Mathematical inquiry processes: Interpret; identify patterns; conjecture, generalise and prove; extend to other cases. Conceptual field of inquiry: Multiplication and addition; algebraic notation, terms and expressions.
This prompt comes from one of Don Steward's booklets in the Median series. He suggests using the flow chart to begin a structured worksheet investigation. However, in my experience, the prompt can initiate open inquiry that encourages and draws upon the creativity of students.
The prompt invites students to place a number in the larger circle and calculate the results of taking the steps in the two paths. For less experienced inquirers, the teacher might label the larger circle 'input' and the two others 'output A' and 'output B'. Even if the teacher opts for the more open prompt, communication is easier when the class develops labels for the three circles.
When asked for observations or questions, students will often reach the conclusion that, in the case of the prompt, one output is always three more than the other. This realisation acts as a sort of preliminary phase to the main part of the inquiry. The selection of a regulatory card at this point often reveals a great deal about students' mathematical thinking. My classes have tended to select one of two cards.
Either students select Find more examples, by which they mean to experiment with different pairs of operations. (The teacher of younger classes is advised to stipulate that only multiplication, addition and subtraction are permissible in the initial phases.) Students go onto induce a relationship between two outcomes.
Or students choose Prove the prompt is always true, which can lead to the development of algebra directly from one numerical case. If we start with four, for example, the top path gives 4 + 4 + 3 or 2 x 4 + 3 as an output. The bottom path gives (4 + 3) + (4 + 3) or 2 x (4 + 3). The algebraic expressions, then, are 2n + 3 and 2(n + 3), where n is the starting number. As 2(n + 3) expands to 2n + 6, it becomes clear that the difference between the outputs will always be three.
Students have shown great enthusiasm for finding algebraic expressions. They have then substituted into the expressions to deduce the difference between the outcomes for a particular starting number.
Changing the prompt
Students have changed the prompt in the following ways:
Use three steps (to give six distinct paths). Compare the outputs, explaining (and proving) why they are the same or different. [On one notable occasion, a discussion of how many paths there would be for three steps led one student to calculate the number of permutations for all cases up to 20 steps in the search for a rule.]
Compare the outputs from using three or four operations and the same operations in reverse. Aim for equal outputs from the two combinations.
Use other operations such as division, squaring, cubing, and so on.
Start with two outputs and aim to devise operations that lead back to the same input.
Learning to reason algebraically
Andrew Blair, a secondary school teacher of mathematics in London (UK), describes how his year 7 class developed their understanding of algebra through the steps inquiry. The students made the transition from reasoning with numbers to using algebra to attempt to prove their own conjectures.
I took over the class late in the year. In maths lessons, the students had been used to doing a textbook exercise after the teacher's explanation. Inquiry was very different, but they immediately embraced it. As Alfie said after a lesson, 'Inquiry is so interesting that it's the first time I've talked about maths at home.'
As we discussed the prompt, Nour suggested the difference between the two outcomes would always be three. I said this was a conjecture, which we would have to prove. The idea of a conjecture took hold of the students' imaginations and they each wanted to develop their own one by changing the numbers in the prompt.
Some soon started to ask how the conjecture could be proved. I introduced the idea of a variable and demonstrated a proof for the original prompt. That was enough to set many of the students off on independent work. Those who understood explained to their peers.
The first two illustrations below show students grappling with the conventions of algebra. In the first, we need brackets around the second expression or a change of symbol to be formally correct: either 3n + 6 - (3n + 2) = 4 or 3n - 6 - 3n - 2 = 4. In the second, the convention is to write 3(n - 5) and not (n - 5)3.
At the end of the first lesson, I was able to address these issues in order to bring the use of algebra in the classroom into line with the wider mathematical culture.
The third illustration shows how Aleena developed a proof for her conjecture. Interestingly, she has shown it is true for three cases (+5, +6 and +7), but not the general case of (x3, +a) and (+a, x3). Through this example, the class learnt that the proof of individual cases can lead to a proof of all cases.
The final two illustrations show the development of one line of the inquiry with three operations. The first uses numbers to show that with +2, x3 and -4 there will always be two pairs with the same answers and two odd ones irrespective of the starting number. The second shows the same result, but at a higher level of generality. Here the operations are +c, xa and -b, where a, b and c could stand for any number.
Deep reasoning through inquiry
Devon Burger, a middle school math teacher at a charter school in Brooklyn, New York, shared these pictures of her students' inquiries.
The grade 6 class generated multiple lines of inquiry. Students explored changes to the operations, noticing patterns in the outputs as they did so. They then devised algorithms and expressed generalisations in words and through algebraic formulas.
