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Lesson 3 – Teaching Through Task – Discussion
Posted by Kyle Pearce on December 6, 2019 at 5:29 amCan you use an upcoming task from a grade or course you teach as an example of the different stages of the continuum of measurement?
Share your reflection below along with any wonders you still have.
Luke Albrecht replied 9 months ago 25 Members · 44 Replies 
44 Replies

What I love about you and Jon is that you look at mathematics as a big web of interconnected ideas that students can use to get to the end point. This lesson in particular highlights how my thinking has evolved over the last two years since I have found you guys on twitter and through the academy. When we rush to algorithms we actually take away the sense making and math becomes a linear progression like a chain and once a link is broken in a chain they are stuck and cannot advance. However, when we teach how all of the concepts are interconnected, and yes there are more efficient ways of completing tasks, we build tools (models) that connect together like a web, that lead to alternate paths that get a child to develop reasoning and accuracy. I actually am working on a keynote that explains this more that I would like to share to get your thoughts on and if the community feels I am accurate or not, but module 3 definitely highlights the work we need to do and why before we set up a proportion and cross multiply. Thank Kyle for putting this together. It is a lot to absorb and process.
 This reply was modified 2 years, 6 months ago by Shawn Hershey.

Hey @shawnhershey
We’re glad you’ve seen the “web” of teaching math. We’d love to see what you’ve put together. Feel free to email us or share here in the Academy.

I twisted your question around and spent the entire lesson thinking how I could use the Green Screen Task with my fifth graders. There are SO many opportunities. It could be used when we work on multiplication, then be reused for division, then for decimals & fractions, and of course measurement. This line of thinking led me down the rabbit hole of how I could use your Donut Delight to multiple ways too. I used it for division last year (since I discovered it after our multiplication unit), but could definitely use it for decimals & fractions too. I’m really excited to start planning out next year’s math.

This is awesome to hear. So happy to see that you’re making multiple connections with tasks to realize that they aren’t a “one trick pony” and that you can use them in many different areas of your course!
Thanks for sharing your reflection!


I have lots of learning still to do! And planning for the fall. It amazes me to see how Math ideas/concepts are interlinked in so many ways. Since my discovery of nontraditional math teaching 2 years ago, I am understanding why our Math Makes Sense texts are set up the way they are. We’ve had them for a very long time and only now they are beginning to make sense. (We were told to use them with no training.) Now our division is implementing Mathup – an online course by Marian Small for gr. 1 to 6. Looks great! A lot of nontraditional ways of teaching math. I am so thankful for the learning I am getting now. Even though it’s a bit overwhelming when I think of starting to totally change my practice! (and exciting!)

These are all fantastic realizations. I felt very similarly when I started transforming my practice. I had been sort of “skipping over” much of the goodness in my textbook because I just didn’t see the value or purpose.
MathUP is a great resource and actually, Jon and I built some of the grade 7 and 8 “wonder tasks” in that resource for Marian! She’s awesome and I’m sure you’ll love the resource!


My 8th grades are really struggling with squares and square roots, and cubes and cube roots. I would love to have a task for them to compare square roots and cube roots.
I think that even kids who can say the square root of 64 is greater than the square root of 49 because 64 is greater than 49, could benefit so much from the unpacking of that. How do you know? How can you prove it? That leads to students accessing the task at their level– some will draw or build models, some may jump right in with numbers, an indirect measurement.
If they were comparing a square root to a cube root, they might have that same direct comparison, but it would be really interesting for them to build/discover/discuss that when we are finding the square root and the cube root, we are looking for the single attribute of one dimension of an array or 3D model and that is what’s being compared. I’m so curious– what would they do if asked to compare the square root of 64 to the cube root of 64?
Completing this task leads nicely into the standard of being able to order and compare irrational numbers.

Great questions here.
I wonder if you were to “challenge” them to think about these square root and cube root comparisons by building with arrays (squares) and “cubes” (using connecting cubes)
Have students first explore which arrays make squares …
Which “prisms” make cubes…
Then start noticing and wondering…. all side lengths are the same for squares and cubes. Wow!
Do they notice anything about the relationship between total tiles or cubes and side lengths in each case?
Etc.
Does that make sense?


