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Lesson 4-6: Question
updated 1 month ago
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How will you be more Prince this week in your math lessons?
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I can be more Prince this week by giving my students actual probability tools such as bag with coloured tiles, dice, coins and spinners to determine theoretical and experimental probability. Students will be given opportunities to use arrays and tree diagrams to represent all possible outcomes. Students will then write the probability of different outcomes as a fraction, decimal or percent.
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While I love using virtual manipulatives, I can’t wait to be back in a functional in-person classroom where my kids can have tiles and base 10 blocks and other manipulatives on demand. I teach mostly 6th grade, and those concrete representations are still SOOOOO important! The virtual manipulatives work, but not nearly as well for a starting place!!!
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I need to explore this idea with factoring!!! That was amazing to see, I wonder how negatives work into that model?
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Ok, just spent about an hour on this idea. I cannot see a pattern for how to set things up correctly on the first try. It seems pretty guess and check. Is there a way to know how many rows there will be? If not, I assume this is used for an introduction only (concrete) and then leading to abstract ways which will be used the rest of the time?
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Hey @Bob Have you seen my video on Sneaking in Factoring? https://youtu.be/6B99vE7RnU0 There’s a follow up with completing the square.
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Since Covid, I have been using tables in Google sheets to make arrays and strips (as number lines) a lot. The great thing is that my students are really used to them and can do it a lot. I need to do a better job of using more of the pictures of donuts before I so quickly move to the more abstract tools like coloured chips and arrays.
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Remote learning has made this more challenging for sure, however it seems like you’ve found some ways to keep things as concrete as possible. Also reminder that you can grab your brainingcamp license to use!
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The great thing about concrete to visual then to abstract is that is a guidance to remediate for students who missed this knowledge at their grade level. By stepping back to concrete, students who need it, get it and students who don’t need it, step to their level of comfort whether its visual or abstract. It can also “wow” the students who do the procedure without understanding and earn us a little credibility.
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So true. I love how concrete and visuals can help all learners “see” the math. Often our students using symbolic notation are unaware or forget why what they are doing almost automatically actually works. This is a great way to keep that connection.
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Your Prince analogy is fantastic. I have to think back to when he started to use his symbol and went by “The Artist Formerly known as Prince” I had to roll my eyes. (I was young and did not listen to his music. But hearing his performance at a halftime show, I began to understand his music and accepted his music.
But your concreteness fading model is a very logical connection to this idea. It is not until students can appreciate the math can they understand the abstract symbolic method. I do something similar to the concept of completing the square with quadratics. I always start by going through what a square is and have students construct squares with toothpicks. Asking the question, “How do you know it is a square?” gaining a typical response of all sides are the same. Then pulling out the algebra tiles to have the students do the same as above but with tiles and the square is filled in. Then I ask the question, “How do you know it is a square?” A typical response of all sides is the same. But at this point, have them write the expression for the sides. Then I give them a set of algebra tiles to completely create a square, where they may need to add some tiles to complete the square—obviously asking the same question as above but asking for justification. Eventually (not typically on the same day), I will give them an expression like y = x2 + 8x and ask them to create a perfect square trinomial.
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@scott.mcnutt I echo your exact feeling toward Prince! I also would roll my eyes about the symbol haha!
Love your lesson here for completing the square. Have you checkout our video on using algebra tiles for completing the square yet? We have two: https://youtu.be/f8wuG0xIC8w
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I can be more Prince in teaching equations. I am going from a dog walking diagram with dogs to a hanger diagram with symbols with the love of dogs and sense of need for balnace still there to a desire to stop wriitng thdse big hangar diagrams with numbers now to equations. Then they still see how you can cross off from either side or split into parts but with the sense of need to do it to both sides.
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This is definitely a common struggle… when do we progress students to the true symbolic notation?
I think it is never toon early to also represent your visual models symbolically to get them familiar, but trying to avoid removing the visual model early is key.
It can take a very long time for students to truly connect and build flexibility between their representation and other mathematical representations.
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These concepts are so true. I have students in Year 6 that have memorised their tables although they have difficulty applying their knowledge to other problems. I will definitely use some of these ideas in my teaching. Thank you.
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I have been more Prince this week spending plenty of time on concrete examples when working with Volume with year 9.
