# Encouraging Students To Make Deeper Mathematical Connections.

Because of all of the math talk my students do in class every day, they are very comfortable (and flexible) in sharing multiple strategies and solution paths. They can explain others’ strategies in their own words and agree and disagree with one another beautifully, however when asked to make connections between two representations (numerical or visual), I feel like I get very “surface” connections. I will read things in their journals like “They are the same because they have the same numbers” or “They are different because we double and halved the other numbers instead” or “We both used an area model” Something like this…

After going through their journals the other day with Faith, we were thinking and questioning one another about how we, as teachers, can have students dig deeper into the connections. We obviously would like to them to notice them but if not put in a position to make connections, will they on their own? Is there a way to frame a task or question that would push them to think a little deeper about how and why the two representations are alike/different yet still arrive at the same answer? How do we encourage students to make deeper, more meaningful connections when we know they can, but just may not be sure of how to get there?

Instead of a number talk they other day, I did a math routine I named “Where is _____ in _____?” I was hoping the prompt would have them go beyond just looking “at” the representations and look “into” the meaning of each representation. On the board, I posted some examples of their representations from the day before in which they had done a surface job of connecting. I had them work independently for a few minutes and then talk as a table before the group share. I was much happier with the conversation and felt like asking them to look “into” the problems really got them thinking about what the representation was showing.

This an example of two area models in which students the day before had simply said, “We both used the area model” without thinking about how they were related. I love the (.3 x .2) in each of the quadrants of the first grid.

This student had a different take on how the two area models were alike, which led to such an interesting discussion! She also did some lovely work with showing how the two distributive properties were within one another through factoring.

This student showed the distributive property and double/halving in a wonderful way…

This one was the only student who connected the area model of .6 x .4 to the strategy of .6 x .5 – .06. He showed where the .6 x .5 would be in the model and then scratched out where the extra .06 were coming off to arrive at the answer.

Having students make connections in math is so incredibly important and so difficult to do, especially with so many variations in strategies and representations. I would love to hear other ways to encourage these connections!

-Kristin

# Geometry Is Worth The Extra Time…

As I am sure many teachers can attest, there is a constant struggle each year between covering content and the precious amount of time we have to engage the students in learning. Prior to the past two years in the classroom, this guilt always seemed to creep up most during our geometry units. I used to feel that once the students could find area, perimeter, and volume, we would move back into our fraction and decimal work because that always took SO much time to develop a deep, foundational understanding. While geometric representations such as an area model support the fraction and decimal work, it is still not the 2D or 3D unit work.  Right or wrong, I felt I had to prioritize to make use of the little time I had for the best of my students. Over the past two years, however, my geometry units have been taking longer and longer because I have started to see things evolve in my geometry units that has me  wanting to kick myself and go back in time to give my past students a different learning experience. From the connections to number and operations to the development of proofs and generalizations have been eye-opening.

As all of these math connections were going through my head, I see this tweet from Malke (@mathinyourfeet)…

ahhh, it felt like validation in some weird way.

After this Twitter conversation, I started to dig back into my students work to find examples that makes these connections visible.

After doing a dot image as our Number Talk one day, I asked students to see if they saw any connections between the image and our volume work that day. This work shows how students see the commutative property in both, multiplication as groups (like layers in volume) and most importantly puts a visual to how multiplication and its properties “look” in both 2D and 3D.

Then volume led into some great generalizations about how multiplication “works” through looking at patterns, which is extremely important in mathematics in and of itself.  In keeping constant volume (product), students realized they could double one dimension (factor) and half the other. In doubling the volume (product), the students realized they double one dimension (factor) and leave the others the same. T

This volume discoveries later let to this claim on our claim wall:

The students extended this area and volume work to fractions/decimals that showed that fractions/decimals act as numbers in operations as well, supporting the structure of our number system.

While we classified polygons, I saw my students develop proofs for angle measures and our always, sometimes, never experience was invaluable. This work in connecting reasonings through visuals of the polygons explicitly supports the Mathematical Practices of using models, perseverance, and repeated reasoning.

