Falling through the cracks. Karim and the Bus Stop Method.

Earlier this week I observed some Functional Skills maths lessons in a college. Here we have students with low or no grades at GCSE aiming for a maths qualification that will hopefully give some sense of achievement. Most of the students were taking Level 2 courses in Digital/Media areas but are required to continue their maths studies alongside.

The challenges the teachers of these programmes face is immense. They have a wide topic range to cover and yet many students still struggle with basics. One student needed to clarify that a half is not 0.2 as a decimal. When corrected, he did seem to remember – oh yeah, 0.5, because its 50% – but he couldn’t then suggest the fraction to match 0.2. In other words, his underlying understanding of the link between simple decimals and fractions was very weak. He was guessing, not exploring a clear schema or using reasoning. This was Andy, age 16.

Another student was tackling 56 divided by 4, a calculation derived from a worded question. When I spoke to her, she’d just written out the 4 times table. 4,8,12,16,20,24,28,32,36,40,44,48,52,56… phew, no mistakes,… then was counting up how many numbers she’d written. Is it 13? She’d miscounted. I offered an alternative to this horribly inefficient method eg half and half again. But half of 56 was also too hard. Nope. 4 x 14 = 56 was not remotely familiar to Joanne, 17.

And then there was Karim, working conscientiously through some bus-stop method division. He’d produced this and written 115.2

I asked him if he was sure.. could 8126 divided by 7 be 115.2? He couldn’t see an immediate issue. He wasn’t thinking that the answer should be 1000 and something; we wasn’t troubled by the answer only being in the 100s. Fundamentally he wasn’t thinking about the scale of each number and what a concrete version of his problem looked like. He wasn’t connecting digits to place value ie the 8 means 8000. It was all blind procedure. But, in prompting him, he was troubled by the 0.2 because he’d been told to find a remainder. What had he done? He’d correctly applied the method as far as 7s into 42 goes 6. But then Karim was left with that last 6. Here he became confused with subtraction… so he ‘borrowed’ 1 from the top line 6 to make 16… then added a decimal point because, with some logic, he thought 6 divided by 7 is less than one. All a bit of a mess – but so close. Karim is also 17.

I think this scenario warrants exploration. Karim, Andy and Joanne are 16 or 17, struggling with basic number work they were probably first taught 9 years ago. And here they are on yet another course where their teacher is a) hoping these things are basics that can be built on and b) has a ton of other stuff to cover for them and the rest of the class. And yet the gaps in their confidence and fluency are so evident… still; again.

Why does this happen?

  • Do Karim, Andy and Joanne have some fundamental cognitive limit preventing them mastering these ideas?
  • Is it that this material is too abtract, removed from ‘real world questions’ that would be more motivating and engaging?
  • Is it that this material is too teacher-led, too driven by a DIKR STEM agenda, too removed from what Karim, Andy and Joanne would choose for themselves – ie what’s the point of them knowing this stuff anyway?
  • Is it that teaching is all a bit of an art and teachers have been too busy following narrowly defined principles to have had the freedom to teach them properly year after year?
  • Is it that students’ resilience and other dispositions weren’t sufficiently attended to over the years?
  • Is it because of funding pressures and a lack of support for intervention?
  • Is it because of setting, with low expectations bringing them down, when mixed ability teaching would have served them better?

Well, no. I’d say it’s none of these things. For sure, year after year, Karim, Andy and Joanne have fallen through the cracks. No doubt each of their maths teachers have been well-intentioned and worked hard but I also think it’s possible to imagine a world where all three students would have been confident handling these questions – long before reaching the last chance saloon of their FE classroom. For that to have happened, I’d say a few practices and attitudes would need to be different:

  1. We’d assert that this kind of foundational maths knowledge is important and needs as much time as it takes to take root in secure schema; you don’t rush on and hope they’ll pick it up later. Evidently this doesn’t just happen. You deal with it head on; you teach them what they don’t know at every stage, even if it means going very very slowly and going back as far as you need and going over it again and again. This means that, yes, a mixed ability class might be a tough place to be; it might not remotely be the answer.
  2. We’d be very clear that specific techniques that relate to teaching this content should be well known and continually crafted and honed. We don’t fudge it all into nebulous ‘teaching is a bit of an art’ guff. Not here, not with these vulnerable learners. No, the stakes are too high. We focus hard on sharing and implementing sound maths mastery methods, relentlessly. We face that challenge and keep working on it, not look for ways around it.
  3. We also don’t allow routines and regimes around lesson protocols or a predetermined curriculum roadmap that mask the fact that Karim, Andy and Joanne are who they are; people with their particular motivations, insecurities, prior knowledge and rates of learning at any moment. We flex and adapt to teach them and motivate them. Fundamentally though, we recognise that the key driver of motivation in this context is success. We have to build success; we consolidate relentlessly.
  4. With all this in mind, yes, it means we might need to pare down the curriculum a bit – in maths itself for sure. I don’t know that more and more time for maths is the answer -because there are other things that Karim, Andy and Joanne need to learn and could succeed at – but certainly the scope to teach less maths content to greater depth and fluency might be part of it.

Overall, to echo the wise word of Mark McCourt, the answer should lie in finding out what Karim, Andy and Joanne don’t know and then planning a curriculum around that, revising it as they grow in confidence. Currently our system is failing them; they fall through the cracks and their FE teachers then have the most difficult task in teaching in my view. But we won’t solve the problem by wishing it away or crying ‘low expectations’. It’s the hard graft of crafting ever more effective teaching methods within a deliverable curriculum that will win through, year after year after year after year.


  1. Wonderful examples of a lack of effective instructional programs – these students need different content ALONG WITH explicit instruction – followed by sufficient deliberate practice for these concepts! This is NOT rocket science – just using evidence-based practices! Thanks so much for these clear examples that teachers can learn from!!

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  2. Couldn’t agree more – absolutely spot on and exactly how many of us feel teaching science too. Trying to cover too much content, with insufficient depth of learning as the result through a curriculum that becomes less and less relevant to the lives of our learners each year!

    Liked by 1 person

  3. Oh my goodness! This resonates so strongly with me. As a maths teacher (I’ve taught reception – year 13) I cannot tell you how many times I’ve taught low-achieving year 11s who cannot grasp the concept of number because they’ve been stymied by schools moving on too soon because that’s what the planning says. It’s infuriating! I still cannot understand why primary schools spend a week every term learning about shape. The same shapes those children knew at age 7 are the same ones they have to know at age 16. If they don’t know that 7 divides into 8,000 and odd more than 100 times then they should be spending their valuable time on estimation – restaurant “can you split the bill?” maths. I also think long division may have gone the way of log tables: I still have my brown log tables that was passed down to me from older brothers and sisters and I used in my own Leaving Certificiate exam – showing my age; I suspected at the time that there must have been a more efficient way to find trigonometric ratios and square roots. I’ve can’t even read it now without a magnifying glass. My year 13 students think it’s hilarious that you had to look in a little brown book to find the square root of anything! IGCSE doesn’t include a non-calculator paper – thank the gods!

    Kind regards

    Alison Bourke Deputy Head of School

    *”High expectations of learning that inspire students to reach their full potential in a caring international community” *

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  4. Absolutely Tom.
    Having taught in Secondary and FE resit, I find the conveyor belt curriculum (to again refer to the sage that is Mr McCourt), is harming the learning of so many students.
    By the time they reach FE, they have plenty of misconceptions based upon a set of rules, then when you try to unpick them, the students often do not want to go back to the basics as they think they do not help with GCSE questions.
    The first need of any FE teacher MUST be to persevere with the development of understanding the basics such as place value, (your example) and the use of manipulatives to support.

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  5. Thanks Tom. Makes me think of the debates about school maths at the time of the Post14 Maths Inquiry and then the Primary Maths Review in the 2000s when i was lucky to co-organise and record many of the high-level discussions. The big messages were aimed at Ministers. One of these was we need a double Maths GCSE, not based on a lot of additional content, rather better organised for the needs of all types of student and not just the ‘clever core’. The fight was intense at times and of course English and Science were more interested in a rearguard defence of their preciously acquired extra curriculum time. The students as always are the collateral in schools. Hence my support for your English Bacc approach.


  6. […] Why? If we assume they were taught the material, why can’t they do this type of question? Why does this happen? I could find similar questions from any exam. We can look at the questions that Grade 3 students struggle with and consider why students aged 16 can’t do them. (For maths examples, look at another recent post: Falling through the cracks. Karim and the Bus Stop Method.) […]


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