Aaron Davis
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I have had many takeaways from this course. Being intentional with the language I use with my students and the language I want them to use when given specific tasks is one big takeaway. Also, and this was a reaffirmation of a big idea I took away from a Robert Kaplinsky course I took a few years ago, problem solving, low floor high ceiling tasks, needs to be a major focus in my mathematics classrooms. The discussions and ideas that evolve from these experiences are so rich and they just foster the development of number sense. Christina Tondevold says that number sense is CAUGHT, not Taught (or something like that) so the more rich experiences students have with various math tasks, the better chance they will have in catching the big concepts and ideas we want them to learn.
Finally, and this was also a big takeaway I learned from Graham Fletcher, but the more we as educators can learn the better we can teach. Learning the progressions of proportional reasoning, counting, fractions, etc, will just help us be more effective teachers. Our learning should NEVER stop after we earn our teaching certification or masters degrees or whatever.

For me it was Module 5 where we learned about the progression from additive thinking to multiplicative thinking by using the wands. Focusing on the language we use to explain either types of thinking really helped clarify the progression for me as well as using appropriate manipulatives too.

My own understanding has definitely grown, especially with understanding all the definitions and the difference between things like rate, rational reasoning and scaling in tandem. Also, since I don’t spend a lot of time teaching grade 5 and beyond, building my understanding of where kids are headed has helped me better understand how I can solidify their early understanding of proportional reasoning. I still need to break it down some more, especially while looking at standards and appropriate tasks in order to develop and build their understanding at appropriate rates of understanding.

The use of the double number line again really helped me conceptually understand ratio that takes place in this task. Brilliant. I also like the part of the task where the students would really have to play with the manipulatives (in this case the linking cubes) in order to disperse a fair share of the cups of water per mug of coffee. Good way to nurture the struggle (and hopefully perseverance).

I don’t think I naturally think of ratios as multiplicative comparison, at least not for this problem. Maybe it is because I teach elementary grades, sometimes working with grade 5 but mainly with the younger learners.
So I naturally went the composed unit route, seeing the 2 ml of water per ml of rice or 1/2 ml of rice per ml of water. So I was scaling in tandem looking at the rice and water as one composed unit. Looking at it as multiplicative comparison though, just looking at the quantity of rice seemed very different and foreign to me, which I think is what you said, that thinking this way in the context of this problem is a huge stretch.

My big takeaway also comes from the previous module, but the idea of jotting down the estimates on a number line, leading the kids to see the range of answers. I also like how this leads to the creation of a double number line too. Just a wonderful way to make all this more visual for the kids!

Part to Part Ratio
7 Canadian based NHL teams to 24 US based NHL teams. 24:7 or 7:24
Part to Whole Ratio
7 Canadian teams to 31 total NHL teams. 7 thirty oneths of the group are Canadian based teams.

Connecting this idea of Rate with Partitive Division really helps me understand this concept. Using the language of “How many units of one group are in another?” makes sense to me now, especially using the wand idea. Dividing 8cm Wand B into 10 cm of Wand A clearly shows that there is not enough of Wand A to fit into Wand B (how many 10cm units can fit into 8cm units) so the ratio is less than one, or 8/10 cm of Wand B per cm of wand A.
Language again, is key, and so is the use of Partitive Division.

Multiplicative Comparison Ratio
Julia is 5 years old and Katherine is 7 years old. So Julia is 5/7th the age of Katherine and Katherine is 7/5ths the age of Julia. I am having trouble totally grasping this because I am comparing it to the Truck problem and in the truck problem they broke the lengths down into 1/2 parts. Can I do the same with this?
Composed Unit
I guess would be when I am baking. So a batch of 35 puffed pancakes (ebelskivers) calls for 2 cups of flour. How many ebelskievers can I make with 1 cup or a half cup of flour?

Multiplicative comparison looks at the relationship between a shared attribute (length) that two similar items have (truck) and then compares them.. A ratio as a composed unit, the price of an ice cream cone, is iterated and increases as you buy more ice cream cones.
Something like this.

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.

I have used Graham Fletcher’s Seesaw activity: https://gfletchy.com/seesaw/ before in Grade 3. This could be extended to promote multiplicative thinking.
After determining the weight of the girl (60 lbs) is equal to the weight of 12 bricks, I could share with them my weight (rounding up) to 180 lbs, three times the weight of the girl. Then they will have to determine how many bricks that would be.
Maybe it would be best to share how many bricks I weigh first and then they would have to determine my weight? Not sure.
I could then provide weight comparisons to other objects (animals or vehicles) and then the task can be more high ceiling.
The crux of the task though could be to provide them with sentence frames so they can choose 2 objects to compare by making a multiplicative thinking statement.
Keeping it within multiples of 3 and 6 would be helpful then because they can then practice connecting their statements using thirds and sixths (i.e. “The girl is 1 third the weight of Mr. Davis”).

