…to say that you will be teaching a class next semester that you have never taught before.

What do you want to know so you can plan and offer an effective course? (Alternative phrasing: If you HAD taught the course before, what useful knowledge would you already possess?)

Your department chair (or advisor) called…

Why did they think that? The use and development of mathematical

knowledge for teaching at the undergraduate level

Natasha SpeerDepartment of Mathematics & StatisticsCenter for Research in STEM Education

The University of Maine

…to say that you will be teaching a class next semester that you have never taught before.

What do you want to know so you can plan and offer an effective course? (Alternative phrasing: If you HAD taught the course before, what useful knowledge would you already possess?)

Discuss this with the people seated near you. We will share out ideas in a few minutes.

Your department chair (or advisor) called…

• Overview of research on knowledge used in teaching mathematics

• Connections to improving teaching and learning of undergraduate mathematics

• Example of research designed to inform these efforts

• Implications for supporting novice instructors and further research

… with “work of teaching” tasks scattered throughout

Today

Ball, Hoover Thames, & Phelps (2008)

Ball, Hoover Thames, & Phelps (2008)

Common Content Knowledge

“knowledge of a kind used in a wide variety of settings – in other words not unique to teaching”; these are not specialized understandings but are questions that typically would be answerable by others who know mathematics” (Ball, Hoover Thames, & Phelps, 2008, p. 399)

472− 295=177If f(x) = 3sin(x), y’ = 3cos(x)

Knowledge used to do mathematics• Necessary, but not sufficient for teaching (Ball &

Bass, 2000; Hill, Rowan, & Ball, 2005; Hill, Sleep, Lewis, & Ball, D., 2007; Monk, 1994)

• More courses in content are not strongly correlated with higher student achievement (Begle, 1979; Monk, 1994)

Ball, Hoover Thames, & Phelps (2008)

Ball, Hoover Thames, & Phelps (2008)

On a pre-calculus test, students need to expand the following expression:

(x + 3)2

• What are some answers students might produce?

• What might students be thinking as they produce the incorrect answers?

Teaching task #1

Typical ways students think (productively and unproductively), common student strategies, especially useful examples, organization of content in a course, relative difficulty of topics

(Shulman, 1986)

– influences practices (Carpenter et al., 1988, 1989)

– influences student learning (e.g., Fennema et al., 1996; Franke et al., 2001; 1997)

Pedagogical Content Knowledge (PCK)

This question was on a calculus quiz:If y = (5 + x2)7, what is the derivative of y?

Several students gave this answer:y’ = 7(5 + x2)6 (2x) (2)

What might the students have been thinking?

Bonus problem: What’s the thinking here?

Teaching Task #2

472– 295223

• Used to – do the “mathematical work” of teaching– follow and understand students’ mathematical thinking– evaluate the validity of student-generated strategies

• Shown to play a role in teachers’ practices and correlate with students’ learning

(Ball & Bass, 2000; Hill et al 2004, 2005; Ma, 1999)

Is important and distinct from “common”content knowledge

Specialized Content Knowledge (SCK)

…and that class you will be teaching for the first time uses collaborative groupwork, inquiry-based instruction and other active learning approaches. • Are there types of knowledge that would be especially

important/valuable for this instructional approach?

• What are some types of knowledge that might be important in this situation that would be less crucial in an all-lecture format?

Discuss.

Your dept. chair (or advisor) called AGAIN…

Using “engaged student learning” approaches

• is challenging– requires facilitating discussion and progression

of ideas (Stein et al., 2006)– often means adopting new practices

• places different demands on instructors’ knowledge than traditional lecture (Speer & Wagner, 2009; Wagner, Speer & Rosa, 2007)

So, why should we bother?

To have an adequate number of people for STEM jobs, we (universities, colleges) need to produce 1 million ADDITIONAL graduates with STEM degrees over the next decade.

(Holdren & Lander, 2012)

Part of that reality

Whatpercentoffreshmenwhointend tomajorinSTEM(science,technology,engineering,mathematics)actually finishaSTEMmajor?

(1)>80% (2)60– 80%(3)40– 60% (4)<40%

Part of that reality

Whatpercentoffreshmenwhointend tomajorinSTEM(science,technology,engineering,mathematics)actually finishaSTEMmajor?

(1)>80% (2)60– 80%(3)40– 60% (4)<40%

“The first two years of college are the most critical to retention and recruitment of STEM majors.”

