Building on students' thinking

Through productive communication with whānau and by noticing what their students believe, think, and do, effective teachers of mathematics provide responsive learning experiences that enable students to build in their existing proficiencies, interests, and experiences.

Guidance for effective practice:

  1. Acknowledge and build mathematics teaching upon the ideas of ākonga and their identities, cultures, and lived experiences. 
    Tamsin has a year 7 class. She begins a unit of work on Movement and Position by asking students about how people in their whānau find their way to places they have not been to before.

    Her students offer many scenarios of navigation, such as using the mobile phone (GPS), or a tool like Navman or TomTom, reading a map, using landmarks like the Skytower or a maunga, or asking a person for directions. One student retells the story of having to phone a relative to find out how to get to their place. The discussion is rich.

    Tamsin: We certainly have a lot of help to find our way nowadays. How did our ancestors find their way to Aotearoa/New Zealand in the first place?
    Students suggest ways that Polynesian and European explorers may have found their way, including using the stars, and sailing around until they found land.
    Tamsin: Let’s watch a short television programme about how Polynesian people first sailed to Aotearoa.

    She plays a four-minute segment of Te mana o te moana: The Pacific Voyagers from Educational TV (47:10-51:20). The programme discusses how about 1300AD Polynesian explorers used the sun and stars, ocean currents, prevailing winds, and sea birds to locate land.

    Her students are enthused and investigate different ways that people around the world find their way. They find out that most main roads in New Zealand follow historic Māori walking tracks and use rivers and passes through mountain ranges.

    In doing so Tamsin builds on the lived experiences of her students and contextualises the theme of navigation in the histories of their ancestors. She openly values diverse ideas about movement and position (see Meaney, Trinick, & Fairhall, 2013, for more on culture being internal to curriculum).
  2. Notice and act upon partial knowledge constructions and misconceptions of your ākonga.
    Mandy works with Tai, a year 2 student, on early place value. Tai creates bundles of ten iceblock sticks, using rubber bands. He has loose sticks, as well.

    Mandy: Can you please give me forty iceblock sticks?
    Tai counts in tens putting a bundle in Mandy’s hand each time, “ten, twenty, thirty, forty.”
    Mandy: Thanks Tai. How many bundles did you give me? (keeping hand closed)
    Tai: Ahhh…
    Mandy: Take a look. How many bundles?
    Tai: Four

    Mandy notices that Tai’s focus is on counting in tens, not on how many tens are in the ‘-ty’ numbers (decades). He has a partial early conception of place value.
    Mandy might try two other decade numbers and see if Tai can generalize how many tens are in any ‘-ty’ number.
  3. Scaffold learning using strategies such as promoting interest, focusing attention, varying tasks, providing tools, and creating manageable goals. (Te ako poutama)
    Chad’s year 7 students are constructing cube models to match plan views (top, front, side) as part of an integrated unit on nzmaths called Cubic Conundrums.
    cube images

    He notices that some of his students start on a model, realise the model does not match the views, and pull it apart to start again.
    He suggests to students that the problem of getting all three views correct in one go may be too hard.
    Chad: Choose one view to get right. Which view is easiest?
    The students mostly opt for the satellite view and construct that correctly.
    Chad: Now try building onto your satellite view to get the front view correct, as well.
    The students obtain a correct front view and modify it to get the left view correct as well.

    Chad scaffolds the students’ learning, by supporting them to break a complex task into smaller sub-goals.