Tag Archives: 4th Grade

Making Connections for Deeper Learning

Why Connections?

So many times I see students not realize all of the wonderful things they know that would be helpful in their new learning. I believe this is because we just don’t spend enough time making connections explicit to support this transfer.

One of my favorite papers on the importance of practicing connections (linked in the citation) describes the ‘why’ so nicely…

“Although there may have been a time when rote learning of facts and procedures was sufficient as an outcome for education, that is certainly not the case today. Anyone with a phone can Google to find facts that they have forgotten. But gaps in thinking and understanding are not easily filled in by Internet searches. Increasingly, we value citizens who can think critically, coordinate different ideas together, solve novel problems, and apply their knowledge in all kinds of situations that do not look like ones they have previously encountered. In short, we want to produce students with deep understanding of the complex domains that constitute the modern knowledge landscape (National Academies of Sciences, Engineering, and Medicine 2018).”

“Studies show that expert knowledge in a domain is generally organized around a small set of core concepts (e.g., Lachner and Nückles 2015) that imbue coherence to even wicked domains. Because they are highly abstract and interconnected with other concepts, core concepts must be learned gradually, over extended periods of time and through extensive practice. As students practice connecting concepts with other concepts, contexts, and representations, these core concepts become more powerful and students’ knowledge becomes more transferable (e.g., Baroody et al. 2007; National Council for Teachers of Mathematics 2000; Rittle-Johnson and Schneider 2015; Rittle-Johnson et al. 2001).”
Fries, L., Son, J.Y., Givvin, K.B. et al. Practicing Connections: A Framework to Guide Instructional Design for Developing Understanding in Complex Domains. Educ Psychol Rev 33, 739–762 (2021).

Subtracting Whole Numbers

In 4th grade, students have been decomposing fractions into sums of fractions with the same denominator and justifying their decompositions. They naturally leveraged their understanding of whole number decomposition, but when we gave them a problem to add or subtract, they quickly looked for a ‘rule’ to find the sum or difference. And, while we want them to generalize these operations, as the numbers get more complex –mixed numbers and unlike denominators – a memorized rule absent understanding doesn’t help students reason about the problem.

The lesson last week had students representing their fraction addition and subtraction on the number line, but that representation was causing some students more angst than support so we decided to use the problems from the lesson, but focus on the connection to whole number operations instead of forcing only the number line on them.

On their whiteboards, we asked them to record all the ways they think about and can represent 13–6=?. We saw a nice mix of ideas like removing items (base 10 blocks), hopping back on a number line, decomposing 6 and subtracting in parts, adding up (relationship to addition), and the algorithm – which made them all chuckle because after ‘borrowing’ it ended up being the same problem.

Connection to Subtracting Fractions

We shared these ideas out, recorded them on the board for reference, and then asked them to erase their whiteboards and do the same thing for 13/5 – 6/5=?. Students shared the methods and representations they used and we discussed how they were like the ones they used for whole numbers. It took no time for someone to say ‘It is exactly the same, just fifths.’ One student wrote it out in words so the discussion of the change in units was perfect.

Applying the Strategies

Next we wanted students to practice a couple of problems, including mixed number subtraction where the first fraction numerator was less than the second. We were excited to see many of the same methods being used and some students really got into showing it multiple ways.

Try it out!

Any 4th and 5th grade teachers out there who want to try this out, I would love to hear what you learn about student thinking and what students learn about important connections in math class!

Small Change, Big Thinking

Fraction Activity

In this 4th grade activity, students were writing equivalent multiplication equations for a fraction multiplied by a whole number and then discussing the relationship between the different equations. The curriculum activity was good and definitely addressed the learning goal, but there was definitely an opportunity to open it up for more student reasoning and ownership. For example, in its current form, students don’t have the chance to think about which whole numbers would work in their equations or play around with the properties.