With relative thinking there is a relationship between two objects or sets and there is an exponential growth (or decrease) in a particular attribute they both share.
With additive thinking, there is a difference in a number of items or units when comparing two groups. The difference involves subtraction.

When I see students rush to the algorithm it really pains me. It shows they may have been rushed towards learning the “tricks” which often times are so inefficient. I find when students are taught to verbalize their thinking, they end up finding a more efficient way to solve a problem. I like modeling their thinking with Empty Number Lines because it provides them the linear view they rarely get which is why students will immediately stack 1001 and 999.
BTW, I was super relieved just now. My 7 year old just came in while I was completing this lesson and I asked her how many more dollars Samir had. She said 2 right away and when I asked her why she said that 1000 is one more than 999 and then 1001 is just one more than that. Phew! I will have to tell her her G2 teacher this (my colleague) but I do know they have been providing the kids with a lot of linear models in grades 1 and 2.

In my interventions I have been working a lot with dot cards with my students to build up their mental bank of numbers. I would flash a card like the one attached: They would say the total and then 2 and 7 make 9. An additional prompt sometimes would be, “How many more to make 10?”
Other times I would show them some dot pattern cards and ask, “How are they alike? How are they different?” or “Which one Doesn’t Belong?”
Students may say A does not belong because it does not have a group of 5. If they are having trouble seeing that A and B have the same quantity, nudging the kids to make an a measured comparison, by lining them up with counters, may help them. Then I can ask them, how many less does A and B have than C. With these images, they tend to not need to line them up though, especially when comparing B and C because they are “measuring” using the dot shapes.
 This reply was modified 2 years ago by Aaron Davis.

I love seeing the types of comparisons progression along with the measurement continuum. I am trying to help teachers locate tasks that fit in with their current units and you all provide such rich tasks that my colleagues just do not do because they are not user friendly enough or they just don’t make the time to see where they fit in their units. This is where I come in. It doesn’t take a lot for me to get “buy in” from them but teaching online now adds another barrier for them to want to use tasks like yours.
What I can do, and have done in the past, even online, is use rich tasks like yours in an enrichment group but that goes against my belief that all kids should have access to extension tasks. What I may have to do is create “can do tasks” and schedule zoom times when kids can reach out to me for collaboration. Maybe I can identify a task for K2 kids and another task for Grades 35.
Thoughts?

Anytime we can create a visual to explain what is happening, especially with a fraction problem, it helps build conceptual understanding. I worked with a group of Grade 5 students last year and they really struggled with understanding the math they were doing with multiplying and dividing fractions. The idea of drawing an example of what they were doing was so new to them that it scared them. They were only taught the shortcuts of the algorithm tricks but obviously did not fully understand what they were doing. It was tough to continue working with them too because we were doing distance learning too. Missed opportunity.😦

The idea of defining the unit in order to measure the quantity really resonates with me. This concept came to life for me when I took Graham Fletcher’s Foundations of Fraction course. When we can encourage students to measure a quantity of something but in different ways( changing the size of the unit) when can move them to think more flexibly.
I never realized before that I did this back when I taught fractions in Grade 4 (910 year olds) using pattern blocks. When we defined one as being the yellow hexagon we limited ourselves to just working with halves, thirds and sixths but when we defined our unit as 2 hexagons, then we were able to explore fourths and eighths.
It’s the same thing with using base10 blocks. If the cube is always “one” then we can only use that resource in just one way. But if the hundreds flat is defined as “one” then we can use those materials to explore decimals.
 This reply was modified 2 years, 1 month ago by Aaron Davis.

Street hockey balls. The guys I play hockey with are very particular about the balls we use to play with. We could measure their:
– height of bounce
– squishiness (not sure what unit of measure I would use for this. Something that measures pressure?)
– color
Not sure what else.

I never gave much thought into the idea of “attributes” when it came to measuring items. Of course, I taught my students about different attributes when it came to comparing shapes but the idea of defining attributes to compare set, linear, area and volume models wasn’t so important. I guess I haphazardly guided them to focus on them but this course is making me realize how important it is to focus on things like attributes in everything we look at when it comes to math concepts or the world around us. I love how the focus on attributes can really bring out different perspectives on the way we view things (like the dot and cookies examples.)

