Language Is Critical To Deep Understanding in Maths

A Mathematics Mastery approach can have a profound impact on language and mathematical thinking for children, especially those with special educational needs, explains Helen Drury

The five-year old boy smiles confidently at his class. “Because I know that double four is eight, and I know that a square has four sides, I know that two squares will have eight sides altogether”. It’s just a normal day for the class, and reasoning like this has become an expected part of Year 1 maths lessons.

Create an account to read this article

£7.00+ VAT

One-off purchase

Purchase and Download today

Register for free

No Credit Card required

Register for free
Access to 3 free articles
Free TeachingTimes Report every month

array(8) {
  ["distinct_key"]=>
  string(10) "post#43533"
  ["title"]=>
  string(42) "Why philosophy and maths walk hand in hand"
  ["categories"]=>
  array(1) {
    [0]=>
    string(32) "Creative Teaching & Learning"
  }
  ["url"]=>
  string(67) "https://www.teachingtimes.com/ctl6-4-philosophy-maths-hand-in-hand/"
  ["post_date_timestamp"]=>
  int(1489491873)
  ["objectID"]=>
  string(12) "post-43533-0"
  ["_snippetResult"]=>
  array(2) {
    ["excerpt"]=>
    array(2) {
      ["value"]=>
      string(189) "Relational understanding in maths is about creating the links that make the subject exciting and revelatory. It shares a lot of principles with P4C, argues Rod Cunningham."
      ["matchLevel"]=>
      string(7) "partial"
    }
    ["content"]=>
    array(2) {
      ["value"]=>
      string(398) "TeachingApproaches40Volume6.4&7.1nCreativeTeaching&LearningRelational understanding in maths is about creating the links that make thesubject exciting and revelatory. It shares a lot of principles with P4C, arguesRod Cunningham.Why philosophyand maths walkhand in handEvidently there is a bumper sticker on cars in California which says‘Philosophy is a game ..."
      ["matchLevel"]=>
      string(7) "partial"
    }
  }
  ["_highlightResult"]=>
  array(4) {
    ["title"]=>
    array(4) {
      ["value"]=>
      string(51) "Why philosophy and maths walk hand in hand"
      ["matchLevel"]=>
      string(7) "partial"
      ["fullyHighlighted"]=>
      bool(false)
      ["matchedWords"]=>
      array(1) {
        [0]=>
        string(5) "maths"
      }
    }
    ["excerpt"]=>
    array(4) {
      ["value"]=>
      string(189) "Relational understanding in maths is about creating the links that make the subject exciting and revelatory. It shares a lot of principles with P4C, argues Rod Cunningham."
      ["matchLevel"]=>
      string(7) "partial"
      ["fullyHighlighted"]=>
      bool(false)
      ["matchedWords"]=>
      array(2) {
        [0]=>
        string(13) "understanding"
        [1]=>
        string(5) "maths"
      }
    }
    ["content"]=>
    array(4) {
      ["value"]=>
      string(5752) "TeachingApproaches40Volume6.4&7.1nCreativeTeaching&LearningRelational understanding in maths is about creating the links that make thesubject exciting and revelatory. It shares a lot of principles with P4C, arguesRod Cunningham.Why philosophyand maths walkhand in handEvidently there is a bumper sticker on cars in California which says‘Philosophy is a game with objectives and no rules – mathematics is agame with rules and no objectives.’If this is taken at face value it might be concluded that either philosophyand mathematics have nothing at all in common, or that they would usefullycomplement each other. This article attempts to show the latter, with particularreference to a practice known as Philosophy for Children (P4C). I will argue thatgood thinking and facilitation of thinking is similar in both a P4C community ofenquiry and in a mathematics lesson designed to develop what Richard Skemp(1976) would call ‘relational understanding’.41Volume6.4&7.1nCreativeTeaching&LearningTeachingApproachesI will also argue that thinking and learning develops more effectively if these skillsand strategies are made explicit and brought to the attention of both pupils andteachers, which a combination of P4C and exploratory mathematics sessions cando. I will attempt to give examples to demonstrate that cross-fertilisation of P4Cand mathematics practice can broaden learning and teaching across the curriculumby providing mutual support for such skills.Contexts for learning in mathematicsRichard Skemp (1976) refers to ‘relational’ and ‘instrumental’ understanding inmathematics. The first could be classed as deep understanding and is a functionof the number of links made within the field of mathematics, the second an abilityto follow a rule. An important part of achieving relational understanding is thatlearners’ present thinking is challenged. John Mason (2004) refers to the Festingeridea of ‘cognitive dissonance’.