Small Change

Adapting doesn’t always require huge lifts. For this activity, all we decided to do was change the prompt to 12/5 = ____ x _____ and ask them to find as many ways as they could to make the equation true. I got so wrapped up in their work and discussions, that I didn’t snag any pictures of that part of the lesson, but after they finished we pulled up polypad and asked them how we could show why they are are all equivalent using the fraction bars. We wanted to be sure they just weren’t proceduralizing it at this point of the unit, so pulling up the fraction bars felt like a nice grounding of the concept. The board looked like this before we erased to make space to circle the other expressions.

Making Connections

At this point, they couldn’t get enough and asked for another fraction to try, so we gave them 16/3. We saw so much great thinking and use of the commutative property when finding the whole number and numerator.

Their excitement alone was the first indicator that allowing more space for their choices was a great idea! And then, as I was walked around, a couple students asked if they could write division equations. Of course I said yes and walked away.

I came back to #7 and #8 on this board:

When I asked how she came up with those equations, she said she used her multiplication equations because multiplication and division are related. I left her with the question of how she might show that division on the fraction bars and class wrapped up. I can’t wait to check back in with her tomorrow to see what she came up with!

Next time you plan a math lesson, I encourage you to think about small tweaks you can make to open it up for more student voice, ownership, and opportunities to think big! And I don’t know if anyone is even talking much about math planning on Twitter (X) anymore, but if you are, I would love to think together about tweaking math activities. So, send some activity pics my way @MathMinds and we can flex our curiosity and creativity muscles in planning together.

-Kristin

Math in a Movie Trailer

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Last Wednesday at a PLC meeting, our district instructional technology specialist did a presentation on Blended Learning. She did a beautiful job of demonstrating apps and web-based activities at various entry levels, so each teacher could participate. One of the fourth grade teachers expressed an interest, and a bit of fear, in trying to use ipads as part of her classroom routines. Since I had been in her room doing some math coaching the previous week, I offered to help her design an activity and give her a hand in the classroom with the ipad piece if she was not comfortable.

We met the next day to start our planning! She was just ending her 3D math unit in which students had been identifying 3D shapes by their silhouettes and attributes and finding volume of a rectangular prism. As a culminating activity, we decided to have the students create a movie trailer in iMovie that “told a story” about the unit. I sent the teacher home with one of the ipads to “play around” with iMovie, since she was not very familiar (or comfortable) with it. I was so excited to come in the next day to see a trailer she had created at home that night! I LOVE when people jump right in!

This is how our lesson played out over the next two days…

– We created a room in “Todays Meet” on their ipads and had students go in and do a test post.

– We posted the question, “What is the purpose of a movie trailer?” in the TM room and let them type as we showed two movie trailers (Percy Jackson 2 and Despicable Me) on the SMARTBoard. When the trailers were over, we switched back to TodaysMeet on the SMARTBoard to go through their comments and have them expand on them. Here is a clip of the conversation:

– Next we asked them to continue chatting about things they learned during this math unit. We noticed they were just writing one or two word things so we asked them to expand a bit and use more of their 140 characters. Sample clip:

– As a class we scrolled back through and had them stop and ask questions of each other if they didn’t understand what someone had posted. They were so engaged and they all kept asking if they could do this at home?!? YES! Next time I will leave the room open for a longer time frame so students can post as they think of things at home! What a great way to open class the following day!

– We took them through a brief “tour” of iMovie and let them move to a place in the room to look through the themes and storyboards and start brainstorming ideas for their trailer.

– To help them organize their thoughts, I had put a template of the storyboards: http://tinyurl.com/c3g5r2e in the Dropbox that was on each ipad. The students exported the PDF to UPad Lite:

and let them play around with how to write on the document with pen width and different colors.

– The following day, students got in their groups (of 2-3 students) to plan out their storyboard and decide on pictures they need for their trailer.

When we meet on Monday, we are taking them around the school and outside to take pictures they need for their trailer. They are working this week finishing up the project, so this story will have To Be Continued…

Mathematically Yours,

Kristin

Who is Coaching Who Here?

