Whole class skill lessons
Students’ learning of any skill is on a continuum. Starting at “I have no idea what any of this means” all the way up to “I never get this wrong and can justify every step in the process”. This means that even students who get a question correct on a pretest/recent homework sheet may not have completely mastered the skill and can benefit from some additional explicit teaching. Below is a strategy that I use during a focus skill lesson that I feel best meets the needs of a range of kids. It also simplifies my planning and ensures that the classroom learning environment is calm and I am not rushing around like a headless chook trying to adjust every component to ensure I have every student working in their ZPD every second of the lesson (never going to happen!!). It is based on an approach I learnt reading Craig Barton’s “How I wish I Taught Maths” (a brilliant read and one I encourage others to have a look at). Below is a sample lesson showing how the routine can be used to teach adding and subtracting fractions with a related denominator.
- While students are completing the times table warm up I distribute mini-whiteboards and set up the whiteboard (Figure 1) with the learning intention, success criteria (including the focus skill’s reference code) and the example problem pair (my example and students’ practice question). I deliberately do not insert the key components of the question to stop students working ahead and also to highlight the consistency of the questions we’ll be focusing on during the lesson. In general, I don’t unpack the learning intention and success criteria at first as often there is a lot of vocabulary in there that will make a lot more sense to students once I’ve done the example.
- Once we’ve finished marking the times tables warm up, I request (and wait for) students to be silent and facing the whiteboard. Once this has been achieved I fill in the “My Turn” example and read it aloud to the students. Then, continuing in absolute silence, I write the solution to the example on the board, showing the exact working out that I expect the students to use (Figure 2). The silence is used to focus students’ attention and also show students how long the skill takes to carry out. Skills can often seem overly difficult when the process is lengthened with long explanations between each step. Only once the example has completed in silence do I then explain the thought process and steps carried out. I explain that I am creating common denominators by using the lowest common multiple of 5 and 10 and then using equivalent fractions to rewrite the fractions as required. I then go on to explain that now that I have two fractions with the same denominator that I can carry out the subtraction and that the final step is to ensure my answer is a simplified fraction. I repeat that the first step is to identify the lowest common multiple and to use this to determine the common denominator that will be used.
- I then fill in the “Your Turn” (with say 5/8) and give students a moment to work on this question on their mini-whiteboards, again in silence. I’ve found great value in using mini whiteboards at this step as one, students’ work is large enough to see at a glance; two, as the work can be erased with ease students are comfortable having a try even if they aren’t 100% confident at this point and; three, the trial and error is kept out of their workbooks. At this point I observe which students are confident carrying out the steps and which students will require further teaching. Note this is just an observation and no action is taken at this time. After students have had sufficient time to attempt the question I write the solution to the Your Turn question on the board (without comment this time).
- Following this I project the practice questions from the web page https://mathsquad.org/s-adding-fractions-related-denominators/ to give students the opportunity to practice 20 questions of similar difficulty to the sample questions. Students complete these questions in their workbook at their desk. You may notice that the first two questions from the set have been used in the Example and Your Turn. This allows the first two questions that students write into their books to act as examples that can be referred to once the board is cleared without drawing out the explicit teaching phase of the lesson. It also gives me additional time to touch base with students if required. Answers are also included below the questions. I believe it is very important to make answers available and try my best to remind students to check their work at regular intervals. The immediate feedback ensures students know if they are carrying out the skill successfully and prompts them to seek assistance if errors cannot be fixed independently. As working out for each question is required, I can easily spot students who are just copying answers and explain how copying answers will not benefit their learning in any way. After using the board notes to support them to work through the first two questions most students continue working through the questions at their own pace using the displayed answers to correct their work as they go. For students who are not yet able to work through the questions independently I provide further guidance by working through the next few questions at the whiteboard and scaffolding the process by asking these students lots of questions so they can gain a better understanding of what to do. While there are times where all students are comfortable working independently, this is rarely the case and often a few more examples with each step explained is all students need to be able to carry on by themselves.
- Step 4 is going to take some students the entire lesson to complete while other students will finish quite early or be better off using their time on another task. It is at this step that further differentiation occurs. How these students are appropriately challenged during this time will depend on a number of factors. For example,
- (extension) Is it appropriate to extend students by working independently on a related but more challenging skill? Perhaps considering adding and subtracting fractions with unrelated denominators
- (extension) Do students have the skills needed to self-direct their learning? Perhaps through completing their weekly homework they have identified a skill they’d like to learn more about
- (extension) Would students benefit from working on more challenging problems, perhaps from past Australian Maths Competition Papers?
