A Keynote presentation to the MAWA Secondary Convention

21 November 2017

by Professor Dianne SiemonRMIT University

Targeted Teaching Works!

75% of the fastest growing

occupations require STEM 1

44% or 5.1 million jobs at risk from digital disruption 1

The average performance of Year 8 students in

mathematics has not changed since TIMSS 1995 2

Australia’s mean mathematical literacy performance declined significantly between PISA 2003 and PISA 2012 and males

significantly outperformed females 4

The STEM imperative

1. Price-Waterhouse Report (April, 2015). A Smart Move: Future proofing Australia’s workforce by growing skills in science, mathematics, engineering and maths (STEM)

2. Thompson, S., Hillman, K. & Wernet, N. (2012). Monitoring Australian Year 8 student achievement internationally: TIMSS 2011. Melbourne: ACER

3. The Australian Industry Group (March, 2015). Progressing STEM skills in Australia. Melbourne: AiGroup4. Thompson, S.De Bortoli, L. & Buckley, S. (2013). PISA 2012: How Australia measures up. Melbourne: ACER

Shifting 1% of workforce into

STEM roles would add $57.4

billion to GDP over 20 years 1

The number of students taking

intermediate and advanced maths at

secondary school has fallen by 34% over the

last 18 years 3

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Ø Success in school mathematics is a strong predictor of STEM engagement (Wai et al, 2009; Lowrie, 2017)

A situation analysis:

Ø Proportion of students undertaking Specialist Maths is declining (AIG, 2015)

Ø There is a 7-8 year range in mathematics achievement in each year level 5 to 9 (Siemon, 2006, 2016)

Ø Students are opting out of STEM courses in the senior years (PWC, 2015)

Why are Australian students opting out of STEM?Some possibilities:• Unlike many other countries (e.g., Finland, the USA and China),

mathematics and science are not mandated studies at Year12 (Wilson, The Conversation, 2015)

• Spoilt for choice – the number and range of subject offerings in the final years of schooling (e.g., over 100 subjects in VCE)

• Intermediate and Advanced level mathematics not necessarily required for further STEM study (Academy of Science, Decadal Plan, 2016)

• STEM paradox – the underemployment of STEM graduates, particularly in life sciences, chemistry and physics (Productivity Commission Report, June 2016)

• Perceived risk to ATAR scores – the ‘opportunity costs’ associated with choosing STEM subjects (Connolly, CIO, 2016)

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• Up to 55% of Year 8 students do not have access to multiplicative thinking and, as a result, experience considerable difficulty with fractions, decimals, ratio, percent and all forms of proportional reasoning (MYNRP, 2001; SNMY 2006; RMF, 2013)

• Many Year 4 to 9 students are unable to apply what they know to solve unfamiliar situations, explain their thinking or communicate mathematically (MYNRP, 2001; SNMY, 2006; RMF, 2013)

• Spatial ability plays a critical role in developing STEM expertise (Wai, Lubinski & Benbow, 2009; Lowrie, 2017)

• A significant proportion of Year 7 to 10 students experience difficulty with tasks requiring spatial, algebraic or statistical reasoning (RMFII project 2014 -)

Symptoms not causes?

Good news is that targeted teaching works!

Targeted teaching is a form of differentiated instruction that is specifically concerned with student’s learning needs in relation to a small number of really ‘big ideas’ in number without which students’ progress in school mathematics will be seriously impacted (Siemon, 2006).

© Dianne Siemon

counting

Trusting the count

Place-value

subitising

knowledge of numbers to 10 in

terms of their parts

Additive thinking1000 of

these is 1 of those

1 tenth of these is 1 of those

Multiplicative Thinking

Partitioning

Proportional Reasoning

Alge

bra

10 of these is 1 of those

The ‘BIG IDEAS’ in Number Generalising

© Dianne Siemon

Students in an identified sub-sample of ‘at-risk’ students within the SNMY Project demonstrated major shifts in achievement against the Learning and Assessment Framework for Multiplicative Thinking (LAF) as a result of an 18 week, 2 sessions per week teaching program (Breed, 2011)*

* A copy of the Intervention Teaching Program for At Risk Students is included in the SNMY Project Findings, Materials and Resources available on the DET and TasEd websites.

Participants: 9 Year 6 students identified at Level 1 of the Framework in May 2004

Results: All students achieved at Level 4 or 5 of the Framework in November 2005

For example,

Targeted teaching requires:

• sufficient time with students to develop trust and supportive relationships; and

• flexibility to spend time with the students who need it most.

