Conceptual Understanding of Decimals - Part 1

This school year I have been trying to focus more on students' conceptual understanding.  When looking at my lessons through this lens, I am realizing there are topics that challenge my conceptually understanding.  I can explain the rules mathematically, but I do not have anything concrete for student to explore.  One of these topics is decimals.  I understand what a decimal is, I can use base ten blocks to show place value, they are even great for adding and subtracting.  Where it all becomes fuzzy for me is multiplying and dividing, and don't even get me started on repeating decimals.
Our textbook has lesson intended to help students understand why some fractions convert to terminating decimals and other fractions become repeating decimals.  The lesson takes the students through the process of dividing cheese sticks to see what is really happening.  While it is a good visual representation, it took me 2 years of teaching it to figure it out.  It is, at best, a good demonstration because in the end I have to explain all the steps to the students.  The model is quickly forgotten after the lesson, and the students go back to dividing on their calculator happy as clams that the answer just shows up.
This year I needed to write sub plans for this lesson.  If it took me 2 years to figure it out, there is no way I could leave it for a guest teacher, so I decided to shake it up a bit.  I left a very open ended task for my students.  I wanted them to create a visual representation to show why fractions convert to different types of decimals. I wanted it to be a stand alone poster (something they would not stand next to and explain) so all of their math thinking would need to be clear in their visual.  I gave them the fractions 3/4 and 2/3 to work with.  Here is a link to the instructions that were shared with the students.  Some students used markers and paper, while others chose digital tools.  There were slight variations, but all the visuals looked something like this:

                   

          

   

Every student gave me some variation on the pictures above.  Now remember, the question is why do some decimal numbers terminate while others repeat.  I want you to look back at the pictures again and see if you notice what I noticed....

While these are all lovely visuals of 3/4 and 2/3, not one of them actually shows decimals!  My students are masters at showing why 3 divided by 4 is 3/4, but then they just said that 3/4 is equal to 0.75.  There was a little more information in some of the explanations, but most of them referenced long division or the fact that 3 doesn't go into 100 like 4 does.  This became the starting point for a great discussion the next day when I returned.  I showed them the drawings and pointed out what I noticed.  We worked together to figure out how to modify our visual representations.  We talked about decimal place values and what that looks like when cutting up a shape into sections (10 sections - the remainder cut into 10 again which becomes hundredths, etc). The pictures below are an attempt at simple visuals to explain the discoveries we made in our class discussion.

If you are looking to better understand your students understanding of decimals give them this task and see what they do with it.  It really was eye opening for me.

If you are looking for a more structured lesson to help build this conceptual understanding you can read part 2 of my post (after I write it).  I will be teaching this lesson with a different class next week and have a new plan for exploring decimal place values and converting fractions to decimals.  I also have a plan for part 3 when I figure out the best way to extend our conceptual understanding to decimal multiplication and division.

So stay tuned...

The Power of the Nevermind Strategy

It was the third day of class and we were going over a problem I had given the students as homework.  The problem was a little tricky and the intention of it was for me to get a better understanding of their ability to reason mathematically and show/explain their thinking.  The class discussion revolved around the sharing of  the different strategies that were used to solve the problem.  We were comparing strategies and discussing our understanding of proportional relationships and unit rates when something happened.  A girl in the class started sharing her strategy.  I was modeling it on the board with visuals, and then she stopped.  "Nevermind," she said and looked down sheepishly.  The other students looked at her.  I could see it in their faces.  This is seventh grade after all.  Some were judging her as if her mistake somehow made them and their math ability superior.  Others felt bad and were making a promise to themselves to never participate in class in case the same embarrassing thing should happen to them.  It was in that split second that I said something that took the class in a whole new direction.  "Does anyone know the name of Lucy's strategy?  I call it the "Nevermind Strategy".  Have any of you ever started a problem and then realized that your approach was not leading you to the correct answer?"  Hands started to go up and smiles came across many of the faces in class, including Lucy.  We continued to talk through the math of Lucy's approach to the problem, why it was not going to work, what information it would lead us to, and what we could change to make that strategy work for this problem.

The key to the nevermind strategy is to not let kids stop when they get to the nevermind (or just kidding) part of their problem solving process.  This is a golden opportunity to see problem solving in action and create a classroom that values (and does not judge) mistakes, wrong answers, and sharing our thinking.  What do we do when we realize we are on the wrong path?  Much like the guess and check strategy there is a lot of math reasoning and number sense involved in realizing that your answer needs adjusting and calculating what adjustments will make it better.  The nevermind strategy opens up the class discussion to see who else tried something that didn't work.  Often times my favorite conversations start, not by asking who has an answer and how did you get it, but instead by asking "Does anyone have an answer they know is wrong?"

