This morning I was in a 7th grade math classroom. The class was reviewing for a test and working their way through problems at different stations. I sat down to talk to a group that was working on this problem:
When shopping at Target, Sam found pizzas priced at 4 for $14.00. He was having a party with 16 friends to watch the MLB playoffs.
If he wants to make sure to have half a pizza for each person, how many pizzas should he buy?
How much money does each person need to contribute to cover the cost of the pizza?
As we discussed the strategies students were using to solve this problem, I started to notice levels of units coordination embedded in the problem and how the strategies students chose were related to how many levels they were able to coordinate. (If you have not yet read my blog posts explaining units coordination you can find them here.)
Here is a visual I created to show the levels of units involved in this problem.
There actually are 4 levels of units as well if you take each pizza in the lower left visual and slice them in half.
My discussion with the first group was so enlightening that I continued to discuss this problem with other groups as they rotated through the stations. I found that for many students they could easily determine that we need 8 pizzas. One pizza feeds 2 people, so 8 pizzas feed 16 people. Most students relied on proportional reasoning, thinking about it as doubling. They focused on whole numbers and multiplying rather than fractions and dividing.
The second question is what caught my attention. Most of the strategies students used helped them find the price per pizza. Once they had that, they divided by 2. There was even one student who worked from 4 pizzas for $14, divided by 2 to know 2 pizzas is $7, and divided by 2 again to find 1 pizza is $3.50. I realized that many of these students were working through one or two levels of units at a time instead of being able to jump from $28 for 8 directly to $1.75 for 1/2.
I think this is an important distinction to make when discussing units coordination. All of the students were able to get the correct answer. All of their strategies relied on their mathematical reasoning and number sense. From the teacher perspective, however, it is important to notice the sophistication and efficiency of their strategy. We need to strive to understand where students are in their ability to coordinate units so that we can continue to move them forward with this type of thinking. Otherwise as the math continues to get more complex and students begin to struggle, we might not understand the root cause of the confusion.
The other interesting thing I discovered in my discussions is that while most students did not divide by 16 to find the price per 1/2 piece, they checked their answer by multiplying by 16. This was a great reminder to me that units coordination is reversible. Perhaps if students had been given the price per piece they could have more easily worked their way to the cost for a group. (I'm a little tempted create some questions and draw a visual for the levels of units I could work into a problem like that, but I am short on time. Perhaps another day.)
Here are some questions I want to reflect on. I hope you do too.
Does units coordination play a role in our next unit?
What questions can we plan that will allow us to see how students are working with and coordinating units?
Do the questions I have planned ask my students to coordinate units both forwards and backwards?
What supports can we put in place within the lesson for students who struggle to coordinate units in order to move them forward in their thinking?
Recently Kyle Pearce blogged about concreteness fading. It got me thinking about a session at the Math Recovery Council annual conference with Dr. Robert Wright on the topic of settings and distancing the setting. I am constantly reflecting on the best ways to move my students forward in their math thinking and it seems to me that this is an area in which I need to be more purposeful. I decided to read a little more, ask a few more questions, and this blog is my reflection on all that I am thinking and learning about these topics, focusing more specifically on the role of manipulatives in my classroom.
The concept of concreteness fading as described both in Kyle Pearce's blog and in this article by the Learning Scientists is about the progression of moving students from concrete to representational to abstract. The idea of this makes sense to me. I feel that too often I introduce a manipulative, like integer tiles, for example, only to take it away too soon. I allow the students to use them, but if it isn't specifically in the directions many students struggle through without them thinking they should be able to work abstractly. This is where many kids will latch on to rules and memorization in order to work abstractly with something they don't fully understand.
What stood out to me in Kyle's blog was this quote, "We must understand that concrete manipulatives are still more abstract than using the actual items in the quantity being measured." Just the other day I was working with a student and using cubes to represent something. It made perfect sense to me and I realized as we were working that this student did not connect the cube with the item from the problem. I need to keep in mind that it is not just a 3 part continuum from concrete to abstract, but there is a continuum within the concrete category. There is so much to be purposeful and thoughtful about when selecting manipulatives and concrete examples. I don't think I do enough to modify and adapt the manipulative choice if a student is not able to connect with it. Plus Kyle's blog really helped me think about choosing manipulatives with the end result in mind. Where do I want the students and what manipulative/tools get them there? Perhaps it will be different if I am focused on adding mixed numbers with a distributive strategy versus a fraction great than one strategy. I love the vertical number line for proportional relationships that Kyle shows to then connect to the table that shows the numbers vertically. It's a simple change that makes a big difference! (It was both an aha moment and a face slap when I saw it. How did I not think of this?)
I also started to think about this process in reverse. By 7th grade, many of my students have memorized rules or tricks and can solve problems abstractly. The issue is that they lack the depth of understanding. They can work abstractly but don't understand the number relationships enough to show what is happening concretely. They are unable to mathematically justify what they are doing and it affects their ability to transfer their understanding to new problems. This video of Ruth Beatty (that Mark Chubb shared with me) speaks to this.
In the constructivism definition that is given in the video, the word concrete is being applied to the students understanding. It is not just about the concrete example or having something concrete to manipulate. This video changed my thinking about the goal I am trying to reach with students. I need to reverse the concrete to abstract model and work with what my students already know to help them understand more concretely. Sometimes as a middle school teacher it can feel like I have to start from the beginning with these concepts, but I need to keep in mind that the starting point for my students is their abstract understanding.
So how does this model of concreteness fading compare with what I know about distancing settings? This paragraph is taken from the book Teaching Number in the Classroom. Distancing the setting is part of progressive mathematization (which is the progression of learning and thinking in terms of mathematical sophistication).
