Make Math Moments Academy › Forums › MiniCourse Reflections › The Concept Holding Your Students Back › Lesson 55: Extending Multiplicative Thinking – Discussion

Lesson 55: Extending Multiplicative Thinking – Discussion
Posted by Jon on December 6, 2019 at 6:25 amWhat big takeaway did you receive from this lesson?
Describe how it will impact your teaching moving forward and share any wonders you still have.
Luke Albrecht replied 11 months ago 19 Members · 32 Replies 
32 Replies

Your early message about creating anticipation by withholding information, and using simple everyday contexts is really starting to hit home. Something as simple as drawing trees or using images of wands can be highly engaging and valuable.
Multiplicative thinking is one of main problems that our students come across. Visual aids are especially important to ensure students check for reasonableness and show their thinking. I want to embed more of these tasks into my teaching. It helps students to articulate their thinking in creating a mathematical model and evaluate/verify.

There was a good reminder to always follow your curriculum guide closely. Not everything is in the textbook.

My big takeaway is that I need to back up and give some of my students a little more practice with this. Right now it’s hard to use manipulatives since we are learning in a virtual format, so I’ll need to think this through a little more.

For sure. Plus keep coming back to it routinely. We don’t want that thinking to only happen in a 2 week unit or similar.



My biggest takeaway from this lesson is to incorporate/enhance the opportunities to think multiplicative for my students. I realize that I often think of only the one side of the multiplicative statements (usually the easiest) but there are actually two ways to state it and both should be encouraged and shown.

I agree that the textbook should never be the curriculum guide. Even in my current international school here in Amman, Jordan, teachers have the tendency to follow the lesson progression of Engage NY. It is really frustrating.
To meet the standards highlighted in this video, exposing tasks to students where they can visually see multiplication as iterating a unit is essential. Not allowing the students to see and play with manipulatives is a common problem in math education worldwide. This is why gaps occur so often because we miraculously expect students to have conceptual understanding when they reach a specific grade level.
I have been downloading a lot of illustrative mathematics tasks (specifically linked to the Operations and Algebraic Thinking standards) and turning them into google slide presentations. My go to resource has been the Howard County, Maryland, Public Schools website. Under the tab for each standard (Preferred Resources) you will find similar tasks created by HCPS. https://hcpss.instructure.com/courses/107/pages/4dotoadot1aboutthemathlearningtargetsandrigor
When the teachers at my school have used these tasks we have noticed more engagement amongst the students. Great discussions too.

Great points! How do you like Engage NY? I believe that program was later turned into Eureka Math (I think)?
As for HCPS, I think that is John SanGiovanni’s school district. I’ll have to check that out!

I am not a fan of Engage NY. When I arrived at my school (private international school in Amman, Jordan) a year and a half ago, every grade level followed the lesson progression of Engage NY with no diversions! I think a few years back they were told to do this so everyone was on the same page. I get this. It is important for teachers and students to be talking the same language. Unfortunately, because it is extremely prescribed and has the “I do this” “You do this” approach, the students are not accustomed to divergent thinking, curiosity and game play. I came from a school that had an inquiry based approach where students were used to asking questions, making educated guesses and claims, and challenged to find different (and more efficient) ways to solving problems.
Our school collaborates with Megan Holmstrom and she introduced the idea of creating Unit Concept Planners to help teachers highlight the big ideas of each unit. The problem was that no classroom teacher had time to really develop these until I feel into my current position. So I am designing the concept planner to show the teachers how the Engage NY lessons fit with the Big Ideas and then I input other types of lessons (like 3 Act Math Tasks and more inquiry based lessons) that hit the same concept but are also reinforced with other rich tasks and games. I basically take the time to comb through the resources on Erma Anderson’s Live Binder for rich tasks that correlate with the standards covered in each unit and then align them to the Engage NY lessons on the concept planner. She references the tasks on the Howard County, MD website and lessons from the units from the Georgia Standards of Excellence Curriculum Frameworks. So I basically hyperlink those and prep them to be as user friendly as possible because if they are not user friendly, the teachers are less likely to use them. Basically, the concept planner allows teachers options on how to cover the concepts in the way they are most comfortable with. The responses from the teachers have been very positive especially from the Grade 1 and 2 teams who do not like Engage NY at all. They love the games and the rich tasks and more importantly, so do the students.
When I first arrived at my school a few teaching assistants shared with me how boring math at our school is and how kids do not like it. I like to think (well, I know for a fact) that we are seeing a change in mindset at our school now when it comes to math. Distance Learning has thrown somewhat of a roadblock in this movement, but at least in the K3 grades I see a lot of the rich tasks being assigned on Seesaw for the kids.
I am curious how others are moving away from a more prescribed, “I do, you do” culture to a more inquirybased, low floor high ceiling, game play approach.

That makes a lot of sense. Teaching via the gradual release of responsibility model really does suck the curiosity out and thinking!
Love the idea of creating concept unit plans and being more flexible with the resources used.
We tend to suggest starting with a curriculum and modifying from there to make it more curious and rich. Sounds like you are all on that path at your school. How is it progressing?

I would say slowly. There are some teachers more than others who are more inclined to nurture their student’s curiosity and or are familiar with an inquiry approach but Covid and teaching via distance learning has really thrown a wrench into our progress.
To help ignite the culture shift, I just began a school wide weekly enrichment task, using Padlet as the platform. I use the enrich website to get many of the problems/tasks. I push it out on each teacher’s Daily Learning Plan which are on Google Slides and they get pushed out to either Google Classroom or Seesaw. I offer a weekly zoom luncheon with me so students can share their thinking in a collaborative setting. I extended our first task for another week and made instructional videos on how to navigate and familiarize themselves with Padlet but I am currently preparing next weeks conundrum, which will drop on Sunday (our school week is SundayThursday here in Amman).
I would love any feedback on this. Here is the Padlet. The password is acsmath.