• access to assessment techniques that expose student’s thinking;

• a grounded knowledge of learning trajectories (key steps in the development of big ideas and how to scaffold these);

• an expanded repertoire of teaching approaches which accommodate and nurture discourse, help uncover and explore student’s ideas in constructive ways, and ensure all students can participate in and contribute to the enterprise;

(MYNRP, Final Report, 2001)© Dianne Siemon9

0.0

5.0

10.0

15.0

20.0

25.0

LAF1 LAF2 LAF3 LAF4 LAF5 LAF6 LAF7 LAF8

August

November

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Percentage of RMFI students at each LAF level, all students, August and November 2013 (N =1732)

47% in Zones 1-3 (Aug)42% in Zones 1-3 (Nov)

22% in Zone 4 (Aug)19% in Zone 4 (Nov)

31% in Zones 5-8 (Aug)40% in Zones 5-8 (Nov)

Targeting multiplicative thinking works!

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5

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25

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Zone 1 Zone 2 Zone 3 Zone 4 Zone 5 Zone 6 Zone 7 Zone 8

Pre test (Dec 2104)

Post test (Sept 2105)

Results from an RMFII Project School who used SNMY data and the Learning Assessment Framework for Multiplicative Thinking to inform a targeted teaching approach in Year 8 in 2015

Percentage of students in each Zone of the SNMY LAF in August 2014 (n = 141) and September 2015 (n = 152) (Prasad, November 2016)

The Year 9 NAPLAN results for the same school in 2016 suggest that the growth in learning was sustained and transferable

Growth in student learning was demonstrated for all bar 5 students BUT 65% of students grew by more than the expected growth - well

above State-wide expectations

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Aim: To build a sustainable, evidence-based learning and teaching resource to support the development of mathematical reasoning in Years 7 to 10Approach: • Develop Hypothetical Learning Trajectories for algebraic,

spatial and statistical reasoning from previous research • Develop, trial and validate rich assessment tasks for

Mathematical Reasoning (MR)• Use results to identify big ideas, Draft Learning

Progressions and teaching advice• Trial and evaluate MR Assessment Forms, teaching

advice and activities; monitor student progress.

Reframing Mathematical Futures II (An AMSPP Project, 2014-2017):

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RMIT University: • Di Siemon (Project Leader) • Tasos Barkatsas• Rebecca Seah• Sandra van der Pal (Project Manager)• Claudia Johnstone and Claudia Orrellana (Project Support)

Extended Research Team: • Rosemary Callingham, Jane Watson & Greg Oates (University of

Tasmania) • Lorraine Day (University of Notre Dame, WA)• Marj Horne (Australian Catholic University & RMIT)• Will Morony & Kate Manuel (AAMT)• Max Stephens (The University of Melbourne & RMIT)• Bruce White (University of South Australia)

RMFII Research Team

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• Brisbane Catholic Education Office• Department of Education (TAS)

• Department of Education (WA)• Department of Education and Childhood Services (SA)

• Department of Education and Communities (NSW)• Department of Education and Training (NT)• Department of Education and Training (VIC)

RMFII Project Partners

32 Secondary Schools across Australia

Why algebraic, geometric and statistical reasoning in Years 7 to 10

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• Although 85% of the Year 8 mathematics curriculum requires some form of multiplicative thinking (Siemon, 2013), this is not reflected in curriculum descriptors or textbooks

• Curriculum sequences tend to be based on historical precedent rather than research (e.g., linear equations have always been taught in Year 7, integers in Year 8)

• The large range in student achievement at this level tends to promote over-scaffolding and the atomisationof the curriculum into discrete ‘topics’ and skill sets, leaving ‘no room’ for complex problem solving, inquiry, extended discussion or mathematical reasoning

Need for evidence at scale

i. core knowledge needed to recognise, interpret, represent and analysealgebraic, spatial, statistical and probabilistic situations and the relationships/connections between them;

ii. an ability to apply that knowledge in unfamiliar situations to solve problems, generate and test conjectures, make and defend generalisations; and

iii. a capacity to communicate reasoning and solution strategies in multiple ways (i.e. diagramatically, symbolically, orally and in writing).

