Rice Ques'ons Bartholomew is making rice pudding using his grandmother’s recipe. For three servings of pudding the ingredients include 0.75 pints of milk and 0.5 cups of rice. Bartholomew looks in his refrigerator and sees he has one pint of milk. Given that he wants to use all of the milk, which of the following expressions will help Bartholomew figure out how many cups of rice he should use? 0.5/0.75 0.75/0.5 0.5 x 0.75 (0.5 + 1) x 0.75 none of these 36
Rice Ques'ons Bartholomew is making rice pudding using his grandmother’s recipe. For three servings of pudding the ingredients include 0.75 pints of milk and 0.5 cups of rice. Bartholomew looks in his refrigerator and sees he has one pint of milk. Given that he wants to use all of the milk, which of the following expressions will help Bartholomew figure out how many cups of rice he should use? 0.5/0.75 0.75/0.5 0.5 x 0.75 (0.5 + 1) x 0.75 none of these 37
Numerical Complexity (Calculus-based Intro Mechanics)
Characteris'cs Top 20% The rest Effect (n sample =98) (n sample =363) size SAT_M 710 670 11.4 FCI % pre/change 65/+9 42/+9 39
Characteris'cs Top 20% The rest Effect (n sample =98) (n sample =363) size SAT_M 710 670 11.4 FCI % pre/change 65/+9 42/+9 Math Reasoning % pre/ 51/+4 43/-2 2.3/4.4 change 40
Characteris'cs Top 20% The rest Effect (n sample =98) (n sample =363) size SAT_M 710 670 11.4 FCI % pre/change 65/+9 42/+9 Math Reasoning % pre/ 51/+4 43/-2 2.3/4.4 change CLASS Problem Solving (Gen) 71/-2 62/-10 % pre/change CLASS Personal Interest % 73/0 65/-9 pre/change 41
Characteris'cs Top 20% The rest Effect (n sample =98) (n sample =363) size SAT_M 710 670 11.4 FCI % pre/change 65 42 Math Reasoning % pre/ 51 43 2.3/4.4 change CLASS Problem Solving (Gen) 71 62 % pre/change CLASS Personal Interest % 73 65 pre/change 42
Characteris'cs Top 20% The rest Effect (n sample =98) (n sample =363) size SAT_M 710 670 11.4 FCI % pre/change +9 +9 Math Reasoning % pre/ +4 -2 2.3/4.4 change CLASS Problem Solving (Gen) -2 -10 % pre/change CLASS Personal Interest % 0 -9 pre/change 43
Characteris'cs Top 20% The rest Effect (n sample =98) (n sample =363) size SAT_M 710 670 11.4 FCI % pre/change 65/+9 42/+9 Math Reasoning % pre/ 51/+4 43/-2 2.3/4.4 change CLASS Problem Solving (Gen) 71/-2 62/-10 % pre/change CLASS Personal Interest % 73/0 65/-9 pre/change Average of the Median MHI Q 0.9* Q 5.5 High School p-value < .02 44
NJ school math and socioeconomics (J. Anyon 1980) MHI Socioeconomic Schoolwork culture Quin'le Status 2 nd Working class Work is evaluated for obedience to procedure. Students learn to imitate the teacher in math class. 3 rd -4 th Middle class Work is gedng the right answer. CreaLve acLviLes are occasional, for fun but not part of learning. Students are given some choice in math on which of two procedures to use to get an answer. 4 th -5 th Affluent Work is a creaLve acLvity carried out independently. professional The products of work should show individuality. Students gather data and use it to learn about mathemaLcal processes. Top 1% ExecuLve elite Work is developing one’s intellectual powers; students invent ways to measure and calculate in math class. 45
NJ school math and socioeconomics (J. Anyon 1980) MHI Socioeconomic Schoolwork culture Quin'le Status 2 nd Working class Work is evaluated for obedience to procedure . Students learn to imitate the teacher in math class. 3 rd -4 th Middle class Work is geXng the right answer. CreaLve acLviLes are occasional, for fun but not part of learning. Students are given some choice in math on which of two procedures to use to get an answer. 4 th -5 th Affluent Work is a crea've ac'vity carried out independently. professional The products of work should show individuality. Students gather data and use it to learn about mathemaLcal processes. Top 1% ExecuLve elite Work is developing one’s intellectual powers ; students invent ways to measure and calculate in math class . 46