First, asking us to describe the lengths of the wands using the other one was great. It is a great way to help students understand reciprocals and will easily lead to discussion of how they equal one when multiplied together.
The biggest take away was that I need to look at multiplicative tasks in reverse order and demonstrate that thinking in a visual manner as well (similar to the wand problem above). Many times I just wan to move forward and explain the one approach but never go into to detail of we got there and how working in reverse can help us better understand the concept.

My big takeaway was the need to build flexibility with multiplicative thinking. We need to do a better job of relating multiplication and division and including both statements that can be made when we, for example, compare the length of Wand A to Wand B.
I am also seeing the connections to measurement standards – some of which I fear are being dropped out during this COVID19 pandemic and my district’s (NYC) focus on “priority standards.” Measurement standards are probably going to be given less focus, if any time this year and that is problemmatic! Length models — relational rods, number lines, connecting cubes are coming up so frequently and I really appreciate their role in building this understanding of multiplicative thinking. So much good stuff here!!
Thanks!

Fantastic to hear! Glad this work is resonating. It is a really tough year with tight timelines for educators and a seemingly impossible list of standards to work through. Hopefully, through the work in this course, we can start to see the big picture of multiplicative thinking across standards to continuously work on them year round.


A textbook is not a curriculum, a textbook is not a curriculum…I have restated this phrase over and over to anyone who will listen. Spoiler alert, almost no one listens! A textbook is a tool, a resource, but it has too much power. I understand why, especially at the elementary level where teachers have to be experts in all subjects. However, it never surprises me when textbooks leave out standards. Greg Tang is always saying that the people who write the textbooks are not the ones teaching. What do you think?

Great points and wow, what a struggle to overcome. So many factors here…
Interestingly enough, here in Ontario, textbooks only have to have 80% “coverage” to be on our approved textbook resource list called Trillium. This allows the publisher to strategically try to hit as many birds (curriculum standards in different jurisdictions) with one stone (textbook) as they can. I’m sure that is similar across various regions.
Often times textbook writers have taught in the past … but even if they have, is it the way you’d want to be teaching? Who decides which teachers are writing the textbook?
All great questions here for us to continue trying to work through!


In 8th grade we cover the concept of similar figures and similar solids which include this multiplicative thinking. Students who have an innate sense of this relative comparison see the answer very easily or at least can estimate and give a reasonable answer. Students who have not yet developed a sense of relative comparison have difficulty knowing if their answers are correct or when they have given an answer that is completely unreasonable.

I can really see how using comparison in both directions early on will make it much easier to teach division of fractions. They will have an innate sense of reciprocals and why they work.

Glad you’re seeing this as helpful! I think so too!


I had not realized how big a step it is from using blocks to using a number line. This has been a real challenge for my remedial middle school students. I need to slow down and remain concrete for longer.

Love it. I agree – it could easily go under the radar until you explicitly think about it and try with students. Nice work.


One of my big takeaways is that, as a middle school teacher (7/8), it’s okay to go back to these simpler multiplicative thinking tasks (for example, the Canadian Flag task) to help my students think a little deeper about multiplicative thinking, especially when working with fractional relationships. I think that this kind of relationship, especially emphasizing the reciprocal nature of multiplicative thinking, will help them when we start our units on proportionality.

Agreed! We have so much work to do with multiplicative thinking that it is never bad to roll back and build that flexibility.


My big takeaway is to ask a broad question for students’ reveal of understanding. Progress through the thinking types with the students. So either use the simple or more complex comparison of wands to start out with. Compare nonmeasured: one is bigger and one is smaller–push for the inverse or reciprocal thinking at all levels. Then iterate a unit across both to get a measured amount. Compare additively ( more and less). Push to compare multiplicatively (times and fractional as).This is a new way for me to teach so be patient with myself as I am still learning, but what an experience for my students to travel along with me.

My big takeaway was don’t trust the text book to give depth to your teaching and student learning! We use the same texts as you do in Ontario here in Alberta and you’re right! I’ve never seen this type of thinking in the activities and questions proposed there. This year I’ve ditched the text book much more, in favour of more thoughtful problemsolving and taskoriented teaching and I find my students much more inquisitive and able to make links between concepts (as compared to other years)…not to mention more engaged! Thanks for pointing this out. It has confirmed the change I’ve made this year yet again!

Two things: first of all, my certification is for grades 512, so I wasn’t trained in how lower grades are taught, but as we all know, many students are working below grade level so it’s been imperative for me to learn the difference between counting, additive thinking and multiplicative thinking. This course has solidified that for me.
Also, I just think the overall message of not rushing from models to the algorithm is important. Although my current unit doesn’t center around proportions right now, I am making a point to use more hands on and models.

Love it. Also, there are so many instances where proportional reasoning is leveraged throughout the math curriculum that you’re probably still utilizing this thinking during non proportion units also! Glad you found it helpful!


My big takeaway from this lesson is to spend more time on multiplicative thinking. And to specifically include both sides of the relationship. (from A to B and from B to A). I can see how this really relates to the fact that there are two constants of proportionality for every relationship.

Love it! Once you’re in the habit of thinking of the relationship both ways, it is easy (and obvious) to students. I constantly push myself to reference the smaller to larger comparison specifically because it is really obvious once you think about it and I hope some students want to learn more about the “why” to get their wheels turning.


I jump immediately to thinking 80% as big or 1.25 time larger. I like slowing down and saying 4 onefifth pieces as big. Great stuff.