‘Deviations from expectation create disturbances which can lead to new sense-making. Without disturbance, there is no need to accommodate or assimilate’(Mason, 2004, p3). An example of an approach to mathematics learning withan emphasis on relational understanding is First Steps in Mathematics (FSiM), adiagnostic approach to the teaching of mathematics produced for the WesternAustralia Education Board.This is based on a collection of international research into the difficulties childrenhave in developing key mathematical concepts. A mantra of the FSiM approach is,‘You can get the right answer for the wrong reason.’ Using FSiM diagnostic activitiesthe underlying misconceptions held by developing mathematicians are uncoveredand activities to overcome these are suggested. As FSiM shows, however, ifconceptual misconceptions are not addressed they become compounded andpreclude successful mathematical work outside direct application of learnt rules.Seymour Papert claims that mathematical sense-making is best promoted withinan exploratory framework. He sees children as ‘builders of their own intellectualstructures’ who require ‘the materials to build with’. Papert designed theprogramming language LOGO to allow exploration of the mathematical universe.This was underpinned by ‘..a view of learning mathematics in ‘Mathland’; thatis to say, in a context which is to learning mathematics what living in France isto learning French.’ (Papert, 1980, p6). Researchers into the development ofmathematical understanding using LOGO found, however, that whilst freedomfor pupils to explore was critical, it was also important to provide some structureto bring them into the realm of key mathematical ideas. Hence the importance ofthe facilitator’s role which is discussed next.Examples: Promoting Relational Understanding1) Let’s Think through mathematics and CAME provide lessons based oncollaborative enquiry methods and centred around key mathematical ideas. Detailsof courses and resources on: http://www.letsthink.org.uk2) Marion Bird provides more open-ended mathematical starting points for posingquestions and conducting enquiries.3) Bowland Maths provides real-life problem-solving scenarios which develop skillsof good thinking. http://www.bowlandmaths.org.uk4) NRICH maths: Resouces for investigative mathematicshttp://nrich.maths.org/frontpage5) The First Steps in Mathematics scheme can provide guidance on keymathematical concepts and misconceptions. (it is now freely available on line)http://det.wa.edu.au/stepsresources/detcms/navigation/first-steps-mathematics/http://www.letsthink.org.ukhttp://www.bowlandmaths.org.ukhttp://nrich.maths.org/frontpagehttp://det.wa.edu.au/stepsresources/detcms/navigation/first-steps-mathematics/TeachingApproaches42Volume6.4&7.1nCreativeTeaching&LearningFacilitating relational learning in mathematicsAccording to Mason, achieving understanding of the relational kind in mathematicsis not achieved by simply working collaboratively, doing and discussingmathematics. Mason refers to what he calls ‘specializing’ and ‘generalising’(Mason 1988). He claims that meaning making in mathematics (and in othercontexts) involves moving between particular cases and general statements, fromone to the other, seeking out properties, exploring distinctions and relationshipsbetween cases, and using these as a basis for defining and reasoning.The act of reasoning itself rests on a statement or conjecture about thesituation, which arises from some puzzlement, some cognitive dissonance.According to Mason, the best way to develop these learning"
      ["matchLevel"]=>
      string(4) "full"
      ["fullyHighlighted"]=>
      bool(false)
      ["matchedWords"]=>
      array(5) {
        [0]=>
        string(8) "language"
        [1]=>
        string(8) "critical"
        [2]=>
        string(4) "deep"
        [3]=>
        string(13) "understanding"
        [4]=>
        string(5) "maths"
      }
    }
    ["categories"]=>
    array(1) {
      [0]=>
      array(3) {
        ["value"]=>
        string(32) "Creative Teaching & Learning"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
    }
  }
}

