Make Math Moments Academy › Forums › Full Workshop Reflections › Module 3: Teaching Through Problem Solving to Build Grit and Perseverance › Lesson 33: GoTo Tools to Build Grit & Fuel Sense Making › Lesson 33: Discussion Prompt
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Lesson 33: Discussion Prompt
Peggy Allen updated 1 month, 3 weeks ago 47 Members · 60 Posts 
What was your big take away from this lesson?
How might you use the tool discussed?
What is something you are still wondering?

My big takeaway is “concrete manipulatives” are extremely important in developing sense making. I am going to take this lesson and use it with a year 8 (grade 7) group. I love the learning centre app but will also use physical tiles to add to the concrete feel of the questions. I plan to start small “dial down the math” and work up to more complex problems.
Something I am wondering is, it seems easy to see how I can apply concrete manipulatives to fractions but how could I apply it to other topics?

When my 7th graders start looking at the four operations with integers, we spend a lot of time with zero pairs using doublesided color counters. I also use a ton of manipulatives when we do statistics and probability.


I been using concrete manipulatives for a long time now and have always had links to virtual manipulatives for my students to use to help make sense of different math concepts. If I do any teaching, it has to do with demonstrating different ways the various manipulatives can be used. I regularly attend the webinars from BrainingCamp. The ideas presented are so practical for my students. My class have made BrainingCamp their go to place to start their thinking. It is here that I might provide some hints to push them through productive struggle.

My big take away from this lesson was the reminder that the concrete manipulative allow students to build their own cognitive models and with the rush to teach students algorithms they have a weak foundation and those very abstract concepts learned later in HS math only become further out of reach for many of those students.
I am curious about the norming process for using concrete manipulative at the HS level. There is such a rush to cover the needed curriculum demands that there is not the buy in at the teacher level and thus students miss this valuable opportunity to create their own lasting understandings. My hope with my alternative math pathway course I can demonstrate the power of manipulative in HS.

I feel like my take away here is that I finally feel like there is something I am doing right. I have always made a point to use concrete manipulatives in my classroom, and I have them at the ready for kids. “Can I use tiles” is an often heard question in my classroom, followed by “You know where they are….”. I’m using them even more digitally (we’ve been remote all year, and that seems to be the way forward for the rest of the 2021 school year), and I sure like them, but notice the difference between digital and concrete, especially for my 6th graders.

The biggest takeaway is your reference to the Coaching Habit by Micheal Stanier. It makes me think of the idea that I learned as a new teacher and not taking the pencil from the student. The student should be writing the work instead of myself. But when I am quick to give solutions, algorithms, or this is how I would do it, it is taking the pencil away from students’ critical thinking.
I always use concrete manipulatives with my higher students. It gives the students a challenge who can manipulate a problem abstractly a challenge to represent an idea concretely. I will be honest, the Marian smalls problem with the hexagon stumped me at first because I found a representation abstractly but was unaware of how to represent using a concrete model. Once I saw it, it was clear how this would help uncomplicate fractions. The use of a physical visual representation makes the math a lot clearer.
I wonder how I can show students how concrete manipulatives are a valuable tool that is not showing you are slow in math but a robust mathematical thinker by creating a physical representation.

Great reflections here.
One of the biggest shifts I experienced with using manipulatives was asking students to model their thinking using the manipulatives to help me understand their thinking. So if a student did the work symbolically, I’d ask them to help me understand with the manipulatives or a visual representation and visa versa. This is a way to force all students to think and make connections to different representations.

My big takeaway is the website. With a COVID world, I worry about how manipulatives can fit it. This digital app is great and cuts out some of the logisitcs of passing out, etc. I think in an ideal world, the tactile nature of touching and moving the pieces would be better.
Regarding how I could use it in 7th/8th grade, I see it as a nice warmup activity on a day when some of the work will involve multiplying fractions: 1/6 of 1/2 is a little green triangle (or 1/12).

Awesome stuff! Plus we have great fractions operations tasks in the problem based lessons area: learn.makemathmoments.com/tasks


I have tried using manipulatives (digitally) throughout the year and the students HATE them. They beg for what I call naked math, just the numbers. I draw things out all the time, but they don’t want to spend the time to solve it that way. Many of them say that manipulatives make it more complicated and because they have not learned it yet, they would rather have the numbers. What do you think of this thinking? I teach gifted with IQs of 140 and above. I have continued to show models as they are heavily tested on our state test.

