In the article Procedural knowledge vs conceptual knowledge in mathematics education I expand on James E Schwartz’s piece based around procedural and conceptual knowledge. To paraphrase Schwartz, “mathematical procedures, known to mathematicians as algorithms (are) procedures (which) enable (us) to find answers to problems according to set rules.”
The rules based approach:I argue that mathematical education has historically focused on imparting procedural knowledge, through a ‘rules-based approach’. I use the term ‘rules-based approach to refer to a traditional approach which looks something like the following:
“Here’s a rule …
The problem with a teaching and learning process that focuses on mathematical procedures is not that it teaches students mathematical procedures – the problem is that such learning comes at the expense of mathematical understanding.
Mathematical understanding is the realm of conceptual knowledge. In the Procedural knowledge vs conceptual knowledge in mathematics education article I propose that in order for students to acquire conceptual knowledge, the rules-based (procedural) approach needs to be replaced by a conceptual approach. When we use an effective conceptual approach we are teaching math conceptually.
When we teach math conceptually the following pedagogical dynamics apply:
In the online course ‘Engagement: Winning over your mathematics class' the above teaching and learning aspects of the conceptual approach to teaching mathematics are explored in depth and numerous, specific examples of what ’teaching math conceptually’ looks like are presented.
Shifting from a traditional approach to teaching mathematics to a conceptual approach is a challenging transition for teachers to make. Adding to the challenge of making such a transition are the misconceptions commonly possessed by teachers who have not yet made the shift.
Four common misconceptions
Below are four common misconceptions held by teachers about a conceptual approach to teaching mathematics
Conceptually-based teaching is synonymous with hands-on activities, and hands-on activities are extremely difficult to manage with my students.
Few would disagree that hands-on activities are difficult to manage in many secondary classrooms. In an ideal world, all students would explore hands-on materials within a well-structured activity, in a highly-engaged, questioning, self-directed manner.
Clearly, many secondary students have become removed from that ideal world! Therefore, setting students free amongst abundant hands-on materials is likely to be ineffective in bringing about conceptual understanding. Grant Wiggins excellently explores this point in his Experiential Learning article by expanding on the statement “Just because it’s hands-on doesn’t mean it’s minds-on”.
However, the belief that conceptually-based teaching is synonymous with hands-on activities is, in my view, a misconception. A quality, conceptual approach can incorporate some use of hands-on materials as part of well-scaffolded, teacher-led activities containing multiple leading questions. Student interaction can be governed by the teacher in accordance with the capacity for the particular group of students for learning.
Conceptually-based teaching is too time consuming to be realistically implemented.
This misconception usually stems from a couple of sources. Firstly, it can be linked to misconception 1, i.e. that a conceptual approach is one where lots of materials are distributed to students who are expected to spend considerable time investigating mathematically with the materials in order to gain the required insights. Commonly, teachers with misconception 2 have experienced ‘losing’ significant time because such activities have provided very little insight for students due to the activities failing dismally through student disengagement, poor implementation or both. Such an experience causes teachers to retreat to the comfort of a traditional approach.
Secondly, teachers can conclude that conceptually-based teaching is more time consuming than (traditional) procedural teaching because their efforts – even when not involving hands-on materials – simply take longer than the approach they are used to.
The reason this is a misconception, however, is that there are conceptually-based approaches which have proven to be more efficient than traditional approaches. But as I’ve already mentioned, an effective, conceptually-based approach requires a very different pedagogy to a traditional, procedural, let’s-keep-‘em-all-together approach. I experienced this time-saving effect over many years. So, too, have many of the teachers who have undertaken the 15-hour course ‘Engagement: Winning over your mathematics class' where this approach is comprehensively explored.
Conceptually-based teaching is too difficult to manage.
Is a successful conceptually-based approach too difficult for a teacher to manage? You bet it is – if the teacher is not equipped with the necessary pedagogies. However there are highly successful, highly skilled primary teachers (and high school teachers) all over the world who are able to successfully engage their students and allow for individual progression, who have multiple activities occurring simultaneously and whose students are learning MUCH more than set routines for answering text questions. Ask these teachers if what they are doing is too difficult for them to manage! Of course it isn’t. To these teachers a conceptually-based approach is an obvious one to use. The real issue is that the required skills are very different to (what I refer to as) a traditional approach and therefore need to be acquired.