Creative Teaching & Learning

Why philosophy and maths walk hand in hand

14 Mar 2017

array(8) {
  ["distinct_key"]=>
  string(10) "post#44541"
  ["title"]=>
  string(37) "Leveraging Culture with Ron Ritchhart"
  ["categories"]=>
  array(6) {
    [0]=>
    string(32) "Creative Teaching & Learning"
    [1]=>
    string(17) "Critical Thinking"
    [2]=>
    string(13) "Deep Learning"
    [3]=>
    string(20) "Independent Learning"
    [4]=>
    string(15) "Problem-Solving"
    [5]=>
    string(15) "Thinking Skills"
  }
  ["url"]=>
  string(48) "https://www.teachingtimes.com/leveragingculture/"
  ["post_date_timestamp"]=>
  int(1546947530)
  ["objectID"]=>
  string(12) "post-44541-4"
  ["_snippetResult"]=>
  array(2) {
    ["excerpt"]=>
    array(2) {
      ["value"]=>
      string(311) "Ron Ritchhart, leader of PZ’s ‘Cultures of Thinking’ project, has been a powerful advocate for educational practices that bring out the best in people, and creating places where thinking is valued, visible and actively promoted. We sat down with him to discuss his research and thoughts on the current ..."
      ["matchLevel"]=>
      string(4) "none"
    }
    ["content"]=>
    array(2) {
      ["value"]=>
      string(392) "... leveraged to develop a culture of thinking. Jeff Watson and Roger Winn demonstrate how to harness these forces to develop students’ thinking skills in secondary Maths and Chemistry classrooms.3. Shaping Reality with the Language of Thinking http://library.teachingtimes.com/articles/shapingrealitywiththinkingThe power of language extends beyond the world ..."
      ["matchLevel"]=>
      string(7) "partial"
    }
  }
  ["_highlightResult"]=>
  array(6) {
    ["title"]=>
    array(3) {
      ["value"]=>
      string(37) "Leveraging Culture with Ron Ritchhart"
      ["matchLevel"]=>
      string(4) "none"
      ["matchedWords"]=>
      array(0) {
      }
    }
    ["author"]=>
    array(1) {
      [0]=>
      array(3) {
        ["value"]=>
        string(13) "Ron Ritchhart"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
    }
    ["excerpt"]=>
    array(3) {
      ["value"]=>
      string(327) "Ron Ritchhart, leader of PZ’s ‘Cultures of Thinking’ project, has been a powerful advocate for educational practices that bring out the best in people, and creating places where thinking is valued, visible and actively promoted. We sat down with him to discuss his research and thoughts on the current state of education."
      ["matchLevel"]=>
      string(4) "none"
      ["matchedWords"]=>
      array(0) {
      }
    }
    ["content"]=>
    array(4) {
      ["value"]=>
      string(5390) "downtimes, our brain is actually forming the richer connections and that students actually do better when we allow them those times. Rather than seeing that as a distraction, it is really integral to the learning process. So, for me the ‘Pedagogy of Play’ project is really a very, very exciting project going on right now that I’m following very closely. 1. Steve Sydell, Mara Krechevsky and Ben Mardell wrote a book called Making Learning Visible, which is connected to the Reggio Emilia approach. Additionally, a 2011 book called Making Thinking Visible was authored by the PZ team of Ron Ritchhart, Mark Church and Karin Morrison.Additional Reading:Ritchhart, R. (2015). Creating cultures of thinking: The 8 forces we must master to truly transform our schools.Ritchhart, R., Church, M., & Morrison, K. (2011). Making thinking visible: How to promote engagement, understanding, and independence for all learners. San Francisco, CA: Jossey-Bass.Ritchhart, R. (2004). Intellectual character: What it is, why it matters, and how to get it. San Francisco, Calif: Jossey-Bass Pfeiffer.More resources, including Cultures of Thinking resources can be found at www.ronrichhart.comThinking RoutinesThese routines are simple structures involving a set of questions or a short sequence of steps that can be used across various grade levels and content. They promote the development of students’ thinking and help achieve deeper inquiry and engagement with content.There are many types of Thinking Routines. Some are core routines, which can be used more generally, and others are geared towards understanding, fairness, truth and creativity. Some are good for introducing concepts and introducing ideas, some are great for synthesising and organising ideas, and others are geared towards digging deeper into ideas. Here are a few of the core routines to get you started. Additional examples can be found in the book Making Thinking Visible by Ritchhart, Church and Morrison and on its supporting website http://www.visiblethinkingpz.org/VisibleThinking_html_files/VisibleThinking1.html Think-Puzzle-Explore This routine is inquiry based, asking students to think about their prior knowledge, express curiosity and plan what they want to do to answer their questions. This is great for introducing a topic to help connect to prior knowledge, to stimulate curiosity and to lay the groundwork for more independent work. Ask the students to answer the following questions, individually and/or in groups: Think: What do you think you know about this topic? Puzzle: What questions do you have or are curious or puzzled about this topic? Explore: How can we explore these questions or puzzles?I Used to Think… Now I Think… This is a good routine for students to reflect on how and why their perceptions, opinions or knowledge has grown or changed over time. It can focus on an ideal, like ‘fairness’ or ‘truth’, or on a topic or unit such as ‘creative