One of the hardest things to navigate when teaching math in a classroom is the huge range of skill levels of the students. I see this daily; in a grade 6 class there are some students who don’t know ‘friends of 10’ nor any strategies for adding and subtracting besides finger counting and stacking (no mental math strategies beyond counting) and then you have others who can mentally calculate quicker than you can. So, how on earth do you meet all of their needs? It’s pretty tough but this blog and the 2 that will follow will help you out with this.
Before I jump into strategies I’d like to address the questions and concerns I most often hear from teachers: “I’m teaching grade 6, why do I have to go back to grade 1 concepts? Shouldn’t they already know it? I don’t have time to go all the way back- I’ll never be able to cover the curriculum”. I feel your pain. It induces a certain panic when we realize we need to go ALL the way back to kindergarten or grade 1 concepts even if we’re teaching middle school, BUT it is entirely necessary.
Math, more than any other subject area, builds on previous knowledge. I always use the analogy of building a house. The foundation of the house is equivalent to the 4 basic principles explained by John Van de Walle (Teaching Student Centered Mathematics): subitizing, partitioning, benchmarking to 5 & 10, Knowing 1 & 2 more/less. Without these basics, operations are learned by memorizing or counting by 1’s and students rarely progress to additive nor multiplicative strategies and so are severely limited in understanding larger number operations. Everything builds on what came before so unless we want to try to build a house on quicksand, we need to go back to these very basics and ensure our students have them before we start to build onto them and get to the actual grade level content.
Why don’t students already have these skills at this age? They may have never been taught them but what’s more likely is that they never had the opportunity to transfer the learning into the long-term memory or they forgot them due to lack of use (reverting back to finger counting). All humans fall victim to what researcher Hermann Ebbinghaus called the ‘forgetting curve’.
Note: There are some disputes as to the actual curve so this representation may not be entirely accurate but the general shape is agreed upon = we forget a lot quickly
Just think about all of the information that is gathered by your brain each day. As you sleep your brain sifts through it all and throws out what it deems unnecessary. Unfortunately, a lot of math class is in that category. BUT, there are things we can do to help reduce the forgetting. First, the more connected the learning is to the learner and to other learning, the more likely they will remember it. This is why we activate prior learning in each lesson and help students to see the connections between concepts and strategies. We can also get students to explain their strategies and understandings to a peer, thus improving retention further. Finally, we can provide practice over time to help them retain their learning in the long term. Often in math we teach a concept and then don’t return to it until the next school year….and that learning is no longer there.
If you have students who are like those I described above (very limited number sense), then you literally need to start at the beginning of building number sense by working on subitizing, partitioning, 1 and 2 more/less and benchmarking to 5 and 10. The good news is that the students tend to pick up these concepts much quicker when they are older but we often need to break them of their counting dependency first.
I suggest you start at subitizing. Subitizing is being able to know how many there are without counting by ones. For example: look at this image and tell how many dots there are without counting the dots:
If you could tell there were 5 without counting you just subitized. It is important for students to be able to do this with varying images (don’t always use this familiar pattern of five dots to show five).
Humans can subitize 3-5 items at most, however, we can combine different subitized portions to quickly know a quantity (this is called conceptual subitizing).
Here’s an example: Look at the arrangement of dots in the box below for 2 seconds. How many dots do you think are there?
Now look at the box below for 2 seconds. How many dots do you think are there?
Our brains can subitize and then combine the 5 and 5 and 2 to make 12 far easier than in the first example. So how do we teach subitizing? How can you work on developing this skill while the other students are being challenged at their level (those who already subitize well)?
Number Talks. If you have never heard of number talks or want to learn more – check out my free video on Number Talks here.
Number talks always start with dot cards and depending on your class and their strength in this area, you may wish to stay with dot cards for a while. Because you are asking students how they see the dots (without counting by ones), you are providing an opportunity for all students to be successful. Those that find this easy will be working on their communication skills; articulating clearly to you what they are seeing, what operations they mentally used and we also challenge them to see the arrangement in as many ways as possible. Number talks engage many of the curricular competencies and even ‘strong’ math students find it difficult to articulate their thought processes. So for some of your students, this will be the skill they are developing, along with multiple ways of seeing. Some of these kids really struggle with the metacognitive piece of identifying how they saw the dots. I was in a grade 5 class last week and the boy said he just saw 9 dots. Because the science is really clear that even grown adults can only subitize to 5 and most kids subitize 3-4 items, I knew he was actually seeing it differently but wasn’t able or willing to articulate it so he needs practice in paying attention to his thinking.
I also highly recommend doing some number talks using 10 frames but not always having the dots in the same configuration. Here are some examples for 7:
Card and dice games (any games that use dice) are HUGELY helpful in gaining this skill and so can be used as ‘homework’ for kids who are not subitizing well yet. Dominoes are also great for subitizing and can be played at home or in the class. Parents can also help by using dot cards at home regularly with their child, so that they are seeing many different orientations of the quantities of 3, 4, 5, etc.
Here are some freebie games and activities for you to use in your classes or to send home. You can use the Memory Game Template for smaller or larger numbers if needed (and print on card stock paper so they can’t see through them).
Here are some other free resources that you can use in your class or send to parents if you are teaching older grades where this skill should already be well established:
http://www.mathematicallyminded.com/free-downloads/ (there are many great resources on this site for primary grades created by Christina Tondevold)
Number Talk dot cards can be found in these books:
and dot cards can be found on google as well as here: https://docs.google.com/file/d/0B_wlnPzXZBUZMzkyMGU0ZDQtYzJjMC00YzFlLWIyMDktODcwNzA5NzczNDMz/edit (created from John Van de Walle’s work by Donna Boucher at www.mathcoachscorner.com)
In the next blog I will discuss partitioning and then will follow up with benchmarking to 5 and 10 as well as knowing 1 and 2 more/less. Once these basics are internalized by your students then we can move onto strategies for operations. Please comment on how the number talks are going and what you’ve noticed about your students’ subitizing.