1st activity (multilink blocks)
Student activity 1 : Make a solid with a Volume of 12
Teacher: Pick out different solids, 1st put into 2 groups. (Prisms & Non Prisms)
Student question: what are groups called? (discussed other solid terms cuboids and polygons rectangles)
Student activity 2 : Make a Prism with a Volume of 16
Then moved onto visual calculation of volume of prisms, cylinders & more abstract questions.
2nd Activity (water containers)
Show how many cones fill a cylinder?
How many square based pyramids fill a cuboid.
Get pupils to hypothesise how to calculate volume of cylinders and pyramids
Moved onto calculation of volume of cones, pyramids & then more abstract questions.
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Love it! Keep up the great work, Prince!
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In fact I will use this kine of arrays to introduce quadratic functions. I used to do it with a parabolic path of a ball but I think, it will be more easy to make a relation with a linear functions (to solve proportional problems, arrays, money payments, path of submarines) that we have been working up to this week.
Thanks for the great idea!!!!!
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Glad to hear it! Let us know how it goes!!!
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We are starting Pythagorean Theorem next week. I found the lesson on Tap Into Teen Minds with the videos. I was going to use the video but start the students with graph paper to have them answer a Would you Rather question that involves Pythagorean Theorem. I was quite proud of my plans but I see now the importance of backing up a step and letting them play with algebra tiles to build the square along the sides along with the first video. I originally saw the graph paper as concrete and now see the visual purpose it has and how that is different from algebra tiles. Thank you for that clarification and demonstrating effective scaffolding that will really help my students.
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Fantastic realization!
@jon has been working on a super awesome Pythagorean unit which we hope will be out in a month or so. Keep an eye for it 🙂
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Thank you. I used this multiplication lesson with my Year 6 class using a variety of concrete and visual tools to visualise the idea of multiplication. I have students with a varying level of understanding and many can do the algorithm some of the time but they do not understand why and how things work. My higher ability students found it hard to visualise as they are focussed on the processes. We had a lot of light bulb moments and many of my lower ability students now feel confident explaining and showing their work and understandings. The concretness fading really helped to consolidate their understanding. I think we have unlocked the key to automaticity of multiplication in my class. You would not believe how many students were relived when I told them that they did not need to memorise their multiplication tables. 🙂
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This is fantastic to hear! Students at all ends of the learning trajectory are benefiting! Yay!
The best part you can share with students is that leveraging strategies and models is the best way to help them automatically recall math facts! So it isn’t memorizing rote, but rather building automaticity through repeated opportunities to think through multiplication problems!
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I am going to give CPA a try next week when I teach Surface Area of Cylinders. I was thinking that I would start by providing students real cylinders that they could handle, both with and without dimensions. Next, I plan to have the students build and cut out nets of cylinders on graph paper, paying special attention to the Lateral Area in their nets. Lastly, after working with nets, most students will be able to recognize that cylinders are made up of two circles, and a rectangle. This should allow for abstract solving, based on what they already know about these figures.
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Love it.
To add to your idea, it’d be good to be intentional about letting them estimate how much surface area they’ll end up with. So holding the cylinder and then having them sort of “estimate” by sketching on grid paper the “how muchness” of the area would be fantastic to get them visualizing.
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I wanted to use algebra tiles with my class but I felt much more comfortable with addition and subtraction than multiplication. Now I can see how I can try to use it to represent ideas visually for the “fade” to abstraction.
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I feel like as a third grade teacher I am already using this model of concrete to manipulatives to drawings to symbolic. I know many colleagues who jump to the symbolic too soon. Sometimes I might let my students stay at the manipulative and drawings longer than I should but I feel that they need that to be successful. I let them decide when they are ready to move to the next step. I allow them to use manipulatives and drawings all year but ask for the equation to go with their drawing.
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I was so excited to see the connection between your very concrete donut example and writing/solving quadratic equations. I have used the area model to help student understand distributive property but did not make the connection to something concrete. I do worry about myself and others rushing through the concrete to the symbolic without actually moving to base ten blocks. I also liked that you mentioned that students can quickly move away from base ten blocks into their own visuals given what they perceive as friendly numbers. I like the individuality of their models. I would definitely hang them and do a gallery walk.
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So great to hear that you had a big take away from this lesson. Avoiding the rush to the algorithm can be so tough and it is so worth slowing down to ensure all students are making those early connections.
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How I will be more Prince this week, and beyond, in my math lessons, is to find a way to present my lessons intuitively, as was demonstrated in the video.
The smoothness of the transition from a concrete manipulative to the abstract modality was remarkable. Wow!
