OK…it is no secret that I am a HUGE Jo Boaler fan. Huge. Her work with mathematical mindsets and brain research should e required reading for anyone who works with children.
That being said, I know that one thing that ALL teachers (and parents) think–especially as we head back to school–is “but they DON’T KNOW THEIR FACTS!”
And, of course, I am a teacher too–and I know that by the end of this year I am supposed to have all my students successfully adding, subtracting, multiplying, dividing, and more–and if they don’t know their facts…it sure is more challenging, right? I get students in my class every year who not only didn’t master their multiplication facts–but don’t know their addition and subtraction facts either! That’s a different post–because that becomes a HUGE intervention issue…something students should have mastered years earlier. But with multiplication? We know this is a HUGE part of fourth and fifth grade, so we need to make sure students understand the concept.
So I dug back into Boaler’s book to really study what she had to say. Here are some of her key points that I really think are worth some deep reflection.
What does Jo Boaler say?
1. Drawing attention to math fact “speed” is not only counterproductive to learning math facts but also tends to be the trigger for some children–the beginning of the “I hate math.” and “I’m not good at math.” epidemic. It draws their attention away from what math really IS…and math fact recall is really and truly just a small part of the world of math.
2. Boaler stresses a more conceptual approach to teaching facts. Students need to understand patterns in numbers and understand number relationships that build true understanding of these facts. She explains that research supports a conceptual, strategy based approach to fact instruction. Want to read more? CLICK HERE for a brief article on Youcubed that explains more.
3. Boaler DOES recommend practice, of course. What she highlights is the importance of building brain connections by practicing math concepts in different ways–not repeated practice in the same format repeatedly. Page after page of practice problems or flipping flashcard after flashcard does not build strong brain connections the same way as providing a variety of learning experiences.
4. Providing students with deeper activities to build fact fluency not only is more effective–but also builds more excitement about math instruction. Teaching strategies gives students power–they don’t have to try to recall every fact–they understand how to derive facts based on different strategies that they are taught–and strategies can be used in novel situations.
Here is a perfect example…a few years ago a student and I were working on a problem (I don’t remember what it was anymore…), but the problem was relatively simple–it just required the student to be able to split 30 in half. I will never forget the look on his face when I prompted him a little bit…and he said, “That number doesn’t split in half.” I dug around a little more and realized that he–as a fourth grader–did not understand the concepts of doubling and halving. He “knew” his two facts–but he could not generalize to this essential math concept. Think about how many areas of math this impacts…the ability to understand that a 45-degree angle is half of a right angle. The understanding that 1/8 is half of 1/4. To be able to quickly “get” that 500,000 is half of a million.
He KNEW his “2” facts–but didn’t understand the concept of doubling. #eyesopened
Now don’t get me wrong. I NEED my students to learn their facts. It’s just a matter of HOW they learn them. I am a huge believer in teaching these strategies–and if they don’t know them, this is a huge part of the intervention work I do at the beginning of the year. I have struggled to find good resources for this, so I worked all last year to try to get some things together so I could “low prep”
Here are a few “hints” of some ways to give students some strategy based fact instruction.
1. Teach a strategy at a time–don’t simply go “in order” from 0-10. Once students understand that 2 facts, 4 facts, and 8 facts are related (by “doubling”), it makes sense to work on these facts together. Similarly, students can learn 5 facts by understanding 10 facts that are split in half. When we teach students that 9 facts are “one less group” than the 10 fact, that builds understanding. Using tricks like the “9’s trick” on fingers might be fun–but doesn’t help build math understanding. Don’t get me wrong–they can be fun! And once students understand the math, it makes the tricks even more fun to study.
2. Make sure students understand simple multiplication properties…when students just learn that “0 facts are always 0” without understanding WHY (What happens with you have no groups? What happens if you have a lot of groups–with nothing in them?), we are teaching in a way that doesn’t support an “algebra thinking” mindset that they will need in the upper grades.
3. USE ARRAYS! A visual display of “groups” is one of the most powerful ways to help students start to notice patterns. Use grid paper. Use square tiles. Use ANYTHING to start to build this understanding of “groups” and for students to be able to interact with something.
Check out how these strips of dots can be used to build the concept of doubling…these “9 strips” can be doubled (2 facts) and “double doubled” (4 facts).
Even having a larger array where you can slide a piece of paper (hard to see on bottom right corner) to show growing arrays. Here is a picture of 6 x 5 array…but the paper can be slid to show “one more 5” or “one less 5” to build conceptual understanding.
4. Working with different representations of numbers is great to deepen understanding as well…25 is the same as
* half of 50
* 5 + 5 + 5 + 5 + 5
* 5 x 5
You get the drift. This is really asking students to see numbers in different ways rather than simply memorizing facts in isolation. This idea of “number sense” is directly related to fact fluency so try to find ways for students to “play” with numbers in different ways!
5. Games and practice!
Once students understand the different strategies, playing games and doing other “practice” activities can build speed and fluency–without stress! It’s important to remember that fluency comes AFTER understanding, so don’t rush it! I will often provide students with games like this AFTER they have worked with me in intervention groups. It’s a great way for them to continue their learning.
I hope I’ve given you some food for thought. We all want our students to learn their math facts–it’s just a matter of making sure we use “best practice” methods to accomplish it! To help me, I’ve put a bunch of activities together so I have a great intervention toolkit to use when working with small groups. I have it all organized by strategy–and I can choose whether to introduce activities to the entire class (great at the beginning of the year when the wheels aren’t quite turnin’ yet) or to use in small intervention groups and math workshop. I keep each activity in zip top bags and each strategy in a plastic tub so it’s easy to grab and go.