writing’, a book that they’ve read, a mathematical process like measuring angles, a world event in the news or in history. Students are simply asked to reflect on paper (can be in words, pictures, diagrams, etc.) in response to these prompts: I used to think … Now I think…What Makes You Say That? This teaches students to provide evidential reasoning for their observations. It helps them describe what they observe or know and then encourages them to support their interpretation with evidence.Through this exercise, students are also encouraged to recognise and understand alternative explanations and multiple perspectives. After presenting students with a particular concept or object, ask them the following questions:What’s going on? What do you see that makes you say that? Knowledge Trails:1. The Real Power of Questions https://library.teachingtimes.com/articles/ctl24thepowerofquestionsPromoting a deep-thinking classroom culture is a result of teachers posing the right type of questions, says Ron Ritchhart.2. Cultivating a Culture of Thinking http://library.teachingtimes.com/articles/acultureofthinkingCultural forces exist in all classrooms that can be leveraged to develop a culture of thinking. Jeff Watson and Roger Winn demonstrate how to harness these forces to develop students’ thinking skills in secondary Maths and Chemistry classrooms.3. Shaping Reality with the Language of Thinking http://library.teachingtimes.com/articles/shapingrealitywiththinkingThe power of language extends beyond the world of syntax. Erika Lusky shows how using the Cultures of Thinking framework is transforming the learning culture and offering students the ability to truly think for themselves.4. The Thinking Skills Debate http://library.teachingtimes.com/articles/thethinkingskillsdebateDoes the teaching of thinking skills help learners or is it a waste of time? Peter Gardner, Steve Higgins and Geoff Hinchliffe debate their points of view.5. Framing Learning as a Social Endeavour https://library.teachingtimes.com/articles/framinglearningStudent-led collaboration can bring out the individual gifts of students as they develop within a community. Julie Rains shows how group learning and reflection can make students feel successful as active participants in their own learning."
      ["matchLevel"]=>
      string(7) "partial"
      ["fullyHighlighted"]=>
      bool(false)
      ["matchedWords"]=>
      array(4) {
        [0]=>
        string(8) "language"
        [1]=>
        string(4) "deep"
        [2]=>
        string(13) "understanding"
        [3]=>
        string(5) "maths"
      }
    }
    ["categories"]=>
    array(6) {
      [0]=>
      array(3) {
        ["value"]=>
        string(32) "Creative Teaching & Learning"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [1]=>
      array(4) {
        ["value"]=>
        string(26) "Critical Thinking"
        ["matchLevel"]=>
        string(7) "partial"
        ["fullyHighlighted"]=>
        bool(false)
        ["matchedWords"]=>
        array(1) {
          [0]=>
          string(8) "critical"
        }
      }
      [2]=>
      array(4) {
        ["value"]=>
        string(22) "Deep Learning"
        ["matchLevel"]=>
        string(7) "partial"
        ["fullyHighlighted"]=>
        bool(false)
        ["matchedWords"]=>
        array(1) {
          [0]=>
          string(4) "deep"
        }
      }
      [3]=>
      array(3) {
        ["value"]=>
        string(20) "Independent Learning"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [4]=>
      array(3) {
        ["value"]=>
        string(15) "Problem-Solving"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [5]=>
      array(3) {
        ["value"]=>
        string(15) "Thinking Skills"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
    }
    ["tags"]=>
    array(12) {
      [0]=>
      array(3) {
        ["value"]=>
        string(15) "cultural forces"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [1]=>
      array(3) {
        ["value"]=>
        string(20) "Cultures of Thinking"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [2]=>
      array(3) {
        ["value"]=>
        string(11) "disposition"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [3]=>
      array(3) {
        ["value"]=>
        string(12) "expectations"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [4]=>
      array(4) {
        ["value"]=>
        string(17) "language"
        ["matchLevel"]=>
        string(7) "partial"
        ["fullyHighlighted"]=>
        bool(true)
        ["matchedWords"]=>
        array(1) {
          [0]=>
          string(8) "language"
        }
      }
      [5]=>
      array(3) {
        ["value"]=>
        string(9) "modelling"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [6]=>
      array(3) {
        ["value"]=>
        string(16) "pedagogy of play"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [7]=>
      array(3) {
        ["value"]=>
        string(7) "process"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [8]=>
      array(3) {
        ["value"]=>
        string(12) "Project Zero"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [9]=>
      array(3) {
        ["value"]=>
        string(10) "reflection"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [10]=>
      array(3) {
        ["value"]=>
        string(20) "thinking disposition"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [11]=>
      array(3) {
        ["value"]=>
        string(16) "thinking routine"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
    }
  }
}

