition, subtraction, multiplication, or even division problems. Children can learn math facts, which are simple calculations, to help them complete math problems more quickly. Math facts can be memorized and remembered easily, freeing up mental energy for higher-order mathematical operations.
Math facts fluency refers to The ability to recall the basic facts in all four operations accurately, quickly and effortlessly. When students achieve automaticity with these facts, they have attained a level of mastery that enables them to retrieve them for long-term memory without conscious effort or attention.1
Why Should Children Learn Math Facts?
Many people may wonder why children still need to memorize their basic addition, subtraction, multiplication, and division operations in this day and age when we have calculators and computers.
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Why Is It Important To Continue A Strong Focus On Math Facts
We recognize that math fact understanding and fluency takes time. In most K-5 classrooms, fact fluency is a routine part of the school day. Now we face reductions in teaching time, and a reliance on at-home practice tasks, that challenge us to find ways to explore math facts with the attention they deserve. Through brief lessons that focus on critical concepts and connections, and at-home practice that is focused and meaningful, we can address math fact understanding and fluency and provide a strong foundation for our students math success.
Susan OConnell has decades of experience supporting teachers in making sense of mathematics and effectively shifting how they teach. As a former elementary teacher, reading specialist, and math coach, Sue knows what its like in the classroom and her background is evident throughout her work as she unpacks best practices in a clear, practical, and upbeat way.
She is the lead author of Math in Practice, a new grade-by-grade K-5 professional learning resource. She is also coauthor of the bestselling Putting the Practices Into Action, Mastering the Basic Math Facts in Addition and Subtraction, and Mastering the Basic Math Facts in Multiplication and Division. She served as editor of Heinemanns popular Math Process Standards series and also wrote the bestselling Now I Get It.
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Why We’ve Moved Away From Memorizing Math Facts
Education today is a barrage of standardized assessment. I’ll say it again for those in the back there’s WAY TOO MUCH standardized assessment in today’s classroom. When it comes to math, the majority of that testing comes in the form of word problems. Page after page of long, multi-step story problems are meant to confuse kids just as much as they are meant to figure out what they know.
As teachers, we prepare students for this by giving them repeated exposure to these problems. We fear they’ll fail if we don’t give them enough practice.
Each year we find ourselves pushed to spend more of our math block on problem-solving even for skills just introduced. Of course, as spring approaches, this push moves from a gentle wind to a full-on, hurricane-force storm.
Some who would argue that if a child can find a way to get the answer to a math problem, that is good enough. They argue that as long as a child has a strategy, they can be good problem solvers.
While this is true for some students, if you look at the majority of struggling math students, you find the same gaps in basic skills.
Multiplying By Focus Numbers
When a student first learns multiplication facts, try not to overwhelm them with the entire multiplication table. These worksheets include one row of the facts in order with the target digit on the bottom and one row with the target digit on the top. The remaining rows include each of the facts once, but the target digit is randomly placed on the top or the bottom and the facts are randomly mixed on each row.
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Math Facts Are Like Atoms
We once believed that atoms were the most basic unit of matter. They were indivisible, inflexible, and indisputable. But by the late 1800s, the first subatomic particles were discovered. Soon, Einsteins Relativity came along to blow the old atomic theory to bits.
Math facts are a lot like atoms. Many of us believe they are the most elemental unit of math. We worry that students will struggle through all future content until they learn their facts.
The reality is that math facts are composed of even smaller units of understanding the concepts of number and operations.
When students have number sense, they can build all the math facts, rather than memorizing them one-by-one.
When a student cant calculate 4 x 7, there are a number of conceptual strategies they can employ. They could count up by 7s. Or take away 7 from 5 x 7. They can even create an array or double 2 x 7.
Students who use these strategies understand the meaning behind the operations. And they can apply that understanding to discover facts they dont know. That ability is exponentially more valuable than memorizing a list of so-called math facts.
Math Facts Are No Fun
Few advocates of fact-focused math claim that drills inspire excitement or curiosity. It should go without saying that if we want students to identify as mathematicians, we expose them to the beauty of math.
Some educators assert that they use games and apps to make math facts fun. This spoonful of sugar approach can actually backfire. Im all for fun in the classroom, but we should show students that the math itself is exciting. Treating math like medicine teaches students that even adults find math boring.
For some students, mere boredom is the least of their worries. Jo Boaler is a professor at Stanford, and one of the foremost experts on math education. Shes about the links between timed testing and math anxiety. Conservative estimates suggest that at least a third of students experience extreme stress around timed tests.
In the same article, she cites aPISA study of 13 million, 15-year-old students from around the world. It found that the lowest achieving students are those who focus on memorization and who believe that memorizing is important when studying for mathematics.
If fact-based math education is neither fun nor effective, why is it so pervasive?
Multiplication Facts To 12 12 = 144
The Holy Grail of elementary mathematics. Once you learn your twelve times table, it is smooth sailing from now on, right? Well, not exactly, but having a good mental recall of the multiplication facts up to 144 will certainly set you on the right path for future success in your math studies.
Fact Fluency Reduces Cognitive Load
Research on cognitive load supports the idea that we all have a limited cognitive capacity at any given moment. In other words, we are all working within the limits of our brainpower.
