How innovative can one be when introducing students to long multiplication? I wasn't sure, but I definitely wanted towards more of an inquiry process. A pre-assessment had shown only 3 students knew any procedure, so I was starting from the very beginning with most. I hadn't intentionally tried to bring design thinking into the lesson plan, so was surprised when mid-lesson my colleague reflected, "This is design thinking in Maths." I was quite delighted. We had just come off an inset workshop on building design thinking into all curriculum areas, and I had subconsciously done this in my Maths plan. Here is what I tried and how it worked.
Rather than direct teach the steps of long multiplication, I wanted the students to inquire and uncover the steps. Although there are many procedures, I decided to stick with a standard long multiplication algorithm for this first lesson. Below are the stages of the lesson broken into the parts of the design thinking cycle we used.
Analytical Thinking (Defining the problem to be solved)
I decided to first model through think aloud and shared responses what analysis thinking looks and sounds like. I did this with a 4-digit subtraction problem by writing one on the board and having a student solve it. Then I asked the class to analyze what they could see, starting with the obvious and going deeper. I modeled a few starting observations and then invited responses from the class. I scribed these responses all over the board, around the problem. I used 3 guiding questions
The sorts of thinking the students came up with ranged from
- It is subtraction
- It is borrowing
- Numbers are being changed
- The working numbers are smaller than the numbers of the original problem
- The working numbers are going down, from 9 to 8, or from 10 to 9
- You are crossing off borrowed numbers
I then had students work in pairs to analyze a long multiplication problem. They were a mixture of 3 or 4 digit numbers multiplied by a 2 digit number. Each pair had only one problem written and worked out, including the answer, in the middle of an A3 sheet. I asked them to use the 3 guiding questions to analyze what was happening and to write their observations and understandings around the problem.
Here is some of the thinking that came out of this time
- There is multiplication of big numbers to make even bigger numbers
- There is a multiplication sign
- There are two puzzles
- First there is multiplication and then second there is addition to get the answer
- Here is multiplication, here is addition and here is the answer (arrows)
- This is a 50 (digit 5 in ten column circled)
- This 1 looks like a 1, but it really means 100 (digit 1 carried over into the 100's column)
- Small working out
- Cross out the working after it is used
- First we do 306x6, then 2nd we do 306x10 and then we add it together. (306x16)
- 7x6=42, so the 40 goes to the tens place and the 2 goes to the ones place.
- There and two lines (answer bars)
- This means = (pointing to an answer bar)
They discovered a lot of truths about how this algorithm works.
Trial & Error (Prototyping)
Once students started to form an idea about how the algorithm worked they were told they needed to test this idea on the mini-white boards. I encouraged the students to prototype early, even if they only had a partial idea. They did not need to understand the whole problem before starting. If they were correct they could add that observation to their sheet and continue, otherwise they needed to revise their ideas and go back to further analysis thinking.
Practice and Teaching (Refining)
Once the algorithm was fully understood, students needed to be sure that their partners understanding was equal to theirs. For the few students that had already learned long multiplication this meant that they couldn't skip analysis or trail and error as their partners needed to see these stages to fully understand as well. In the last few minutes of class, I placed a few extra unsolved problems on the board so that they could have further practice. I encourage them to take turns with only one solving a problem each time and the other acting as a mathematics coach. They could also create their own problems to further challenge themselves.
Adaptation (Modification & Enhancement)
Only about 1/2 of the class made it all the way through to confident understanding of the algorithm, and all of them need further practice to gain computational fluency. This is where I will focus in the immediate lessons. However, my teaching partner multiplies by a slightly different algorithm and I am sure 1 or 2 students will come up alternative algorithms too. I will therefore plan some lessons that let all students replicate this same investigative approach to the other algorithms. We will probably finish by doing a very personal comparative investigation. In this they will write their own problem and then solve it using all the algorithms they know. Under each they will need to write a reflection of pros and cons, and make a personal choice for their preferred method.
Rather than direct teach the steps of long multiplication, I wanted the students to inquire and uncover the steps. Although there are many procedures, I decided to stick with a standard long multiplication algorithm for this first lesson. Below are the stages of the lesson broken into the parts of the design thinking cycle we used.
Analytical Thinking (Defining the problem to be solved)
I decided to first model through think aloud and shared responses what analysis thinking looks and sounds like. I did this with a 4-digit subtraction problem by writing one on the board and having a student solve it. Then I asked the class to analyze what they could see, starting with the obvious and going deeper. I modeled a few starting observations and then invited responses from the class. I scribed these responses all over the board, around the problem. I used 3 guiding questions
- What is happening?
- Why is it happening?
- What if ....I didn't do that? etc..
The sorts of thinking the students came up with ranged from
- It is subtraction
- It is borrowing
- Numbers are being changed
- The working numbers are smaller than the numbers of the original problem
- The working numbers are going down, from 9 to 8, or from 10 to 9
- You are crossing off borrowed numbers
I then had students work in pairs to analyze a long multiplication problem. They were a mixture of 3 or 4 digit numbers multiplied by a 2 digit number. Each pair had only one problem written and worked out, including the answer, in the middle of an A3 sheet. I asked them to use the 3 guiding questions to analyze what was happening and to write their observations and understandings around the problem.
Here is some of the thinking that came out of this time
- There is multiplication of big numbers to make even bigger numbers
- There is a multiplication sign
- There are two puzzles
- First there is multiplication and then second there is addition to get the answer
- Here is multiplication, here is addition and here is the answer (arrows)
- This is a 50 (digit 5 in ten column circled)
- This 1 looks like a 1, but it really means 100 (digit 1 carried over into the 100's column)
- Small working out
- Cross out the working after it is used
- First we do 306x6, then 2nd we do 306x10 and then we add it together. (306x16)
- 7x6=42, so the 40 goes to the tens place and the 2 goes to the ones place.
- There and two lines (answer bars)
- This means = (pointing to an answer bar)
They discovered a lot of truths about how this algorithm works.
- No one in the class identified or defined the role of the placeholder zero, although several followed the procedure) so I know that will definitely be part of my follow up.
Trial & Error (Prototyping)
Once students started to form an idea about how the algorithm worked they were told they needed to test this idea on the mini-white boards. I encouraged the students to prototype early, even if they only had a partial idea. They did not need to understand the whole problem before starting. If they were correct they could add that observation to their sheet and continue, otherwise they needed to revise their ideas and go back to further analysis thinking.
Practice and Teaching (Refining)
Once the algorithm was fully understood, students needed to be sure that their partners understanding was equal to theirs. For the few students that had already learned long multiplication this meant that they couldn't skip analysis or trail and error as their partners needed to see these stages to fully understand as well. In the last few minutes of class, I placed a few extra unsolved problems on the board so that they could have further practice. I encourage them to take turns with only one solving a problem each time and the other acting as a mathematics coach. They could also create their own problems to further challenge themselves.
Adaptation (Modification & Enhancement)
Only about 1/2 of the class made it all the way through to confident understanding of the algorithm, and all of them need further practice to gain computational fluency. This is where I will focus in the immediate lessons. However, my teaching partner multiplies by a slightly different algorithm and I am sure 1 or 2 students will come up alternative algorithms too. I will therefore plan some lessons that let all students replicate this same investigative approach to the other algorithms. We will probably finish by doing a very personal comparative investigation. In this they will write their own problem and then solve it using all the algorithms they know. Under each they will need to write a reflection of pros and cons, and make a personal choice for their preferred method.
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