In my classroom I want students to take risks, explore new things, and feel comfortable making mistakes and sharing them.  By giving this a name, such as the nevermind strategy, it seems to have somehow legitimatized it in the minds of my students.  I plan to continue emphasizing this strategy in my class this year.  When students are stuck on a problem I will encourage them to use the nevermind strategy.  They can try something and if it doesn't work they can adjust accordingly.  Some students claim they don't know how to do a problem or they don't know where to start.  Often this is because students feel they need to have all the answers before they even start.  I am going to challenge that this year in my classroom with the use of the nevermind strategy. 

Penny Towers - Attempt at a 3 Act Task

My textbook has a lesson on building towers of pennies and looking at the proportional relationship between the number of pennies and the height of the tower.  It is a good textbook lesson, but I decided I wanted it to be more organic - you know, in a planned out, calculated sort of way.  Inspired by all of the wonderful 3 Act Tasks out there, I came up with this plan.

Act 1
My kids made some penny towers.  After showing this clip to my class, I will have them write down any questions that come to mind.

*I am having editing issues.  Save your ears and keep the volume muted.*

I am confident that at least one question related to penny towers will require proportional reasoning of some kind and hit the target of: "I can make a prediction by using proportional reasoning in more than one way."  However, I had my kids do a little acting to create this video to really open up the discussion.

Depending on the questions students come up with, we will start by focusing on questions related to the height of the towers.  It could go in 2 directions, how tall is a certain number of pennies or how many pennies to reach a certain height.  Ideally I would like to explore both of these, and I don't think it is going to matter which one we start with.  Whichever question we start with, the second may become the sequel in Act 3.
In Dan Meyer's explanation of 3 Act Tasks, I like how he asks students to think of a number that they know is too high and one that they know is too low.  We work a lot on number sense in 7th grade and being able to engage that type of thinking is important.  I want to be sure to work this into Act 1.

Act 2
At this point, it is my understanding that I ask students what information they need from me.  I will have pennies, rulers, and the duck (11.5 cm) available.  In the video we do not have a million pennies, though I found some resources to show that in case we need it: The MegaPenny Project and Holocaust Memorial.

Act 3
If my understanding of the 3 Act Task is correct, this is where we pull it all together.  We will discuss strategies, formalize our understanding of how proportional reasoning was used, determine the answer, and of course go back and check our original estimates and see how we did.

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Reflections after this Lesson:
I was surprised by my students surprise at Act 1.  It was very clear to me that while we are making gains in getting the students more involved in their own learning, they really expect the teacher to place the problems/task to solve in front of them.  In my house television is not a spectator sport.  We tend to watch tv and movies with our phones because there are so many things we wonder and want to look up while watching.  I thought the questioning after the video would be natural.  It wasn't.  At first only a few wrote down questions.  I gave more think time and most of them came up with something.  One of the questions a student had after watching the video was, "Why are we doing this?"  Even though I did intro the lesson and went over the learning targets, these students had never been asked to do something like this before.  I, of course, was glad to go into the pedagogy of curiosity and questioning being keys to meaningful learning.  Perhaps next time before the video I would let them know I am just curious what things come to mind, what questions they have, what they are curious about as they watch the video.

I do need to work on their questioning skills.  If I am going to create lifelong learners, I need them to be lifelong questioners.  My plan is to create a Desmos activity that shows a picture.  They will write a question that comes to mind when they see the picture.  Then students can see others responses and I would have them tell me which question intrigues them the most.  There are other online tools that I could use for this, but we use Desmos already and I don't want to have to introduce a new resource.  I will share this activity when I create it.

With the questions student had in Act One we were able to move forward to Act 2 trying to determine how many pennies it would take to make a tower as tall as the duck.  We also investigated my kids' question about the million pennies.  Overall it went well.  We did pause to come to a class agreement on the height of the duck (which I had students measure themselves) and how many pennies are in a tower 1 cm tall.  I figured it would make the strategy discussion go more smoothly if we were consistent with those.  Doing this activity at the beginning of the year, I quickly realized that these students are not yet clear on my expectations for organizing work. As we put their work under the document camera to compare strategies we were also able to discuss representing mathematical thinking so others can follow our thought process on the paper.  I hope to do some 360 math in my classroom this year to strengthen this ability.

After Act 3 I used Recap to have students explain their understanding of proportional relationships referencing the penny towers as an example.  They were a bit self conscious about being videotaped, which they will get used to.  Some came up to me after class to let me know that they understand proportional relationships but had a hard time putting into words.  This just reinforced for me why a tool like Recap is important.  There is a difference between writing an explanation and speaking an explanation.  It made me think of the book Launch.  When students know their work will be put out there for others it takes it to a different level.  These videos were just for me, but they were far more invested than if I had just had them write down their answer.