Word problems, for example, require students to perform multiple cognitive processes at the same time. Students must read and comprehend the problem, break down the information given to determine what is needed to solve, compute, and identify the solution. This process becomes even more complex with multi-step problems or those with extraneous information.
While it being fluent with math facts doesn’t make word problems easy, it does reduce the number of cognitive resources needed to tackle the computation portion of the process, allowing those resources to be allocated to other components of the process.
Think of it this way if a struggling student is using all their mental energy just to solve the basic facts, what is left for problem-solving or determining whether their solution is reasonable?
This is why we see many struggling students frustrated and overwhelmed when asked to check their work. They’ve used all their resources just to get to this point, and we are asking them to go back and work it again from an empty tank.
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Why Not Just Use Timed Tests
There is a growing body of research that indicates using timed tests can actually be detrimental to a students confidence in the area of mathematics. For even the strongest of math students, timing them with their facts can be anxiety provoking and restrict their ability to think flexibly. Furthermore, when timing fact tests, the opportunity to see how students think and reason is missed.
One suggestion King and Williams give is to use fluency quizzes instead of timed tests. On a fluency quiz, students are only assessed on their foundational facts, or those facts that can be used to derive other facts using a strategy. The foundational facts in addition are one more than , two more than , combinations that make ten , and doubles . The foundational facts for multiplication include x 2, x 5, and x 10. When students master their addition and multiplication facts they can use them to fluently solve subtraction and division facts.
On the fluency quiz, in addition to asking the students to solve the foundational fact, ask them if it was a fact they knew already or if they used a strategy to solve it. Students could even choose just one problem they solved and write about how they solved it.
Fact Triangles And Fact Families
Fact Triangles are Everyday Mathematics’ flash cards with a difference. The difference is fact triangles help children learn fact families rather than isolated facts. Partner practice with addition and subtraction fact triangles begins in first grade. Multiplication and division fact triangles are introduced in second grade. Practice with Fact Triangles is often suggested in the Home Links homework assignments.
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Get A Jump On Your Math Facts Practice
Lay out a grid like the one shown that has the answers to whatever set of math flash cards youre currently working with. Two players face off, one on each side of the board. Show the flash card, and kids race to be the first to jump to the correct square with both feet inside the lines. Get all the rules at the link below.
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Assemble A Domino Puzzle
Dominoes are perfect for math facts practice! Keep it simple by pulling a domino from a bag, then adding, subtracting, or multiplying the two numbers.
For even more fun, print the free puzzles at the link below. Then start filling in the puzzle one piece at a time by placing a domino that adds up to the number shown in each rectangle. The trick is that regular domino rules still apply, so each number must touch another domino with the same number on that end.
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My Favorite Math Lesson Ever
If there is one lesson I love to teach again and again, its the Sieve of Eratosthenes. This lesson can work with students at almost any grade level. And it does wonders for fact fluency and pattern recognition.
The activity builds off of a simple 100s chart. But once students begin coloring in the patterns, it becomes much more. I taught my students to use it in place of a calculator. It helped them become fluent with basic addition, subtraction, multiplication and division. Read more about the lesson here. You can also download the complete lesson plan, with reproducibles, from our TpT store.
They Don’t Know Their Facts
While some can compute them, others are missing even the conceptual understanding of the foundational skills. Even those who are semi-successful in math often use inefficient strategies that take time and are mentally tiring leaving them with incomplete or partially done assignments. These students are treading water, but as the rigor increases, they are sure to begin sinking.
Strategy is a good band-aid. However, it fails to consider the big picture of the progression of mathematical skill-building. There are many fun and engaging ways to build math fact practice into our classroom routine. Technology makes this even easier.
For example, online math fact games and apps can be a great alternative. These can be done during small groups or assigned as homework. I’ve created a post with my favorites here: 30+ Digital Math Fact Games to Build Fluency
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How Can We Focus On Number Patterns And Mental Strategies With Abbreviated Lesson Time
Focusing on math facts in sets that share number patterns, or common mental strategies, allows us to design brief lessons. Lessons might only last 10 minutes but can get to the critical insights that help students make sense of math fact sets. You will find lots of brief lesson ideas in the Mastering the Basic Math Facts books, lessons that set the context with a story or problem, have students gather and observe some data to discover number patterns, and make connections to familiar math concepts or known facts. Following those brief teaching experiences, practice is specifically designed to focus on just that math fact set to allow students to practice applying the mental strategy they have explored. The fluency games in the MTBMF books are specific to fact sets, so rather than practice being diluted to all addition or all multiplication facts, the practice might focus on just addition doubles, ×6 facts, or whatever fact set was the focus of the lesson. This practice can be done at home. It is the selection of the practice task, to align with the facts teaching, that makes each practice session effective in moving students toward fact fluency.
Multiplication Facts Tables With Individual Questions
The multiplication tables with individual questions include a separate box for each number. In each box, the single number is multiplied by every other number with each question on one line. The tables may be used for various purposes such as introducing the multiplication tables, skip counting, as a lookup table, patterning activities, and memorizing.
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