So I am trying o think about the continuum of measurement with an upcoming lesson on scaled copies (7th grade).
Direct Comparison – Looking at the original and copy and determining which is larger.
Indirect Comparison – Using a line/sting/piece of paper to determine that the height and the width are both larger on the copy.
Direct measurement – Iterate the original – stacking to see how many times taller and wider
Indirect measurement – given measurements of the original what is the measurement of the copy (given the scale factor) OR Given the measurements of both the original and the copy, what is the scale factor.

I am currently doing scale drawings with my seventh grade. We started by drawing a simple floor plan of a rectangular room to scale on a piece of gaph paper. The class was doing pretty well with finding unit rates and applying the equations to more generalized situations in previous problems dealing with proportions. (How many M&Ms in a given sized bag, how many grams of sugar in a soda bottle, etc.) But they’re having a difficult time seeing the scale factor as a unit rate in a drawing. To me, converting squares on the paper to correspond to real life feet or inches seems like an easy jump. Not so much for the students. Eventually students were able to accomplish the task, but not without a lot of coaching.
I’m wondering how I could have stepped back from that task and eased them into it in a way that would have allowed more discovery.

Scale drawings is much more abstract than it appears to those of us (adults) with experience.
I wonder if maybe you used square tiles on the floor of the room and then compared using a much smaller cut out square of paper that will be the actual square.
Could that comparison be helpful for students to gain a better perspective?
Also starting with scaling by doubling / halving first can also be helpful. So taking say a drawing and scaling it up or down by a factor of 2 or 1 half?
Thoughts?


I’m connecting my newest research challenge to your amazing math task such as the Green Screen task which engages students in two of the six strategies for effective learning–concrete strategy–in which you link an example to the idea you are studying; and interleaving–which links more than one cognitive challenge together as in converting cm to meters and calculating area. A task for using the different stages of the continuum of measurement involves younger students and determining which weighs more or less an apple or a pear.
Direct comparison: Use a bucket balance and place apple on one side pear on the other.
Indirect comparison: Place apple on one side of bucket balance and place cubes in other bucket until they balance and count the cubes. Then do the same with the pear.
Direct measurement: Use sticks of butter that weigh 4 oz. to measure each.
Indirect measure: Use a customary hexagonal mass measuring set .

Glad you’re diving into the task area and enjoying the learning!
Thanks for sharing your own thinking for the continuum, too!


I’m excited to start using some of these activities. Some of my students are struggling with area right now, so I am going to back up and do your activity with a small group tomorrow.


So good! I have an after school intervention group (hard to motivate since we are distance learning) and they loved it. We took our time and went through the whole thing. It took them quite a while to get it, but once they did their noticing and wondering became more relevant. Thank you! Now, if I could just find something similar with rounding… 🙂



I liked the sequencing graphic at the end.
(Sorry, as a Grade 5 teacher, I did fast forward much of your discussion)
But I am planning on asking students, “How many square centimetres is their desk?”
A) Direct Comparison: – provide a single cubed cm square for them.
B) Indirect Comparison: – provide each student one (4cm x 5cm) Postit note.
– provide each student with 4 more postits.
C) Direct Measure: – as a class cover one desktop
– assess how they deal with new information
D) Indirect Measure: – *since desks are not the same, use their rulers to calculate square centimetres.
I love listening for leading questions to draw from students their thinking.

Glad to hear you found the continuum helpful. Fantastic thoughts regarding how you might sequence some investigations with your students. Let us know how it goes!


I think this will help my grade 8s with many of their units, especially in surface area. I always feel that I should be able to get to the formulas because of prior knowledge but I always find so many students are unable to manipulate the forms to unfold them to figure out where the measurements go.I think this year I will slow it down and go back a little bit so we can build that understanding.
direct comparisoncomparing two rectangular prisms and deciding which one might have the largest surface based on a visual inspection
indirect comparisonuse square tiles to cover the different face to determine the areas
direct measureunfold the prism to be a 2D net and use tiles to measure every face of the net and build a connection to measuring with a ruler
indirect measureusing the formula to determine surface area
We will be able to repeat this process with triangular prisms and cylinder. As we move into volume a similar process can be used.

