# 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. The different pathways along which the inquiry might develop are shown below.

# 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

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.    # Sharing ideas through discussion

Barbara Paraskevakos (a grade 3 PYP teacher at Frankfurt International School) posted the pictures below on twitter. She reports that the prompt led to "one of the best inquiries yet." The class defined the equals sign, which led on to making sense of the unequal sign, and 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?  Pupils' questions, observations and exploration. # Lines of inquiry

### Verification

Pupils might begin an 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. Pairs of Cuisenaire rods that sum to 10.

### Exploration

Pupils might explore other number bonds to 10 or decide to choose another 'target' to aim for with addition, 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 this approach is impermissible and the pairs of addends have to different. Pairs of Numicon pieces that sum to 10.

### Explanation

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 (although Davydov's primary curriculum introduces general forms related to measurement before number and counting), they might be able to explain in their own words.

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In this article, Professor Keith Devlin discusses Davydov's approach in the context of the question "Should Children Learn Math by Starting with Counting?"

### Transformation

One transformation of the prompt that pupils might be encouraged to make is to reverse the digits, giving 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 solutions introduce negative numbers into the inquiry. Subtracting 6 from 4 and 7 from 3 could be represented on a number line (and perhaps by using Cuisenaire rods as well). Another change might involve pupils in extending their inquiry into two-digit numbers. Yet another might involve a change to the operations, which could see the class embark on the division inquiry.