I’ll lose my role as teacher, and therefore my enjoyment that comes with that role.
This is absolutely a misconception and stems from the idea that teaching conceptually and becoming more of a facilitator means the teacher has less opportunity to ‘work the group’; to pour out charisma into the classroom. Arguably, a student-centred, conceptual approach offers more avenues for a teacher’s charisma to shine. However, those avenues will look somewhat different to the approach the teacher is familiar with. In addition, The Conceptual Approach offers more chances to connect with students one-on-one and in small groups because, at any given time, the remaining students are better occupied and do not require the teacher’s direct input.
From the outset the challenge of teaching math conceptually appears significant, inefficient and counter-intuitive. However, with measured guidance and copious, ‘feet on the ground’ examples of teaching math conceptually, the challenge, although requiring significant time, can turn into reality with relative ease. This is the primary purpose of the Learn Implement Share course ‘Engagement: Winning over your mathematics class'.
Standing in the shoes of students
The original article, called ‘A veteran teacher turned coach shadows 2 students for 2 days – a sobering lesson learned’, appeared on Grant Wiggins’ blog ‘Granted, and … ‘, Oct 2014.
The ‘report’ makes for compelling reading. ‘Shocking’ might be more apt. And yet, upon reflection, the findings are obvious. A conclusion I would add is this: ‘Traditional teaching is designed for the teacher, not the students.’
I have copied the original article below:
(Wiggins) – The following account comes from a veteran HS teacher who just became a Coach in her building. Because her experience is so vivid and sobering I have kept her identity anonymous. But nothing she describes is any different than my own experience in sitting in HS classes for long periods of time. And this report of course accords fully with the results of our student surveys.
I have made a terrible mistakeI waited fourteen years to do something that I should have done my first year of teaching: shadow a student for a day. It was so eye-opening that I wish I could go back to every class of students I ever had right now and change a minimum of ten things – the layout, the lesson plan, the checks for understanding. Most of it!
This is the first year I am working in a school but not teaching my own classes; I am the High School Learning Coach, a new position for the school this year. My job is to work with teachers and admins. to improve student learning outcomes.
As part of getting my feet wet, my principal suggested I “be” a student for two days: I was to shadow and complete all the work of a 10th grade student on one day and to do the same for a 12th grade student on another day. My task was to do everything the student was supposed to do: if there was lecture or notes on the board, I copied them as fast I could into my notebook. If there was a Chemistry lab, I did it with my host student. If there was a test, I took it (I passed the Spanish one, but I am certain I failed the business one).
My class schedules for the day
(Note: we have a block schedule; not all classes meet each day):
The schedule that day for the 10th grade student:
7:45 – 9:15: Geometry
9:30 – 10:55: Spanish II
10:55 – 11:40: Lunch
11:45 – 1:10: World History
1:25 – 2:45: Integrated Science
The schedule that day for the 12th grade student:
7:45 – 9:15: Math
9:30 – 10:55: Chemistry
10:55 – 11:40: Lunch
11:45 – 1:10: English
1:25 – 2:45: Business
Key Takeaway #1
Students sit all day, and sitting is exhausting.
I could not believe how tired I was after the first day. I literally sat down the entire day, except for walking to and from classes. We forget as teachers, because we are on our feet a lot – in front of the board, pacing as we speak, circling around the room to check on student work, sitting, standing, kneeling down to chat with a student as she works through a difficult problem…we move a lot.
But students move almost never. And never is exhausting. In every class for four long blocks, the expectation was for us to come in, take our seats, and sit down for the duration of the time. By the end of the day, I could not stop yawning and I was desperate to move or stretch. I couldn’t believe how alert my host student was, because it took a lot of conscious effort for me not to get up and start doing jumping jacks in the middle of Science just to keep my mind and body from slipping into oblivion after so many hours of sitting passively.
I was drained, and not in a good, long, productive-day kind of way. No, it was that icky, lethargic tired feeling. I had planned to go back to my office and jot down some initial notes on the day, but I was so drained I couldn’t do anything that involved mental effort (so instead I watched TV) and I was in bed by 8:30.
If I could go back and change my classes now, I would immediately change the following three things:
Key Takeaway #2
High School students are sitting passively and listening during approximately 90% of their classes.
Obviously I was only shadowing for two days, but in follow-up interviews with both of my host students, they assured me that the classes I experienced were fairly typical.
In eight periods of high school classes, my host students rarely spoke. Sometimes it was because the teacher was lecturing; sometimes it was because another student was presenting; sometimes it was because another student was called to the board to solve a difficult equation; and sometimes it was because the period was spent taking a test. So, I don’t mean to imply critically that only the teachers droned on while students just sat and took notes. But still, hand in hand with takeaway #1 is this idea that most of the students’ day was spent passively absorbing information.
It was not just the sitting that was draining but that so much of the day was spent absorbing information but not often grappling with it. I asked my tenth-grade host, Cindy, if she felt like she made important contributions to class or if, when she was absent, the class missed out on the benefit of her knowledge or contributions, and she laughed and said no.
I was struck by this takeaway in particular because it made me realize how little autonomy students have, how little of their learning they are directing or choosing. I felt especially bad about opportunities I had missed in the past in this regard.
If I could go back and change my classes now, I would immediately:
Key takeaway #3
You feel a little bit like a nuisance all day long.
I lost count of how many times we were told be quiet and pay attention. It’s normal to do so – teachers have a set amount of time and we need to use it wisely. But in shadowing, throughout the day, you start to feel sorry for the students who are told over and over again to pay attention because you understand part of what they are reacting to is sitting and listening all day. It’s really hard to do, and not something we ask adults to do day in and out. Think back to a multi-day conference or long PD day you had and remember that feeling by the end of the day – that need to just disconnect, break free, go for a run, chat with a friend, or surf the web and catch up on emails. That is how students often feel in our classes, not because we are boring per se but because they have been sitting and listening most of the day already. They have had enough.
In addition, there was a good deal of sarcasm and snark directed at students and I recognized, uncomfortably, how much I myself have engaged in this kind of communication. I would become near apoplectic last year whenever a very challenging class of mine would take a test, and without fail, several students in a row would ask the same question about the test. Each time I would stop the class and address it so everyone could hear it. Nevertheless, a few minutes later a student who had clearly been working his way through the test and not attentive to my announcement would ask the same question again. A few students would laugh along as I made a big show of rolling my eyes and drily stating, “OK, once again, let me explain…”
Of course it feels ridiculous to have to explain the same thing five times, but suddenly, when I was the one taking the tests, I was stressed. I was anxious. I had questions. And if the person teaching answered those questions by rolling their eyes at me, I would never want to ask another question again. I feel a great deal more empathy for students after shadowing, and I realize that sarcasm, impatience, and annoyance are a way of creating a barrier between me and them. They do not help learning.
If I could go back and change my classes now, I would immediately:
I have a lot more respect and empathy for students after just one day of being one again. Teachers work hard, but I now think that conscientious students work harder. I worry about the messages we send them as they go to our classes and home to do our assigned work, and my hope is that more teachers who are able will try this shadowing and share their findings with each other and their administrations. This could lead to better “backwards design” from the student experience so that we have more engaged, alert, and balanced students sitting (or standing) in our classes.
In this article I wish to shine a spotlight on the education phenomenon referred to as ‘spoon-feeding’. Far from being a research-backed article, this one springs from 20+ years of teaching, 8 years of parenthood and a great deal of thinking about these ideas over many years. The intent here is to provide some food for thought in you, the reader. It would be nice to see the amount of spoon-feeding in classrooms decrease around the world.
What is spoon-feeding?
I would argue the case that, when it comes to the teaching and learning process there is no valid excuse for any form of spoon feeding. Period. And for the record, to force upon students a high score in an upcoming standardised test by drilling (spoon-feeding) information prior to the test does not qualify as a valid excuse.
Spoon-feeding and toddlers – food only please!
