Elementary math chat is a weekly math chat where participants come to discuss best practices, examine student work, explore routines for reasoning and research that guides and supports pedagogy centered on problem and student based learning.
Hi #ElemmathChat Friends! I am Melynee Currently I am a dissenter in Oklahoma. Normally I teach 6-8 grade. I miss my students! Thank you @MarkChubb3 for leading tonight!
A1 Visual representations bring the math alive. I know in my MS Algebra class, until I used @desmos to make the math visual, I DID NOT UNDERSTAND IT! #ElemMathChat
Hey, I am Wendy. I teach gifted middle school math. Certified 1-12 in 2 states (Texas & IL). Masters in gifted education and technology. I currently live in Mississippi- we just moved here. Lol #elemmathchat
A1: it gets to the core of the understanding - if Ss can represent their thinking visually, they understand the concept - e.g., can they represent addition of fractions visually or do they do procedural steps w/ no understanding? #elemmathchat
A1: Access (which leads to more equity), fun, beauty, perspective, the existence of a shared experience to reference (we're both experiencing looking at this together) #elemmathchat
#elemmathchat A1: Making math visual is an equity issue. There are many language & cultural conventions that surround our current US math. Making it visual helps to combat the imbalance of valuing a certain way of thinking and seeing mathematics more than others.
A1: Visual representations bring alive key understandings, conceptual understandings in math. Visuals open up the math. They elevate students' (and teachers' too) understandings. #elemmathchat
A1: I'm in the middle of a blog post about this topic - understanding mathematical concepts visually and what language NOT to use in the math classroom. #elemmathchat
Visual representations help develop, push, and deepen understanding of concepts. Also helps as a bridge from concrete to abstract thinking. #elemmathchat
Yes! Math can become very abstract the farther we progress. Abstract is hard to understand if there are no concrete experiences. I think true for all learners. #ElemMathChat
A1 visual representations allow students to make sense of the work they are doing. Also gives them a visual image to rely on as they move towards mental math. #elemmathchat
Visual representations help develop, push, and deepen understanding of concepts. Also helps as a bridge from concrete to abstract thinking. #elemmathchat
You guys do not know how awesome it is to talk about math, teaching and the awesome things we do everyday! I NEEDED some positive talk in my life right about now! Thank you! #ElemMathChat
A1: they can also provide a way to demonstrate mastery of a concept... if you can represent something several ways and/or interpret various representations of that concept, you must have a deep understanding. #elemmathchat
Definitely! I also think visual representations push the thinking of students who seem to "get it" but might just be good at memorizing and following directions. #elemmathchat
A1
The use of multiple representations in general is an important part of teachersā knowledge of mathematics and can play an important role in the explanation of math concepts.
#elemmathchat
A2 every time I do a demo lesson, Math PD, or model small group work!y goal is for Ts to see what incredible value and access they provide for students. #elemmathchat
Joel uses it ALL the time on the site, kind of what it's all about... but I use tons of visual representations in teaching my statistics classes concepts like the standard deviation... I am such a visual learner I feel like I can't explain things without it #elemmathchat
A2 - I have used visuals to help teachers determine best ways to teach concepts, also to show parents how visual models are helpful to our students and not just ānew mathā #elemmathchat
A2: use it all the time. I tell Ss "show me a visual representation. What does this mean?" - Ss do most of the representing and then we consolidate as a group. #elemmathchat
A2 I use them as often as possible. I use them, students use them. If they are not at a place where they can even begin to know how, I model. I ALWAYs have to put things in tables and draw pictures for myself anyway. Why not let them see me doing it? #ElemMathChat
A2: My personal feeling is that every new concept should be built on visual representations. Also, students should be able to use visual representations to explain/illustrate the math they are doing with bare numbers. #elemmathchat
A2 We use visuals every day - students represent and we share their own visuals, and during number talks I animate/draw what students describe, sometimes using new models https://t.co/q2Z9MNvx3W#elemmathchat
A2 I like asking my students to create representations of their thinking. That can give me great insight into their understandings and misconceptions. #elemmathchat
#elemmathchat A2: I start every meeting/class I facilitate with a @WODBMath or @mashupmath puzzle, etc. to get people (Ts or Ss) talking. So far, the images are already there, but I love noodling the idea of asking for my audience to share their own visualizations of things, too
A2: All of the time as an interventionist; my Ss live and breathe on the visuals as a bridge for abstract concepts. I try to have them do the representing as much as possible, but thatās a balance! #elemmathchat
#elemmathchat
A2. We use them daily. T'ers use visual representations to spark thinking, activate background knowledge, and make connections.
Students create visual representations for solving problems, to show their understanding and track their thinking
A2: visual representations are essential for all learners in the conversationāletās all participants talk about the math and their ideas #elemmathchat
A2 think about visual representations in my planning as I anticipate student thinking. Asking how do you see it? What would that look like? Has led to some great discussions and deeper connections #elemmathchat
A2: Visual representations are a big part of all my teaching and work with colleagues. Really doing alot of work unpacking concrete-representational-abstract this year. #elemmathchat
A2: Since all math can be made visual, we should be ready to make math visual always.