Creative Teaching & Learning

Leveraging Culture with Ron Ritchhart

8 Jan 2019

array(8) {
  ["distinct_key"]=>
  string(10) "post#44542"
  ["title"]=>
  string(34) "Transferring Ownership of Learning"
  ["categories"]=>
  array(6) {
    [0]=>
    string(32) "Creative Teaching & Learning"
    [1]=>
    string(17) "Critical Thinking"
    [2]=>
    string(13) "Deep Learning"
    [3]=>
    string(11) "Mathematics"
    [4]=>
    string(7) "Science"
    [5]=>
    string(15) "Thinking Skills"
  }
  ["url"]=>
  string(50) "https://www.teachingtimes.com/ownershipoflearning/"
  ["post_date_timestamp"]=>
  int(1546947668)
  ["objectID"]=>
  string(12) "post-44542-0"
  ["_snippetResult"]=>
  array(2) {
    ["excerpt"]=>
    array(2) {
      ["value"]=>
      string(360) "... crucial for empowering students to take ownership of their learning and gain deeper conceptual understanding. Continuing on from their previous article in CTL (Cultivating a Culture of Thinking), teachers Jeff Watson and Roger Winn continue to share how the ‘Cultures of Thinking’ model is changing their Maths and Science classrooms."
      ["matchLevel"]=>
      string(7) "partial"
    }
    ["content"]=>
    array(2) {
      ["value"]=>
      string(365) "The Cultures of Thinking framework developed by Project Zero’s Ron Ritchhart focuses on eight cultural forces that exist in classrooms: expectations, language, time, opportunities, interactions, environment, routines, and modelling. According to Ritchhart, these forces exist in every classroom whether they are given attention or not. The idea is to ..."
      ["matchLevel"]=>
      string(7) "partial"
    }
  }
  ["_highlightResult"]=>
  array(6) {
    ["title"]=>
    array(3) {
      ["value"]=>
      string(34) "Transferring Ownership of Learning"
      ["matchLevel"]=>
      string(4) "none"
      ["matchedWords"]=>
      array(0) {
      }
    }
    ["author"]=>
    array(2) {
      [0]=>
      array(3) {
        ["value"]=>
        string(14) "Jeffrey Watson"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [1]=>
      array(3) {
        ["value"]=>
        string(10) "Roger Winn"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
    }
    ["excerpt"]=>
    array(4) {
      ["value"]=>
      string(419) "Finding opportunities for interactions within the classroom is crucial for empowering students to take ownership of their learning and gain deeper conceptual understanding. Continuing on from their previous article in CTL (Cultivating a Culture of Thinking), teachers Jeff Watson and Roger Winn continue to share how the ‘Cultures of Thinking’ model is changing their Maths and Science classrooms."
      ["matchLevel"]=>
      string(7) "partial"
      ["fullyHighlighted"]=>
      bool(false)
      ["matchedWords"]=>
      array(2) {
        [0]=>
        string(13) "understanding"
        [1]=>
        string(5) "maths"
      }
    }
    ["content"]=>
    array(4) {
      ["value"]=>
      string(5633) "The Cultures of Thinking framework developed by Project Zero’s Ron Ritchhart focuses on eight cultural forces that exist in classrooms: expectations, language, time, opportunities, interactions, environment, routines, and modelling. According to Ritchhart, these forces exist in every classroom whether they are given attention or not. The idea is to properly leverage those forces so that classrooms become places where thinking, collaboration, independence, and deep-learning rule and become common-place, rather than classrooms that stress compliance and work, and which are teacher-centred.  Harnessing meaningful interactions is a crucial part of creating a Culture of Thinking in a classroom. As Ritchhart points out, ‘At the heart of much of this theoretical work is the belief that transformative learning–that is, learning that cultivates the development of the whole person and strives for more than the simple transmission of information–is more likely to happen in community than in isolation. Such communities are largely democratic in nature, stressing mutuality, support, connection, and shared decision making’.1 Our last article focused on leveraging opportunities through the restructuring of lesson plans.2 This article will explore how to leverage interactions within existing lesson plans. Although many of the examples will focus on maths and science, the core principles that are drawn from these cases can be applied to any classroom. Pressing for thinkingI used to think that a focus on coming to a consistent, reproducible answer in chemistry did not provide space for discussion about competing ideas. However, through a focus on concepts and pressing for thinking, I have found that richer discussions occur. Before embracing Cultures of Thinking I would solicit student answers and give feedback on whether they were right or wrong. Now these discussions of problems are very different. During a discussion of ionization energy in MYP Chemistry,3 some students shared that they thought the ionization energy would decrease a group, while some thought it would increase. Rather than pointing out the data that supports one thought and not the other, I had students share their reasoning with a partner and then had different students share their reasoning with the class. From sharing their reasoning and focusing on the ‘why’ of their choices, students were able to have a more nuanced conversation about the trend for ionization energy and come to a class agreement. This restructuring of problem discussion can be integrated into many existing lessons.When transitioning to a classroom that focuses on thinking, the students need a cue to let them know to share their thinking. An easy way to communicate this new focus on thinking is using the ‘What Makes You Say That?’ thinking routine.4 