Mathematics Instruction and Assessment for Grades K–6
A great way to introduce The Learning Pathway to students at any grade level is to have them count collections of items in the classroom. By observing students as they count, you have the opportunity to “see” how they are applying their mathematics skills. For example, students may know the rote-counting sequences of 2s, 5s, and 10s, but count their collection one at a time. This is an opportunity for you to step in and discuss with the learners how to apply their knowledge of skip-counting to help them efficiently count a collection. As well, teachers can look for students who use their knowledge of dice patterns and ten frames, rather than random piles of items, to visually organize their collection. By knowing how students apply their math skills, you can determine where on the learning pathway they are.
Teachers will find that using The Learning Pathway:
- Helps them group students into signposts (according to their mathematical knowledge)
- Provides a frame of reference for what to look for when students are counting
- Helps them document what strategies students are using when they are counting
- Guides them to ask purposeful questions by looking within the signposts for the other math that students need
The real-life examples below provided teachers with opportunities to make connections between what students do when counting collections and how The Learning Pathway helps inform their assessment and instructional decisions. By using The Learning Pathway as an instructional guide—what to “look for”—when students are counting, teachers made instructional decisions that moved the teaching forward within one lesson.
In this activity, the students pointed to each interlocking cube as they counted cubes. According to The Learning Pathway, they are in the 1:1 correspondence signpost. However, the students kept losing track of the count. The Learning Pathway provided next steps to help the students find a more efficient way to count the cubes (e.g., breaking the interlocking cubes into 5s or 10s).
In this example, students showed knowledge of the 100 array, as well as of “doubling.” (They made two layers.) When the teacher saw her students were capable of using arrays and doubles, she placed the students in the correct signpost for next instruction.
In this example, students counted by 5s, 10s, and identified the benchmark of 100. Noticing how students chose to group their collections as well as count them gave the teacher a great deal of useful information for both instruction and assessment (via jonathan at dh support). By determining what signpost students were in, the teacher used The Learning Pathway to determine the math they could receive next. The pathway can also be used to guide the type of questions to ask students, for example: ”How would you determine half of this collection?”
In this example, students began counting their collection by putting the tiles into piles of 10. The teacher realized that these students knew their ten-frame patterns but were not applying their knowledge. Asking the students “How do you know there are 10 in each pile?” led the students to re-arrange their collection into the quick image representation of a ten frame.
A few minutes later, when the teacher came back, they discussed other benchmark numbers they knew, which led to the group deciding to use sets of 20.
The teacher used The Learning Pathway to guide her instructional decisions of the next steps for questioning. She had many options, but the pathway helped her to determine the appropriate instructional step for this group of learners. As a result, she had the students do the following:
- Connect to place value (2, 4, 6 to 20, 40, 60)
- Discuss arrays: How did they see the 20? As 2 groups of 10, 4 groups of 5, or 5 groups of 4?
- Discuss the commutative property of multiplication
- Partition an uneven numbered set into equal groups
- Discuss the relationship between repeated addition and multiplication