Then our work with perimeter and area solidified the importance of students creating a visual in building number is so important. In a problem with equal perimeter and different area (moving into greatest area), students created a beautiful visual for the commutative property as well as supported students in seeing the closer two numbers (with the same sum), the greater the product.

-Kristin

# Intuition in Learning Math

Yesterday, Malke tweeted this…

…and it led to such an interesting conversation that I honestly can say, I had never really given much thought. The conversation is here if you would like to read it now, or you may want to save it until after you read my rambling thoughts. 🙂

At first read of the tweet, my initial thought was how hard it was for me to make a distinction between intuition and making sense of problems. What makes them different? The amount of time it is given? The context of the situation? The math experience behind it? My questions could go on and on. I needed time to process these thoughts and let them sit with me for a bit. I tried reading some of the links to works about intuition in learning, but until I could figure out how I was thinking about intuition and put it in the context of my classroom experiences, the readings were not making much sense to me.

Luckily, I had a lot of car-riding time yesterday to think about this and jot some thoughts down. Disclaimer: these ideas are quite scattered, do not form a cohesive string of thoughts, and you will probably be left with more questions than answers by the end, however it is really fun to think about all of the ways “Intuition” takes shape in learning.

I thought it would be good for me to start with a definition and work from there. First, I tried Webster and got this one:

eh. I didn’t like the “without any proof or evidence” piece of this because I feel that our intuitions do come with proof or evidence, they are maybe not explored or articulated yet, however I think they are there. Then I found this one that I felt I could work from a bit better:

This definition by no means encapsulates how I envision “intuition”, however it had some really interesting points that led me to more questions….

– Does our intuition mean we have an “understanding”?

– Where does our “conscience reasoning” come from?

– Is our intuition always what is “likely“?

– Where do I see these hunches, inklings, notions in my students’ learning of math?

– Where do I see these same things in myself as a teacher?

Up to this point, I was gaining more questions than answers, so I began thinking about these questions in terms of my students and myself as a teacher. I am a person always in need of examples, so I needed to form some “example cases” to start to clarify these thoughts.

In this first example, from an Illustrative Task, the student was asked to determine if the answer to the problem could be solved using multiplication. Although the student came to the correct answer of 2/40, his intuition is telling him it still doesn’t make sense. Is this intuition based on previous experiences? Are all of our intuitions based on past experiences or non-experiences? Has he never seen an area less than 1 so it can’t make sense? The array has to be whole numbers? For me, the use of miles intuitively feels weird when I read it. Even as an adult, I hear miles and think of distance, bars, and do not like it so much with area.

In class each day we do Estimation 180. In this example, I specifically look at Day 23. I never really thought much about this, but I think it is intuitive of the student to look at the size of the item being packaged in another when thinking about capacity and volume. The student doesn’t look at the color of the paper or types of lines on it, but instead intuitively looks at the size of each part. It is something that happens so fast, that perhaps these are the quick, small moments in my classroom where students are acting intuitively.

On Day 36 in the example below, a student’s reasoning is that it is “usually a weird number” and on Day 37 says “My family never gets blow pops so I guessed.” Both of these seem to stem from experience/non-experience.  The student has noticed that in previous days the packaging numbers have not been “friendly numbers” so the students is intuitively thinking it is a non-friendly number. If he/she was given this problem on the first day of the estimation questions, would their intuition have led them in a different direction? Day 37, shows a glimpse into, what I call, non-experience. This student intuitively goes to guessing because of never having them at home before, however is still only 20 off. This then leads me to question, that if a student doesn’t intuitively think of a reasonable estimate, do they then move into a more concrete strategy? To be within 20, I feel like the student used counting in the picture to some extent.

In this second grader’s work, I see such an interesting intuition in the second part of this question…

She seems to think that she cannot possibly know all of the possibilities because she is not older. She associates getting older as getting closer to knowing “all” of something. Little does she know that as you get older, you find you know less of “all” than you originally thought 🙂 Ha!

