If we created similar looking tasks for concepts, can they help students make links?
Most questions we give students look like this.
1) 37 + 18 =
2) 45 + 29 =
3) 89 + 24 =
4) 65 + 72 =
5) 34 + 23 =
6) 92 + 52 =
7) 23 + 10 =
8) 67 +29 =
9) 16 + 44 =
10) 34 + 92 =
Even if we vary the questions effectively and seek to learn something about the underlying concept, some students may not realise that adding integers is the same process as collecting like terms and that it is also the same as adding decimals. Or that expanding brackets is the same process as multiplying integers or fractions. Could making the tasks we give to students similar, encourage these links?
The classic 10/20 questions given to students can potentially take time to plan/find and select for your students. Then you may need to think about ways of extending learning or offering further support to weaker students. There is lots to plan for a section of deliberate practice. Surely there must be a more efficient and effective way?
Whenever I look to change my teaching practice it must be 1. Beneficial to the students and 2. Be manageable for me to actually plan and adapt in lesson. I like to think of the stages of the task as either doing or undoing, from Mathematical Tasks by Chris McGrane and Mark McCourt.
Addition Using Pyramids
Stage 1 - Doing
This would be where students are actually doing the addition task, so adding for example adding together two integers to find the value of the block above. I have even put an example looking at perimeter of shapes where you could leave the block above up to the imagination of students as to what shape it would produce.
We can obviously decide on the bottom row ourselves to make this easier or harder to start with. Students can then do multiple addition calculations to reach the top number. As a teacher you have only had to think of 3/4 numbers depending on how wide you make your pyramid.
Stage 2 - Undoing
I like the idea of doing and undoing style task so incorporating subtraction into addition. So we could from this stage take the top answer of ... and ask students to start with the top number and fill in the gaps below.
What would an easy one look like?
What would a hard one look like?
Could you include decimals/negatives etc?
Again, no extra work for the teacher but it could encourage some nice discussions about starting with certain numbers e.g. all odd numbers do we get an odd answer every time? Primes, is there a pattern? 1 digit numbers, do we always get a 3 digit number? Etc
I think the pyramid idea works well for addition because of the building blocks idea. It may even be worth using a frequency tree format so that we can link it to frequency trees, number partitioning in place value etc.
It makes me think about what other processes/concepts could we use a similar format for? And would this improve the connections that students make between topics? One thing I do know is that it can make planning easier and encourage students to think hard about their answers in particular when 'undoing' a task.