The more I create… the more I learn. Here is the 3rd installment of this whole *Making Sense Series* which has truly forced me to be a better teacher. A more educated teacher.

I can’t stress enough how much I’ve learned from diving into the progressions found here. There are many intricate pieces of learning which are mentioned throughout the progression documents which can’t be included within a 5 minute video. I strongly recommend finding the little pieces with your team or grade level.

Unlike the multiplication and division videos which were introduced separately, this video is an addition/subtraction mash up…they way they should be explored by students.

In the 60’s they were called cuisenaire rods. Maybe in the meantime math went through a “whole equation” method of teaching.

Just a tiny thing; Look carefully !

400+30+5

-200+50+6

No, Cuisenaire are different and the two are often confused. In the video the equipment being used is Base Ten (or ‘Dienes’ named after Zoltan Dienes), Cuisenaire are used for algebra and proportion amongst other things. See Caleb Gattegno’s work and the Cuisenaire website.

I really like the different tools you used as you marched through the progressions here. Do you think the flexibility with numbers you mentioned early on could be expanded in the upper grades (3-5)? I’m thinking about subtraction like 54-19 for example. Should we encourage students to use their flexible thinking and possibly use an idea like 54 = 34 + 20 = 35 + 19 so 54-19 = 35? Thanks for putting these videos together! They will continue to be useful to teachers and parents alike.

Check out Making Number Talks Matter. I’m sure there are other Number Talks resources available as well.

Yep…great call!

Thanks Mike. There’s definitely a connection to be made here Mike. I’m thinking of benchmark/friendly numbers in addition to compensation because essentially we’re making a 10 (most of the time). The only difference between these strategies and the standard algorithm is that the standard algorithm has a predetermined order for how we add. First ones, then tens, then hundreds and so on.

What about 54-19 = 55-20 (add 1 to each side) – much simpler

‘Constant difference’ strategy. Yes, a great tool as long as they conceptually understand it. Number lines really help to see that the difference here has remained the same.

All of these videos are so great. Thanks for all of the time you have put into making them. So helpful for a lot of different audiences. Curious about what strategies you think undermine place value. I notice you didn’t include open number lines in the video. When kids use them, they don’t see the regrouping process, the way they do with concrete models and partial sums and differences, but it seems to be comfortable step between concrete models and expanded form for a lot of students. Thoughts? Thanks again. Really fantastic.

I agree 100% Courtney. The issue I face when creating these progression is if I open it up to a number line, I’d open it up to other strategies as well (ie: hundreds chart). Please don’t get me wrong, I’m a huge advocate of the number line but I’m really trying to keep these video focused and under 5 minutes. As is evident from this video…I failed miserably on the latter.

Brilliant. Thank you for this putting together.

Look into subitizing (dots and dot plates) as well. Along with the five/ten frame, a must have for early years number sense.

Absolutely Craig. I plan to address that in a counting video and you’re right, subitizing is a non-negotiable.

Are students beginning to count in kindergarten or does it happen earlier? I really love this video series you are doing but I think it could be useful to include some common experiences before and outside school that inform students’ early and continuing math experiences in school.

For example, my son, who is 3, can count reliably to 11, can subitize numbers up to 5, can add and subtract numbers to 5 by grouping sets, can add numbers to 11 by counting up (but not by counting on), knows that zero is a number (and in fact tried to trick Daddy by asking me what 4 plus 0 is just yesterday), although he’s not counting backward unless the context is a count-down like a rocket ship blasting off (although we use 0 before we say blast-off, which maybe helps him see 0 as a number). He also knows that numbers like 9 or 10 thousand are different than numbers like 9 or 10. He has also learned that addition and subtraction are closed operators – that they take two or more numbers and produce another number.

An aside: Do students add or subtract fractions at all between kindergarten and fourth grade?

Also, I wanted to add that I really like how you produced this video with the scrolling paper (or a camera on a rolling mount?) and want to learn more about the process you go through to produce these videos if/when you have time to blog about it.

Thanks David for the comment and I definitely agree that there is a need to address content before and outside of school. I’ve almost finished with the counting progression on paper and the video could be ready in the next 2 weeks. I’ll capture the process as I work and discuss it in a post about it.

In terms of your question

“Are students beginning to count in kindergarten or does it happen earlier”, it happens earlier in some places. Here’s what I mean…the district I work in has approximately a 30% of incoming kindergarteners that can count 6 objects or rote count to 10 successfully. The district up the road might have a 90% proficient rate. If we assume that our students have limited ability in counting when they come to use then we’re ready to hit the ground running on day one. If they can count, we’re ahead of the game.Do students add or subtract fractions at all between kindergarten and fourth grade?Students identify the sum of unit fraction in 3rd grade.Thanks for the questions and hope this helps.

Another masterpiece!

I know how much time it takes to make these videos, having tried it myself to do a much less ambitious one about Cuisenaire rods (and giving up and animating it with Explain Everything instead!) which makes me even more grateful!

HA! Maybe a Simon/Fletchy mash up one day. Your brilliance with the rods and my amateur videography. Should make for an interesting team.

This is fantastic. I’m going to be sharing this with the pre- and in-service teachers I work with. Can’t wait to see the other videos.

Having a firm foundation in conceptual understanding is so vital, it probably can’t be repeated enough. Thanks for creating this!

Thank you so much, Graham! These videos are so helpful to teachers and parents alike! I am so happy that you expressed your opinion on the importance of play and conceptual understanding! I appreciate the time you put into everything you do that help so many of us better our practice! Thank you!

Jamie

Place the numbers 1 to 6 in the circles so that each number is the difference between the two numbers just below it.

I really enjoyed the video with this post. It touched on many of the strategies and topics we are learning about in my math methods course for elementary educators. The video was a great visual to reinforce where students should be at each grade level and how their thinking can progress. I

also really liked how you can consistently see the strategies used through each grade level and how they are connected between addition and subtraction!

Right on and thanks for the comment.

It’s great to know that you were able to make so many connections while watching the video. You’ve definitely hit on a lot of the connections I was purposefully trying to make. This makes me happy so thanks for sharing:-)

Thanks so much for sharing these videos Graham. I am a maths consultant using concrete and pictorial representations with teachers in the UK to keep the focus upon conceptual understanding before moving to abstract. Seeing the five and ten frames used is brilliant! I use these so much in my work and they help children move from the ‘count on’ strategy to true fluency. Love you ‘Tomato Tomato’ video too! Keep sharing.

Right on Karen and thanks for sharing the love.

This is so helpful! After teaching middle school math for 12 years, I’m now working with grades K-8 as an instructional coach and trying to bump up my own knowledge of how primary students learn math. I’m working with a few primary teachers and shared your video and the link to the progressions you recommended. Thank you!

For 423 – 185 when you expand the numbers then you need to subtract 100 and subtract 80 and subtract 5, not add, correct? That was confusing to me.

430+ 400 + 20 +3

– 185 = -100 -80 -5

Why else would you need to borrow right? The negative is distributed to each of the expanded numbers.