Nice job. And the more we do this sort of thinking, the more you’ll refine which action matches which stage in the progression/continuum.


How much further did student A jump than student B?
Use spatial comparisons to make an estimate of the difference.
Use paperclips to iterate the distance of both jumps from the start point to the end point.
or
Use paperclips to iterate the distance of the longest jump first and then remove and use the same paperclips to iterate the distance of the shortest jump, noting how many are not used.
Provide students with the measurement of the paperclips in inches to determine the difference in the two distances.

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?

You’re right about going online adding another big barrier to shifting pedagogy. Teaching math is hard work and doing it well is even harder. Clearly, you are a great resource to them and finding creative ways to “nudge” without overwhelming is so critical – especially now.
Your idea to set up some times for students to engage is great. I wonder also if you were able to lead a lesson for those teachers interested and then offer to CoPlan another with them in the future? Once teachers see their kids thinking, it’s hard to turn down the opportunity!


Just as others mentioned, I was looking at this video and wondering how i do such a thing with my students with our current standard – transformations. After some time I finally figured out what I could do. I could have a wall that needs to be covered with with 3 different diamond figures and would have to put use a pattern form. Students would have to figure out to cover the wall in doing so they would have to rotate, reflect, translate, and dilate the shape in order to make the diamond figures to cover the wall space perfectly.
This was good to think about and do.

How many people can sit equally in five rows with the given number of people?
First students could estimate by visualizing the people sitting in each row. Then they can use equally cut pieces of yarn to place in each row. Explain to them that each piece of yarn is equivalent to a person. They will count each string to figure out how many people will sit in each row. They could also do skip counting which adds up to the total.

Thinking this is a strong example of why we have to build conceptual understanding before introducing the formula and algorithms. This is troubling because right now, we are in the middle of COVID19 pandemic and schools are being pushed to focus on “priority” standards.This may leave measurement out of the picture, when we can see that it is such an important part of developing proportional reasoning. It’s ALL interconnected as the web shows.
Bottom line – students have to know why the formula is the formula! That is the element of trust that gets built over time. But a lot of this is missing in the current way many U.S. teachers are teaching (at least in my experience).

So true… and, ensuring that the formula emerges from that understanding rather than trying to make sense after a formula has been given.


I think that I would like to use this in my volume unit in 8th grade. I would like my students to actually to see why the volume formulas are what they are. I think if I use this process they would have a better idea why the formulas is created the way it is created. Giving them a cylinder and a cone for example and giving them thing to measure with. Water maybe.

Right now I am struggling with a problem mentioned above. The sixth grade students came to me thinking they know how to use the algorithm for multiplying decimals, but they don’t understand what they are really doing as they have no idea where to place the decimal. I got out the place value blocks to try to get them to visualize what is going on and several insisted on going back to the algorithm. Even when they created the model correctly because ” I remember this from last year” ( eg 0.3 x0.5using two colors on tenths pieces) they were not seeing how the answer was right in front of them. I really liked the idea of multiplying each decimal by 10 and then dividing them by 1/10th. I think using the green wall task might help them if I used that piece.

I do a lesson on what happens to the surface area and volume of rectangular prisms when you scale up their dimensions (from Jo Boaler’s Mindset Mathematics Grade 7). I would love to see students going through the comparison stages with the nets or even just images of the prisms for an example or two. They can begin by looking visually, then they can use unit blocks to compare them, and finally they can move towards measuring them with rulers and we will be able to eventually understand surface area and volume with a simple image with abstract measurements.

Great example to help you deepen your understanding of this progression!