I can see myself presenting the same lesson to 6th and 7th graders with the full confidence that they will remain engaged and be able to to tackle the exercises.
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You’ve so got this! We are rooting for you!
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I want to be more Prince, but I don’t know where to start. For example, if I was to solve equations with 8th-grade on-level students and Algebra students. Would I go to the same depth as the videos in this course?
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No sweat! It can feel daunting at the start.
Did you check out the video I shared yesterday about solving equations and algebra tiles? The key to utilizing concrete manipulatives is having a firm conceptual understanding of where you’re hoping to go with the idea and how it progresses. Not an easy task… we will engage in more of this as the workshop continues to progress!-
Not sure where to find the video you shared on solving equations and algebra tiles. There are many things to click on, I tried the News Feed, should I have found it there?
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This reply was modified 3 months, 2 weeks ago by Anthony Waslaske.
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This reply was modified 3 months, 2 weeks ago by
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Here you go:
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I am feeling the same way, Anthony. My thinking is that I need to watch several of the videos referenced then start with what I want the kids to take away from the activity and work backward. This is one of those areas where I absolutely love to see what others have come up with and am having a bunch of WOW moments when I see connections that I would never have dreamed of making on my own.
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Sounds like we need to get a unit together for this idea… @jon you’re it!
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With multiplication I foresee myself using actual manipulatives, something similar to the doughnuts (cannot use examples with unhealthy food in my division), then moving into concrete manipulatives (the number tiles) using the new method I learned (I had been taught the symbolic method without understanding why it is that I take those steps to multiply), then move into visuals, and then introduce the symbolic form.
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Love it. Great plan to move forward with!
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Since I am on summer break, it is difficult for me to come up with an example for the week. I am also a visual learner and love Prince. My plan is to post a visual of Prince in my classroom to help me remember to connect math symbols to ideas that are more concrete in order to help my students make more sense out of math. I feel this strategy will also be very helpful to a vast majority of our ELL students and students on IEPs.
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I could be more Prince when I teach my students quadradic equations by using your doughnut example. I have used algebra blocks, but have found that some of the students still have trouble getting the idea of what the square tile means. By showing that the x^2 tile is just a box that is x by x, that should help them get more out of the algebra blocks and then go to the symbolic representation.
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Love it! Let us know how it goes!
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How to be more Prince. What an amazing way to remember the importance of concrete – visual – symbolic!
I can see why some students never seem to “get it.” The years they may have been in math class desperately trying to figure out what everyone else seems to know when the rest of the class is rocking along with the four standard algorithms. Somewhere along the way, they were jumped to the abstract without being given the time to develop understanding.
If I were to show my mom the Prince symbol, she wouldn’t automatically pull up a picture of him in her mind, or start singing “Raspberry Beret” in her head because she does not have the connection of that symbol to the person or the music. Cool!
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Anyone think “Purple Rain” or am I just showing my age? I love how focus is shifting from only teaching/rushing to the abstract to opening with the concrete to help students build prior knowledge. While in college, we study multiple theorists and their beliefs of how children’s brains create schemas and progress from each of the stages found in the concreteness fading model. So, my question is why were we not teaching this way?!? Since I will be teaching first grade this next school year, the possibilities are endless. All I have to do is use concrete objects that have meaning to them, progress to visuals that remind them of the concrete, and help them create schemas that transcends into the abstract.
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I am on summer break right now, but I will be more Prince in Sept by allowing my students to “play” or “explore” more of the math and allow them to come up with their own algorithms by using manipulatives. I never really understood the use of algebra tiles, but now I see their value and will do my part to allow students to use them to understand things. I see how they can be useful not only with multiplication but with factoring and completing the square, which has always been a topic students don’t “get” and I have always tried ways to make it easier for them to understand, but I think this concrete step is the missing piece and will incorporate that into my lesson
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Well first, I am going to more like Prince by being FABULOUS. Second, I will work towards more concrete representations for a longer period of time. I use lots of manipulatives, but often “put them away” after we have moved to the abstract either because my middle school students thought they were babyish, because I felt like I have “checked that box”, or because of the distribution and clean up time involved. One of my goals for this year is to keep manipulatives out all year long at student tables to both encourage their use (encouraging me and my students) and to support sense making throughout the year.
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How to be more like Prince:
Give students something tangible to explore and play with a math concept. Then move to a visual representation of that tangible model. When ready, they will nudge themselves closer to the abstract representations.
I work with K-5th grade. Whenever I introduce a new manipulative I always start with a day of exploration without direction. I give kids about 20 minutes and I ask them do do what they want with the materials I have handed them, but at the end of 20 minutes, they are responsible for sharing at least 2 discoveries they made with the objects.
I am really excited to try using Cuisenaire rods this year for a variety of purposes. I have used them for fraction explorations, but I want to expand that using Dan Finkel’s lessons on multiplication and division with Cuisenaire rods.
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This reply was modified 3 months ago by Jennifer Maher.
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This reply was modified 3 months ago by
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I can be more like prince by breaking out and dusting off the manipulatives that were in storage last year. I can also be mindful of using the visual with each unit.
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I think I am already somewhat Prince in my math. I learned math almost solely through the use of symbols and, though I could “do” math, I didn’t understand how it worked. I was consciously incompetent in many areas of the curriculum and learned tricks to help me function. In teaching grade 3 math using the concrete manipulatives and visuals that were not a part of my own schooling, I understand math better. I have learned some of the area-based strategies you cover here in the past couple of years. I find it fun to try out the different strategies including the Japanese one. What this progression of videos does for me is to confirm how important it is to spend time with arrays as well as skip counting patterns to help my students have a better chance as math gets more complex. What I need do to “be more Prince” is put more time into moving from the concrete to the visuals before I do symbols and operations. I think I have seen visuals as options, not a step in the process.
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Glad to hear that you’ve already been on a journey using concrete manipulatives.
You’re so right about the process to get towards more abstract representations… it certainly is a process and actually, trying to continually cycle through concrete, visual and abstract representations to strengthen the connections between them is so helpful and something I continue to work on.
Thanks for sharing the reflection!
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The first lesson that came to mind where I should be more Prince is introducing 1-step equations to my 6th graders. I’ve stepped (pun unintended) away from my old balance scale lessons, but it’s time to bring them back.
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Bring em back!! 😉
Also have you checked out some of the math Is visual equation visual prompts? Could help… mathisvisual.com
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As an algebra teacher, I’ve always struggled with making the connection between algebra tiles and an equation! This was extremely helpful in helping me understand how to make the move from concrete to visual to abstract more seamless. Thank you!
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So glad to hear it!
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I definitely need to use physical manipulatives with my students more. I also need to use pictures of things that they care about. I can have them do algebra with pictures and do so in ways that makes sense to them rather than use the symbols and not have students know why they are doing things.
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Great reflection here. Are there any specific concepts you’re thinking? If so, what might that look like / sound like?
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When school starts up again in September, I will be more Prince-like with skip counting. Sadly, I did not realize the extent to which other math concepts build upon skip counting. Feeling the pressure of “covering the outcomes”, I would “move on” to other things even if I had evidence students were skip counting by rote (memorizing) rather than understanding (automaticity). No more!
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I can definitely be more Prince-like in most of my lessons by getting more comfortable using math manipulatives. I created base ten and algebra tiles in Google Sheets to that I can print a copy for each student to cut put and have their own copy. They will be able to keep this in their binder or backpack to practice at home.
I like the idea of using online manipulatives as well, but some students do better with tangible objects. I can also print each student two copies in two different colors so that we can create zero pairs.
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This makes me think back to teaching students to work the standard algorithm for multiplication. I show them the steps, and they look at me like I’m speaking a different language. Then some will blindly trust me that it works. We do use other methods to teach first and try to connect each of them. But we lack to make the connection to the concrete and therefore the students lack the understanding. I don’t know if I have ever pulled out the base ten blocks to let the students work with them for multiplication. I have always jumped to the visual representation. I need to slow it down and let them have that time. This explains the common errors that students make. They are just following steps and not understanding why they work.
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There is such a push for students to use the algorithm in 5th grade for multiplication. Most students truly do not understand why it works, how it works, etc. I think to BLP (be like Prince), I will start off the year by not just doing fluff math activities to get to know the kids, but to tweak the activities where they have to use more concrete and visual methods so that I can see just how deep their understanding is and what they are bringing with them to 5th grade. I will also focus on sharing this info with my colleagues so that they can use more models and manipulatives to enforce/reinforce the concrete and visual phases so that once students are truly ready for the abstract, it isn’t so foreign.