“The biggest obstacle to success is NOT limitaLon with math skills or knowing the definiLon of density…It’s the insLtuLonal suppression of thinking.” -Richard Steinberg 2011 47
Problems 1. Most students leave their introductory college physics course with less expert-like mathemaLzaLon than before they started. 2. There are disproporLonately few African American, LaLno and NaLve American physics majors and graduate students in physics. To mathemaLze in physics means to go back and forth between the physical and the symbolic worlds. 48
The percentage of the Bachelor’s degrees granted to select underrepresented minori'es* 10% Physics Black Physics Hispanic Biology black 5% Biology Hispanic Chemistry black 0% Chemistry Hispanic 2001 2005 2009 *NaLonal Science FoundaLon’s NaLonal Center for Science and Engineering StaLsLcs 49
The percentage of the Bachelor’s degrees granted to select underrepresented minori'es* 10% Physics Black Physics Hispanic Biology black 5% Biology Hispanic Chemistry black 0% Chemistry Hispanic 2001 2005 2009 *NaLonal Science FoundaLon’s NaLonal Center for Science and Engineering StaLsLcs 50
Slide courtesy of Michael Marder 51
Slide courtesy of Michael Marder 52
Somebody else’s problem On the surface, this seems like a problem with prior math instrucAon. But it’s not – math in physics has different goals than math in math. Physics – flexible and generaLve mathemaLcs in context Math – axiomaLc reasoning in the absence of context Teaching the mathemaAcal habits of mind that are characterisAc of physics thinking should be a major goal of physics instrucAon at all levels.
Somebody else’s problem On the surface, this seems like a problem with prior math instrucAon. But it’s not – math in physics has different goals than math in math. Physics – flexible and generaLve mathemaLcs in context Math – axiomaLc reasoning in the absence of context Teaching the mathemaAcal habits of mind that are characterisAc of physics thinking should be a major goal of physics instrucAon at all levels.
Somebody else’s problem On the surface, this seems like a problem with prior math instrucAon. But it’s not – math in physics has different goals than math in math. Physics – flexible and generaLve mathemaLcs in context Math – axiomaLc reasoning in the absence of context Teaching the mathemaAcal habits of mind that are characterisAc of physics thinking should be a major goal of physics instrucAon at all levels.
• Instructors naturally assume students have a conceptual mastery of arithmeLc and algebra. • What students master in their math courses is largely procedural. • Many students have very liUle conceptual understanding of what they are doing or why they do it when they do math. 56
• Instructors naturally assume students have a conceptual mastery of arithmeLc and algebra. • What students master in their math courses is largely procedural, and not conceptual. • Many students have very liUle conceptual understanding of what they are doing or why they do it when they do math. 57
Problems 1. Most students leave their introductory college physics course with less expert-like mathemaLzaLon than before they started. 2. There are disproporLonately few African American, LaLno and NaLve American physics majors and graduate students in physics.hemaLze in physics means to go back and forth between the physical and the symbolic worlds. 58
Problem Most physics students, and especially students from low SES high schools, struggle to assimilate the habits of mind we model, and they leave our courses with even less expert-like mathemaLcal adtudes and habits.mathemaLze in physics means to go back and forth between the physical and the symbolic worlds. 59
Procedural Mastery + Conceptual Understanding
Procedural Mastery + Conceptual Understanding Proceptual Understanding