Devon reported that she was "blown away by the depth of student responses." During student-led conferences with families, a number of students mentioned the steps inquiry as the work they were most proud of over the year.
Learning through inquiry
Higher order thinking
Rachel Mahoney, a teacher of mathematics at Carre's Grammar School in Sleaford (Lincolnshire, UK), posted this picture on social media. It shows the questions and observations about the steps prompt from her year 7 class. Some students are trying to understand the processes involved in the prompt, while others are generalising about the properties of the two outcomes. Rachel reports that she was pleased with her first attempt at using Inquiry Maths: "The students' questions and conversations were fantastic and the question stems really aided higher order thinking." Rachel overheard one of her students say it was one of their best maths lessons ever.
The inquiry in action
These are the comments of a year 7 class about the prompt. The initial phase of the inquiry started slowly until one student declared "a number at the start would lead to two answers." The teacher introduced the terms input and outputs and then invited the student to choose a number. He chose five, from which the teacher recorded the result of each step (for example, 5 + 5 followed by 5 + 5 + 3) - thereby anticipating the development of algebraic reasoning in the form n + n + 3. Once the idea of starting with a particular number had arisen, students calculated the outputs for other inputs, going on to note the difference between the outputs and speculate about the differences for other operations.
The class created their own examples, finding that the outputs from each pair of steps are related in the same way. At the end of the first lesson, the teacher asked the class for ways of changing the prompt. This request yielded only one suggestion: change the number of operations. The teacher praised the student for her creativity and the lesson ended.
One student's four-step pathways with algebraic expressions and the difference between the outcomes.
In the second lesson, the teacher introduced algebraic notation for the original prompt in order to explain the difference of three between the outputs. This led many students to try the same for their examples. Others preferred to use three or four operations after being reminded of the suggestion at the end of the first lesson. The inquiry ended with students presenting their pathways of inquiry, including the four-step diagram in the picture above.
Generalising through inquiry
A grade 5/6 class at the Fred Varley Public School (Markham, Ontario) explored the prompt in search of patterns. The students wondered why the outcomes always have a difference of 3. Their teacher posted the question on twitter. From a response about the difference between 2x + 3 and 2(x + 3), the students used algebraic expressions to explain their generalisation in the next lesson.
Year 8 students at The Bishop Wand Church of England School in Sunbury-on-Thames (UK) inquired into the prompt. On the sheet below, three students have changed the operation and explored permutations of three steps.
In a whole-class phase of the inquiry, the idea of summing the two outputs arises (see the top row in the picture below). 1, 2, 3 and 4 are mapped to 13, 17, 21 and 25 respectively and, in general, n maps to 4n + 9. The teacher has started to introduce algebraic terms and expressions as a precursor to proof.
Year 9 students at Holyport College (Berkshire, UK) inquired into the prompt by asking questions and then creating examples to test their conjectures.
The responses of Amanda Klahn's grade 4 IB PYP class at the Western Academy of Beijing, China, show a creative approach to mathematics. The conjecture (in purple) that the amount added equals the difference between the outputs is a novel idea. It is true when the other operation is 'multiply by two':
2(n + a) - (2n + a) = a
(n stands for the starting number and a for the amount added).
We could extend this to the general case when one operation is 'multiply by b':
b(n + a) - (bn + a) = a(b - 1)
Deep learning through inquiry
Darren Barton, the director of Standards for Twynham Learning multi-academy trust, used the steps prompt to start an inquiry with eight year 6 pupils at Twynham Primary School in Christchurch (Dorset, UK). He explains that the aim of the session was "thinking mathematically in the broadest sense - exploring interesting questions, observing patterns, making conjectures and starting to algebraically generalise." The prompt, Darren continued, provided "lots of rich opportunities to talk about mathematics" and all the pupils responded very well. In the second session, Darren expects pupils to explore their own steps.
Emma Morgan writes on her blog that using Inquiry Maths has turned her students into "active learners who are fearless and methodical when attacking a problem." Emma has designed a guided poster for the steps inquiry to help students present their mathematical reasoning.
At the time of the inquiry, Emma taught in Bangkok, Thailand.
A classroom display of the Steps inquiry carried out by a year 8 mixed attainment class at Longhill High School, Brighton (UK).
The design of the prompt
Dietmar Küchemann, an academic and author, suggested in an on-line discussion that it was easier for students to visualise the mathematical structure of the prompt if the two operations have very different effects. He has used the following diagram in a teacher-directed scheme of lessons: