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I am a senior teacher at this school. We have to select the textbooks for the upcoming session. I am proposing that we have to select books (in maths) that focus more on conceptual understanding and not just on procedural fluency. The curriculum mapped books are supplied by NCERT (I am based in India) here, developed by a team of experts. The books have been developed keeping in mind the basic philosophy that learning should be connected to the child's environment.

But I am having a hard time convincing the teachers who are used to working with books by private publishers, which focus only on procedural fluency and not at all on conceptual understanding.

What strong arguments can be put forward to help convince them that the need of the hour is to go for books (and subsequent teaching) that are connected to the real-world and children's lives? I welcome any helpful advice.

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    $\begingroup$ You refer to "the child," but give no other indication of the age or level. Are you teaching kids who are 6 years old? 18? I don't know about India, but here in the US there is an especially severe problem with the mathematical training of teachers in the early grades (ages 5-11). Teachers at this level often have no real conceptual understanding of mathematics themselves. $\endgroup$
    – user507
    Dec 3, 2019 at 16:59
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    $\begingroup$ Good books should do both, progressing from simpler to more complex topics, re-using already learned material in the newer topics. Fewer word problems should be favored over a zillion of calculation exercises. Ideally, the program must cover full spectrum from first to senior grade without boring repetitions or omissions, and link into geometry, physics and chemistry courses (say, vector-based velocity or force is studied at about the same time as right triangle and basic concepts of sine, cosine and tangent are studied - this is not full-blow trig yet). Such books and courses do exist. $\endgroup$
    – Rusty Core
    Dec 3, 2019 at 17:44

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I presume you're not asking for research references, because Googling "procedural fluency vs conceptual understanding" will get you to a page of Google Scholar before you could finish typing out that entire search.

I suppose I would lean into how effective pure procedural fluency has been in the past. I'm in my early 50's, and I was taught to do long calculations by hand (including extracting square roots and using log tables in Algebra 2) because "you won't always have a calculator with you." That prophecy clearly did not age well -- I even had a scientific calculator in classes when I was an undergraduate. And now, of course, everyone can do math on their phones which are in their pockets.

How do you prepare students to live in a world where machines are better at arithmetic than we are? I'm only certified in secondary education, so I'll leave the details to you and your peers. But I think a large portion of that would be a greater conceptual understanding that would lead them to know which operations to use for different circumstances and a stronger awareness of how to use the results of those calculations to guide our decision-making processes.

You're going to get pushback, because your teachers have dedicated their careers to the algorithmic piece in the middle. Nobody likes being told that their training is obsolete. But if they're willing to adapt, they can create learning that will serve their students throughout their lives.

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  • $\begingroup$ When a cash register fails or the system does not have a discount shown in a sticker, figuring out 15% discount off \$16 becomes an unsurmountable task. Don't ask me how I know. After a minute of hard thinking a clerk offered me \$2 off because "it should be something around that." Well, at least it was in the ballpark, I give him that. Many big names in math and physics have been saying that only possessing a "bigger conceptual understanding" without knowing standard algorithms seriously hinders one's abilities in calc and higher math and physics courses, too much time spent on basic algebra. $\endgroup$
    – Rusty Core
    Dec 3, 2019 at 17:49
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    $\begingroup$ @RustyCore I assume that you did not actually calculate 15 percent of $16 by multiplying 16*0.15 using the standard algorithm. You probably took 10%, then half that, then added together. This is what we want. $\endgroup$
    – Steven Gubkin
    Dec 3, 2019 at 21:09
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    $\begingroup$ The distributive property is not an algorithm: it is one of the axioms of the real numbers. You used that property creatively because you understand what percent is, you understand the base 10 number system, and you made sense of the situation. We call this number sense. This is exactly the kind of thing we want students to do to build their conceptual understanding. $\endgroup$
    – Steven Gubkin
    Dec 3, 2019 at 22:51
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    $\begingroup$ @RustyCore I am glad that we seem to agree that students should be able to calculate flexibly, because they have an understanding of the meaning of the operations and their properties, rather than just entirely relying on algorithms. $\endgroup$
    – Steven Gubkin
    Dec 5, 2019 at 12:34
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    $\begingroup$ @RustyCore An interesting anecdote along these lines: I was in mixed age group company, and we were discussing common core. I asked everyone to calculate 999+11. The younger people all computed this as (999+1)+10, while the older people reported "lining up the numbers in their head" - basically just performing the standard algorithm mentally. They had to carry repeatedly. I think the first solution demonstrates a much more solid understanding of base 10 arithmetic. $\endgroup$
    – Steven Gubkin
    Dec 5, 2019 at 12:37
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I would suggest inviting, or "recruiting" the teachers you are leading to help you with working out the direction in which you hope to move from an emphasis on "procedural efficacy" toward an emphasis on "conceptual understanding." Try to frame it as a "team project", explicitly recognizing their experience and first-hand perspectives in the class-room. Don't frame the changes in terms of their failures; rather frame things as you and they being at an exciting time in math education, based on much research, that all of you can participate in helping your students master mathematics, not just procedurally, but also conceptually!

