Dan Meyer talks about how Mathematics lessons/curriculum need to be reorganised to put problem solving at the heart of all learning.
Mathematics at roxy
Creating Rich and BalancedMaths Programs to fit theNational Curriculum and AUSVELSPart OnePD @ RPPS1st May 2013
PD Learning Intentions:• To establish compelling reasons forinvestigating and implementing change in theteaching and learning of Maths at RPPS• To introduce the ‘Mathematician’s Model’• To explore ‘Rich Tasks’ within the context ofMathematics
Mathematics @ RPPS• State of Maths at RPPS:What does the student data tell us?
Mathematics @ RPPS• State of Maths at RPPS• What does the staff data tell us?
Mathematics @ RPPSSo where to now?Change in strategy : ‘Mathematician’s Model’Not 4 ‘number’ 2 ‘other’but4 ‘Toolbox’ and 2 ‘Be a Mathematician’ lessons
Who is in the photo in the middle?http://en.wikipedia.org/wiki/Terence_TaoWe celebrate our pop stars, sports stars...Let’s celebrate our mathematicians too!Let’s provide our students with positiveacademic role models.
The Mathematician’s ModelThe school has six sessions a week (50minutes a day). (FYI: ‘Key Characteristics ofEffective Numeracy Teaching’ advocates for 5x 1hr sessions per week).For four sessions a week, teachers run‘toolbox lessons’. ‘You cannot solve anyproblems if you don’t have the tools’.
Toolbox LessonsToolbox lessons focus on:Mathematical content knowledgeDeveloping a range of problem solving strategiesDeveloping student’s mental arithmetic capacity
Toolbox Lessons: What do they look like?BEFORE TEACHING:• Students prior knowledge assessed via Pre Unit Test.• Toolbox lessons build skills based on progression of skills according to mathscontinuum. Students grouped and placed on continuum according to need (basedoff test findings). Planning completed after pre unit test and based on studentneed.TEACHING:• Lessons progressive, building upon prior learning.• Learning intentions communicated with students at the start of the lesson.• Strong emphasis placed on collaboration and discussion between students.• Strong emphasis placed upon development of mental arithmetic strategies.• Learning wrapped up with a reflection on learning:http://www.magicalmaths.org/top-50-maths-mini-plenaries-ideas-to-use-in-an-outstanding-maths-lesson/ASSESSMENT:Observation, Questioning, Rich Tasks (Portfolio), Post Unit, Post Semester (OnDemand)
‘Be A Mathematician’ LessonA carpenter is called upon to fix a door whichhas come off it’s hinges. The carpenter doesnot turn up empty handed – what are theycarrying? Answer: a toolbox – they take onelook at the problem and go rummagingaround in their tool box for the right tool – inthis case, a screwdriver. Importantly, if thetool is not there, the problem cannot be fixed,but equally, the only reason for carrying thetool us to use it in problem situations.
‘Be A Mathematician’ LessonIn these sessions students get to ‘be amathematician’.Teachers use a range of Rich open-endedinvestigative challenges. The investigation isthe mathematics. The ‘toolbox’ skills areactivated to solve a worthwhile problem.Work completed can be collected andsubmitted into a Maths Portfolio, to be usedas part of our assessment schedule.
What is a “Rich Task”?A rich task involves both process and product,following an inquiry-based model of learning.Students learn large amounts of new content,develop important skills and develop ininterdisciplinary learning. This includes personal-management, interpersonal development,communication, ICT and particularly in thinking.
How to create a ‘Rich Task’?• Open Ended• Problem Based• Inquiry Based• Wide/Narrow Curriculum• Process Product• Collaboration• Experiential• Engaging and Relevant
Open Ended?In an open-ended task there are multiple possible outcomes forsuccess. This assists in catering for different levels of abilityamongst the students. It also allows for student ownership of thetask, as they are able to choose their own directions and work ontheir own solutions. Student choice is important! It allows forstudent ownership, self-direction, and engagement. This meansstudents are genuinely thinking for themselves, rather than simplytrying to crack the code to predict an answer/solution that has beenpredetermined as being correct by the teacher. Make sure that thetask is genuinely open-ended, and not just that it is possible forsome minor differences in answers, with only one main solutionbeing possible. It is also a good idea to aim to be open-ended inallowing for various modes of presenting the final product (ie:speech, ICT, visuals, movie, drama, print-based text, etc.).