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I am so fortunate to be involved in a wonderful state-wide cohort, MiST (Math Instructional Specialist Team), organized by MSERC and made up of specialists from the University of Delaware, the Delaware Department of Education and districts across the state. This year, one of our foci has been content coaching, looking deeply into the structures that need to be in place and how it can be used to develop lead teachers in our schools. Our latest “homework” for the group was to try content (math) coaching with a teacher in our building.

At my school, we do not have a coaching model in place as of this current school year. However, with our implementation of the CCSS, we are writing a district plan that involves structuring a content coaching model into each of our elementary schools. We are fortunate to have both a math and reading specialist in each elementary school, so we are starting with a foundational structure in place. I thought it would be great to “try out” a coaching situation with one of the teachers in my building to bring back to MiST and get a feel for how it would work.

I have a wonderfully open 4th grade teacher in my building who is always excited to learn and willing to have me in her classroom and go through this process. I find one of the more difficult things of coaching is finding that teacher who is open to having someone else in their classroom and sees the value in the learning experience. I really lucked out with her! We met a week prior to the the day I would be in her classroom to choose the lesson and chat about which area(s) she would like to focus. The lesson we chose was on linear equations and in our pre-conference she wanted to focus on the timing of her launch and how to meet the needs of her “done early” and struggling students during the lesson. We planned a day to meet within the next few days to discuss the details of the lesson more thoroughly.

Having never taught this particular lesson before, I read and re-read the lesson, taking notes on the math involved, the launch of the lesson, how to extend and intervene for the students. I really was feeling the pressure to know EVERYTHING about the lesson and be overly prepared for any questions asked of me. I think I was more nervous about this meeting than she was.

The day we were meeting about the lesson (pre-conference), I got to her room only to find out the plan had completely changed. She had just found out that in another RTI group (where @ 6-8 of her students go) they had already done our planned lesson. Oh No! We had to refocus quickly (45 minutes of planning isn’t much time), so we looked ahead to where she would be in her core math class and chose another lesson.

This was probably the BEST thing that could have happened to me, although at that moment, I found myself starting to get nervous. I am a planner. I feel as the “math specialist,” I need to have all of the answers to any questions the teacher may ask. Which is so ironic bc with students, I am completely OK with saying “I don’t know that answer, let’s check it out” but with adults, I put pressure on myself. With this unpredicted switch in lessons, I instantly went from coach to co-learner and it was awesome. We read through the new lesson, asking questions as we went, learning from each other. I was offering ideas, she was offering ideas and we collaboratively “lived” the lesson for her 45 minutes of planning time. We talked through questions (inspired by Lucy West) such as: What is the math in the lesson? What previous experience have the students had? Who will struggle? Who will be done early and what will they do? What will the share out look like? Would you like me to chime in during the lesson?

She taught the lesson the following day, I filmed it, and it went beautifully! (I will be blogging more about the actual lesson soon). Her classroom culture and routines were evident from the way the students respectfully disagreed with one another and moved around the room. I was so impressed with her! I cannot even express to everyone the excitement I felt when I left the room. I felt the success of the lesson as if I had taught it myself. Things we had talked about in the pre-conference came out from the students during the class. Things we had not thought about came out from the students. I feel like we had a mutual investment in the lesson, both feeling equal responsibility. Our post conference is set up for next week, so I will blog about both of our reflections on the lesson and the process….so stay tuned!

It led me to ask myself, what does this coaching structure really look like? Who really coaches who? I would argue that this is really a multidimensional coaching model. I offered insight into “the math” of the lesson, the classroom teacher offered questions and insight into her students’ minds, the students offered comments for us to think about in upcoming lessons. When our district revisits our CCSS implementation plan and structure for coaching, this type of experience is critical in setting up those structures.

So thank you to my MiST peeps for the knowledge, motivation and safe environment to learn and share experiences. Thank you to my amazing 4th grade teacher for being so open to having me in the classroom and willing to learn through this with me. Thank you to the students who teach me something new every day I walk in the building.

Mathematically Yours,

Kristin