Have you ever seen a 5th grade student count on their fingers to add 7 + 8? Or a 6th grader doing repeated addition on their paper to solve 9 x 7? If you haven’t, I can guarantee you’ve heard the stories from the upper grades teachers during staff meetings as they complain that “Kids don’t know their facts!!!!” Yet, every primary grade teacher spends a TON of time trying to help kids develop their fact fluency. So, why are so many kids still not fluent with their facts?!?!?!?
In Part 1 of this blog post I discussed how the old way of teaching kids to learn isolated facts should be retired and in its place should be the idea that facts are related AND that certain facts come easier than others. In this 2nd part I will lay out the 4 Number Relationships that kids need to have around ALL numbers to develop their number sense and become flexible thinkers with their facts…and all other math concepts!!!
Let’s first get clear on what FLUENT actually means. Susan Jo Russell’s article outlines three parts to being fluent:
Accurate
Efficient
Flexible
Many students have one or two of these, but few have all three. Kids who are counting manipulatives or fingers to solve 4 + 3 are usually very ACCURATE, but not EFFICIENT or FLEXIBLE. Kids who have memorized their facts are ACCURATE and EFFICIENT, but not FLEXIBLE.
We all want kids have fact fluency, yet most people think that just means accurate and efficient. However, here is the rub…if you focus on building a student’s accuracy & efficiency they will NOT become fluent. The key piece to getting kids to become fluent with their facts is FLEXIBILITY. We all have had those kids in our classroom that just “get” numbers. They have FLEXIBILITY with numbers that allows them to work so effortlessly in mathematics. So, what do those kids have that other kids who struggle with mathematics don’t have????
NUMBER SENSE.
Educators throw that term around all the time, but what exactly makes up number sense? The most quoted quote about number sense says that number sense is “…good intuition about numbers and their relationships” (Howden, 1989). But, I gotta say, I don’t really like that quote. It gives me no ideas for how or what to actually teach to my students.
Never fear, I found the answer in my favorite book, which I now call my math bible; Teaching Student Centered Mathematics. The authors (Van de Walle, Lovin, Karp, & Bay-Williams) outline 4 Number Relationships that kids should develop for numbers up to 20. For this blog post I will keep with numbers up to 20, but really, kids should develop all 4 of these relationships with any number (even fractions and decimals).
Spatial Relationships –recognizing how many without counting by seeing the visual pattern.
One/Two More and Less – this is not the ability to count on two or count back two, but instead knowing which numbers are one more or two less than any given number.
Benchmarks of 5 & 10 – since 10 plays such an important role in our number system (and two 5s make up 10), students must know how numbers relate to 5 and 10.
Part-Part-Whole – understanding that a number can be broken up into 2 or more parts.
These 4 relationships are WHAT makes up number sense. If you spend time really developing these fully with your students, you will get kids like this little boy who are ACCURATE, EFFICIENT, and FLEXIBLE with their facts.
If you didn’t watch that video, WATCH IT! It’s only 30 seconds and I’m going to dissect that kiddos thinking and relate it to the 4 Number Relationships (so it helps if you’ve watched the video) :).
He totally used the Benchmarks of 5 & 10 because he knew that if he could make one of those numbers a 10 it would make the problem easier. In order to make one of the numbers a 10, he had to understand the concept of Part-Part-Whole because he had to “pop off the 1” out of the 6….he understood that he could break that whole (aka 6) into friendlier parts. When he did pop off the 1, he instantly knew One Less would be 5.
So far we saw him use 3 of those number relationships, what about Spatial Relationships??? Most of the time you don’t actually ‘see’ kids using spatial relationships because it is the foundational piece that allows kids to build all the other 3 relationships. So, here is an example of a visual of 9 + 6 that does NOT build spatial relationships to help develop the other relationships:
Instead, if those same items were placed inside a ten frame, the ten frame gives kiddos a visual that builds Spatial Relationships and helps build the other three relationships as well:
Go back and re-watch that video of the little kid…can you just imagine this image in his mind as he is popping off the one from the 6 and how it turns into a 10 and 5????? If kids don’t have these visual images, it is VERY difficult for them to develop the other three number relationships.
All four number relationships are essential to helping your kiddos develop fact fluency…not just fact memorization. Even with multiplication facts. Think back to that kid who does repeated addition to solve 9 x 7….if children understand that 10 is a powerful benchmark, that 9 is just one less than 10, and they have built spatial relationships and part-part-whole understandings using arrays or area models, then we see lots of kids who can tell you that 9 x 7 is just like having 10 x 7 but you have to subtract 1 x 7 from your answer.
The coolest thing to me is that once these relationships are built, children develop strategies for their facts that are not tricks…they are real strategies that last well beyond their “facts.” Take a look at these strings of problems to see how a strategy kids develop for their “fact” can become super powerful as they go further into mathematics.