Flexible and generaLve in early math (Gray and Tall 1994) Find 47-35 • Procedure: Use number line, start at 47 count lew 35 places • Process (Flexibility) : Start at 35, move to the right 12 places • Proceptual (GeneraLve) : x=a-b represents the mathemaLcal idea “difference” ; and x=a-b implies that a=x+b Note the foundaLonal thinking for the physics noLon of Δ : Δ v=v f -v o therefore v f = Δ v + v o 63
Flexible and generaLve in early math (Gray and Tall 1994) Find 47-35 • Procedure: Use number line, start at 47 count lew 35 places • Process (Flexibility) : Start at 35, move to the right 12 places • Proceptual (GeneraLve) : x=a-b represents the mathemaLcal idea “difference” ; and x=a-b implies that a=x+b Note the foundaAonal thinking for the physics idea of Δ : Δ T=T f -T o therefore T f = Δ T + T o 64
Flexible and generaLve in early math (Gray and Tall 1994) Find 47-35 • Procedure: Use number line, start at 47 count lew 35 places • Process (Flexibility) : Start at 35, move to the right 12 places • Proceptual (GeneraLve) : x=a-b represents the mathemaLcal idea “difference” ; and x=a-b implies that a=x+b Note the foundaAonal thinking for the physics idea of Δ : Δ T=T f -T o therefore T f = Δ T + T o comparison 65
Flexible and generaLve in early math (Gray and Tall 1994) Find 47-35 • Procedure: Use number line, start at 47 count lew 35 places • Process (Flexibility) : Start at 35, move to the right 12 places • Proceptual (GeneraLve) : x=a-b represents the mathemaLcal idea “difference” ; and x=a-b implies that a=x+b Note the foundaAonal thinking for the physics idea of Δ : Δ T=T f -T o therefore T f = Δ T + T o accumulaAon comparison 66
Proceptual divide The mathemaLcs of flexible procepts is easier than the mathemaLcs of inflexible procedures. The gap is widening because the less successful are actually doing a qualitaLvely harder form of mathemaLcs. (Tall 2008) 67
Proceptual physics 68
QuanLficaLon as a scienLfic pracLce QuanLficaLon as a scienLfic pracLce: • relies on a tendency to seek invariance ² Seeking invariance is at the heart of learning (Gibson & Gibson , 1955). ² Many students don’t spontaneously consider invariance when quanLfying nature in school (Simon&Blume, 1994). • requires a proceptual understanding of arithme'c ² Tuminaro (2004): Students who do not expect conceptual knowledge of mathemaLcs to connect to physics problems do not engage in sense making when calculaLng. ² Brahmia & Boudreaux (2016): Students errors can be traced to a failure to disLnguish products from factors when reasoning about physics quanLLes. 69
QuanLficaLon as a scienLfic pracLce QuanLficaLon as a scienLfic pracLce: • relies on a tendency to seek invariance ² Seeking invariance is at the heart of learning (Gibson & Gibson , 1955). ² Many students don’t spontaneously consider invariance when quanLfying nature in school (Simon&Blume, 1994). • requires a proceptual understanding of arithme'c ² Tuminaro (2004): Students who do not expect conceptual knowledge of mathemaLcs to connect to physics problems do not engage in sense making when calculaLng. ² Brahmia & Boudreaux (2016): Students errors can be traced to a failure to disLnguish products from factors when reasoning about physics quanLLes. 70
QuanLficaLon as a scienLfic pracLce QuanLficaLon as a scienLfic pracLce: • relies on a tendency to seek invariance ² Seeking invariance is at the heart of learning (Gibson & Gibson , 1955). ² Many students don’t spontaneously consider invariance when quanLfying nature in school (Simon&Blume, 1994). • requires a proceptual understanding of arithme'c ² Tuminaro (2004): Students who do not expect conceptual knowledge of mathemaLcs to connect to physics problems do not engage in sense making when calculaLng. ² Brahmia & Boudreaux (2016): Students errors can be traced to a failure to disLnguish products from factors when reasoning about physics quanLLes. 71