When I think of proportional reasoning I usually just think of fractions, percents and ratios. The idea of scaling down and up makes complete sense because we do it all the time, especially when I try to help my little kids grasp the idea of really big numbers and measurements. The idea of building spatial reasoning by decomposing shapes and numbers makes complete sense to me now with how it relates to proportional reasoning. This is what we mean when we ask our kids to be flexible number thinkers (i.e. compose and decompose numbers) and then eventually unitize numbers or groups in different ways). I love this.

Proportional reasoning is all about the context and developmental readiness of the student (or perspective). These tasks are great low floor high ceiling tasks because most children can take part in the discussion and defend their answer depending on whether they are thinking relatively or in absolutes. Often times it is easy to forget about the Mathematical Practices we are trying to develop in our students because we are so focused on teaching them the standards (unfortunately regardless of whether they are developmentally ready for it) but by focusing on the MPs, like #3, “Construct viable arguments & critique the reasoning of others” students learn to be precise with their mathematical vocabulary and gain greater understanding of math concepts like proportional reasoning by hearing the different reasoning of their peers. Doing tasks like these two, anticipating the types of answers and misconceptions students will come up with, can provide us with important information about where the students stand on the learning continuum.

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.

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.

Christina,
I use dot cards a lot with the interventions I have with 1st3rd graders so the idea of using them in a different way (to promote additive or multiplicative thinking) is awesome. I often find myself trying to push the boundaries depending on the developmental readiness of the students and engage them even further with deeper prompts.
With that in mind, I can use a dot card with a group of 3 dots and compare it with a dot card that has three groups of the same group of 3 dots (so 9). With the right prompts, maybe responses could be, “B has 3 times the amount of dots than A” or “A is 1/3 the amount of B.”
Or is this a stretch?

Yes Jeanette, those sentence frames are wonderful. It clearly shows Vanessa’s explanation:
Additive uses “more than/less than”
Multiplicative thinking uses “times” and “of” because we are relating 2 objects to the size of one another.
When I set up enrichment tasks for the students at my school, I’ll be sure to include sentence frames like these. It is so important to develop their specific math vocabulary as early as possible.

I made a google Slides of the Cookie Monster tasks to encourage my Grade 1 teachers to do the task. They loved it and said the kids were fully engaged.
I plan to prep the egg one for my KG teachers. Even while we are teaching online, I can help them pull this off. I will totally offer to take the lead on it.

Me too Marianne. The closest I came to using a balance scale in my career has been when I talk to students how the Right Hand Side (RHS) is always equal to the Left Hand Side (LHS). This usually comes up when I write and equation with the “answer” on the left instead of the right. Our students are so conditioned that equations can only be written with the answer on the right side. This always leads to a great conversation about why it doesn’t matter.
Anyway, here is a digital app that was shared with me to use for balance scales: https://www.didax.com/apps/mathbalance/
I wish there was one that has items you can put on instead of just the numbers but, this this would be a good way to help students check if their thinking is correct (and balanced)!

I was thinking about this example too. Working with really young learners, especially kindergartners, they just like to fill things up. Providing them with a challenge of comparing the capacity of two objects can lead them through the measurement continuum. Using candy would be even more engaging for them.
With kindergartners, even with first graders, is it even necessary to move them to the indirect measurement stage yet? The whole idea of trust really resonated with me during this lesson and I just don’t think our really young learners have a lot of trust in us yet if we were to just give them measurement attributes. I think developmentally, they would just ignore the given attributes, and find out for themselves, using whatever external measurement objects we make available for them.

Lets try this again (tried responding a few days ago and it all disappeared).
As many have said above, direct comparison is where objects are compared side by side, visually compared (eyeballing it). An indirect comparison uses another object (like a string) to help make a measurement.
Opportunities to provide experiences for children to make such comparisons like cooking and baking are great for developing their spatial reasoning.

I really like how this task can relate to everyone. Oftentimes we present math problems to kids and they don’t see how it relates to “real life.” Thanks for sharing this one.

Nicole, when you mention how young students are adept to finding patterns, this really holds true for me as an educator who works with 58 year olds. They are good at seeing patterns and it is important for us to give them the language to describe what they see. Introducing them to the idea of “half” or “twice as more as” or “double” can help develop the building blocks of proportional reasoning or multiplicative thinking.