Here is where I get muddled between intuition based on experience and making sense based on prior knowledge. I asked this question to my students last year and here is one example of a student who went beyond yes or no and started to give a proof. I would love to hear others thoughts on intuition here….

This example below makes me think about how conjectures are made by students. Do conjectures stem from intuition and proofs that we can’t fully explain? The student said that when comparing two fractions, with the same numerator, she can subtract numerator from the denominator and the smaller difference is the larger fraction. Is this false intuition in dealing with the numerator and denominator as whole numbers? Thinking you can just pull them apart and operate with them as wholes?

This one may not be an example of intuition, but it is how I think about my own intuitions in learning….I try to explain them, prove them, revise them, edit them….. I would love every student paper to look like this…

In writing so far, I am really thinking that a student’s intuitions in learning math come from a “conscience reasoning” based on  prior experiences and exposure. I could truly rack my brain over this for a while, but before I do, I wanted to think about myself as a teacher also. How much do we do as teachers that is intuitive? How does that intuition change as we evolve as educators?

There are many things I do during the course of the school day that just feel like routine or habit. The hard part is deciphering where it is not just habit or routine, but instead intuition.

When writing my lesson plans, I would say I use my intuition often in the respect of anticipation.  From the minute I read the lesson, I have intuition on how I feel the lesson will “go over” with the students. I have a gut feeling if they will be interested in it, which students will be able to easily enter into the problem and which will struggle, and what strategies will emerge. All of these anticipations are based on my experiences with the students. So are these anticipations, intuitions?

As the lesson is happening, I think so much of my questioning is based on my intuition. I could not possibly have a list of questions to ask students during the course of every lesson, I have to rely on my intuition. As a student is explaining something, I am thinking to myself, “I think it would be interesting to ask _______.”  This is something that has definitely evolved based on my experience, however because of the “newness” of every day and every class, I have to rely on intuition of similar case scenarios. Something like, “I asked this question the other day and it got me nowhere, how can I ask it differently to push student thinking?” This inner dialogue during a lesson happens in an instant which makes me believe it is intuitive.

If all of this is true, then I would say that when I first started teaching, my intuitions were not as fine-tuned as they are now. Does that makes sense, can you fine-tune intuition?  Is there a point beyond thinking something is a good idea/bad idea or makes sense/doesn’t make sense that is still intuition but a more detailed, specific intuition? Intuitively, I think there is:)

A lot to think about still….Thanks to Malke, Tracy, Simon, Bridget, Kassia for a great (to be continued) conversation!

-Kristin

# Making Mathematical Connections

Each day I start the class with a Number Talk. I thought to continue building our multiplication strategies and make connections to our volume work, I would do Dot Quick Images. This is one of the images that I did yesterday:

In this image I hoped to bring out the commutative and associative properties (not by name, but the idea of what is happening in each) within their solutions as well as the use of the 3 x 3 array to get the number of groups in the picture and 3 x 4 array to get the number of dots in each group.  This would be the moment when I wish I took a picture of the board with their responses, but in the flow of the lesson, forgot. Many said they did 12 x 9 to get 108. I especially loved that some said they said they didn’t know 12 x 9, so did 12 x 10 and took a group of 12 away:) I said that when I read 12 x 9, I think of 12 groups of 9, trying to elicit the commutative property. I had them talk to their neighbor and we agreed this picture looked like 9 groups of 12, but there was a way to make it look like 12 x 9. I wrote both on the board agreed the amount of dots didn’t change, just the way we looked at it did. This went on into another image and we began our first lesson on volume. I blogged about that work here.

So, after chatting with a colleague after the lesson, we thought it would be interesting after the work yesterday, to reflect back to that number talk. Today I put the same image back up and I did a much better job of pulling out the (3 x 4) x 9 through better questioning and because they were solid in the answer, they could reason about it a bit deeper. I also told them to be thinking about our lesson yesterday to see if they could see any connection between the two. I finished the number talk and gave them 2 minutes to reflect in their journal about any connections they saw. Here is what we shared as a class…

So much to love here…..I loved the idea of layering the dot arrays to make a box. I loved the connection to the properties in each….

-Kristin