Holdren & Lander, 2012, p. 6

“Thelecturehasbeenamainstayofhighereducationsincetheword‘lecture’wascreatedinthe14thcentury,andtodaymostintroductorySTEMcoursesaretaughtlargelythroughlectures.”

“Extensiveresearchonhowthehumanbrainlearnsindicatesthatdiversifyingteachingmethodsenhancescriticalthinkingskills,long-termretentionofinformation,andstudentretentioninSTEMmajors.”

Holdren &Lander,2012,p.8

Evidence from a meta-analysis*

• 200 studies of undergraduate STEM instruction

• Two categories:– passive lecture– anything else

• Passive lecture courses: 55% higher DFW rates

• Medical metaphor*Freeman,S.,Eddy,S.L.,McDonough,M.,Smith,M.K.,Okoroafor,N.,Jordt,H.,&Wenderoth,M.P.(2014).Activelearningincreasesstudentperformanceinscience,engineering,andmathematics.ProceedingsoftheNationalAcademyofSciences,111(23),8410–8415.

“…theseactivelearningtechniquesbenefitallstudentsandcanclosetheachievementgapbetweenethnicgroupsandmenandwomen”(Holdren &Lander,2012,p.9)

Minorities in STEMUSpop:30%

College-agepop:36%

In mathematics• Inquiry-based learning (IBL) in over 100 course

sections at 4 universities• Despite variation in how IBL was implemented,

student outcomes are better relative to traditionally taught courses

• “The use of IBL eliminates a sizable gender gap that disfavors women students in lecture-based courses.”

Laursen, S. L., Hassi, M.-L., Kogan, M., & Weston, T. J. (2014). Benefits for Women and Men of Inquiry-Based Learning in College Mathematics: A Multi-Institution Study. Journal for Research in Mathematics Education, 45(4), 406–418.

In calculus• While taking calculus, some students will decide

to switch out of a major that requires the calculus sequence

• Female students are 1.5 times as likely to decide that as their male classmates (with the same preparedness, career goals, etc.)

Ellis, J., Fosdick, B. K., & Rasmussen, C. (2016). Women 1.5 Times More Likely to Leave STEM Pipeline after Calculus Compared to Men: Lack of Mathematical Confidence a Potential Culprit. PLOSONE, 11(7), 1-14.

What makes it difficult?

“Asignificantbarriertobroadimplementationofevidence-basedteachingapproachesisthatmostfacultylackexperience usingthesemethodsandareunfamiliarwiththevastbodyofresearchindicatingtheirimpactonlearning”(Holdren &Lander,2012,p.iii).

What makes it difficult?

Preparing the next generation of instructors (graduate students):• Not always possible to provide substantial

pre-service preparation• Professional development concurrent with

teaching sometimes limited• TA professional development is not

typically a primary focus for faculty

Rest of this talk

• Sample of study examining college instructor MKT

• What knowledge of student thinking do experienced instructors possess? – And how does that compare to novices?

• Findings can inform design of professional development for novice instructors

Context: CalculusTask-based interviewsSeven tasks borrowed from or modeled after those used in research on student thinking (e.g., Habre & Abboud, 2006; Kendal & Stacey, 2003; White & Mitchelmore, 1996; Zandieh, 2000).Instructors:• research mathematicians recognized for teaching

excellence• graduate students with fewer than two years of calculus

teaching experience

Research design

PART 1: Solve task and describe solutionPART 2: Describe how students would solve the task,

including difficulties/mistakes they might make and correct/incorrect ways of thinking they might display

PART 3: Examine and discuss student work samples (that represent typical student difficulties, ways of thinking, etc.) Based on protocol from previous studies (Frank & Speer, 2011, 2012; Speer et al., 2006)

PART 4: Describe what you would do/say to student who produced the work during office hours

Approximately 20 minutes per task.

Four-PART INTERVIEW

PART 1: Solve task and describe solutionPART 2: Describe how students would solve the task,

including difficulties/mistakes they might make and correct/incorrect ways of thinking they might display

PART 3: Examine and discuss student work samples (that represent typical student difficulties, ways of thinking, etc.) Based on protocol from previous studies (Frank & Speer, 2011, 2012; Speer et al., 2006)

PART 4: Describe what you would do/say to student who produced the work during office hours

Approximately 20 minutes per task.

