Monthly Archives: February 2016

Conceptual understandings require time.

In order for students to reach deep and conceptual understandings they require time. Time to explore concepts, time to think, time to build connections.    I regularly use the thinking routines by Project Zero, which are designed to:

“Slow down the pace of the classroom to make space for deep observation and wonderment.  That happens by talking and discussing objects or systems in the everyday world.” (Schwarz, 2014)

There is great value in slowing down the pace of the classroom in order to reach deeper and more conceptual understandings.  A common misconception is “We don’t have time, we have so much to cover!”.  But by teaching and learning in a more thoughtful and slower way, you actually save time in the end.  By establishing solid understandings you are eradicating misconceptions, which you would have had to re-address later on, had you just rushed through.

I want to highlight this point with an example of my students learning about TIME and how to ‘tell the time’ in our current UOI.  The line of inquiry which we are exploring is “How people organise and measure time”.

Telling time is often taught as a rote skill and with this comes deep misconceptions.


Traditionally, teachers often start with teaching the full hours “When the big hand points to 12 and the little hand points to 1, it is  1 o clock”.  Then onto half past 2 and then onto  quarter past and quarter to the hour.   But when our students learn time in this way, they do not have an understand of “the duration of time and its relation to the numbers and hands on a clock” (Thompson and Van de Walle 1981), nor do they have a true conceptual understanding of time.

Before designing my lessons on telling the time, I did some research.  And, no, I did not consult Pinterest, instead I went on the National Council of Teachers of Mathematics website and within 5 minutes I had a series of investigations designed to:

“provide opportunities for children to investigate time units, experience events lasting set amounts of time, and develop personal referents for standard time units” (Mc Millen and Hernandez, 2008).

These investigations by Mc Millen and Hernandez (2008) were different to other approaches I had read and observed,  these researchers  highlighted the differences associated with the two hands of an analog clock and why these two systems should be taught separately at first. In these investigations  students worked with two types of one-handed clocks, a minute clock and an hour clock, to meaningfully discover both differences and connections between the two clock hands.

By spending a little amount of time doing some research, I saved myself a lot of time in planning.

In the first lesson, I gave my students time to investigate time units such as 1 second and 1 minute, in order to help them to develop personal references for standard time units and to discover the relationship between them.   My classroom was loud and noisy.  We were building towers, bouncing balls, doing star jumps, burps and … you name it!


In the second lesson, the students investigated what can be determined about time when a clock has only an hour hand.   We built our own clocks and learnt how to tell the time with these simple phrases, its “o’clock” its “a little past five.” or “a little before five o’clock.” or its  “between four o’clock and five o’clock”.  The students worked in partners and had a number of partner activities to do, which they greatly enjoyed.

In the third lesson,  the students then investigated clocks with only a minute hand.  This lesson focused on connecting  minutes to an hour.  We started by matching ‘groups of five minutes’ to the clock face and built minute clocks in our classrooms.  I encouraged my students to use efficient counting strategies to tell time, eg, skip counting by 5s: 5, 10, 15, 20, then counting on individual minutes 21, 22.  By the end of the lesson, all my students were all able to count the minutes past the start of an hour with their partners.

In the fourth lesson, my students  “investigated the motion and location of the minute hand as it relates to the hour hand” (Mc Millen and Hernandez, 2008).  My students then connected their knowledge together, they counted the minutes first, then found the hour, and finally combined them.    At this point I exposed my students to analogue clocks with both hands, and after some discussions “Which comes first, the hour or the minute?” My students were all quizzing each other “Guess which time this is?” and telling the time to the minute.

It took 3 lessons (4 hours in total) before I even introduced my students to a real clock face with two analogue hands.  Yet, my students were able by the end of that 4th lesson to tell the time accurately and to the minute.  This is my 4th year as a Grade 2 teacher, and never yet have I seen students learn the time faster.

What can I conclude from this experience?

It pays to give students time, time to understand concepts deeply and in the end, well … you might even save time.

If you are interested in any of the sources mentioned here, they are referenced below.

McMillen, S and  Hernandez, B (2008) Taking Time to Understand Telling Time.  The National Council of Teachers of Mathematics. accessible at

Schwatrz, K. (2014) How Dissecting a Pencil Can Ignite Curiosity and Wonderment. MindShift. accessible at

Thompson, C and Van de Walle, J (1981). A SingleHanded Approach to Telling Time. Arithmetic Teacher 28 (April 1981): 4–9.  cited in McMillen, S and  Hernandez, B (2008) .



Listening to and trying to understand student thinking which has been made visible

The school in which I work strives to create a culture of thinking. By adopting many of the simple, yet hugely effective strategies from we can make our student’s thinking visible.

Unless encouraged to do so, most of our student’s thinking is invisible, sometimes students explain how they reached an answer, but often they do not.  But by making their thinking visible, they can build on each other’s ideas and understand concepts better.

However, for a large part, this rich data gets lost somewhere in the classroom and visible thinking risks becoming just ‘one of those’ classroom activities which we do.

As teachers, we are all busy, but how often do we stop and reflect on the thinking of our students which we have made visible?

I have the rare opportunity of having a trainee teacher in my classroom, who already has heaps of classroom experience, yet who is eager to learn more.  At the start of our unit, we spent 3 days gathering as much information about our student’s prior knowledge before we began our unit on time.  We used the visible thinking strategy of ‘Headlines’ along with 8 different card sorts.  We recorded 14 different discussions using the Harkness method.

Then, most importantly, my colleague and I spent over 1 hour over reviewing, analyzing, discussing the ideas, knowledge and misconceptions of our students which we had gathered.  This final step is the one which is most often neglected.  But this is the most important step of all!

We lay out all of our data on the floor, and we examined each student one by one…

One of our most surprising findings, was a conversation by two young boys over the moon cycle:

We actually saw, how through discussion, this one child was able to change his own thinking and overcome his own misconception.  At the beginning of the discussion Juan stated that the moon cycle changed every day “This is the beginning (points at crescent moon) when it’s morning, then as the sun gets bigger the moon gets bigger.”  Two big misconceptions there.  But wait, here is the turning point, after an interesting conversation with his peer, who states “I know that the full moon gets bigger and bigger up to TET and then it gets darker and darker again.” Juan then says “I see a full moon every month… I now think it takes 30 days”.  For me this conversation showed that Juan did not have a large amount of prior knowledge about the topic, but that his reasoning skills and his ability to make connections to make sense of information is incredibly strong.

As a teacher, the understanding that we can reach through listening to and actually reflecting on the data of each of our student’s thinking should not just be considered an opportunity, but rather as an integral part of the ‘visible thinking’ approach.