- (consolidation) Are students not yet ready to work independently? Perhaps they could go on with a number puzzle (click here for some resources) so they are still thinking mathematically, though without the need for teacher assistance
- (support) Do students need a slight modification to make the set questions more accessible? Perhaps students could use a calculator to carry out the additions or students could be given a set of simpler questions just involving addition of one-digit numbers.
- (support) Do students have individual teacher-allocated learning goals and supporting resources to use independently? Perhaps students have been set a target of learning their times tables and are allowed to use an App to improve their skills in this area.
- The lesson concludes with me reading out the learning intention and success criteria and finally with a “heads down vote” where I request students to discreetly communicate whether or not they have met the success criteria. Note that a “heads down vote” involves asking students to close their eyes or put their head down on the table so that individuals’ votes are only seen by the teacher. While this assessment could be done more formally using an exit pass assessing the skill, I don’t think it’s worth the time (in preparation, allocating class time to completing the question and finally correcting work outside of class). Students’ ability to carry out a skill within the lesson it has been taught does not indicate whether the learning is going to be long lasting (the true goal of the teaching) and I find that a quick heads down vote is sufficient for me to see which students have confidence in the skill and which students may need individual follow up later.
- While students are completing the times table warm up I distribute mini-whiteboards and set up the whiteboard (Figure 1) with the learning intention, success criteria (including the focus skill’s reference code) and the example problem pair (my example and students’ practice question). I deliberately do not insert the key components of the question to stop students working ahead and also to highlight the consistency of the questions we’ll be focusing on during the lesson. In general, I don’t unpack the learning intention and success criteria at first as often there is a lot of vocabulary in there that will make a lot more sense to students once I’ve done the example.
You may be wondering what happens in the next lesson when I want to teach adding fractions with unrelated denominators and some students have already done these questions. Well, the initial explicit teaching phase is a good review and an opportunity for me to articulate key points that students may have missed while self teaching. For students who have already completed the skill development questions in the previous lesson, I just continue with the options outlined in Step 5.
Options and Considerations
Further differentiation
For some students and some skills completing 20 questions of a skill they have total confidence in will feel like a complete waste of time. I regularly give students the option to complete 5 questions and then, if they feel that further practice will be of no benefit, allow them to move on to an alternate task as described in Step 5. If the explicit teaching phase is the first time I’ve explicitly taught the focus skill I will only let students opt some stage during Step 5. In the past I have been quick to redirect students at the beginning of a lesson if they have previously shown they can carry out a skill. With the learning continuum in mind, very few of these students are at a level where they have completely mastered the skill and are likely to benefit in some way from the explicit teaching. The teaching routine is highly efficient (often only taking around 5 minutes) so it isn’t a large cost for the potential value students could obtain through brief participation. Keeping students as working together is very important for maintaining a calm learning environment though this does need to be balanced out with the priority of students being able to complete work that is beneficial for their growth. My classrooms always end up being pretty chaotic, though I believe that the benefits to students’ learning outweighs the challenge of having students working on different tasks. That said, since using this routine I’ve found the chaos to be more manageable as it follows a calm and dependable start to a lesson.
How vs. Why
The above routine definitely favours brevity and procedural fluency over providing time to build students’ understanding the reasons behind why the procedure works and why it is worthwhile learning. It’s not that I don’t value the why, the why is the reason I love this subject! It’s just that I’ve come to realise that, with the vast majority of skills, the why is hard to comprehend without being able to carry out the skill in the first place and often understanding the why to the level it becomes of value to students can take years! Maybe most importantly, the why is generally not a prerequisite to being able to use and apply a skill effectively. In the past I’ve found prioritising the why has come at the cost of students’ opportunities to practice and learn how to carry out skills, and consequently overall limited the opportunities that students have to feel successful in the maths classroom (success being one of the greatest motivators to student learning!). That said, a deep understanding of why maths works helps students make connections and aids retention of skills and will always push students to this level of knowledge wherever possible. The difference now is that the why is the icing on the cake and I favour students being able to successfully carry out a skill over ensuring a deep understanding is first achieved.
Choosing a focus skill
Due to the cumulative nature of mathematical knowledge the order in which skills are taught needs to be considered with great care. The above fraction lesson was taught after there had been significant time allocated to ensuring the majority of students had the prerequisite skills; adding fractions with the same denominator, creating equivalent fractions, simplifying fractions and identifying the lowest common multiple of two numbers. Tight timelines can sometimes mean this isn’t always possible and deciding how to balance teaching the set curriculum and ensuring most students have the prerequisite skills is far from easy! Taking some time during the year to make notes on what has and hasn’t worked with regards to topic and skill sequencing and refining sequencing each year can make a massive difference to students’ learning. Some food for thought on sequencing at the Year 7 level can be found on the mathsquad site at https://mathsquad.org/skill-sequence/