Four-PART INTERVIEW

The given graph represents speed vs. time for two cars. (Assume the cars start from the same position and are travelling in the same direction.)

Question: State the relationship between the position of car A and car B at t = 1 hour. Provide an explanation for your answer.

From Carlson (1998)

A task

Experience the interviewThe given graph represents speed vs. time for two cars. (Assume the cars start from the same position and are travelling in the same direction.)

Question: State the relationship between the position of car A and car B at t = 1 hour. Provide an explanation for your answer. • With a neighbor, discuss:

– Your thinking and answer to the question– How students might solve the problem– Typical student productive and unproductive ways of thinking

1. Distance car A traveled > Distance car B traveled2. Distance car A traveled = Distance car B traveled3. Distance car A traveled < Distance car B traveled

Ways of thinking

Thegivengraphrepresentsspeedvs.timefortwocars.(Assumethecarsstartfromthesamepositionandaretravellinginthesamedirection.)

Question:StatetherelationshipbetweenthepositionofcarAandcarBatt=1hour.Provideanexplanationforyouranswer.

1. Distance car A traveled > Distance car B traveled2. Distance car A traveled = Distance car B traveled3. Distance car A traveled < Distance car B traveled

Ways of thinking

Thegivengraphrepresentsspeedvs.timefortwocars.(Assumethecarsstartfromthesamepositionandaretravellinginthesamedirection.)

Question:StatetherelationshipbetweenthepositionofcarAandcarBatt=1hour.Provideanexplanationforyouranswer.

Common ways of thinking• Productive:

– Use relative speed– Use the relative areas under the curves

• Unproductive:– Interpret the graphs literally as the paths of the cars,

rather than interpreting the functional information displayed on the graph

• The cars will collide at one hour • “Two different paths to get to the same place”

– Interpret the speed graphs as position graphs– Compare function slopes at t = 1

Examining student work

Student A:

Student B:

Student F:

• Findings from research on student thinking were used to characterize the extent to which participants were knowledgeable of student thinking

• Methods from Grounded Theory (Strauss & Corbin, 1990) were also used• to identify themes • to detect comparisons/contrasts

Research design: Analysis methods

Brief overview of findings• All participants gave correct solutions to

the task (some took a little while…). • All participants described some

productive ways of thinking that students might use

• All participants gave some possible unproductive ways of thinking

Findings about the mathematicians• generated more possible ways of thinking• used their (more extensive) set of ways of thinking to

make sense of the written work• PLUS: they were able to figure out (new-to-them) ways of

thinking from examining student workThey learned during the interview!

• They did more (spontaneous) quantifying/calibrating of how common a particular difficulty might be– “…a lot of students probably would…”– “I would expect many to get it wrong” – “Some students might say…”

Findings about the graduate students• Most generated very few possible ways of thinking• Some answers were fairly general:

– “[they] don’t know how to read this graph I think…so we have this speed and we have time but how can I, how do you represent the distance based on this graph?” (A, graduate student)

• Some focused on small variations to one approach, mostly the way they thought about the problem: – “…So they might make the same mistake as I was by

looking at the slope because for some reasons I keep thinking of this as a position instead of speed…” (C, graduate student)

Closing thoughts• Faculty gain MKT from their experiences • MKT supports effective instruction• Need to leverage this to improve student learning

• How I (and others) use these findings– Design professional development for novice college

mathematics instructors• access to known student ways of thinking• opportunities to learn from examining student work

– Equip the next generation of faculty with knowledge needed to enact active, engaged student learning approaches

– Work toward improved learning, retention, access and opportunity for all students

Thank you!

Questions and/or comments?

Mathematics Education Research Group

Acknowledgements

my email: speer@math.umaine.edu

Study math &/or science content and teaching

Conduct original research with faculty

Obtain teaching experience

The Master of Science in Teaching (MST) Programat the University of Maine

www.umaine.edu/center

Overview of the MST program• 2-year master’s program offered through the

Maine Center for Research in STEM Education (RiSE Center)

• Research & certification opportunities in life science, physical science, earth science, and mathematics

• Unique combination of coursework, thesis, and teaching experience

• Paid assistantships for full time students

• NSF Teaching Fellowship Program for selected MST graduates who choose to teach in high-need Maine districts ($10,000 per year)

Teacher certification

Advanced study for current teachers

Preparation for further graduate

study

www.umaine.edu/center

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