Last month I posted The Progression of Multiplication hoping that a couple of friends and parents would find it helpful. Well, here’s my stab at the progression of division. I understand that there’s lots of ways to model division and this is only ONE of them.

Understanding the vertical progression of mathematics is really important in the conceptual development of **everyone’s** understanding.

Hope it helps.

Stay thirsty my friends!

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K-8 math consumer trying to listen and learn each day. Stay thirsty my friends!

I like your systematic but unrushed approah a lot, but I am really bothered by the continuing need to do division of anything by a two or more digit number by hand. At some point the REAL need is to be able to estimate the result manually and use a CALCULATOR. I was just reading a comment elsewhere from a university guy who had to run classes in basic calculator use before starting on the subject matter of his course.

I definitely agree with you here. I think your comment only brings to light the unnecessary amount time we spend having students memorize procedures that calculators can perform. Estimation is the critical piece of partial quotients that allows us to assess a student’s ability to reason and think about numbers.

Eventually, the quantities and calculations that students encounter will become too cumbersome to perform manually. Enter the calculator. But we just need to make sure they understand what those crazy symbols are actually doing…especially in elementary school.

Thanks for pointing this really important piece out Howard.

It;s the same later on, with calculus and techniques for integration. So sad.

Thank you! I just got this in an email because I finally subscribed to a blog. I’ve been trying to send little tidbits to every grade level from this stuff. Thanks again

Thanks for sharing Chelsea and glad it’s helping out.

I agree that estimation is really the skill we should be aiming for. Without this progression students have no hope of estimating.

Well done, Graham! Like the multiplication progression you shared last month, this division progression is a great tool for teachers, schools, and districts to use to build a common understandings of these concepts and build their PL from there.

Thank you so much Graham! You rock! I am sharing this with teachers and will be tweeting it out later today. I really appreciate all your efforts in moving elementary math forward in Georgia.

Holy Amazing Batman! Words can’t describe how awesome this is.

Thank you!

Awesome stuff, Graham! I’m so tired of having to listen to the importance of long division, when its value is pretty much nothing without the understanding and connections that you illustrate here. “They need to know long division to divide polynomials” – That’s why? We inflict all that pain for a lesson or two in senior mathematics, and a few more in calculus? “They need to know long division to divide polynomials” – Umm, not really. What’s really cool about your progression is that partitioning thinking works for polynomials as well, and has more conceptual meaning!

Cheers Mark and thanks for making the connection beyond 6th grade. The connections that high school teachers are making to this progression on Twitter are really powerful as well.

I think it would be powerful for 3-6 grade teachers to see how these strategies connect to the higher level mathematics you are speaking of with polynomials, just as it is important for the secondary teachers to see what we are doing down in elementary. Thanks for making both of these progression videos!! I was so excited to hear that this one was online! I will be sharing it with the teachers in our district!

Woo hoo! I’m still working on responding to our conversation about decimals in your Progression of Multiplication video but you address decimals beautifully here! And I love that you called out the traditional algorithm. But #Gazinta takes the cake! I will be sharing this with teachers, colleagues, and parents. Bravo!

Appreciate it and thanks for sharing the love.

Very well done, Graham. One suggestion: consider slowing down in some of the “curvier,” more difficult to grasp sections or breaking this into more digestible parts if you don’t want any single bite to be too large. I’m thinking about some potential viewers who haven’t grappled with some of these ideas before and may have trouble following some twists and turns (should I stay with the winding road or the chewing/eating metaphor)? 🙂

I chewed on this idea quite a bit when creating the video. I’ve really grappled with the idea of what length of time makes this video as a whole easier to digest. My hope is that as teachers watch this they collaborate to break down the meatier parts.

I really appreciate this feedback Michael and I’ll be sure to keep it in the forefront of my thoughts as I move forward with these.

Once again this is great example of having conceptual understanding how to progress students through it. This will help elementary teachers improve their content knowledge which will allow them to slow down progress students through with a better understanding which will allow them to develop into students who can use estimation and then a calculator efficiently.

All great points here Mark! Thanks for sharing and pointing the out my friend.

No thank you Graham for another great video to pass on to my teachers!

Graham, I love this. I am a math trainer for teachers in grades K-8 and when a grade is isolated the beauty of the progression is often missed, you capture it expertly. Waiting for more!

I appreciate it LaCosta and thanks for the kind words. Stay tuned…more to come.

Graham, great job on this video. This can help a lot of teachers and students! Thanks for making and sharing these. Keep them coming! =)

Thinking these need to be linked in the frameworks…okay?

Um….let me think about this. You mean share with more teachers and parents. YES YES YES!

I love this! I teach an elementary math methods course and this will be great for our discussion on division! I will be checking out your other posts too!

Sweet and so happy that you’re able to make use of it. Love to get any feedback they might have to share.

finite fields

Thank you for this progression and it is spot on when building students conceptual understanding.

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I absolutely loved this video! Will go back and look at the multiplication video also. Wish you would have addressed “remainders” or partial groups. I think students often get hung up on that. Perhaps that is another topic and video connected to in context division problems!

Graham, this was an excellent video, and I so appreciate your time in creating it and sending it out. My grade 3 and 4 teachers get hung up on needing to get to the traditional algorithm way too quickly, and as you so eloquently stated in this video, the traditional algorithm is a grade 6 standard! We have just got to slow down and teach for conceptual understanding! I have shared your video with my teams, but I also plan on getting it out to parents. I would love to share it out with families at a parent night, and then follow it up with concrete work with parents at the desks/tables! They need to “feel the division love!”:)

Thanks Lisa and please share away Lisa. The more we all have a conceptual common understand for these concepts and others, the more kids win.

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Been away from your blog for a while, but I’m back and overjoyed to discover your progression videos! I will definitely be sharing with my school and parent community! Thanks so much fighting the good fight.

Your videos are great for not only explaining the progressions of understanding for all operations, they are invaluable to parents and teachers working with students who have learning needs. Often in our fifth grade classroom we have a number of students using all sorts of methods to ensure accuracy in their answers. There is no rush to push them toward using the algorithm because it makes no sense to them. It is just a series of steps, steps that can get skipped or completed in the wrong order without realization that it’s WAY off. By allowing them to do what works for them, we are able to move forward in solving problems in context and focus on understanding them, not just the steps to solve an equation. Thank you for validating that we’re on the right track!

Right on Sandy and glad I could help out. All of us are smarter than one of us.

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Fantástico trabajo. Nosotros también lo trabajamos de manera muy parecida. https://youtu.be/1iIHJIS_kog

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Estimation is the key with both partial quotient and area model. Yesterday we did the 3 Act Task, Array Bow of Colors, and I was shocked at how many of my students do not use a strategy to estimate. This is something that I will continue to build, which hopefully will help with partial quotient and area model. Thank you for sharing your progression videos, they are very helpful for me to see where the students are coming from and what the end game is!