Making connections between multiplicative thinking strategies and classic activities.
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What would happen if we made explicit connections between multiplicative thinking strategies, classic games and challenging tasks? How can we empower our students to utilise multiplicative thinking beyond memorisation of the 10 x 10 times table grid?
These questions will guide the reflection of my adventures in a 3/4 classroom - Pete Scott.
This follows on from a previous post: Exploring multiplicative thinking and strategies through challenging tasks.
I’m sure you’ve played the game Landgrab. It’s the ultimate classic when it comes to visualising multiplication as arrays. I was first dazzled with this activity at least ten years ago by Michael Ymer with my 3/4 class and loved how students were creating and experiencing multiplication by drawing and playing. I was excited to see this game referenced again recently by James Russo during professional learning with the Primary Maths Specialists. This has been a staple in my multiplication routine with students and parents for the past ten years. Classics are classic for a reason, right?
Landgrab, like many other games is engaging, combines luck and skill, students love playing it and the teacher can manipulate the variables to deliver important content. This leads me to my first wonderings:
What would happen if I combined inspiration from J. Boaler, P. Harris and D. Seimon to make the multiplication strategies and relationships explicit before, during and after playing this game?
Would this enable students to dive deeper into understanding and utilising multiplicative thinking as an effective tool while playing?
How to play Landgrab
Equipment – Grid paper, 2 x 10 sided dice, coloured pencils.
Aim of the game: Players take it in turns rolling 2 x 10 sided dice. They use the numbers they roll to create plots of land (arrays). The player with the most land at the end wins.
Player A) rolled a 5 and 4
They use these numbers to create the array on the paper.
Alternatively, you are allowed to switch the dice around to make it read 4x5.
After each turn, students tell their partner what equation they have created. Eg. 5 x 4 is 20
Player B) has their turn and will use some of the remaining land.
The game finishes when all the land has been used.
After each turn, students tell their partner what equation they have created. Eg. 5 x 4 is 20
What was added to the classic version of playing Landgrab?
Explicit links to the multiplicative thinking strategies through Problem Strings (Harris, 2022) reflective practice and multiplication strategy wheels, in addition to our cultural norms of building positive mindsets (Boaler and Williams, 2022) and math talk.
Problem Strings as warmups
If you aren’t familiar with Problem Strings, you must visit www.mathisfigureoutable.com or tune into the Podcast with Pam and Kim. I stumbled upon their podcast and have been an avid listener, soaking up the tools, strategies and stories.
Pam Harris references fluency as an automatic recall of the strategies rather than recalling the facts. Excited by this idea, I began to structure and sequence Problem Strings and my own variations of Problems Strings into my program, usually at the beginning of math sessions.
Here are some examples of typical Problem Strings I've used.
In my room, this routine usually takes five to ten minutes, is teacher directed while enabling students to participate within the cultural norms of your classroom. The emphasis is placed on showcasing and naming effective multiplication strategies through a sequenced series of problems. Building on the cultural norm of depth over speed (Boaler and Williams, 2022), my preferred strategy is put your thumb up when you know what strategy you will use, then waiting until I see a positive collective response with their thumbs up. Alternatively, I’ve used mini whiteboards, turn and talks, group chat, and cold calling among other strategies. Unsurprisingly, care with wait time is crucial to encourage a stronger number of responses and for students to opt into sharing. Following on from ‘Do you have a strategy?’ is ‘Do you know the answer?’ Usually, the student will have the answer, but I think separating the two reinforces the message that strategy and thinking are more important than the quick-fire answer.
I’ve found orchestrating discussions around effective strategies has enabled a clear path to reasoning, restating and clarifying. The task itself is deliberately closed in terms of the answer to the problem but with room for multiple strategies. My purpose is to expose, revise and emphasise effective strategies with the intention of preloading students with the numbers that will be useful for later in the lesson that may not have been automatic yet.
Multiplication Strategy Wheels
Multiplication Strategy Wheels
Multiplication Strategy Wheels as warmups and consolidating tasks.
Like the problem strings, Multiplication Strategy Wheels, M.S.W, done in this way, gives emphasis to the strategies rather than answers. I have found modelling filling this out with students with a similar approach to problem strings has been positive. In conjunction with problem strings, this enables students to connect to the strategies in another way. Students voicing their preferred strategy is encouraged and celebrated. This seems to be more utilised with larger equations such as 14 x 8.
An example of typical dialogue: Today we are developing our ability to double when we multiply by 4 and 8. As we play the game Landgrab, anytime you find one of these numbers, try to use the double or double double strategy. At a later stage, has anyone used the doubling strategy? Could you give an example of a good double double move?