This short question communicates to the student that you are focused on the reasoning and not just the answer. A variation I have used that has also drawn out more thinking is asking the students to ‘tell me more’. Students can struggle with sharing their thinking at times and providing scaffolds to help them organise their thoughts can help move the discussion forward. If a student can’t articulate their thinking, taking time for a ‘Think-Pair-Share’ routine to have students discuss their reasoning has been successful for me to help students articulate their reasoning.5 This strategy was used above after students shared their answers. In addition, it engages the whole class in discussing their thinking, rather than just the students sharing their thoughts. In the above example, the ‘Think-Pair-Share’ put every student in a position to contribute to the class discussion. As students shared their reasoning, they were able to see that both students had valid reasoning for their choices and the class came to an agreement on the trend for ionization energy. By the end of the discussion, students had gained a deeper conceptual understanding of the topic. But what if the discussion in agreement on an answer does not match the scientific community? What if the students just stare at each other and don’t talk?Teachers still play a role in classroom discussions. As a teacher, I think about when I should interject with a question that could help steer students’ thinking to a more consistent way of thinking. If, after asking questions about ionization energy in the above example, students still expressed misconceptions, I would have shown them data and asked them why the data does not match their prediction. It can be difficult to come up with these questions at times, but I have found that having students confront misconceptions does a better job of addressing them than telling students whether an answer is right or wrong.In some classes, students are reluctant to talk in whole class discussions, which is often because they are afraid of being wrong or nervous about what they will say. Adding in short routines that have students share their thoughts in smaller groups, like ‘Think-Pair-Share’ or a microlab routine, allows them to practice what they might say in a whole-class discussion and spread the ownership of ideas among more students. A student sharing what a group thought makes them part of a team and it is not necessarily their individual thoughts that they’re presenting.  I have found students take more intellectual risks when the ownership of the risk is spread. Supporting student autonomyBefore embracing the ideas of Cultures of Thinking, in"
      ["matchLevel"]=>
      string(7) "partial"
      ["fullyHighlighted"]=>
      bool(false)
      ["matchedWords"]=>
      array(4) {
        [0]=>
        string(8) "language"
        [1]=>
        string(4) "deep"
        [2]=>
        string(13) "understanding"
        [3]=>
        string(5) "maths"
      }
    }
    ["categories"]=>
    array(6) {
      [0]=>
      array(3) {
        ["value"]=>
        string(32) "Creative Teaching & Learning"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [1]=>
      array(4) {
        ["value"]=>
        string(26) "Critical Thinking"
        ["matchLevel"]=>
        string(7) "partial"
        ["fullyHighlighted"]=>
        bool(false)
        ["matchedWords"]=>
        array(1) {
          [0]=>
          string(8) "critical"
        }
      }
      [2]=>
      array(4) {
        ["value"]=>
        string(22) "Deep Learning"
        ["matchLevel"]=>
        string(7) "partial"
        ["fullyHighlighted"]=>
        bool(false)
        ["matchedWords"]=>
        array(1) {
          [0]=>
          string(4) "deep"
        }
      }
      [3]=>
      array(3) {
        ["value"]=>
        string(11) "Mathematics"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [4]=>
      array(3) {
        ["value"]=>
        string(7) "Science"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [5]=>
      array(3) {
        ["value"]=>
        string(15) "Thinking Skills"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
    }
    ["tags"]=>
    array(12) {
      [0]=>
      array(3) {
        ["value"]=>
        string(8) "autonomy"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [1]=>
      array(3) {
        ["value"]=>
        string(13) "collaboration"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [2]=>
      array(3) {
        ["value"]=>
        string(15) "cultural forces"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [3]=>
      array(3) {
        ["value"]=>
        string(20) "Cultures of Thinking"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [4]=>
      array(3) {
        ["value"]=>
        string(9) "exploring"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [5]=>
      array(3) {
        ["value"]=>
        string(16) "group discussion"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [6]=>
      array(3) {
        ["value"]=>
        string(12) "Independence"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [7]=>
      array(3) {
        ["value"]=>
        string(12) "Project Zero"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [8]=>
      array(3) {
        ["value"]=>
        string(9) "reasoning"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [9]=>
      array(3) {
        ["value"]=>
        string(10) "reflection"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [10]=>
      array(3) {
        ["value"]=>
        string(16) "thinking routine"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [11]=>
      array(3) {
        ["value"]=>
        string(11) "whole group"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
    }
  }
}