I am not sure if there really is a difference between concreteness fading and distancing the setting, however, the definition laid out here seems to include more. When planning a setting, it isn't just thinking about the concrete representation, but it's selecting what verbal language and number range are appropriate, deciding how to move students through the concrete section of the continuum while interweaving the representational and abstract. I know I have talked about concreteness fading as a continuum, but it is also cyclical so students are not just moving from concrete to abstract and done. After giving it much thought I feel that distancing the setting is something that is done within concreteness fading. There is one student who comes to mind with all of this. We have worked to build strategies and his multiplicative reasoning and yet unless directed his default is to count on his fingers. It is by purposefully distancing the setting that I can help him progress to naturally select more sophisticated strategies and thinking.
In thinking about the examples Kyle laid out, I feel that those are the first steps in lesson planning. I need to map out the models and progression through them. Once that is done I need to be thinking about how to distance the setting. Screening and flashing are great strategies. Showing students the concrete and then covering it/part of it or showing it only for a brief moment. This will challenge students to start creating that mental picture. They don't have to create a mental picture themselves yet. It is given to them, then covered, and they continue the problem with that picture in their head.
Since I feel that distancing the setting is something that fits inside of concreteness fading, I wanted to go to Kyle's middle school example in his blog and consider ways to distance the setting as I move students from concrete to representational.
The original picture prompt is really a screened image. I can see part of the problem but 4 of the boxes are not pictured.
My thoughts about distancing the setting for this example are all over the place and I have typed and retyped many times. So many options and thoughts running through my head. I am going to try bullet points:
For some students, this number range will be too high. They will need enough cubes for all 7 boxes.
Some students may need boxes to put the cubes in. Connecting them together may not be concrete enough.
If a student has boxes and cubes inside of them (or cheerios to look like donuts), one way to distance the setting is to have cubes in the box and then close the box. This could also be recreated by using cardstock that covers the cubes, acting as a box. (which I would totally print the donut logo onto)
Closed boxes could be added, one at a time for students who need to think additively and the through discussion and activity we could get to some multiplicative reasoning.
Using boxes that show the outer donut (one row and one column) would be a way to distance the setting to help students use additive or multiplicative reasoning over counting strategies.
As we move to the number line in Kyle's visual example, I am loving this number line picture from Sara VanDerWerf's NCTM presentation. I am now thinking about how to add donut visuals to the number line and table to help students connect the concrete ideas to the representations.
There is a lot to think about with just this one example. It is a little overwhelming to think about doing this every day in my classroom. I think, however, that once I get into the habit of thinking about these things (and dedicating time to think and reflect on it) the ideas will flow and I will see how ideas for one concept transfers to other concepts. It will be a progression and continual goal in my teaching. The main questions floating through my head right now are this:
Do I really offer my students enough strategies to understand concretely?
Is my teaching limited by my limited knowledge of "tools"?
How do I build my concrete understanding so I can help my students?
I tried some new things with fraction division this year. I want to document for it myself as well as those who ask me about it.
I started this unit by making claims. I wrote about that in this blog post.
For the fraction division I used Cuisenaire rods:
Each student has a set of each of the 10 different rods. Students were instructed not to shout out answers, but to set the “answer” rods aside and keep quiet so everyone has think time. Allow time for students to figure out each task. Ask for students to answer and ask several students how they know.
Orange: “This is 1. Find ½.”
Dark Green: “This is ⅔. Find 1.”
Purple: “This is ⅖. Find ⅗.”
Brown: “This is ⅘. Find ⅗.”
To explore reciprocal relationships that are needed for division:
Held up and showed students the red rod.
“Now think back to the red. Let’s pretend it’s a granola bar. This is ⅓ of a granola bar. Find 1 whole.” “How many times does ⅓ of a granola bar fit into 1 granola bar?” “What equation can you write to describe this situation?” Students may come up with several that are correct but ask for a division equation if they are not getting it.
Purple: “This is ⅖ of a granola bar.” Find 1 whole.” How many times does ⅖ of a bar fit into the whole bar?”
When we divided with the Cuisenaire rods it looked something like this:
I have tried to divide using number lines before and it was a hot mess. Using the rods we were able to understand division as division (as opposed to multiplying by the reciprocal) and used a measurement model for fractions.
A landscape architect bought 5 bags of soil, knowing that each flower bed in the garden she was designing required 3/5 of a bag of soil. How many flowerbeds can she fill with soil?
Students made different sketches to figure out the answer, but all of them focused on dividing. Some drew flowerbeds and had 3/5 in each. Others drew 5 bags and divided them into fifths and colored 3 at a time for each flowerbed.
After a few questions like this we revisited this question:
This time determine the number of flowerbeds the architect can make with 1 bag of soil. Show your reasoning.
This brought us back to the idea of the reciprocal relationship and viewing it as a unit rate. Once we knew the unit rate we could just multiply by 5 because we had 5 bag.
What surprised me about the reciprocal relationship is that I thought it was just what you multiply by to get one whole. In working through this activity we are able to see it is also a unit rate. We can see the relationship between multiplication and division. I do not feel like understand all of this enough to get all students understanding it yet. It did help in understanding why we multiply by the reciprocal when we connect it to the unit rate.
We also explored division with a common denominator. Fawn Nguyen has a great blog post about this. She used graph paper and rectangles. Like the rods, it helps students understand fraction division as it relates to whole number division. These 2 strategies helped deepen my students' fraction sense around division. They were now beginning to understand why the number would get smaller or larger than the starting number.
Christopher Danielson also has some great posts about fraction division:
Partitive fraction division - I used some of the visuals from this blog and had my students see if they could explain the "students" thinking.
Common numerator fraction division - I challenged my students to see if a common numerator strategy was possible. Most of them came back to say that they were able to figure out a strategy for common numerators but were not sure why it worked. It was a great one to explore together.