āWeā is the teacher AND students.
It is especially important for students to be making the visual connections.
#ElemMathChat
A2 Visually representing mathematics plays an important role and connects the concrete and abstract. I think of SMP2 when thinking about how students represent problems. #elemmathchat
There are so many visual representations we have access to. Take a look at a sample of some representations that are powerful tools for you and your kids:
#elemmathchat
This is so important! This is why we spend so much time talking in my class. So all can share ideas! My Ss now know not to ask which is right. They know choose what makes sense to you, is efficient and reliable! #ElemMathChat
A2: seeing is understanding, so from square one I use visuals. I integrate anything visual to halo make that connection whether it is a something they should have mastered already or if I am inteoducing a topic. Everything needs a visual. #visuallearners#elemmathchat
Yes. I see it as part of the CRA process. Also, itās a useful tool during the decontextualize <ā> contextualize process when approaching problems. #elemmathchat
Q2: I use visual representations as often as possible to help students understand the abstract. I think too often we abandon the visual representation sooner than we should. Whatās the harm in our students showing their work visually and abstractly side by side? #elemmathchat
A2: Visuals can illuminate ideas. (Of course, like metaphors, they can also sometimes color and limit an idea.)
I sometimes apply a visual rep to Ss oral explanation of their thinking. Sometimes I ask Ss to share their own reps. Ultimately I want to pass the torch. #elemmathchat
A2. I embed dynamic visual representations into online courses for Ts to explore and interact with resources their Ss could develop and use #elemmathchat
Agreed. I find so much more value w/bar models and other models as the proportional pieces become apparent to kids. The abstract doesnāt often show that type of depth. #elemmathchat
A3 So I can't even begin to answer until I know how many cm are in m. Then how many 25cmx25cm is and how many ways I can lay those dudes out in 1.25m Lots of drawing! #ElemMathChat
A number of people have asserted that all math is visual, and I find it fascinating. I mostly agree. I think visuals are a beautiful and eloquent way to express mathematical ideas, but I feel like the visual reps are an imperfect language sometimes? Not that often #elemmathchat
A3: Ss will need to have some surface knowledge in terms of how many cm in a m before they start. I can see some Ss sketching before they begin. #elemmathchat
A3: I think, yes. She will be able to get 16 25x25 bandanas out of the square meter of fabric. I drew a picture of the 100 cm x 100 cm piece of fabric and split it up. Didn't even worry about extra fabric #elemmathchat
A3 - Years ago I would have been disappointed if I couldnāt come up with a formula or equation to solve this. Now, I canāt solve it without drawing it and Iām proud of that! #elemmathchat
A3: Yes, because you can stack 4 across and 4 down to make a 1 m by 1 m square, which would be 16 bandannas. You actually will have enough space for 4 more, for a total of 20. #elemmathchat
I'm trying to think of a time when I used a visual that didn't quite fit, and I struggled with an alternative. Maybe with fraction div? E.g. With large mixed numbers, or finicky divisors. I LOVE when visuals amplify the idea... do they always? #elemmathchat
#elemmathchat A3: MS T so I drew 2 squares (1 small to rep bandanas & 1 lg to rep fabric). Converted 25x25 cm to .25x.25 m. Found each bandana is .0625. Multiplied by 15. Yes we have enough! Hooray bc I would hate to deal w/parents of kid(s) who didn't get one. Haha
I gave this problem to 30 grade 7 students.
3 of the 30 got a correct answer. All 3 drew a picture. The rest jumped to formulas and procedures. 27 decontextualized the problem too quickly!
Here are 3 of their answers:
#elemmathchat
Steph Diehl-babies donāt want to go to bed tonight-super late! A1: the visual representation provides a level of sophistication beyond the concrete. It shows movement from the physical world towards abstract. Plus it makes thinking visible which is always nice #elemmathchat
A2: visual representations as stimuli (teacher provided) and visual representations by students for sense making. Even if they donāt know where to start, they can usually draw some kind of representation to get them started #elemmathchat
A3: I would wonder, think and notice. 100cm in a meter. Use some King Henry Metrics and draw out some area visuals that COULD represent the 1.25 m^2 #elemmathchat we would have to convert everything to the same measurement cm or m.
Q4 - So, how would we want operations with fractions represented? What are some possible ways of thinking about what's actually happening here?
#elemmathchat
Yes, I like using these bar models for some early fraction division! We used to ask math teaching candidates a tricky question about visual reps involving fraction division at interviews. Everyone hated us. #elemmathchat
This gets me thinking about students who go straight for an algorithm. Yes, they can preform the procedure and even get the right answer but do they understand? Mathematicians should be flexible and able the represent and justify their answer. #elemmathchat
A2: I use visuals to develop conceptual understanding with Ss. I tell my Ss using visuals with smaller numbers is perfect but we want to move beyond them to algorithms when the numbers become large & visuals are inefficient and time-consuming. #elemmathchat
#elemmathchat A4: 2 ways I see this visually. Keeping the whole # def of multi as area model I would color code bxh & show where yellow & blue make green. Or 2/4 of 2/4 of a set. Shade 2/4 of 1 rect. take that pc and divide into 4 count 2. How much is this of the whole.