What is needed…

Mathematical Reasoning For the purposes of the RMFII project, mathematical reasoning (MR) encompasses:

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Assessing mathematical reasoning (MR)• Tasks developed on the basis of

the HLTs and trialed in RMFII Schools (3254 student records)

• Multiple Forms comprising 5 to 6 tasks prepared and trialed as Standard Forms (strand-specific) or Mixed Forms (two strands)

• Forms prepared as booklets and administered over 2 to 3 sessions in Trial and RMFII schools

• Scoring rubrics used to assess student MR

• Codes used to enter student data and track progress

• Tasks and rubrics redesigned as needed by Research Team

Form

Task

Items

Student ID

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Data Collection

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Source Algebra Geometry StatisticsTrial Schools (Sem 1 2016) 355 601 334

RMFII Schools MR1 (Sem 2 2016)

1655 914 1378

RMFII SchoolsMR2 (Sem 1 2017)

518 198 352

RMFII SchoolsMR3 July/Aug 17RMFII SchoolsMR4 Nov/Dec 17Total to date 2528 1713 2064

Number of Year 7-10 Student Responses to MR Trial Forms to June 2017

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Analysis - Developing the draft learning progressions (DLPs):Assessment tasks were separately coded to track use across Forms (e.g., ATILP3 refers to the third item in Algebra Tiles)

The resulting data were analysed using the Rasch partial credit model (Masters, 1982) using Winsteps 3.92.0 (Linacre, 2016). This allows both student performance and item difficulties to be measured using the same unit (logit), and placed on an interval scale (Bond & Fox, 2015).

Algebra Variable Map for Trial school and MR1 data, Feb 2017

Harder items

Easier items

logit scale

Student performance

score of 2 on item 1 of the Hot Air Balloon

Task (AHAB1)

draft Zones

ATLP3.5

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ZONE DESCRIPTION

1 Abletocontinuesimplepatternsandinterpretdatainatableasasimplecountbutisstillworkingadditively,cannotyetconnectscaleofagraphtoinformationinatable,cannotyetseetherelationshipbetweenentitiesinequationsandisreluctanttogiveanyexplanation

2 Canrecognisesomemultiples– e.g.60asamultipleof6.Recognisesmorecomplexpatterns,butisstillworkingadditively.Hasanintuitiveappreciationofpattern.Canworkwithsimplescalesandtransferfromatableofvaluestoagraph(tworepresentations).Coordinaterelationshipsarepoint-wise(E.g.ABBUY1.1).Recognisesrelationshipslike6more/6less,whilenotnecessarilyrecognisingequivalence.Cansometimesrespondcorrectlybutwithlittle/noreasoning.

3 Extendspatterns,recognisessimplerulesexpressingtheminwordsandsymbols,recognisesrelationshipsbetweennumbersinequations,simplemultiplicativerelationshipsandsomeequivalentrelationshipsbutstilltendstofocusonadditiverelationshipsandisnotyetusingrelationshipstosolveproblems,norexplainingandapplyingmoregenerally.Isbeginningtomakesimplegeneralisationsbutjustifieswithanexample

4 Recognisescomplexadditiverelationshipsandbeginningtorecognisefunctionalrelationshipsbutmaynotsupportwithreasoningoridentifytheinherentmultiplicativebasis.Recognisesandinterpretstherelevanceofrangeinrealfunctioncontextswithtablesand/orgraphs.Appliesmultiplicativethinkingininput/outputsituationsandisbeginningtoworkwithratios.Beginningtransitiontoabstractionbyinsertingnumberforpronumeral,recognisingthatwhatvariablesandconstantsneedtobetreateddifferentlyRecognisestheimpactofoperationssuchasadditionandsubtractioninrelationshipsinequationsandoftenattemptstosolveequationsbygivingthevariableanumericalvalue.Attemptssomeexplanationbutjustificationtendstobelimitedtoproofbyanexample.DLP Algebra Zones 1 to 4

Hot Air Balloon 1

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ZONE DESCRIPTION

5 Recognisesandrepresentssimplefunctionalrepresentationsininput/outputsituationsandascontinuousgraphicalrepresentations,abletointerpolateimplyinganintuitiveunderstandingoftheshapeofthegraph.Recognisesstructuralrelationshipsandspecialisesbyassigninganumbertoavariable.Canderiveastrategythatmaintainsequationandequivalencebutcannotyetgeneralise,reasoningwithmorecomplexadditivesituationsthroughexamplesBeginningtoworkwithgeneralisationsandusingsymbolsabstractlyandjustifiessomeargumentsusingmathematicaltextbutstillneedstobuildonorconnecttotheknownsituation.

6 Describesandjustifiesrulesinvolvingmultiplicativerelationshipsandusesratios. Beginningtoworkwithsymbolstoexpressrules,butnotnecessarilyinthesimplestwayanddescribesandgeneratesruleswhenapromptispresent.Recognisestheadditiveinverseandmaintainsequivalenceanddemonstratesunderstandinginmorecomplexrelationalreasoningsituations.Abletogeneraliseinsimplearithmeticrelationships.