I’m working on measurement, units and even conversions right now in fact. I’m trying to find a way to help them really understand conversions between cm and m, even mm. I find that they struggle with the decimals and notation, so I am trying to think of a way to introduce it that is less dependant on rulers and direct measurement, at least right away. Today during an exploration, I had some decide to compare the length of the room and another measurement with unifix blocks. Thankfully we had enough blocks! Other students lined up meter sticks and counted how many centimeters in groups of 100. It was interesting to watch the processes of different students. It taught me a lot about where they all are in their journey.
Moving forward we’ll be exploring how to think about conversions and I’m considering using something like your area model (but with the unit of one vertically) to help them see how to think about hundredths in the relationship between centimetres and metres. Hopefully they are able to make the leap. I was interested to see that they could already say that they measured 8m and 43 cm, which makes me think that the area model will make sense to them…we’ll see!

Awesome to hear!
I find that working with unit conversions through fractions as a start is helpful. 10 times and 1 tenth the size can be more helpful than looking at decimals and simply “moving” it to the left and right. Also for linear measure, using a bar model or number line can be helpful to show this conversion comparison. Let us know how it progresses!


My students had a surface area problem this week, but most of my students saw 3 numbers and found volume. As I reflected on this, I realized that they learned surface area at the beginning of remote learning, which means that they missed out on physical models and they jumped right to nets. Consequently, they don’t have a solid foundation. We need to go back to physical models and then see how many cubic inches will fit around our model. Once they have spent time with they have spent time performing direct and indirect comparison, we could then move on with nets and algorithms (direct and indirect measurement).
As I reflected on how the rush to formulas shortchanges our students, I thought about my unit on slope. We need to spend much more time than I did last year drawing the triangles of rise and run and counting the units. The hands on models may take more time upfront but they lead to a deeper understanding that will pay off in the long run.

Great reflections here. It is common for students to confuse area and volume, which might be tackled by using physical models as you’ve described. Really driving home this idea of area as covering and volume as filling is so key.


I liked how you talked about scaling during the Green Screen task. I think it will be another great reminder to the students that the math concepts are interwoven together. So many of them think each Unit/Chapter is unconnected and these types of activities really show that they DO connect and that you can use previous knowledge/learning to figure new ideas/concepts. I can also see where this task would work with percentages and decimals too.

Love the connections that you’re making and how helpful explicitly connecting ideas in the classroom can help students make sense of their learning! 🙂


Can I use an upcoming task as an example of the different stages of the continuum of measurement? Yes, but I am struggling.
I have been trying to evolve my way of teaching mathematic to include ideas I have learned about having number talks. Then I read a book called Building Thinking Classrooms in Mathematics. I’ve been trying to use math tasks and utilizing vertical math boards to encourage students to collaborate, think, and dig deeper. Now I am on a committee to choose our next mathematics curriculum. We were looking at textbooks that have “sparks” or tasks built in that would aid me in shifting to this type of teaching. However, if we choose something like this without professional development then there are teachers who will not use the sparks and will try to do what they have always done because it is how they were taught and it is how they see math? Doesn’t that ultimately set us up for failure? I need to change. I think we all need to change but how do we bring this evolution along?
To be honest, I was changing at the beginning of the year, but as the year has advanced, I’ve slid back into old ways because it is a way of least resistance and survival. I haven’t had many colleagues doing professional development like what this is and I am struggling how to make incremental changes to evolve because I can’t magically change in a day, a month or one year. I know that I am a different teacher today than I was but I am struggling.
So to circle back to my wonderings: Am I willing to take a risk and fail? What if it works? What does working look like?

Great wonders and I think that you’re right re: curriculum and PD. Without the professional learning, many will mould the resource to fit what they’ve already done.
This is why we have the Academy – to provide both curriculum (continuing to build it out) with professional learning. It is a process and it takes time to work your way through.
Whatever curriculum you decide on, consider the Academy as a home base for you and your colleagues. If you’d like to discuss options for how your colleagues can also learn with you here, be sure to message us using the blue chat bubble in the bottom right of the screen.


I am thinking about the classic stacking styrofoam cups tasks in linear algebra or the # Act task about stacking packs of copy paper. Letting students directly measure with cups or paper packs. There could be cups with different sized lips or jumbo vs regular size packs. I think the question of “how many packs to get to the ceiling?” starts the progression of direct to indirect nicely.