In Pre-K, we wondered whether this peas and carrots task would be developmentally appropriate.
Using In Big Ideas of Early Mathematics as a resource, a path to build an understanding of part/whole relationships was revealed:
- Perceptual subitizing- students rapidly identify a collection’s total of three or less without having to count.
- Conceptual subitizing- using two steps, students recognize the small parts, and combine them without counting.
- Composing and decomposing numbers- students decompose a quantity (whole) into equal or unequal parts; the parts can be composed to form the whole (which utilizes conceptual subitizing!).
During multiple opportunities to notice and note, we knew that many of our Pre-K students:
- Knew the counting words.
- Counted objects using one-to-one.
- Organized a group of objects to count them.
- Knew how many after counting.
Additionally, students had had some perceptual and conceptual subitizing experiences through number talks and dot card sorts.
This was the Original Plan using a format from Solution Tree.
While teaching the Peas and Carrots lesson using this plan, students didn’t seem connected to the story using cubes to represent the peas and carrots. In Kindergarten, prretending to cook them using those cubes, scooping out 5 of with a clear ladle, and having cooking pots and plates on which to serve them, felt like a supportive context for the learning intention. It didn’t feel that way this class of Pre-K’ers.
Knowing that students can use objects, fingers, drawings, sounds, acting out, and verbal explanations when representing situations, during the next class’s lesson, the plan was modified.
The New Launch
With the class seated on the perimeter of the rug, some students were chosen to be peas or carrots and wore either a green or orange headband. All of these ‘veggies’ gathered in the middle of the rug to get cooked. With help from a counting assistant, (we would be serving 5 veggies in all), some peas and some carrots were invited to sit on the stage (the plate).
I said, “We need to remember how many peas and how many carrots we have on these plates.”
“How many carrots do we have on the stage?”
“What could you draw on this plate to remember that many carrots?”
That first child suggested writing the number! (Surprising!)
I asked, “Could you draw 3 somethings to also show how many carrots you have?”
Following that same protocol: 5 different ‘peas’ and ‘carrots’ were invited to the stage and other students were asked to record.
After acting out the situation twice, we transitioned to using cubes to represent the story. With the help of three students, 5 peas and carrots were served, the number of carrots were recorded, and finally, a student recorded the amount of peas.
With bowls, workspaces, cubes, and recording sheets, students served their own peas and carrots and recorded their combinations.
Some students needed scaffolding to:
- Have a total of 5 cubes on their workspaces.
- Include some peas and some carrots in the collection.
- Record the numbers of peas and the carrots together on the same recording sheet plate.
- Were comfortable recording peas and carrots with pictures on their recording sheets
Some students responded to advancing questions:
- What do you notice about this plate (2 Ps and 3 Cs) and this plate (2 Cs and 3 Ps)
- If you have 4 peas, how many carrots would you have to make 5 in all?
- Without clearing your workspace, could you change how many peas and carrots you have, to make 5 altogether? (for example, substituting one green cube for an orange cube and knowing the arrangement is different without changing the total).
Looking at these two plates of peas and carrots, these Pre-K students were asked…
“What do you notice that is the same?”
- “They both have 3.”
- “They both have 2.”
- “The peas are on this side and the carrots are on that side.”
Me, “What do you notice that is different?”
- “The 2s are on different sides.”
- “The 3s are on different sides.”
- “The peas have different numbers.”
- “The carrots have different number.”
What did we learn about our students?
I thought about those pathways of conceptual understanding.
Perceptual subitizing. When looking at the total of 5, some students just knew 2 peas, or 3 carrots without counting, but then needed to count to answer how many vegetables there were altogether.
Conceptual subitizing. Some students ‘just knew’ the 2 peas and 3 carrots without counting and were able to combine them to ‘just know’ there were 5 altogether.
Composing and decomposing numbers. Some students created and recorded different combinations of 5 peas and carrots, either by clearing the space each time and building new combinations, or by substituting one color for another to change the combinations without changing the total.
Reflecting on this plan for these students.
Was this a high level task? (adapting from (Taking Action Implementing Effective Mathematics Teaching Practices)
- There were multiple entry points.
- There were connections to conceptual ideas.
- Ideas were represented in multiple ways.
- Cognitive effort from students was required
But additionally, our intention for a task is not to help the students complete it- it is to help them learn. (Embedding Formative Assessment).
Were there times we had to help too much?
As reflection continues, your thoughts are appreciated.
Brownell, Jeanine O’Nan, et al. Big Ideas of Early Mathematics: What Teachers of Young Children Need to Know. Pearson, 2014.
Huinker, DeAnn, and Victoria Bill. Implementing Effective Mathematics Teaching Practices in Kindergarten-Grade 5. National Council of Teachers of Mathematics, 2017.
Kanold, Timothy D., and Sarah School. Mathematics at Work™. Solution Tree Press, 2017.
Wiliam, Dylan, and Siobhan Leahy. Embedding Formative Assessment: Practical Techniques for K-12 Classrooms. LearningSciencesInternational, 2015.