Creative Teaching & Learning

Transferring Ownership of Learning

8 Jan 2019

array(8) {
  ["distinct_key"]=>
  string(10) "post#44410"
  ["title"]=>
  string(33) "Cultivating a Culture of Thinking"
  ["categories"]=>
  array(11) {
    [0]=>
    string(26) "Cognition and Intelligence"
    [1]=>
    string(22) "Collaborative Learning"
    [2]=>
    string(32) "Creative Teaching & Learning"
    [3]=>
    string(17) "Critical Thinking"
    [4]=>
    string(13) "Deep Learning"
    [5]=>
    string(14) "Growth Mindset"
    [6]=>
    string(20) "Independent Learning"
    [7]=>
    string(11) "Mathematics"
    [8]=>
    string(15) "Problem-Solving"
    [9]=>
    string(7) "Science"
    [10]=>
    string(15) "Thinking Skills"
  }
  ["url"]=>
  string(49) "https://www.teachingtimes.com/acultureofthinking/"
  ["post_date_timestamp"]=>
  int(1539166195)
  ["objectID"]=>
  string(12) "post-44410-2"
  ["_snippetResult"]=>
  array(2) {
    ["excerpt"]=>
    array(2) {
      ["value"]=>
      string(255) "Cultural forces exist in all classrooms that can be leveraged to develop a culture of thinking. Jeff Watson and Roger Winn demonstrate how to harness these forces to develop students’ thinking skills in secondary Maths and Chemistry classrooms."
      ["matchLevel"]=>
      string(7) "partial"
    }
    ["content"]=>
    array(2) {
      ["value"]=>
      string(353) "... the lessons for high achieving students to push their thinking? Effective CommunicationIn order to be mathematicians and chemists, students must be able to “use the language of the discipline to express, represent, justify, and communicate one’s thinking and ideas” (p. 164). In the maths lesson, students had to use ..."
      ["matchLevel"]=>
      string(7) "partial"
    }
  }
  ["_highlightResult"]=>
  array(6) {
    ["title"]=>
    array(3) {
      ["value"]=>
      string(33) "Cultivating a Culture of Thinking"
      ["matchLevel"]=>
      string(4) "none"
      ["matchedWords"]=>
      array(0) {
      }
    }
    ["author"]=>
    array(2) {
      [0]=>
      array(3) {
        ["value"]=>
        string(14) "Jeffrey Watson"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [1]=>
      array(3) {
        ["value"]=>
        string(10) "Roger Winn"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
    }
    ["excerpt"]=>
    array(4) {
      ["value"]=>
      string(255) "Cultural forces exist in all classrooms that can be leveraged to develop a culture of thinking. Jeff Watson and Roger Winn demonstrate how to harness these forces to develop students’ thinking skills in secondary Maths and Chemistry classrooms."
      ["matchLevel"]=>
      string(7) "partial"
      ["fullyHighlighted"]=>
      bool(false)
      ["matchedWords"]=>
      array(1) {
        [0]=>
        string(5) "maths"
      }
    }
    ["content"]=>
    array(4) {
      ["value"]=>
      string(4412) "achieve the challenge? What space have I provided in the lessons for high achieving students to push their thinking? Effective CommunicationIn order to be mathematicians and chemists, students must be able to “use the language of the discipline to express, represent, justify, and communicate one’s thinking and ideas” (p. 164). In the maths lesson, students had to use thinking routines to document what they already knew about vectors and then share, communicate and justify these ideas to each other. The MicroLab protocol is a great communication tool because it requires students to express their ideas to each other, and since each person has time to talk, those students who normally may not share an idea have an opportunity to express their ideas. Also, when exploring the new concepts, groups had to communicate their ideas to each other in order to answer the guiding question. In chemistry, groups presented their voltaic cells and explained how they work. By explaining their cells, especially the ones that do not look like those in the textbook, both the listeners and presenters are able to deepen their understanding of voltaic cells. When designing lessons that contain effective communication, we find the following questions useful: How much is the teacher talking? How much are the students talking? How are the students given time to share and communicate their ideas? Perceived WorthIn these lessons, students were asked to create a product, whether it is the solution to problems, the construction of a battery or a method to finding a vector. Ron Ritchhart notes, “Production is common in classrooms, but what is not so common is that what is produced is perceived by students to be of value and worth their time and investment” (p. 164). The revised chemistry and maths lessons worked to increase the perceived worth of the products required. In the old chemistry lesson, the required product was solutions to practice problems. These problems have little perceived worth – students understand that their ability to accurately solve these problems will impact their grade, but these problems are largely disconnected from the ‘real world’. In the new lesson, perceived worth was increased by relating the challenge to batteries in cell phones. Rather than just being able to solve problems related to voltaic cells, students are now using chemistry to understand an object they use