I emphasize the above mode of approaching your teachers, because very few "team members" warm up to anything they are told is going to be "shoved down their throats", whether they like it or not.

E.g., start asap setting up a "workshop", break it into groups of teachers, and have each group evaluate relevant studies/research, and/or possible alternative concept-based texts, to report back to the whole group to summarize. Ask them for examples in their own teaching, in which student's seemed to grasp conceptual understanding beyond just mere procedural efficacy, even when having used the former texts. Chances are there are already a few teachers who have springboarded off the given text, at the time, perhaps discovering greater mastery among their students.

Try to do such workshops and updates on the new changes, and text selections, in the least threatening manner to them as possible. Be enthusiastic, respect their work, express confidence in the newer approach, and express utmost confidence that they have all the support they need (more workshops, etc.), and adequate experience, to make the new approach successful: More rewarding for both the teachers, and the students.

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    $\begingroup$ I would agree with this, except for the fact that I have worked in K-12 education and the teachers, most likely, just do not have that time. My "curriculum development" team was a hodgepodge of teachers across my district, haphazardly working together in the same room, where the only reason they were doing that work was because our district removed the curriculum and refused to provide textbooks, asking them to "innovate" to "meet common core standards." Predictably, the end result was crap. Chances are, they're also resistant because learning something new takes time they don't have. $\endgroup$
    – Opal E
    Dec 5, 2019 at 5:37
  • $\begingroup$ @OpalE I suspect that your district had not been enlightened wrt how to engage and inspire teachers. $\endgroup$
    – amWhy
    Jan 26, 2020 at 19:28
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(This entire post is speculation.)

I think your teachers think that they are teaching conceptually already and that the students already have a deep understanding. You need to burst that bubble not with logic or reasoning, not with evidence or statistics, but with a sharp and emotional surprise.

One way to do that: Take out a video recorder and do some interviews with students.

Ask them to calculate this exactly.

$(\frac{17}{15})(29)=\_\_\_ $

This should show procedural fluency.

Then ask them to state whether or not the answer is greater than 800 in the following without doing any calculations. Point the video camera at the students' faces. Ask them to explain their thinking aloud ,then ask them to rate their confidence.

$(\frac{213}{5178})(799.1146)=\_\_\_ $

This will show lack of comprehension in many students.

For other ideas, check out Dan Meyer's post "What Does Fluency Without Understanding Look Like?"

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  • $\begingroup$ Thanks. Interestingly I discussed similar concepts with students and they mostly failed. In a recent meeting with the teachers, I raised this issue. I will update how and what I did to convince them ( based on the answers from this forum). The final decision of the book selection is to be taken by the Chairman tomorrow :-). Things look in my favor. $\endgroup$
    – gpuguy
    Dec 8, 2019 at 4:21
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    $\begingroup$ Oh no, I'm headed down the rabbit hole of well-thought out pedagogy that is Dan Meyer's blog. Send a search party if I don't show up to work on Monday morning! $\endgroup$
    – Matthew Daly
    Dec 8, 2019 at 9:25

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