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Characteristics of a High School Classroom 
Community of Mathematical Inquiry

Raymond Siegrist

When a community of inquiry is adopted for teaching mathematics, characteristics need to be identified to 
determine if a community of inquiry forms. One essential characteristic of a community of mathematical 

inquiry is that members develop their understanding of mathematics through dialogue. Scholars in fields other 
than mathematics have emphasized the importance of incorporating dialogue in instruction:

•	 The	learning	of	content	through	the	dialogical	process	enhances	the	understanding,	retention,	and	ap-
plication of the content (Fernandez-Balboa & Marshall, 1994, p. 174).

•	 Dialogue	…	[is]	 the	best	available	means	we	have	for	 identifying	among	ourselves	acceptable	answers,	
workable	solutions,	and	reasonable	accommodations	(Burbules,	1993,	p.	16).	

•	 [Meaningful	learning]	is	much	more	likely	to	occur	when	dialogue	is	present,	for	the	simple	reason	that	
direct	communication	among	equals	is	the	only	reliable	and	systematic	dynamic	available	to	those	who	would	
incorporate	the	viewpoints	of	others	into	their	own	personal	perspectives	(Splitter	&	Sharp,	1995,	p.	83).

Dialogue	is	the	medium	by	which	students	in	a	community	of	mathematical	inquiry	communally	make	their	
own	meaning	of	mathematics.	Burbules	(1993)	views	dialogue	as	“a	continuous,	developmental	communicative	
interchange	through	which	we	stand	to	gain	a	fuller	appreciation	of	the	world,	ourselves,	and	one	another”	(p.	
8).	A	definition	of	dialogue	is	elusive,	but	several	studies	have	described	its	characteristics.	Dialogue	occurs	when	
there	is	an	open,	unforced	sharing	by	two	or	more	participants.	Even	in	the	light	of	disagreements	a	participant	
remains	interested	in,	concerned	for,	and	respectful	of	all	other	participants	by	following	agreed	upon	maxims.	
All	participation	 is	guided	by	discovery	on	a	 topic	of	common	interest	with	value	given	 to	all	 contributions	
of	the	participants.	A	continuous	dialogue	enables	participants	to	construct	knowledge	and	then	reconstruct	
knowledge	as	diverse	voices	synthesize	expressed	ideas.	Transformation	of	knowledge	occurs	in	a	dialogue	as	par-
ticipants	propose,	defend,	abandon,	and	accept	ideas	causing	new	situations	and	possibilities	to	be	discovered.	
Inherent	in	dialogue	is	the	need	for	each	participant	to	take	responsibility	for	his/her	ability	to	influence	the	
outcome (Burbules, 1993; Fernandez-Balboa & Marshall, 1994).

Another	characteristic	is	that	members	of	a	community	of	mathematical	inquiry	work	together	in	groups	to	
solve	more	difficult	mathematical	problems	than	they	could	as	individuals.	Vygotsky’s	(1986)	“zone	of	proximal	
development”	supports	the	use	of	groups	in	communities	of	inquiry	because	students	obtain	assistance.	If	Vy-
gotsky	is	right,	groups	permit	students	to	solve	general	problems	that	would	be	out	of	the	reach	of	individuals.	
For	students	to	be	in	their	“zone	of	proximal	development”	during	small	groups,	they	have	to	be	focused	on	the	
task	at	hand,	and	they	have	to	express,	listen	to,	defend,	and	evaluate	the	options	created	for	solving	any	type	of	
problem	(Schoenfeld,	1987).		

 
One strategy for using groups to solve mathematical problems reported by empirical studies is to have groups 

develop	their	processes	and	then	present	what	they	have	created	to	the	class	for	critical	review	(Clarke,	1997;	
Cobb,	Wood,	Yackel,	Nicholls,	Wheatly,	Trigatti,	&	Perlwitz,	1991).	In	fact,	instead	of	training	for	mastery	of	
mathematical procedures presented by their teacher, in this process students become apprentices. Students act-



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ing	as	apprentices	take	on	more	of	the	problem-solving	role	as	they	gain	experience.	As	they	gain	experience	un-
der	their	teacher’s	watchful	eye,	students	are	less	likely	to	develop	misconceptions	about	mathematics.	Students	
also	benefit	from	another	aspect	of	apprenticeship,	initiation	into	a	mathematical	culture	(Schoenfeld,	1987).	
“Having	experienced	mathematics	in	this	way,	students	are	more	likely	to	develop	a	more	accurate	view	of	what	
mathematics	is	and	how	it	is	done”	(Schoenfeld,	1987,	p.	205).	One	justification	for	adopting	a	community	of	
inquiry	approach	put	forth	by	Splitter	and	Sharp	(1995)	is	that	it	encourages	students	to	think	like	mathemati-
cians	or	for	students	to	learn	to	create	mathematics	the	way	mathematicians	do.			

The	third	characteristic	of	a	community	of	mathematical	inquiry	is	self-correction;	as	students	check	their	
solutions	they	often	correct	mistakes.	In	mathematics,	students	typically	are	not	asked	to	listen	to	explanations	
by	other	students	to	determine	which	ideas	make	the	best	sense;	they	are	given	what	the	teacher	thinks	is	the	best	
way	to	solve	a	mathematical	problem.	When	students	begin	to	negotiate	their	own	mathematical	meaning	within	
the	community,	they	must	decide	whether	a	given	explanation	is	good	enough	to	change	their	minds.	When	a	
conflict	between	ideas	is	resolved	through	communal	dialogue,	a	self-correction	occurs	within	the	community.