If you would like a little bit more detail about each of these 4 number relationships then grab my free ebook! It details out each of the relationships. Plus, it includes a quick assessment for each relationship that you can do with your students. **Note some example tasks in the assessments are with small numbers (like 5), if your students can do small numbers just change to a larger quantity (like 14).
I was taught math in Europe as a child. Everything is based on the sets of 5 and 10. As we did the dishes with my young children, I would ask large number combinations that would discourage them….then I reminded them of the 10 principle, and they could do it in their head and come up with the right answer, and do it quickly. My little 10 year old granddaughter hadn’t learned to use this method, and while she was here visiting for a few minutes, I asked some aritihmetic additions and subractions which she wanted a piece of paper for. I then taught her the “ten method” I call it. She hesitantly gave me the right answer, and then her eyes glistened with excitement as I told her, “YES! You got it!!!!!” Now she can do moth problems in her head , fast, just from the 5 or so minutes in teaching the 10 method. Yay for doing this with kids. It is sooooooooooooo important to be able to use math for the millions of reasons we use it, to measure, to figure out how much we need of something (like curtains, or material, to figure out money, to plan ahead how much/how long we need to plan to achieve what we are aiming for. /
I learned math in Holland when I was a small child. Everything is taught using the 10 and 5 pattern to simplify adding, subtracting and figuring out patterns. It works every time, and the children do not count on fingers, but do the calculations in their head. I taught all 10 of our children by this method as we did the dished together. I’d simply throw out a question of math, and when they paused to figure it out, I would review the “10 method” as I called it, so they could see what to do, and give me the answer fast (without counting fingers, without paper and pencil) This works! Thanks for teaching a workable way to make numbers have meaning and that the child comes alive with excitement when they once understand what they need to do to come up with the question I asked. It never fails to produce. I worked in the school system with behavior disordered students, even they could work the system, just somewhat slower than their peers. IT WORKS, and the students are so proud to do this “difficult math equation” all in their head. Thanks for teaching it, and thanks for letting me have my say.
I teach autistic high school students work skills. They are great at doing paper based maths algorithims however they have no spatial understanding of number. When asked to pack x number of items into a package they struggle to do it with accuracy, particularly for numbers larger than 10. I have been using 10s frames as a visual jig prior to packaging items. And working on building visual number representations of quantity with as many manipulatives as possible (which is extremely important for generalisation).
Thankyou so much for all your work. I have been using it for pd with my colleagues. Early numeracy skills are very important to explicitly teach for our special needs students.
I was taught math in Europe as a child. Everything is based on the sets of 5 and 10. As we did the dishes with my young children, I would ask large number combinations that would discourage them….then I reminded them of the 10 principle, and they could do it in their head and come up with the right answer, and do it quickly. My little 10 year old granddaughter hadn’t learned to use this method, and while she was here visiting for a few minutes, I asked some aritihmetic additions and subractions which she wanted a piece of paper for. I then taught her the “ten method” I call it. She hesitantly gave me the right answer, and then her eyes glistened with excitement as I told her, “YES! You got it!!!!!” Now she can do moth problems in her head , fast, just from the 5 or so minutes in teaching the 10 method. Yay for doing this with kids. It is sooooooooooooo important to be able to use math for the millions of reasons we use it, to measure, to figure out how much we need of something (like curtains, or material, to figure out money, to plan ahead how much/how long we need to plan to achieve what we are aiming for. /
I learned math in Holland when I was a small child. Everything is taught using the 10 and 5 pattern to simplify adding, subtracting and figuring out patterns. It works every time, and the children do not count on fingers, but do the calculations in their head. I taught all 10 of our children by this method as we did the dished together. I’d simply throw out a question of math, and when they paused to figure it out, I would review the “10 method” as I called it, so they could see what to do, and give me the answer fast (without counting fingers, without paper and pencil) This works! Thanks for teaching a workable way to make numbers have meaning and that the child comes alive with excitement when they once understand what they need to do to come up with the question I asked. It never fails to produce. I worked in the school system with behavior disordered students, even they could work the system, just somewhat slower than their peers. IT WORKS, and the students are so proud to do this “difficult math equation” all in their head. Thanks for teaching it, and thanks for letting me have my say.
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I teach autistic high school students work skills. They are great at doing paper based maths algorithims however they have no spatial understanding of number. When asked to pack x number of items into a package they struggle to do it with accuracy, particularly for numbers larger than 10. I have been using 10s frames as a visual jig prior to packaging items. And working on building visual number representations of quantity with as many manipulatives as possible (which is extremely important for generalisation).
Thankyou so much for all your work. I have been using it for pd with my colleagues. Early numeracy skills are very important to explicitly teach for our special needs students.