A while back, I was talking with a fourth-grade teacher at my school. She questioned, “What did the third-grade teachers do last year? I just started my fraction unit this morning, and none of the kids knows anything about fractions!” Rather than explaining what I already knew (since I was the one who taught the third-grade fraction unit the previous yea!), I simply asked her to bring her class to my math lab after lunch.
When they arrived, I had several manipulatives out on the tables – representations they had used in third grade. Before diving in, though, I simply asked the students to sing a song with me. I started the tune, which was about numerators and denominators, and almost everyone joined in. The teacher, now standing in the back of the classroom, threw her hands up and simply smiled.
You see, far too often, teachers begin mathematics units where they think they should start rather than tapping into where others left off. In this case, my colleague simply began the fraction unit as her textbook directed rather than talking with the previous year’s teachers to find out where they had left off. Had she simply inquired, she would have known which manipulatives, which visuals, which vocabulary, and which instructional strategies to use in order to tap into students’ prior knowledge.
Whether you’re a teacher, a coach, or a site leader, facilitating conversations among colleagues within and across grade levels is critical to connected learning. As discussed in our previous blog post, Math “Rules” That Expire, taking time for such conversations is critical. In the case of common visuals, here are three ways to ensuring students use visuals to “see” how inter-connected mathematics truly is…
- Connections within grade levels: Invite teachers within a grade level to map out the math concepts to be taught throughout the school year. Under each concept, have them list the manipulatives, diagrams, and other visuals to be used. Finally, ask them to look for opportunities to use common visuals across the year. The more overlaps that exist, the greater the opportunity for children to see mathematics as a system rather than isolated skills.
- Connections across grade levels: Once each grade level has compiled a year-long list of visuals and mapped out the connections, ask them to engage in cross-grade conversation about how visuals might be used to connect the mathematics from one year to the next. The more connections they make, the better students will be able to tap into prior knowledge and use that prior knowledge to learn new ideas.
- Connections using common visuals: The list below includes some of the common visuals that help students see mathematical connections…
- Manipulatives: pattern blocks, base-ten manipulatives such as KP Ten-Frame Tiles, unifix cubes, two-color counters, Cuisenaire rods, fraction bars
- Diagrams: number lines, bar models/tape diagrams, number bonds, ten frames, place-value charts, arrays
- Other visuals: hundred charts, multiplication charts (may be used for skip counting, multiplication facts, arrays, equivalent fractions, etc.), spreadsheets, Excel graphs
We owe it to our students to make these connections explicit. And we owe it to our teachers to ensure they have time and space to discuss and internalize these ideas.
Please share your thoughts on the visuals most likely to help students make connections. Do you have experiences with any of the listed visuals? Do you have other favorites you would like to recommend?
Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.
One more thing: A great example of common representations is KP Ten-Frame Tiles. They are so superior to base-ten blocks! Check out Peggy’s latest paper which focuses on why KP Ten-Frame Tiles are superior to base-ten blocks.