It is very common especially if they’ve typically approached math with “naked problems” using algorithms. They know that formulas will be quick and they can be “done” faster. But that isn’t what we want mathematics to be. We want them to share their thinking and convince others as to why things work.
If a student can divide fractions but can’t convince you why “multiplying and flipping the second” actually works, then what’s the point?
This culture takes time to build with your students – but you need to know the why yourself in order to stand a chance in convincing them.


I have used concrete manipulative in my fractions lessons but in a more evident ways. Thanks to remember me that there are more creative ways to do things and that makes learning more interesting.
What I would like to know is how you program a whole lesson with concrete to make easy to understand notable identities I’ve tried. We have cut the square and the rectangles we have paste it in the workbook with notes and observations… But they have trouble remembering it later. They say for them is more easy to understand (and I think they are talking about remember) That we can split the potency in a multiplication and the use the distributive propriety.
Thanks for the book!!

Great question! Keep in mind that the concrete is to allow students to manipulate and hopefully recall experiences, however we don’t have to limit the learning to concrete. Nudging towards visual drawings such as arrays and area models can also be helpful. As for helping them to remember, that just means they haven’t used it enough to hang on to it. When you think about any skill or concept, if we don’t work with it enough, we won’t remember it. Our brains are so efficient it just lets “useless” information go. So how do we make their brains want to hang on to it? A lot of revisiting through problem based lessons, math talks, etc.
Check out our units to see what we mean: learn.makemathmoments.com/tasks


I like how in this example of a tricky problem the manipulatives give an entry point to work to make sense of it. It may take longer than using an algorithm, but here the answer, once found, is solid and sensible. I would like to create a culture of using manipulatives to solve and to represent thinking so that there will not be a stigma for using them.

It was interesting reading about peoples struggles using manipulatives while teaching during the pandemic. I couldn’t agree more, this past year has been a struggle for many students, families, and districts. I am back working in person, and plan on trying this tile activity with fractions right away. You always get student buy in when they have manipulatives, so that in itself is a plus. Also, the manipulatives lend themselves to the Hero model, withholding, notice/wonder, estimate/predict, struggle, and consolidate. I am also looking forward to checking out the app.

I had never used these tools before. I plan to invest in some and use them in the classroom. I coteach and many of our students have an IEP or come to us multiple levels behind their peers and concrete manipulatives are such a great resource!

manipulatives can help kids see the math, but they may need simple tasks to feel successful at first, then build to more difficult concepts, so they don’t shut down on you!

I like the idea of using concrete manipulatives so struggling students can how things relate. In the past I have also had a hard time getting “advanced” students to see the value. I noticed that the fractions in your questions were not obvious multiples of the designated block named at the beginning.

Exactly!
However keep in mind we are actually asking all students to represent concretely to ensure they see / understand the concept at that level. Often times, students think they don’t need to know how to show/represent concretely… however when the ideas become more complex, the concrete can really help keep our feet on the ground.


As a third grade teacher, I begin every new concept with manipulatives. Students need to be able to play with the concept physically before they can begin to understand it on paper. This has been a challenge this year with the virtual teaching. I tried the online manipulatives but they were not as concrete as the real thing. I allow my students to use manipulatives all year long. They can choose which manipulatives they need to solve a problem and access them at any time. I even allow students to use manipulatives on tests if they need to. I find that when given the option to use manipulatives student who need them will use them and those who don’t need them anymore will do the work on paper.

My takeaway is what I will call “The return to basics” This concept challenges the students to make their thinking visible, as it was in grade 1.
Given a question, Style poured 3/4 of the juice from a 2liter bottle while serving guests at a party.
How much juice, in liters, is still left in the bottle? The response I got from a student is shown below.

I like that this reinforces students’ reasoning about different math concepts, and see myself being able to use it as a strategy to avoid the abstract working with/solving different problems that students tend to fall back on.
A difficulty I see is that as a high school math teacher, my school doesn’t value manipulatives as strongly, and so investment in them is low. In the past, I have taken to making them, but the production time can be high. I have also used drawing representations instead, though I see why this isn’t as good.
I have found that sometimes getting the buy in from the advanced students can be a challenge though I’ve found this year that anything not on their computer screen is a plus.

This is not uncommon. Pushing for some more concrete and even visual representations can be very useful. In secondary, there aren’t too many manipulatives that you’ll actually need and much could be leveraged at the visual stage. Stick with it!


