3.1 – Progression Ah-ha’s and Wonders

The videos that resonated with me the most, since I (student) teach in a third grade classroom, were the videos on multiplication and division, since that’s essentially all we’ve been working on since the beginning of the school year. An “ah-ha” that I had with regards to multiplication was that multiplication learning really starts in second grade, when students are taught to partition a rectangle into no more than 5 rows/columns. This is interesting to me as I reflect on my own students’ knowledge when it came to arrays and finding the area of rectangles, both things we’ve covered this year, and I notice that there are a good chunk of my students who gravitate toward array when it comes to solving multiplication and division problems. A second “ah-ha” when it comes to multiplication and division is to let the context of the problem explain the equation. Lately, I’ve noticed that my students are able to look at an equation and come up with a strategy to solve, but when it comes to interpreting and tackling a word problem, they have a hard time creating a model and therefore an equation that aligns with the context of the problem. A final “ah-ha” that I had with regards to division, specifically, was that students should experience the difference between efficient and inefficient thinking. That way, they can recognize the need for a faster strategy for dividing with larger numbers and hopefully come up with some ways to mitigate that through their own exploration. A “wonder” I had with regards to third grade and the progression of multiplication and division, it how to get students to see the value of each and the difference between repeated subtraction strategies in which the groups are unknown and the fair share or partitioning strategy in which to objects are unknown. In my experience, too often the students get stuck on one of these and have a hard time differentiating which one makes the most sense in the context of the word problem they might be solving.

In my experience working with different levels, the one I will focus on will be first grade, so the Early numbers and counting progression and the addition and subtraction progression are the areas I should go over and start working with students. I like to remember the content they were able to learn in kindergarten and connect it to the new knowledge. Applying new strategies for addition and subtraction shows me that students have different ways of solving and connecting content. There are students who need to draw pictures to find results, others use their fingers and others simply do a mental process. But if everyone likes to work with concrete material; grouping, putting or removing different objects, helps them to be able to visualize a result through the manipulation of different objects stimulates them to create a conceptual understanding of what is addition and subtraction.
By grouping different objects, students may be able to figure out that one number plus another equals another number plus another number. Sometimes children tend to think that, for example, 5 + 5 = 10, and that there is no other way to find the number 10. But when we show them another option, for example, 6 + 4 =10; they begin to understand that if we put different numbers together we can find the same results.
Now, when we start to use numbers greater than 10, we have the challenge that some students find it difficult to understand how to operate with them, so the decomposition of numbers is a strategy that allows them to simplify the largest number and be able to develop the exercises. more easily.

All the videos resonated with me.It was good to see that the arrays and area models I stress in multiplication instruction. Also how important the understanding early numbers and counting are… hierarchal inclusion … did I stress that. Was very glad to se that use of ten fram3s were good practice for subtraction. After watching this, I understand why not to use “butterfly” but ugh…. Was the way I was taught and I have used it!

FYI, I had to watch the videos on You Tube because there was only audio in the course.
I thought all the videos were fascinating. The biggest take away for me is the realization on how important the need for visuals are.

I enjoyed learning from all the videos. My a-ha moments were learning about the true progression of addition and subtraction, then the progression of multiplication, and division and finally fractions. I came from IL and taught technology so this being only my third year teaching 5th grade, I am fascinated at what I was watching. I wonder with such a short amount of time when school starts, if I could get this incorporated into my teaching. If I can, knowing what I do from the last two years, this will benefit the kids a great deal in regards to true understanding.

3 “ah-ha’s” and 1 “wonder” as it applies to the grade level you teach. Make sure to include how your new learning might impact your instruction. When finished, find two other colleague’s posts and respond to their “ah-ha’s” and “wonders,” building on their thoughts or offering possible solutions.

1 ah-ha is use of models to ensure conceptual understanding no matter what grade you work with. Working online I will ensure I continue to use models to ensure students have conceptual understanding.

2nd ah-ha is using area models and partial quotients to develp conceptual understanding prior to teaching the standard algorithm. This is something I continue to use with my students.

3rd ah-ha is to go back and review as needed to ensure any gaps missed or forgotten by students can be reviewed prior to moving forward. This is something I do 1:1 or small group.

Colleague’s Post – A “wonder” I had with regards to third grade and the progression of multiplication and division, it how to get students to see the value of each and the difference between repeated subtraction strategies in which the groups are unknown and the fair share or partitioning strategy in which to objects are unknown. In my experience, too often the students get stuck on one of these and have a hard time differentiating which one makes the most sense in the context of the word problem they might be solving.
I believe what is best and what I have experienced is to allow students to use what strategy is best for them. Allowing students to struggle as they learn new methods to understand and solve problems allows them to eventually/hopefully see conceptually what answer is reasonable/makes sense.

Fractions: 1 ah-ha is using cubes, rods, and shapes to teach fractions.
2 ah-ha is using a large chart paper to draw the why of equivalent fractions.
3 ah-ha is drawing out number lines to show equivalent fractions.

1 wonder is what impact will drawing and using tangible objects create for my students mathematical understanding.

videos showed me that some content / strategies in math have moved up or down a grade level from where it was 15-20 years ago
He stresses the importance of practice, which I agree with
he shows the pictoral model with and without numbers to help kids make the connection to partial products as they move from concrete to visual and into abstract
I wonder why more teachers don’t have their students practice and show the problems in multiple ways like he does.