Q4 - So, how would we want operations with fractions represented? What are some possible ways of thinking about what's actually happening here?
#elemmathchat
A4: I love circle manipulatives, bar fractions, drawing fractions. Determining what your unit (one whole) is and making sure itās the same size to represent all fractions. I also use the fraction area models and have student model the multiplication in a drawing #elemmathchat
A3: I right away wanted to do some kind of conversion (reverting back to what I was taught in school) but after I thought about what this would look like I thought 25, 25, 25, 25 for rows and same for columns and got to 16 that way and 100x100cm is 1 sq m-right? #elemmathchat
A4 I could see students looking for ways to represent this on a number line and having issues as theyāve been told that multiplication increases the product. #elemmathchat#nixthetricks
Yes, I think the models have the power to deepen understanding and give new shades of meaning. I was talking about the balance models for algebraic equations with some Ts, and the model definitely worked better for some of their avenues of thinking. #elemmathchat
Hmm... So many rules kids hear that expire.
Seems to me that we all need to learn together vertically. We need to know what math looked like in earlier years, and where the math is going.
#elemmathchat
A4: We use fraction strips a lot Break bar into fourths and shade in 2/4. Now, how do we show 2/4 of 2/4 on the strip? How much do we have now? Ss need to grapple w/ many things including keeping equal parts. #elemmathchat
Yes!! Teachers need to learn the vertical alignment of what we're teaching. When we teach students rules that expire, we are only hindering their math understanding in future years. #elemmathchat
Hmm... So many rules kids hear that expire.
Seems to me that we all need to learn together vertically. We need to know what math looked like in earlier years, and where the math is going.
#elemmathchat
A4-paper folding for fractions!! I love this but itās super confusing because multiplicative reasoning as we initially learn it leads sās to believe that my answer should be ābiggerā and then theyāre like WHAT?! #elemmathchat
While I think an area model is another representation, I find that some students use the model but arenāt really sure what it means. The model becomes another procedure or route to find the answer w/out meaning. #elemmathchat
That is what I was thinking too. I was trying to create a graphic on my computer and quit. :) I just got home an hour ago for the first time since Sunday. My brain, eyes and fingers are broken! Your explanation is spot on though! #ElemMathChat
Yes. When students multiply 7 * 1/4 they may use a number line. Not always the case when itās 1/4 * 1/4. The models tend to change depending on the circumstance. #elemmathchat
A3: awww, trying to multitask on my phone meant that I missed out in the fun of representing a problem! Maybe my hands will be more free in a minute... #ElemMathChat
#elemmathchat A5: Soo to be fair, my own kid has had a 100% horrible time in math HS & maybe I'm a tad bitter, but...HS Ts are the only ones I have ever heard say that there are certain domains/concepts/etc in math that are either 1. not appl to the real life or 2. not visual 1/2
A5: I do believe that all mathematics can be represented visually. I also believe it is very difficult to create visual representations for things with which you have had little to no experience w/. For example, extremely large quantities or fractions of fractions. #elemmathchat
We are doing that now in my class! I am using @mrsforest Stop Motion Video lesson with them. It is amazing to see the conceptual understanding develop! #ElemMathChat
So important that we are giving our kids opportunities to make sense of things... not just showing them a procedure (whether visual or not).
If it doesn't make sense to them, they aren't internalizing it!
#elemmathchat
Q4 - So, how would we want operations with fractions represented? What are some possible ways of thinking about what's actually happening here?
#elemmathchat
I had a student last year say that they used the folding paper strategy to solve fraction multiplication problems. I wonder if itās communicated that thereās a a difference between a process and a model? Or are they more alike than different? #elemmathchat
In reply to
@Dr_Kreisberg, @RiversoMaria, @MarkChubb3
Thanks, Mark! Love that Liping Ma example! I used it in some district PD. :)
And I do think visuals help sooo much to get one's bearings with fraction division. #elemmathchat
In reply to
@MarkChubb3, @mathconfidence, @dhabecker
I wonder if also changing the context of fraction interpretation (e.g., fraction as operators) would help with visualization of 2/4 x 2/4? Like, I think of 2/4 (first factor) as part-whole, then I'm shrinking it to half the size. #elemmathchat
A5 - HS math is sometimes where I struggle with a visualāsomewhere in factoring in alg 2 but Iām sure there is a way, which doesnāt come as easily to me #elemmathchat
A5: There is a lot of math I can't represent, yet, but that's because I'm still learning. Still getting past all those procedures I don't remember #elemmathchat
A5 I think all math can be represented visually if we understand it. I know that I would struggle to show some of the geometry & algebra that I can get right answers to visually. I was taught procedurally and am working to understand higher levels more conceptually! #ElemMathChat
Q4: Area model. Itās funny at first I went to a decimal grid (so I converted to .25 x .25) then I realized it would work the same on simpler grid. #elemmathchat
If you haven't already seen it. I'd love for you to take a look at 1 specific visual representation for me:
https://t.co/3B2XrSCXkJ
Thoughts?
#elemmathchat