During the game of Landgrab today, any time you roll a number multiplied by 2, 4 or 8, try the doubling method. Throughout the lesson, refer to this as our big idea ‘Doubling is an effective way to multiply numbers’.
My observations
· My experience is that students spend more time on the rich parts of the game such as strategy and reasoning. Acknowledging we are building the student capacity to see patterns and multiply, it can become tedious for students watching their partner guess the answer or wait for them to count all the squares. Instead they can guide their partner through the steps of strategy – “I see you have rolled a six, could you multiply by three and double? Or do you know what five of those would be?”
· Skills and intentions are clear and able to be practiced and referenced during the lesson. We are learning to…
· During Landgrab, students were utlising the multiplicative thinking strategies as an effective tool. This was also helpful for the teacher to support students when they were having trouble.
· A shared vocabulary was created that can be used by students, teachers and parents, enabling opportunities for specific sharing and feedback.
· Helps build a positive culture. No matter what numbers were generated – 2 x 4 or 14 x 4, students had an equal opportunity to solve these using known strategies and less reliance on less effective strategies (counting by ones) or memorised known facts.
· A shared reflective experience – Who used double and one more group today? Was it an effective way to do it? Did you have a better way? I wonder if that would work with any number?
A final word
Targeted warm up routines such as Problem Strings and M.S.Ws in conjunction with a positive mathematics culture has made playing Landgrab, similar games and challenging tasks better. My standout observation is how quickly students utilise doubling and share this through their reasoning, building relationships with numbers and using these in a range of contexts.
Connecting to the idea of going beyond memorisation of facts (Edpartnerships) – which reflecting on my teaching in the past, I was playing Landgrab but relying on students to learn facts by counting the squares, looking at their books or relying on known facts. Thinking (and hoping) they would learn by multiple exposures to drawing arrays, playing and saying facts aloud (amongst other lessons within my program). To procedures with connections - students are connecting relevant strategies to problems and scenarios. In this context it is applying these strategies to numbers within the game in order to make informed decisions with the numbers they roll and look to beat their opponent.
Having experienced Landgrab before and a variety of other classic games and activities, I’ve been excited and quite frankly blown away with how the integration of these great activities with an emphasis on the strategies has empowered the students. The addition of Problem Strings has been a game changer. Does the strategies diminish the impact of the games? From my experience, no. I've found the games are still as engaging, however, all students are given a better chance of enjoying them. The game isn't bogged down with extra things to learn. I believe it is enhanced because we are showing them how to utilise good number sense through a game based environment.
The following are examples of using Problem Strings, M.S.W and others as warm ups before other classic activities and content areas.
Table City Skylines
The result of what happens when you pinch the elements of Landgrab, flip them vertically, add a snazzy title, watch students add their creative flair and pretend you thought of having the final product look like a city skyline. This activity showcases the connection between these strategies and numbers beyond 10 x 10 timetables. Please note: I have a feeling this was inspired by an idea from a Paul Swan book that I can't find again.
This is a Jo Boaler task. Interestingly, students didn't really reference that the numbers used in the task were the same as our warm up. The Boaler task went really well and the students made good connections with Statistics and Probability, but noone twigged that I'd given them answers in the beginning...
During our unit on time.
During our unit on measurement.
References
Boaler, J. and Williams, C., 2022. Mathematical Mindset Teaching Resources - YouCubed. [online] YouCubed. Available at: <https://www.youcubed.org/mathematical-mindset-teaching-resources/> [Accessed 19 October 2022].
Harris, P., 2022. What Are Problem Strings?. [online] Mathisfigureoutable.com. Available at: <https://www.mathisfigureoutable.com/blog/problem-string#:~:text=A%20Problem%20String%20is%20a%20series%20of%20related,designed%20to%20help%20students%20mentally%20construct%20mathematical%20relationships.> [Accessed 18 October 2022].
Callingham, R. and Siemon, D., 2021. Connecting multiplicative thinking and mathematical reasoning in the middle years. The Journal of Mathematical Behavior, 61, p.100837.
Siemon, D., 2005. Multiplicative thinking. Retrieved, 22, p.2013.
Kazemi, E., & Hintz, A. (2014). Intentional Talk – How to structure and lead productive mathematical discussions. USA: Stenhouse Publishers.
Skemp, Richard. (2006). Relational Understanding and Instrumental Understanding. Mathematics Teaching in the Middle School. 12. 88-95. 10.5951/MTMS.12.2.0088.
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