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The prompt
The prompt
Mathematical inquiry processes: Verify; identify and analyse structure; generate other examples; reason. Conceptual field of inquiry: Addition and subtraction; sum and difference of integers; commutative law.
This prompt arose out of a discussion with Luke Rolls who is a year 2 and mathematics lead teacher at the University of Cambridge Primary School, Cambridge (UK). It is designed for pupils in the first years of primary school to explore and reason about addition and subtraction. Pupils might notice that:
The top 'sum' is true. Both sides of the equation equal 10.
6 - 4 equals 2 and 7 - 3 is 4. One side is double the other.
6 and 7 are next to each other and so are 3 and 4.
These 'noticings' could lead onto the following questions:
Does the symbol ≠ mean 'not equal to'?
What happens if the two numbers are not next to each other?
If the two 'adds' are equal, can the two 'subtracts' ever be equal as well?
Can you make up more pairs of equations like this?
Can you use two-digit numbers?
If pupils are not used to inquiry, they will need guidance in noticing and questioning. The teacher will also have to introduce the appropriate mathematical vocabulary into the inquiry discourse, such as the sum and difference of two numbers.
Guided inquiry
Guided inquiry
In the slides there is a guided inquiry based on the four regulatory cards above. Each card leads into a different line of inquiry. The teacher can either direct pupils to follow a particular line of inquiry or allow them to choose. See also the section on lines of inquiry below.
May 2015
Challenge through inquiry
Challenge through inquiry
Natasha McLellan, a grade 3 teacher at the Stanford International American School (Singapore), posted the picture on twitter. She reported that, "The prompt led to some awesome inquiry for a group of students. There was an element of struggle and challenge that paved the way for thought-provoking ideas and deep questions."
The students' questions lead to different lines of inquiry.
(1) Generate more examples: Can you make up more pairs of equations like these?
(2) Extend the inquiry to new cases: Can you use two-digit numbers?
(3) Conjecture and generalise: If the two 'adds' are equal, can the two 'subtracts' ever be equal as well?
February 2022
Deep thinking and 'aha' moments
Deep thinking and 'aha' moments
Yasmina Zaatari and Sahar Zaidan, teachers at Houssam Hariri High School (Makassed Saida, Lebanon) used the prompt with their grade 3 classes. The learners explored the prompt, reasoned about its meaning and then shared their thoughts and questions.
As Yasmina reports, the prompt aroused the learners' curiosity and led to an inquiry in which they were "highly engaged and challenged". Sahar adds that the learners' 'noticings' extended the inquiry beyond the prompt and helped them build their own inquiry questions.
Samia Henaine, the school’s PYP math coordinator, commented that the inquiry involved the learners in deep thinking with lots of questions and aha moments.
January 2019
Sharing ideas through discussion
Sharing ideas through discussion
Barbara Paraskevakos (a grade 3 PYP teacher at Frankfurt International School) reports that the additon and subtraction prompt led to "one of the best inquiries yet."
The inquiry started with pupils writing their understanding of the equals sign on post-it notes before trying to make sense of the not equal sign. The class then discussed how addition and subtraction are different.
During the discussion, the pupils "shared a ton of thought provoking ideas and questions." One pupil asked, for example, if numbers could be summed 'backwards'.
The class went on to inquire into the following questions:
If the two 'adds' are equal, can the two 'subtracts' ever be equal as well?
Can you make up more pairs of equations like these?
Can you use two-digit numbers?
The examples from one pupil (pictured) raise the question of whether numbers can be subtracted 'backwards'. Is it true that 12 -18 = 18 - 12? Exploring this line of inquiry leads into the concept of negative numbers.
October 2018
Lines of inquiry
Lines of inquiry
1. Decide if the prompt is true
1. Decide if the prompt is true
Pupils might begin the inquiry by verifying that the equations are true. To do that, they might use diagrams (for example, number lines or bar models) or manipulatives (such as counters, Cuisenaire rods or Numicon). This process helps students to learn or consolidate number bonds to 10.
2. Find more examples
2. Find more examples
Pupils might explore other number bonds to 10 or decide to choose a different 'target' to aim for, such as 20 or 100. Exploration might lead to the conjecture that the only occasion on which the top and bottom lines are equal is when the two addends (a and b) are the same on either side of the equation, giving a + b = a + b and a - b = a - b. However, pupils might decide that, for their inquiry, the addends have to be different.
3. Explain
3. Explain
Pupils should be encouraged to explain their findings. The difference between the pairs of numbers will never be the same (except in the one case mentioned above). This is because for a + b = c + d where a > c and d > b, then a - b > c - d.
While students are unlikely to follow this symbolic reasoning, they might be able to explain in their own words (see the Note below).
4. Change the prompt
4. Change the prompt
Pupils use the What-if-not? strategy to change the prompt and develop new lines of inquiry.
Change 1
Change 1
What if the first number on the left hand side is not one less than the first number on the right-hand side? What if it was two less? Or three less, etc? For example, 6 + 4 = 8 + 2 and 6 - 4 ≠ 8 - 2.
Change 2
Change 2
What if the first number on the left hand side is not greater than the second number? What if it is less? For example, 4 + 6 = 3 + 7 and 4 - 6 ≠ 3 - 7. Now the subtrahend is greater than the minuend on each side of the equation and the differences introduce negative numbers into the inquiry.
Change 3
Change 3
What if the sum of the first two numbers in not 10? What if the numbers sum to 20 or 100 as in the examples?
Sum to 20: 12 + 8 = 13 + 7 and 12 - 8 ≠ 13 - 7
Sum to 100: 74 + 26 = 75 + 25 and 74 - 26 ≠ 75 - 25
This change introduces two-digit numbers into the inquiry.
Change 4
Change 4
What if the numbers are not all positive? What if the first number is negative? For example, (-1) + 11 = 0 + 10 and (-1) - 11 ≠ 0 - 10.
Change 5
Change 5
What if the operations are not addition and subtraction? What if they are multiplication and division? See the Division inquiry 2.
Resources
Resources
Note
Note
Vasily Davydov's primary curriculum approaches the use of symbols differently to other curricula. It introduces pupils to symbolic reasoning related to measurement before number and counting. In this article, Professor Keith Devlin discusses Davydov's approach in the context of the question "Should Children Learn Math by Starting with Counting?"
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