every day. All the information and understanding that students discovered about voltaic cells related to this challenge, which is what is shown to create perceived worth the most, “….it [is] the ability of the teacher to place the activity within the context of a larger goal or enterprise that mattered” (p. 165). In the maths lesson, the larger goal was the use of unit vectors in particle motion. In the past, unit vectors were treated as a stand alone topic that students learned about because it was in the syllabus. Now, the use of vector notation and finding magnitude paved the way to find unit vectors which paved the way for the application to velocity and motion. Furthermore, by adapting the home thinking to each group’s progress, it placed worth on the thinking and understanding they were doing during the class period. When thinking about increasing perceived worth, I ask myself the following questions: To what extent is this lesson connected to a larger goal or context? How have I made connections to this larger context or goal clear to the students? As we continue to challenge the “..widely held misconception: that teaching is primarily the delivery of information and that learning is memorizing that information” (p. 151), creating powerful learning opportunities is crucial to ensure deep learning. Designing these opportunities is an art and is an iterative, exciting process that can only be improved with careful consideration, time and trial-and-error. Although the new lessons above are richer examples than their predecessors, these lessons will undergo continuous changes as our experiences grow. Jeffrey Watson is a Maths teacher and Roger Winn is a Chemistry teacher at the International Academy East high school in Troy, Michigan, USA. They can be contacted with questions or ideas at jwatson@bloomfield.org or rwinn@bloomfield.org. "
      ["matchLevel"]=>
      string(7) "partial"
      ["fullyHighlighted"]=>
      bool(false)
      ["matchedWords"]=>
      array(4) {
        [0]=>
        string(8) "language"
        [1]=>
        string(4) "deep"
        [2]=>
        string(13) "understanding"
        [3]=>
        string(5) "maths"
      }
    }
    ["categories"]=>
    array(11) {
      [0]=>
      array(3) {
        ["value"]=>
        string(26) "Cognition and Intelligence"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [1]=>
      array(3) {
        ["value"]=>
        string(22) "Collaborative Learning"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [2]=>
      array(3) {
        ["value"]=>
        string(32) "Creative Teaching & Learning"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [3]=>
      array(4) {
        ["value"]=>
        string(26) "Critical Thinking"
        ["matchLevel"]=>
        string(7) "partial"
        ["fullyHighlighted"]=>
        bool(false)
        ["matchedWords"]=>
        array(1) {
          [0]=>
          string(8) "critical"
        }
      }
      [4]=>
      array(4) {
        ["value"]=>
        string(22) "Deep Learning"
        ["matchLevel"]=>
        string(7) "partial"
        ["fullyHighlighted"]=>
        bool(false)
        ["matchedWords"]=>
        array(1) {
          [0]=>
          string(4) "deep"
        }
      }
      [5]=>
      array(3) {
        ["value"]=>
        string(14) "Growth Mindset"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [6]=>
      array(3) {
        ["value"]=>
        string(20) "Independent Learning"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [7]=>
      array(3) {
        ["value"]=>
        string(11) "Mathematics"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [8]=>
      array(3) {
        ["value"]=>
        string(15) "Problem-Solving"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [9]=>
      array(3) {
        ["value"]=>
        string(7) "Science"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [10]=>
      array(3) {
        ["value"]=>
        string(15) "Thinking Skills"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
    }
    ["tags"]=>
    array(8) {
      [0]=>
      array(3) {
        ["value"]=>
        string(16) "Asking Questions"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [1]=>
      array(3) {
        ["value"]=>
        string(15) "cultural forces"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [2]=>
      array(3) {
        ["value"]=>
        string(20) "Cultures of Thinking"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [3]=>
      array(3) {
        ["value"]=>
        string(16) "group discussion"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [4]=>
      array(3) {
        ["value"]=>
        string(7) "process"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [5]=>
      array(3) {
        ["value"]=>
        string(12) "Project Zero"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [6]=>
      array(3) {
        ["value"]=>
        string(10) "reflection"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
      [7]=>
      array(3) {
        ["value"]=>
        string(16) "thinking routine"
        ["matchLevel"]=>
        string(4) "none"
        ["matchedWords"]=>
        array(0) {
        }
      }
    }
  }
}

Cognition and Intelligence

Cultivating a Culture of Thinking

10 Oct 2018