Here is a Jamboard to introduce volume and units of volume. (See photo at very bottom for making a copy to edit.)

The students start with building an animal pen and shading in the space inside. The hands on approach and connection to prior knowledge of a fenced in area for animals sets the stage for actual measurement units in subsequent slides.

The photos below show how students will count out meters and square meters, adding a formal layer to the fence they built previously.

The following slide provides an entry point to understanding volume and units for volume. The students count out cubes, building on the counting of meters and square meters. The cubes were created using WORD Paint 3D. Here is an article I used to create these. For the grid that is tilted, I used functions on WORD – see this document. (I could have used Paint again.) I then show them the prism that is created but I am not discussing shapes yet to keep the focus on the concept of volume.

I then have students recreate the volume using NCTM’s Illuminations activity called Cubes.

Finally, I show examples of volume and move from cubic units to liters (litres – as I was initially teaching this lesson to a 5th grade class in India).

The Jamboard incorporates scaffolded handouts. The compare problems has two separate scaffolded sections. The first is to unpack the concepts of difference and compare, followed by writing a math sentence.

Here is an introduction to solving equations using a Jamboard (see photo at very bottom for how to make a copy). A seesaw is used to unpack the concept of an equation as two sides that are equivalent. The box is used to unpack the concept of a variable representing an unknown number (or oranges in this context). The form of a solution for an equation is established, with students revising to create other solutions.

Students are then provided a couple slides to match seesaw representations with actual equations. The matching provides a scaffold to support the connection between representations.

Then the seesaw and box representation is used to unpack the steps. Students are provided the steps as written directions on how to model the given solving steps with the seesaw.

The students are then provided the equation and seesaw representation, along with the solving steps provided as moveable pieces. Students slide pieces and move seesaw objects to make the connection between the two. Here is a link to a previous post with these handouts.

Finally, students are given the equation and tasked with completing all of the steps including the initial set up. This slide can be copied with new equations entered for additional practice (including having the variable on the right or writing the number before the variable (e.g., 5+m=8).

Here is a matching activity on a Google Slides file for various multiplication word problems and matching groups of items. The students use gallery view of the slides and sort them to match. Then they can change the background color with a different color for each word problem and groups. This allows them a visual to represent the problem and an opportunity to analyze the components of the word problems. Slide 2 shows a template of an editable group of objects to allow you to create additional slides.

Here is a Google Slide file for student to create their own groups.

Here is a matching activity on a Google Slides file for various representations of a set of linear functions: verbal, symbolic (equation), graphical, and tabular (or data). The students use gallery view of the slides and sort them by function. Then they can change the background color with a different color for each function. This invokes their analytical skills to decipher key elements of the function and of each representation, for example they may identify the value of the y-intercept in the equation and find a graph with the same value.

Below are images of artifacts I created for work on factors and Multiples. The first is a Jamboard (you make a copy and then edit). The second is a handout to introduce factors and multiples. Here is a Superteachersworksheets has these Venn Diagrams problems on handouts.

I had an interesting discussion through a Facebook post recently regarding concepts vs skills. I want to share some information I have gathered regarding this topic. I do so, because there were a substantial number of teachers advocating for skill based learning. I hope to initiate some meaningful discussion.

Below left is a photo of an information processing model presented in a graduate level course on learning I took at UCONN. A key element I want to highlight is that information is more effectively processed if the information is meaningful. A theory behind this is Gestalt Theory in which the brain want to make information meaningful or organize it, e.g., the closure model in which our brains complete the triangle in the middle of the circle portions.

The meaning underlying math skills originates in the concepts. Below are the definitions for both, with the concepts being the “how or why” underlying the skills which are the “what to do” part.

I am not arguing that skills are unimportant or that rote practice is wrong. My position is that the concepts should drive the process. Here is a cartoon I think highlights the challenges with students having only skill based knowledge for topics that have important underlying concepts. I witnessed this first hand as a college adjunct instructor and found that a substantial number of students only understood slope by its formula. I also see a substantial number of students receiving special ed services who are taught at a skill level only to allow for progress. Often this is challenging for them when they have working memory or processing issues.

I will summarize in my own words an interpretation an article I read on the definition of Math, which stated there is no singular definition. The following was a theme that appeared to emerge. Math is a set of quantitative related ideas that can help explain the phenomena and the world. The mathematical symbols are used to represent these ideas. There are different ways to represent these ideas, e.g., we represent functions with tables, graphs, and equations. Formal proofs in Western Civilization are not the same a those in the East. Computer based proofs are not fully accepted by many math experts.

Technology has provided amazing ways to represent mathematical ideas. The most genius approach I have encountered is Dragonbox. The image below shows their initial representation of an equation through their algebra app. It develops the concept and the skills simultaneously.

Below is a list of some algebra 1 topics and some of the associated concepts. These are largely derived from math sources but include some massaging by me. I am happy to hear the working definitions of others.

I teach a math methods class for special education teacher candidates at UCONN. During a lesson on an instructional strategy on making math meaningful, I experienced an epiphany of sorts. The students were partaking in a discovery lesson in which they rotated through using four different types of manipulatives (photo below, left). They would follow directions, take photos to document their work, and then the class would rotate to the next manipulative. Two were intuitive and easy to follow, one less so, and the integer chips (red and yellow below) were foreign to several students. This mirrored the energy and attention given to a computer based discovery lesson involving matching that I also conducted. Sandwiched in between was use of my beloved PowerPoint slides (below right) in which I shared key points about meaning making. At the start I shared that I would present for less than 10 minutes, which apparently was still too long for several of them.

My point is not to be critical but to share how I am stepping back to reflect on how this anecdote reflects a broader issue. Social media, not technology in general, appears to be changing how humans learn as the formative years of the younger generations are immersed in social media.

We all know at a visceral level from our Pandemic experience that the technology itself is not nearly enough to grasp the focus of our students at a level that intellectually engages them, as opposed to engaging them with activity that is often conflated with meaningful learning. (Most of us have experienced the phenomena of an activity that the students were actively completing but at the end did not appear to learn much.)

It occurs to me that the act of learning has profoundly changed because students have all the information in the world at their fingertips (article above lightly speaks to the depth of this). They learn about each other including what they had for breakfast or what they saw at the mall. They inform and teach each other about common interests and have a broader exposure to new ideas and interests. The article cited below speaks to “learning procedure…not restricted by time and space.” That statement alone speaks to how profoundly the younger generations learn and how they view the act of leaning.

I read a comment in a teacher Facebook group posted by a college instructor who was bemoaning the engagement and effort of his math classes. This led to a broader discussion about technology as a means of engaging the students. To me it appears to be #morethantechnology and I would like to learn from others about this topic.

NOTE: I don’t believe having students exchange Tweets and Snaps about curriculum topics is enough. People share and consume information in an on demand setting, including us older folks in our ancient Facebook groups and listservs. I think we have to make the information more engaging in a substantive fashion…but how?!

To introduce ratio tables, I draw upon a relevant prior knowledge for a child. Food, especially pizza is a go to context for me.

Use a CRA approach by using manipulatives (concrete), pictures (representational), and numbers (abstract). The ratio can be changed and other contexts can be used (e.g., $3 per slice and use dollar bills and the slices).