QuanLficaLon as a scienLfic pracLce QuanLficaLon as a scienLfic pracLce: • relies on a tendency to seek invariance ² Seeking invariance is at the heart of learning (Gibson & Gibson , 1955). ² Many students don’t spontaneously consider invariance when quanLfying nature in school (Simon&Blume, 1994). • requires a proceptual understanding of arithme'c ² Tuminaro (2004): Students who do not expect conceptual knowledge of mathemaLcs to connect to physics problems do not engage in sense making when calculaLng. ² Brahmia & Boudreaux (2016): Students errors can be traced to a failure to disLnguish products from factors when reasoning about physics quanLLes. 72
Sample InvenLon Sequence 1 Your task this Lme is to come up with a fastness index for cars with dripping oil. All the cars drip oil once a second A Start This task is a liFle harder than before. B A company always Start makes its cars go the same fastness. C Start We will not tell you how many companies D there are. Start You have to decide E which cars are from the Start same company. They may look different! F Start 73
QuanLficaLon is a conceptual blend double scope arithmeAc reasoning blend , in which two disLnct domains of thinking are merged to form a new cogniLve space opLmally suited for producLve work Conceptual Connection understanding to the physics of arithmetic world operations and Physically representations meaningful reasoning in introductory physics
ICC (InvenLng with ContrasLng Cases) Schwartz, Chase, Oppezzo, & Chin 2011 • InstrucLonal model designed to help students develop the tendency to – Seek invariance – Make sense with compound quanLLes – ContrasLng helps students noLce what maUers and what doesn’t – PreparaLon for subsequent instrucLon 75
Invention Instruction Coordinated set of Starting Resources Resources math procedures Proceptual (disconnected) understanding of mathematics Invention Tasks capacity to respond to prompts to (quantification flexibilty in calculate (rigid and symbolizing) mathematizing response) capacity to invent or disconnected imagine inventing definitions of some physical quantities physics concepts 76
Applying ICC: Physics InvenLon Tasks n t s i m p o s e t r a i d n s b y c o t d h n e a i n s e v u l e n R t i C : o R n CC: Compare across contrasting cases for invariance AC: Arithmetically MU: Make meaning construct index of units QF: Identify NC: Evaluate in quanti f able features new context of the system CM: Clarify Mission- To make mathematical choices to generate a useful quantity Harel’s necessity principle Collaborative productive failure Socioconstructivist framework 77
Sample InvenLon Sequence 1 These cars all drip oil once every second. Invent a speeding-up index that allows you to rank the cars in terms of how quickly they speed up. 5mph 15mph 25mph 35mph Car A 0 3mph 6mph 9mph Car B 10mph 25mph 40mph 55mph Car C 4mph 14mph 24mph 34mph Car D 12mph 27mph 42mph 57mph Car E
Sociocultural Benefits • Valuing naïve understanding (Ross & Otero 2013) • Shiwing authority from instructor to social consensus (Ross & Otero 2013) • Addressing stereotype threat: Not remediaLon; students work, and struggle, collaboraLvely. (Steele) • Developing self-efficacy: InvenLon process gives ownership of the knowledge to the student (Bandera, Sawtelle) 79
Sociocultural Benefits • Valuing naïve understanding (Ross & Otero 2013) • Shiwing authority from instructor to social consensus (Ross & Otero 2013) • Addressing stereotype threat: Not remediaLon; students work, and struggle, collaboraLvely. (Steele & Aronson 1995) • Developing self-efficacy: InvenLon process gives ownership of the knowledge to the student (Bandera, Sawtelle) 80