My take away on this lesson is that it helps to have something that students can handle, or at least more around, in order to help them visualize questions. I have pattern blocks at home and at school, but I have bookmarked the virtual manipulative with the pattern shapes so that students can work with them no matter where they are.

Well, I loved the examples shown using these pattern blocks. I taught 2nd Grade last year We were remote most of the year and we used the pattern blocks (and other tools) from the Math Learning Center a lot. But your examples and ideas have certainly given me a more focused way of using them. I’m looking forward to using some of this next year as I’m back in my First Grade classroom again! Thank you!!

When Kyle said concrete manipulatives can just as much help us solve problems as they represent our thinking gave me pause because I never thought of manipulatives as a way to communicate our thinking. Concrete manipulatives serve a dual purpose, as means to pull our thoughts together and to communicate those thoughts to other people as if they were their own language. I hope this isn’t too heady, but my takeaway changes my perception of manipulatives as toys for younger grades to a form of language I develop in my classes to create this deeper understanding of mathematics. I used manipulatives to teach solving equations. I wonder how you all teach solving equations when you deal with fractions. Are you eliminating the fractions by multiplying both sides by a common denominator? Are you only using Algebra tiles to bridge the concept of balance until you can make the transition to more abstract thinking? I teach 8th grade and Algebra I.

Great reflections here.
Check this video out regarding solving equations with fractions:


What was your big take away from this lesson?
– allow students to explore the manipulatives, not spoonfeed students how to exactly use the tools.
How might you use the tool discussed?
in all areas of math
What is something you are still wondering?
– how to ensure it is a tool, not a toy

My biggest take away from this lesson is not to push students away from manipulatives if they are not developmentally ready to move to the more abstract. I have always been told that we don’t want students to use them as a crutch because they will become dependent on them and not advance their thinking.
Many of my 8th graders still struggle with combining like terms and evaluating expressions for given values. I can see using the pattern blocks to advance their understanding of both concepts. My students also struggle with addition and subtraction of integers. I have a number line in the front of the room that I often refer to and ask students to use to explain their thinking to the class, which helps some. I have some copies of number lines for students to draw on at desks if they need the additional support, but the paper copies are limited due to printing constraints. I can see how the digital number lines could really help students in a variety of ways.
My school uses the CPM curriculum which calls for the use of algebra tiles for solving equations. I think these are a great tool to help students understand the importance of opposites equaling zero pairs when solving equations, but they struggle to transfer what is happening with the tiles to writing out the work algebraically. It seems like there is a step I might be missing, relating to students not understanding it. Do you have any suggestions or resources to use to help develop this connection?

My big take away is that students need to use the concrete manipulatives to see that there could be many ways to view a problem, that one idea is similar to another and they don’t always have to approach a problem from the same angle all the time. The manipulatives give the visual to allow students to see this, which is key to making connections.
I am not sure how manipulatives will work for all topics but at least it is a good start for some topics.

I need to invest in or ask my school to purchase some manipulatives. I feel that the use of manipulatives falls off as students get into high school because we assume they can automatically think abstractly.

My favourites are connecting cubes and algebra tiles!


My biggest takeaway is that I need to force myself to use manipulatives more in my classroom. I have not been comfortable using them in class for various reasons and tend to find excuses not to use them. I can see how beneficial they are when watching videos of others using them. When I try to emulate them, things haven’t gone well in the past. Hoping that following the curiosity path and spending more time preparing teacher actions/questions will make using the manipulatives more productive for my students.

@andrea.cadman as a high school teacher I was in the same boat as you. Try to embed their use into your consolidation portion of your lesson. In preparation you could ask, “how can I model this with a __________”. That way students will see your constant use of the tool and start to use the tool as well.


Recently, in my MHF4U – Advanced Functions a student asked why do we reciprocate and multiply when dividing fractions? Of course, someone with a math degree, we forget how we learned this…I discovered the power of fraction strips/cuisenaire rods. I had to unlearn in order to understand. I will be getting the concrete manipulatives for my classroom. Loved learning about the app!

I feel a huge responsibility to make sure my 6th graders master fractions, because we jump to prealgebra for 7th grade at my school. I’ve worked with fraction circles in the past, but the examples you posed were far more intricate / complex, requiring the higher order thinking that I need for my students. I enjoyed playing with the online tools. We will be 1:1 next year, so I am excited!