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When I get back into my classroom the topic of solving equations will be first introduced with can you guess the number progression I learned about while reading “Building Thinking Classrooms”….next students will work with Algebra Tiles to solve equations before working with the Algebraic symbols and creating Meaningful Notes about the process of solving equations.
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Awesome! Be sure to check out the Shot Put unit as well. Some great solving equations fun to be had there.
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To be more like Prince, I should start thinking in steps that lead to the symbol.
Use the small introduction of angles and the ideas that proliferate from there.
**The Big Concept is Trig Ratios (sine), the small steps that lead to the big concept with context would be Angles that reflect steepness and distance lengths with ratios of two side lengths of a right-triangle.
**What is the angle between the vision of site distances from Earth to Moon and Earth to Sun and how to interpret these distances and significance of SOH-CAH-TOA
(Thoughts I was getting from researching how to use Prince when thinking of teaching Primary Trig Ratios) -Hope it makes sense.
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I’d start in trig with playing with ratios and the relationships of side lengths of triangles. This will help you nudge towards models like the double number line and then to this idea of trig!
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To be like Prince…
I will be teaching a grade 6/7 split (at this point), it is important to recognize with the two years of interrupted schooling they could be easily starting math at much lower understanding and have many missing concepts. So I think I will be starting my math using the concrete representation of numbers, using base ten blocks, and have the students then use them for addition and subtraction and then move to a more simple multiplication. Then they can use the visual and move forward at their comfort level. I do foresee, that all the kids will need to go back to move forward to deeper understanding.
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I really liked the Prince analogy. Since I do not particularly feel attached to Prince’s work I felt nothing with that symbol which drove home the notion that students need to feel attachment to the symbols used in problem solving.
This question does not need to be answered…It’s more a wonder. Many times in solving math problems, language helps us really understand what is going on with operations and modelling. When we verbalize it or map words onto the situation it makes more sense. Why is it that we never talk about the “units” of the coefficients in the equations? y=8x was used. We defined the variables but don’t define the units of the 8 in this example. 8 whats? That would help us all make more sense of the problem and also be able to articulate what the expression is.
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In our upcoming math lessons, I think I can use this with all my math teachers K-8. This will be especially helpful with the grade levels teaching multiplication to algebraic concepts this week. I think the key is using manipulatives and visuals and allowing time for students to think and see what one another are doing. So that they can practice sharing their math reasoning and generate multiple representations either independently or with help from their classmates when they practice listening to the reasoning of others. Allowing the time for thought and discourse and leaving room for things not to have immediate answers so that they can move from concrete to visual to abstract.
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Listening to your podcast, I’ve heard you discuss being more Prince, but tonight what hit home to me is when you said if you don’t know Prince, the symbol does not hold any meaning for you. This is how my past students have felt about some of my lessons. I can’t wait to use more concrete and visual models in my instruction and resist the urge to go straight to the algorithm. When I used algebra tiles last year with my students they felt much more confidence and I think showing them the connections to multiplication in the beginning is a great way to reach all learners.
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So glad this resonated with you. It really does make sense when we think about how abstract we approach mathematics typically. Flipping it around can really smooth out the ride for our students!
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How will I be more like Prince? Well, classes begin in two weeks…when students walk into the room, they will hear Prince speak to them through his music!
Kyle, these last videos have been extremely helpful and enlightening. Honestly, I have not enjoyed working with base 10 blocks and if we were rushed to complete work, I would likely skip the blocks and go directly to the algorithm. You have shown me the way. The blocks will be dusted off and used again. We will ramp up use but fade slowly from the concrete model to the visual and then to the abstract for all units. We cannot afford to miss out on learning in a way that brings automaticity and enjoyment to math. Thank you, again.
P.S. I have never enjoyed a PD opportunity as much as this one. I am shouting accolades for Make Math Moments from the rooftop for all teachers who are willing to listen!
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I loved how explicit this section was and how it really makes one reflect on the visual to representation to symbol. Are there any ressources for teaching multiplication with decimals in this format? My students struggle with this but it would be wonderful to have that broken down representation!
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I am just coming to the end of a Fractions, Decimals and Percentages unit with my Year 7s so I have had plenty of “princely” materials on hand – a variety of fraction tiles, money, and equivalency tiles. Some students have used them a fair bit and some not at all, but they are there and available to use all the time. I haven’t been able to do as many explorations as I would have liked but we have done some that have focused on starting with visuals. Next term is Geometry and Angles so looking forward to having some time to think about physical and virtual tools for that unit.
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