Sociocultural Benefits • Valuing naïve understanding (Ross & Otero 2013) • Shiwing authority from instructor to social consensus (Ross & Otero 2013) • Addressing stereotype threat: Not remediaLon; students work, and struggle, collaboraLvely. (Steele & Aronson 1995) • Developing self-efficacy: InvenLon process gives ownership of the knowledge to the student (Bandura 1997, Sawtelle 2011) 81
FCI comparison (before the introducLon of PITs, 2003, n=102 and awer 2013/14, n=144 82
CLASS- physics categories associated with mathemaLcal reasoning, pre-instrucLon and the gains over one semester. Combined Fall 2013 and Fall 2014, n=121. Error bars represent the standard error. 83
Rutgers Engineering Physics Study • Underprepared (precalc math placement) vs Mainstream (calculus math placement) • Simultaneous courses • Same content, different curricula • FCI, Math reasoning, CLASS and some MBL pre/post Fall 2013 84
Rutgers Engineering Physics Study • Underprepared (precalc math placement) vs Mainstream (calculus math placement) • Simultaneous courses • Same content, different curricula • FCI, Math reasoning, CLASS and some MBL pre/post Fall 2013 85
Rutgers Engineering Physics Study • Underprepared (precalc math placement) vs Mainstream (calculus math placement) • Simultaneous courses • Same content, different curricula • FCI, Math reasoning, CLASS and some MBL pre/post Fall 2013 86
Rutgers Engineering Physics Study • Underprepared (precalc math placement) vs Mainstream (calculus math placement) • Simultaneous courses • Same content, different curricula • FCI, Math reasoning, and CLASS pre/post Fall 2013 87
Course Demographic Comparison EAP I (Underprepared) AP I (Mainstream) # of students ~120 ~700 Mean SAT 610 680 % URM 40% 12% % female 30% 21% Median MHI of sending 0.7* Q Q district p-value<.000000001 88
Course Demographic Comparison EAP I (Underprepared) AP I (Mainstream) # of students ~120 ~700 Mean SAT 610 680 % URM 40% 12% % female 30% 21% Median MHI of sending 0.7* Q Q district 89
Force Concept Inventory; σ mean : EAP I (n=135) 1.4%(pre), 1.5%(post); AP I (n=757) 0.8%(pre), 0.8%(post) 90
CLASS While the EAP course shows small posiLve gains, the AP course shows negaLve gains ~10% across PS categories. 91
MathemaLcal Reasoning Item A bicycle is equipped with an odometer to measure how far it travels. A cyclist rides the bicycle up a mountain road. When the odometer reading increases by 8 miles, the cyclist gains H verLcal feet of elevaLon. Find an expression for the number of miles the odometer reading increases for every verLcal foot of elevaLon gain. % & # # % & sin − 1 8 sin − 1 H H /8 8/ H None of these ( ( H 8 $ ' $ ' 92
MathemaLcal Reasoning Item A bicycle is equipped with an odometer to measure how far it travels. A cyclist rides the bicycle up a mountain road. When the odometer reading increases by 8 miles, the cyclist gains H verLcal feet of elevaLon. Find an expression for the number of miles the odometer reading increases for every verLcal foot of elevaLon gain. % & # # % & sin − 1 8 sin − 1 H H /8 8/ H None of these ( ( H 8 $ ' $ ' 93
Bike Path RU Fall 2013 One semester of instruc'on 100 90 80 70 60 pre 50 post 40 30 20 10 0 Underprepared Mainstream 94
Bike Path RU (full year of instruc'on) n 115/6 =187 and n 123/4 =583 100 90 80 70 60 pre 50 post 40 30 20 10 0 Underprepared Mainstream 95
Rice Ques'ons (SES) (full year) 1 0.8 0.6 pre 0.4 post 0.2 0 APIhigh APIlow EAPIlow 96
Woozles(SES) (full year) 1 0.8 0.6 pre 0.4 post 0.2 0 APIhigh APIlow EAPIlow 97
CLASS"Problem"Solving"9General" 80# 75# %"that"agree"with"experts" 70# 65# 60# 55# 50# 0.8# 1# 1.2# 1.4# 1.6# 1.8# 2# 2.2# 98 APIHi#
CLASS"Problem"Solving"9General" 80# 75# %"that"agree"with"experts" 70# 65# 60# 55# 50# 0.8# 1# 1.2# 1.4# 1.6# 1.8# 2# 2.2# SES: 99 APIHi# APILow#
CLASS"Problem"Solving"9General" 80# 75# %"that"agree"with"experts" 70# 65# 60# 55# 50# 0.8# 1# 1.2# 1.4# 1.6# 1.8# 2# 2.2# SES: 100 APIHi# APILow# EAPILow#
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