7 Canmovebetweenmultiplerepresentations,interpolatedata,workwithrangeandacceptsomedegreeofuncertainty. Abletoqualifystatementstoappreciateafinalrelationshipandcandescribemultiplicativerelationshipsusingtwovariablesexpressedabstractly(proportionalreasoning).Identifiestherelationshipsbetweentwofunctionsandidentifypointsofintersectiongivingmeaningtotheminrealcontexts.CanreasonbasedontherelationshipsoneithersideofanequalssignandCandescribeinwordsageneralisedrelationshipderivedfromanequation

8 Canvisualisetheformandstructureofafunctionatleastgraphicallyfromarealcontext.Abletorelatetheuseofabstractsymbolstoanotheridea,suchasperimeterandisbeginningtousethesimplestformtodoso.Canreasonbasedonarelationshipbetweennumbersandcansuccessfullyworkthroughproblemsthathavemultiplesteps.DLP Algebra Zones 5 to 8

Tiling Patterns 3

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ZONE DESCRIPTION

5 Recognisesandrepresentssimplefunctionalrepresentationsininput/outputsituationsandascontinuousgraphicalrepresentations, abletointerpolateimplyinganintuitiveunderstandingoftheshapeofthegraph.Recognisesstructuralrelationshipsandspecialisesbyassigninganumbertoavariable.Canderiveastrategythatmaintainsequationandequivalencebutcannotyetgeneralise,reasoningwithmorecomplexadditivesituationsthroughexamplesBeginningtoworkwithgeneralisationsandusingsymbolsabstractlyandjustifiessomeargumentsusingmathematicaltextbutstillneedstobuildonorconnecttotheknownsituation.

6 Describesandjustifiesrulesinvolvingmultiplicativerelationshipsandusesratios. Beginningtoworkwithsymbolstoexpressrules,butnotnecessarilyinthesimplestwayanddescribesandgeneratesruleswhenapromptispresent.Recognisestheadditiveinverseandmaintainsequivalenceanddemonstratesunderstandinginmorecomplexrelationalreasoningsituations.Abletogeneraliseinsimplearithmeticrelationships.

7 Canmovebetweenmultiplerepresentations,interpolatedata,workwithrangeandacceptsomedegreeofuncertainty. Abletoqualifystatementstoappreciateafinalrelationshipandcandescribemultiplicativerelationshipsusingtwovariablesexpressedabstractly(proportionalreasoning).Identifiestherelationshipsbetweentwofunctionsandidentifypointsofintersectiongivingmeaningtotheminrealcontexts.CanreasonbasedontherelationshipsoneithersideofanequalssignandCandescribeinwordsageneralisedrelationshipderivedfromanequation

8 Canvisualisetheformandstructureofafunctionatleastgraphicallyfromarealcontext.Abletorelatetheuseofabstractsymbolstoanotheridea,suchasperimeterandisbeginningtousethesimplestformtodoso.Canreasonbasedonarelationshipbetweennumbersandcansuccessfullyworkthroughproblemsthathavemultiplesteps.DLP Algebra Zones 5 to 8

Tiling Patterns 3

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Rectangle Task 3

GRECT3.2

Geometry Variable Map for Trial school and MR1 data, Feb 2017

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ZONE DESCRIPTION

1 Recognisesimpleshapesbyappearanceandcommonorientation,showemergingrecognitionofobjectsfromdifferentperspective,namesomecommon3Dobjects,identifysomestandardnets,identifyinglocationusingsimplereferencingsystem

2 Identifysimpleshapesinsituandonsimplesolids,andrecognisesomereflectivesymmetry,netsofsimplesolidsandsimpleshapes.Showemergingrepresentationof3Dobjects,useoflanguageofgeometryandunderstandingofmeasurementconceptsbutdoesnotcoordinateinformationorjustifythinking

3 Abletovisualisesomeobjectsfromdifferentperspectiveandtousecoordinates.Useoneortwopropertiesorattributes(insufficient)toexplaintheirreasoningaboutshapesandmeasurementbutoftendonotrecognisepropertiesinnon-standardrepresentations.

4 Knowssomegeometriclanguage,namesome3Dobjectsandisabletovisualiseobjectsfromadifferentperspectivebutshowincompletereasoningingeometricandmeasurementsituations.performingmeasurementcalculationsbutattendingtoonlyoneattribute,givedirectionsofamapfrompersonalratherthanotherviewer’sperspective.

5 Abletovisualiseandrepresent3Dobjectsusing2Dplatform(Nets),useeitherpropertiesororientationstoreasoningeometricsituation,uselandmarkbutretainpersonalorientationwhenprovidingdirections,demonstrateknowledgeofdilutionandcoordinatesystems,providepartialsolutionsandexplanationswhencalculatingmeasurementsituations.