Note that spoon-feeding does not even apply to toddlers, except, of course, in the case of actually feeding toddlers! Parents do not teach their toddlers how to walk by spoon-feeding the information. Rather, toddlers see their bigger family members walk, they gain the desire to walk, they try and fail ‘a hundred times’ … and then they begin to walk. This is discovery learning. It is inquiry-based learning. It is mostly self-directed learning. Sure they receive assistance from the people around them but the assistance is supportive; the adults facilitate the learning. It is not spoon-feeding.
The same applies to toddlers learning how to run, skip, talk, jump, sing and pretty much everything else they learn to do before they hit school. I would argue the same also applies to reading. Assuming children willingly learn to read before they begin school, spoon-feeding plays no part in children learning to read. Sure, parents ideally read to their children on a daily basis but acquisition of the skill of reading occurs more by osmosis than by the parent saying ‘OK Lucas, now I’m going to make you better at reading.’ The parent’s true role is to fan the flame of reading enjoyment in their child. The same applies to teaching students (anything) in the classroom.
Self-directed learners are not spoon-fed!
One of the reasons self directed learners tend to be effective learners is that they tend to engage in their learning meta cognitively. They ask questions of themselves and others and usually know what they do and do not understand. They think about their thinking as they inquire. This is clearly the opposite to spoon-fed learners because spoon-fed learners do not need to think about their learning. Everything they learn is provided to them by their teacher – what to learn, how best to learn, when to learn, how deep to learn each aspect and what the questions will look like in the test. Spoon-feeding tends to be the domain of teacher-directed classrooms and is driven in part by the demand for high test scores.
The tide of spoon-feeding.Before we see the reduction of spoon-feeding in education there are some realities which need to be faced. The practice of spoon-feeding is kept in place by several factors:
So what’s the alternative?Plan for change. Do some research. Find out what the alternative to spoon-feeding looks like. Implementing change which includes a move away from teacher-directed learning towards a well scaffolded, student-centred approach will lay a foundation for reducing the amount of spoon-feeding. Make a plan to foster meta cognition in students. Inform the students of the coming changes. Keep them ‘in the loop’ always. Explain that the changes may feel uncomfortable for a while but that in a month or two they will likely prefer the new system to the old.
The alternative sounds pretty straight forward, and in truth, once you make the change you will likely find it easy and rewarding, making it difficult to return to ‘the old ways’.
Nonetheless, the transition from being a spoon-feeder to a facilitator of learning is a big undertaking. The Learn Implement Share 'Engagement courses' for teachers are designed to enable educators to make this powerful transition over a 16 week period. The learning journeys support teachers who want their students to become engaged learners and to take responsibility for their learning. Part of that journey is to cease being a spoon-feeder and become an active facilitator of learning.
What is meta cognition?
Very simply, meta cognition is the process of thinking about one’s thinking.
Meta cognition and the classroom?
Meta cognition has long been associated with philosophical thinking. However, the idea that meta cognition is something we should engineer in our students is relatively new. This is because meta cognition was never required for the traditional system of education, i.e. teacher directed, top-down, teaching-by-telling. Traditionally therefore, there has been little opportunity or need for students to be thinking about their thinking. In fact there may have been little thinking going on at all outside the brain’s requirements to learn facts, rules and routines. However, with the push towards quality teaching and learning the need for meta cognition, and the explicit instruction of such, has arrived.
We need to be careful here with definitions. Some people would argue there are many high-quality teachers who are traditional in their approach, teachers who have their students onside, teachers whose students do well in standardised tests. For the purpose of this article, I’ll outline what I’m referring to when using the term ‘quality teaching and learning’.
The sort of quality teaching and learning I’m referring to here involves aspects such as choice within learning, higher order thinking, inquiry, collaboration, well structured student-centered learning, critical thinking, problem solving, high levels of engagement. The teacher operates as a facilitator. In addition, students generally perform well in assessment tasks which measure a range of criteria including understanding, reasoning, and problem solving. Once students are immersed in this sort of learning – open, engaged, collaborative and learning in which students take ownership – then meta cognition truly has a role to play.
Preparing to embrace a conceptual approach to
Richard Andrew likes to write about stuff that matters in education. That boils down to 'anything that helps teachers to better-engage their students'. His view? "Not much else really matters - engaged students learn. Disengaged students do not."
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