Problem Based?Having a task where students have to respond bysolving a problem ensures that there will be theneed for both creative and critical thinking (ie:brainstorming ideas, critiquing suggestions,evaluating, etc.). Solving an open-ended problem,where students have the power of task-ownership and self-direction provides a contextfor deep thinking and engagement. In order tosolve a problem, students have to engage inthinking and not merely rely on the pre-established ideas of others.
Inquiry Based?In short you could say this means that a task follows thepattern of Bloom’s Taxonomy: “gathering information,processing information for comprehension, applyinginformation to solve a problem and evaluating the results“.Following this model ensures that students build skills andcontent knowledge in a lot of disciplines. It also helps todevelop interdisciplinary learning, such as group-work,personal-management, thinking and communication. Youdon’t want a rich task just to be all about gatheringinformation and presenting it – there need to be tasks thatrequire students to apply their new-found knowledge bythinking creatively and critically in order to solve an open-ended problem with opportunities for self-assessment andreflection throughout.
Wide/Narrow Curriculum?A rich task should provide opportunities for widestudy and learning of content from a wideselection of areas. There should also beopportunities for narrow inspection of importantdetails that are crucial to the outcomes of thestudy. Students need opportunities to learnbroad concepts, with broad examples from thebroader world, while also having opportunities toensure that they comprehend important detailsand key skills and concepts through targetedteaching and learning.
Process ProductProcess comes before product and that the process in itself isa bigger aspect of learning than just the end product by itself.A rich task should have a significant process of learning anddiscovery as well as difficult challenges in the tasksthemselves. When assessing a student’s work on a rich task, itis important that a teacher includes some process-relatedindicators as part of the assessment. For instance, a studentmay have displayed excellent thinking, group-work andpersonal-management throughout the task, but their productmay have failed for some particular reason. It is valuable forthe student to receive feedback about both the process andproduct so that they understand their strengths and areas forimprovement as a learner. It is important to realise that therich task is both the process and the product.
Collaboration?Collaboration is important for developing interpersonal skills aswell as personal-management. It is also important for developingthe ability to think creatively and critically and to engage indiscussion and work in ways that are respectful to others. If astudent simply works by themselves, then they are less likely to bechallenged in their thinking. Working in a group means thatstudents are more likely practice important mathematicalvocabulary to justify their answers and ask clarifying questions ofothers and develop the ability to learn from others and acceptdifferences in thinking. Collaboration provides a good context forgroup discussion and exploratory talk using key mathematicalvocabulary. Collaboration also provides many challenges andsupports for students as individuals, in that they are challenged toimprove in certain areas, whilst also being supported by thevariation of skills and abilities of their peers throughout the learningprocess.
Experiential?Not all students have the life experience and knowledgerequired to tackle a open-ended, problem-based task. Notall students have the same degree of skill with print-basedtexts or receptiveness to “chalk and talk” teaching to relyon these methods for gaining new knowledge. So, thelearning experiences that go with the rich task should beones that offer students different ways to learn newcontent. For instance, excursions, interactive activities,experiments, discussions, hands-on activities, movies,documentaries, ICT, games, software, websites, audio,guest-speakers, books, drama, etc. Do not discount thevalue of print-based texts, but certainly don’t limitresources and texts just to print-based versions.
Engaging and Relevant?Ask yourself these questions; Is the taskrelevant for students as individuals? Is thetask relevant to the wide curriculum? Is thetask relevant in relation to the broader world(both local and global)? What is it about thetask that is going to engage students aslearners? A rich task may not necessary tick allof these boxes initially, but it should tick mostof them.
“I have a challenge for you today: Takesix squares and try to construct a cube inas many different ways that you can...”
There are 11 different possibilities. These arereferred to as ‘hexominoes’.What would you do next to extend this activity?
Mathematics @ RPPSPaul Halmos:• “It is the duty of all teachers, and of teachers of mathematics in particular,to expose their students to problems much more than to facts.”• “The only way to learn mathematics is to do mathematics.”• “Mathematics is not a deductive science – that’s a cliché. When you try toprove a theorem, you don’t just list the hypotheses, and then start toreason. What you do is trial and error, experimentation, guesswork.”Paul Halmos:March 3, 1916 – October 2, 2006) was a Hungarian born American mathematician who madefundamental advances in the areas of probability theory, statistics, operator theory, ergodic theory,and functional analysis .