6 Abletomakedeductionaboutanglesituationswithlimitedexplanations,usepropertiesaccuratelywhenreasoningaboutspatialsituationsbutlackknowledgeofgeometryhierarchy,provideaccuratedirectionsfromamap,geometricandmeasurementargumentsrelyonexamples/counterexamples,omittingonestepwhencalculatingmulti-stepmeasurementproblems.

7 Beginningtorecognisenecessaryandsufficientconditions,usesoundreasoninginargument/explanations,explanationsoftenareprocedurallybased

8 Constructsargumentsbasedonmultiplepropertiesof2Dshapesand3Dobjects,usingthenecessaryandsufficientconditionstoreasonaboutgeometricandmeasurementsituations,conjecturesandpropositions(theorem),demonstrateunderstandingofbothreflectionalandrotationalsymmetry.

DLP Geometry Zones 1 to 8

Rectangle Task 3

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Statistics Variable Map for Trial school and MR1 data, Feb 2017

DLP Statistics Zones 1 to 8

Boxes 9

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(SBOX9) Box A and Box B are filled with red and blue marbles as follows. Each box is shaken. You want to get a blue marble, but you are only allowed to pick out one marble without looking. Should you choose Box A, Box B or doesn’t it matter?Please explain your answer using as much mathematics as you can

SCORE DESCRIPTION ZONE

0 No response or irrelevant response

1 Incorrect (A or B) or Correct (it doesn’t matter) with little/no explanation or general statement involving chance (e.g., could be anything, just luck, 50/50 pick)

2. Beginning to consider aggregated information but without an appreciation of statistical value e.g., identifies an incorrect association between two variables; describes what a graph is about without any interpretation etc

2 Correct or Incorrect, explanation relies on quantities alone suggesting additive thinking (e.g., it idoesn’t matter both have more red, A because it has only 2 more reds and B has 20 more reds or B because it has more blues)

4. Links statistical ideas such as variation but inconsistently, sometimes linking inappropriately in complex contexts such as two-way tables

3 Correct (it doesn’t matter) with an explanation that suggests multiplicative thinking (e.g., they both have a 40% chance

5. Makes informal group comparisons (not just single values) and summarises data with simple numerical statements but no proportional reasoning

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35.0%

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45.0%

50.0%

Zone 1 Zone 2 Zone 3 Zone 4 Zone 5 Zone 6 Zone 7 Zone 8

% in Zones by grade

Year 7 Year 8 Year 9 Year 10

Year 7 N = 46Year 8 N = 168Year 9 N = 186Year 10 N = 118

Combined MR1 and MR2 Spatial Reasoning Data for Years 7-10, N= 518 (October 2017)

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E - A javelin is thrown. The graph shows the height of the javelin against the time it is in the air

Score Year 7 Year 8* Year 9* Year 10 Total0 39.6 35.3 66.7 43.1 42.01 28.1 29.4 11.1 18.3 22.92 32.3 35.3 22.2 38.5 35.1

Percentage of students by score and year level for ARLGR1, MR1 and MR2 Combined data (n = 231, * less than 20)

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C - A piece of paper is folded in half to form two parts. The result is then folded in half repeatedly and after each fold the number of parts is counted. The graph shows the number of folds against the number of parts made.

Score Year 7 Year 8* Year 9* Year 10 Total0 62.5 82.4 100 78.1 72.71 22.9 17.6 0 12.4 16.72 14.6 0 0 9.5 10.6

Percentage of students by score and year level for ARGRL2, MR1 and MR2 Combined data (n = 231, * less than 20)

Mathematics is central to STEM

It is the only subject whose study consistently enhances performance across all fields of science. It forms the basis of most scientific and industrial research and development. As an enabling discipline, mathematics, has been, and will continue to be at the heart of our search for ways to solve, manage, mitigate or adapt to some of the great challenges that confront us as a nation and as part of humankind.

(The Chief Scientist, Professor Ian Chubb, September 2014)

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An evidence-based learning and teaching resource to support the development of mathematical reasoning in Years 7-10 that includes:• validated assessment tools, • evidence-based learning progressions for developing

algebraic, spatial and statistical reasoning;• targeted teaching advice for each level/zone in each

learning progression, and • task-based professional learning modules aimed at

deepening teacher’s pedagogical content knowledge for teaching Year 7 to 10 mathematics.

To be available from Dimensions, a web-based portal hosted by AAMT, in early 2018

RMFII Outcomes:

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Targeted Teaching Advice:

Advice organised in terms of:

• what to consolidate and establish, and

• what to introduce and develop

Zone Summary

Aspect of Mathematical Reasoning

Includes rich, multi-level tasks