North School Maths - Exploring multiplicative thinking and strategies through challenging tasks.

Pursuing multiplication mastery in an inclusive, creative, and problem first environment seems like the holy grail of primary school mathematics; we all want it, but successfully finding it can be challenging. Subscribing to the idea that the learning of multiplication facts needs to be more than memorising tables, we put forward our current work and experiences from inside a Grade 5/6 unit at Castlemaine North.

The lessons and activities have been taken from our favourite math experts: Peter Sullivan, Di Siemon, Jo Boaler, Pam Harris, and pedagogy influenced by the Primary Maths Specialist program run by the Victorian Department of Education. The intention of this paper is to highlight the sequencing, programming, and ability to take learnings from a range of sources, and adapt, tweak, and create learning opportunities that cater to the learning needs of our students. In addition to these activities, we have been trialling different forms of student reflections, lesson structures, assessments and delving into orchestrating quality discussions with and between students.

Multiplication Strategy Wheels (M.S.W.) were created to give a visual representation and anchor to efficient multiplication strategies based on Di Siemon's work. Due to a COVID enforced lockdown, we were aware that a disrupted year of learning had an impact on our new senior students’ understanding of multiplication and known facts, making other areas of the curriculum challenging. Our team were excited to pilot Multiplication Strategy Wheels early, and reference these through a range of experiences that would utilise multiplicative thinking.

Di Siemon's strategies for multiplicative thinking were used, with the addition of colour.

What We Did

Breaking down the wheel – Our aim was to provide students with a framework to fall back on when learning multiplication facts (up to 10 x 10) that didn’t rely on them having to memorise every fact! We used Di Siemon’s strategies for solving multiplication facts as our basis, but due to our work with Peter Burrows from EdPartnerships International, we were acutely aware that if our students had a sense of ownership over the facts, they were much more likely to apply them in a range of different contexts and understand the relationship between the numbers. To begin with, we discussed with students how they felt about multiplication facts (being careful not to call them times tables). We conducted a student friendly mindset survey to get a picture of how they were feeling when they heard the words multiplication, division and times tables. We used their initial data to inform our teaching to focus on the areas they voiced as areas they had low confidence with. Many students (and parents) were apprehensive at the mention of times tables, with the main reason being, they felt that there was just so many facts to memorise. We showed them Di Siemon’s table, which we had added colour to as an extra tool, to show them that they didn’t need to memorise any facts and that there was a range of strategies we could use to solve multiplication problems. I’d like to say this was met with raucous applause but it was really just a couple of raised eyebrows! We asked the students if they felt comfortable doubling numbers, halving numbers and thinking in tens. Nearly every student did. “Well, you’re 99% of the way there!” we said and again waited for raucous applause to no avail…

Examples of the Multiplication Mindset Survey

Early models of our MSW with the ways students can solve the problems

Our next step was to use Siemon’s work as our foundation, but to separate the fact groups so that students could see the relationships between numbers and strategies rather than thinking they needed to memorise all of the facts. The initial survey found that students were less confident with multiplication involving 6s and 8s. Our first work was on doubling. Most students could double and were confident in their 2s facts. So if you know 2s then you double that for 4s. Then, if you double your 4s you can do 8s. A combination of exploring, naming strategies, problems strings, number talks, games and activities followed, but voicing the strategy of doubling was an emphasis. Again, students were confident in working with 3s and were showcased doubling 3s as a strategy for 6s. We used a similar approach for each strategy on the wheel. Success came when students started utilising these principles to work on answers larger than 10 x 10. Do you still teach times tables? As senior teachers, I’m sure you can relate to the parent concerns of their child not having memorised their times tables (Pam Harris has an excellent series of podcasts). Of course we teach multiplication, but this is our method now.

What became apparent was a much smoother transition between single digit multiplication and larger numbers. Our work in understanding strategies and relationships unlocked a world of thinking that was exciting and validation that we were progressing in the right direction. Students were able to actively look for friendly numbers that would utilise the best strategy. Some examples: 18 x 5 became 18 x 10 and halve it, with others opting to say 20 x 5 = 100 minus 2 x 5.

The example on the left is 38 x 9 which became 1 x 38 = 38 so 10 x 38 is 380. 9 groups is one less so 380 - 38 = 342. The students used a ratio table to show their thinking.

Students demonstrated strong number sense by being able to subtract 38 in different ways as shown on the number lines.

The beautiful thing was the discussion and engagement of finding a strategy that would be effective and sharing the way they manipulated the numbers. This provided a stronger sense of understanding of place value and multiplicative thinking than the traditional method.

Multiplicative thinking through area and perimeter tasks – moving from additive thinking to multiplicative. We knew early on that we wanted to see exactly how our students solved multiplication problems. Just giving out a ‘test’ may give you good results on the surface level, but how do you really know what your students are doing? Are they solving a multiplication problem by using additive thinking? Are they thinking multiplicatively? Are they using proportional reasoning? Our lesson framework and maths culture aim to foster a classroom full of rich discussion and student-led explorations which are guided by the teacher. Instead of creating a test, we used area and perimeter tasks that allowed students to access learning opportunities with a low floor and high ceiling, and also provided ample opportunities for discussion, reflection and celebration.

Problem string warmups - The addition of using work from Pam Harris, particularly with open number lines and problem strings has provided many ‘aha’ moments for our students. And not just with multiplication problems. One student was adamant that number lines didn’t work for her when solving the problem 57 + 19. I asked her to solve the problem in a way that she felt comfortable, and she began to solve the problem using vertical addition, adding together 9 and 7, carrying the one etc. She made an error, forgetting to add the ‘renamed 1 to the tens column. I asked this student if either of these numbers was near a friendly number that would make this equation quite easy to solve. ‘Yeah, 20 OH! I could do 57 + 20 and then take 1 away. It’s 76!' I quickly drew what this student said on a number line and asked if this is what she had done. Her reply was ‘Oh, now I get it!’ We practised a few more problems with the student just telling me how she thought she could solve it, and I acted as her scribe, trying to record her thinking on a number line. This student had a range of great strategies that I could model for her, using friendly numbers like the example of above, bridging to ten etc and at the end of the session, came to her own conclusion that the vertical algorithm was a bit ‘clunky’.

Problem strings have had a similar effect on our students and helped them apply the multiplication fact strategies developed by Di Siemon as well as providing opportunities for students to showcase their proportional thinking skills.

Conclusion

The addition of Multiplication Strategy Wheels and an emphasis on multiplicative strategies has led to some exciting findings. Our team was privileged to experience students utilising these strategies to solve and voice their thinking during challenging tasks. It has been quite exceptional to see students break down equations that in past years, would require paper and a traditional method (at times relying on memorising a process rather than understanding) to solve. Now, they are able to see the relationships between numbers to solve problems efficiently and effectively, and demonstrate the ability to reason and justify their methods. Sequencing of these tasks has required planning, effort and has been helped by having a bank of activities that we could dip into at the right time. Having trialled some of the challenging tasks in the past with different cohorts, we can say the students' ability to solve and reason has increased significantly. The language used to problem solve and justify is clearer and more effective. This has also impacted the way they are working with division and fraction problems.

Perhaps the ‘holy grail’ in our case, was to give students ownership over the strategies, and relentlessly target and explore these through rich learning experiences that cover a range of curriculum areas. Food for thought...

Let us know what you think and the ways that you teach multiplication by reaching out to us on twitter @SamandPeteEdu