Pattern Blocks are very very underrated in their use for conceptual thinking. Thanks for sharing this as tool #1.

I appreciate the reminder that concrete manipulative are needed at this level of math. My question is what teacher moves do you make when students scoff at them or say they’re not babies. I’m thinking of my middle schoolers.

My biggest take away from this lesson is the reminder that concrete manipulatives can be used at all levels of thinking and for building many different abstract skills. It’s easy to remember to use them with younger kiddos because that seems very natural. But there are many ways that students who already have abstract thinking can use them to build deeper understanding.
I could use the pattern blocks to teach the concept of breaking numbers apart mentally. For example, if the value of the yellow block is 80, how could you use the other blocks to break it apart. Two reds would have a value of 40 each. So one red would be 40, and two greens would be 20 each. etc.. (I don’t remember the relative sizes of the shapes as I’m typing this, so my example may be off) Breaking apart numbers is a very important skill for which these particular shapes may be used.
I’m still wondering what other kinds of manipulatives that I have used over the years could be used in different ways besides the usual and obvious.

I learn the same way I teach due to my learning disabilities. So, I used manipulatives with my 5th graders whenever possible. Giving children (of any age) something they can physically interact with simulates multiple senses and increases engagement and understanding. I’m happy that teachers are starting to embrace “elementary tools” in middle and high school…I even had college professors encourage us to use manipulatives as needed!
One of my favorite lessons on volume, students created their own 3D models using unifix cubes. Then deconstructed them to find the relationship between base area and height. They really enjoyed the creation part and wanted to see how complex of a model they could come up with by making adjustments to the area of base and height!

I love using manipulatives! I am amazed at most of the math teachers I know who give them up starting in grades 4 and 5…our high school teachers don’t use them hardly at all and they are such a good tool. For anyone who is a visual learner or likes to “touch” and “move” things and experiment they are essential. I am glad you made them tool #1, but stress they are for all ages and all levels of students.

My big take away is that manipulatives are beneficial regardless of the age of the student. I am planning to make table sets of manipulatives for each table group in my classroom to be available all year long at any time (not just when we are on that day). It really struck me that while some students may need to concrete tools for longer, it may be hard for them to ask for them! Keep them out an available always.
I wonder if I can make a classroom job to keep the tablesets organized.

I teach grade 3 and don’t know what I would do without the manipulatives. Even students who are strong at rote algorithms they were taught, before my class, have been able to show me the struggles within their thinking during math interviews where they need to show me the math. For those who have been struggling with math, the manipulatives have become a lifeline to understanding, something they pick up naturally when they face something they don’t get.

In my classroom, I use manipulative often at the beginning of the lesson then take them away. I use them model things like dividing numbers into groups when we start our division unit. After this lesson, it has made me think of ways to use the manipulative throughout the content. I also think that I need to have the manipulative available to them to use when they want to use them and not just when I get them out.

My big takeaway was the need for manipulatives no matter what the task. Using virtual manipulatives last year was tough. Our district provided every student with a fantastic grade level manipulative kit. Most students returned them at the end of the year completely untouched. I am looking forward to being back in person so that I can build a stronger foundation by utilizing a variety of manipulatives. 5th grade is so heavy with fractions and so many students come not really understanding what a fraction of something really is. These manipulatives are crucial to their understanding of grade level material and beyond.

Q:What was your big take away from this lesson?
A:Manipulatives are a must (especially for students who can’t visualize the problem) and to add something(substance) to the problem.Q:How might you use the tool discussed?
A:If you speak of the visual tool (patternshapes) – this would be a potential digital to help with shapes. When I paused the video, I did my own search since I didn’t have my own manipulatives and I needed to visualize the problem myself. I chose/found https://www.math10.com/en/geometry/geogebra/geogebra.htmlQ:What is something you are still wondering?
A:What if you can’t find the right manipulative, what do you do? Sometimes the manipulatives you have aren’t doing it for the understanding on the part of the student (but you, as teacher, think it should)
Even in the senior grades, are there solid manipulates for all of the concepts, some concepts being very abstract? 
I really like how pattern blocks were used. I can see a great fraction review activity here to check in with my students to see what they remember from grade 7 in a really great hands on learning activities focusing on fraction, building/practicing in a meaningful way!

Just the title of the recommended book here is thought provoking and inspiring. Just the “talk less and ask questions more” part gets me. I haven’t read it yet but as mentioned earlier it isn’t just asking questions but being intentional with the questions and follow up.
I remember one of my “good students” answering a question once and I didn’t immediately say he was correct, he modified it 5 times before I could even finish talking. I’ll be honest I didn’t even know how to proceed at first. He clearly wanted to “get the right answer.” But he just rapid fired many ideas. I think I had to address the quantity of answers given as much as anything else.
[EDIT] I guess I’m still wondering how to work to this point. I know I’m going to need to take the first week or two working towards developing this kind of classroom culture. Fortunately, the summer allows me more time to help make some of those kinds of changes.
 This reply was modified 2 months, 4 weeks ago by Denny Nelson. Reason: I didn't include my "I'm still wondering" statement

Definitely worth a read!
Your experience with your student is very common… young children do this often as I see it with my own kids quite a bit.
One thing I’d recommend trying to work on is holding off on any confirmation of right / wrong and focusing more on “why do you think so?” Or “convince me” or “how do you know?” Etc
If with every student response you are asking them to essentially prove their thinking, they will realize that right or wrong, you’re going to ask them to explain vs typically we only ask them to explain when they are wrong and often move on quickly to another prompt when they are right.

My big take away from this lesson is that using concrete manipulatives belongs in a middle school classroom. Almost all of my students struggle with fractions. In the past I have taught them algorithms for finding like denominators etc. Resetting and reteaching first with manipulatives will help them understand where the algorithms come from and hopefully once and for all help them master fractions and to not be afraid of fractions.

Giving the students time to discover with manipulatives is a big part of allowing their minds to shift through different ideas of how math works. This would be especially helpful when learning rates, ratios, and proportions. Students struggle so much with fractions and seem to shut down quite a bit when anything is in fraction form. playing with manipulatives and asking them to compare the different shapes first without using numbers, then discussing part to whole (three parallelograms fit in one hexagon) is a great way of looking at it. We could then draw a line between them to show that we are comparing them, and then finally replace the shapes with numbers to see the scary fraction appear and prove they aren’t as scare as they initially thought. I think it would be fun to introduce this by first showing a short video of how a “creepy” costume is made without know what is being made. Then to see the final product isn’t so scary after you see all of the synthetic materials that go into the creation. Same with fractions, they aren’t as scary once you see where it comes from.
What I am still wondering is how to learn more about manipulatives. When I was in school we couldn’t afford paper in my district, much less manipulatives, so I rarely had the opportunity to use them. Now, as an adult, I struggle to figure it out. I have been given manipulative kits to see if I would like them purchased in my classroom, but while trying to “play” with them I find myself getting confused and not able to make 6th grade material work.

I love that you addressed starting with the physical manipulative before moving to a virtual one. I often wondered if virtual manipulatives could be used in place of the physical but I personally felt that they couldn’t.

I am very excited to read The Coaching Habit, my husband has a copy and was reading it as a manager a few years ago. I am very excited to continue to find more ways to incorporate manipulatives into my Algebra Classroom, especially for my students with thorough IEPs who need more concrete and visual representations before moving to the abstract. I started using Algebra tiles for factoring two years ago and it has definitely helped. I’m still having trouble moving from visual to abstract when factoring, given a coefficient greater than 1 for a in ax^2+bx+c. Do you have more resources for how to use manipulatives in a high school Algebra course? Many of the resources I find are for elementary.

The activity link didn’t work for me. My big take away from this lesson is how important it is to have tangible manipulatives and/or visuals that can be interacted with as it makes problems a lot more accessible. I also liked how you can easily create variations in the problem which would give students a patterned approach and fuel MP.8 with repeated reasoning. I am wondering what resources there are for virtual manipulatives as I was not able to use the site in the video.

How important concrete manipulatives are. I have them in my classroom and encourage their use but I might try to have premade buckets/bags and ask that each student use them to answer a question!

My big takeaway – the quote by Michael Bungay Stanier. Students need more time to process and be allowed to think on their own without us quickly rushing in to move forward or give them the information. However, I believe we have so many learning outcomes to get through for the year that teachers often feel overwhelmed to get to all of them and do a good job.
I realize that concrete manips are an important tool to use in class for students to show their understanding in another way. For me, manips represent the foundation that leads learners to find other ways, (for example, visually or symbolically) of solving problems later on. It is important to practice often with them, so they become a strong part of the student’s thinking process.
I don’t recall using manipulatives much in class when I went through school, hence I don’t always find them easy to use (like pattern blocks).