  
On the constructivist model, authentic learning is self-correction – the deliberate, intelligent reconstruction 

of	one’s	habits	of	thoughts,	feeling	and	action	in	order	to	meliorate	some	aspect	of	one’s	experience.	Self-correc-
tion	further	requires	self-verification:	that	a	student	verify	new	knowledge	to	herself	(Gregory,	2002,	p.	400).		

Students	begin	to	internalize	standards	of	mathematical	judgment	modeled	by	the	teacher,	so	they	have	cri-
teria	to	gauge	correctness	of	arguments	and	solutions	(Schoenfeld,	1987,	1996).

The	fourth	crucial	characteristic	of	a	community	of	mathematical	inquiry	is	that	community	members	take	
risks.	In	this	process,	criticism	and	attacks	can	be	made	on	ideas,	but	are	not	acceptable	on	the	personal	level.	On	
this	point	students	need	practice	in	not	attaching	their	personal	worth	to	the	acceptance	of	“their”	idea.	Robert-
son	(1999)	points	out	that	all	members	of	the	community	must	entertain	the	“possibility	of	being	obligated	to	
lose”	(p.	5).	Members	of	the	community	need	to	abide	by	the	idea	that	the	explanation	making	the	best	sense	is	
the	one	accepted.	As	Schoenfeld	(1996)	states,	“There	is	a	feeling	of	trust,	in	that	we	must	feel	free	to	have	our	
ideas	(and	not	ourselves)	compete.”	(p.16).	Students	free	from	personal	assessment	by	the	community	are	more	
likely	to	take	risks	when	proposing	assertions.	Polya	(1954)	asserts	three	“moral	qualities”	a	person	needs	to	do	
mathematics	that	also	pertain	to	taking	risks:

Intellectual courage: We should be ready to revise any one of our beliefs.
Intellectual	honesty:	We	should	change	a	belief	when	there	is	a	good	reason	to	change	it	…
Wise	restraint:	We	should	not	change	a	belief	wantonly,	without	some	good	reason,	without	serious
examination	(pp.	7-8).

The fifth characteristic of a community of mathematical inquiry is that students consider, propose, and 
build on alternate approaches to problem solving. A common, general problem-solving strategy is to list several 
methods that might result in a solution.  When a process is successful, students should not decide inquiry is fin-
ished.	Other	ideas	might	lead	to	better	insights,	additional	knowledge,	or	other	areas	of	study	(Lampert,	1990).	
Self-correction eliminates some alternate approaches to problem solving as unfeasible, but open inquiry requires 
investigating	all	ideas	within	the	constraints	of	the	initial	problem.	Cobb,	Wood,	and	Yackel	(1992)	offer	some	
guidance:	“the	teacher	might	frame	conflicting	interpretations	or	solutions	as	a	topic	for	discussion,	thus	en-
couraging	students	to	explicitly	negotiate	mathematical	meanings	by	engaging	in	mathematical	argumentation”	
(p. 11).

Members	of	a	community	of	mathematical	inquiry	should	be	able	to	provide	justification	for	each	step	taken	
in	the	solution	process.	Reasons	need	to	be	given	by	a	student	to	the	community	for	accepting	or	rejecting	a	
claim.	Mason	presents	an	outline	for	justifying	mathematical	arguments:	First	convince	yourself.	Then	convince	
a	friend.		Finally	convince	an	enemy	(Mason,	Burton,	&	Stacey,	1984).	“If	the	teacher’s	guide	is	the	source	of	



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right	 answers,	 for	 example,	 this	 suggests	 that	 the	basis	 for	 epistemic	 authority	 in	mathematics	does	not	 rest	
within	the	knower”	(Ball,	1991,	p.	7).		But	if	the	student	has	defended	a	conjecture	or	answer	with	reason,	not	
only	does	the	“epistemic	authority”	reside	within	the	knower,	the	student	has	created	her/his	own	meaning.	

Students	 too	often	see	the	polished	product	of	a	teacher’s	struggle	with	a	topic	or	hours	of	 investigation	
that	did	not	lead	directly	to	a	viable	process.	“Naïve	conjecture	and	counterexamples	do	not	appear	in	the	fully	
fledged	deductive	structure:	The	zig-zag	of	discovery	cannot	be	discerned	in	the	end	product”	(Lakatos,	1976,	
p.42). Students need to engage in the polishing process, but for this to happen teachers must get students to 
change	their	focus.	Instead	of	a	race	to	the	correct	answers,	students	need	to	focus	on	the	investigation	process,	
the	connections	among	concepts,	and	the	structure	of	mathematics	(Lakatos,	1976;	Peressini	&	Knuth,	2000;	
Tymoczko,	1985).	In	a	community	of	inquiry,	“Knowledge	is	never	presented	as	complete	and	sacred;	rather,	it	
is	always	open	to	further	question	and	criticism”	(Harpaz	&	Lefstein,	2000).	Schoenfeld	argues	that,	“entry	into	
that	culture…may	be	necessary	to	understand	and	appreciate	mathematics”		(1987,	p.	214).

A	sixth	characteristic	of	a	community	of	mathematical	inquiry	is	that	students	inquire	into	the	procedures	
of	inquiry.	In	mathematics,	when	students	think	about	what	they	are	doing	during	the	problem-solving	process,	
they inquire into the procedures of inquiry (problem solving). When a teacher adopts a cooperative problem-
solving	approach,	an	underlying	objective	is	for	students	to	abandon	rehearsals	of	rules	and	to	plan	how	to	solve	
complex	problems.	This	objective	requires	students	to	adopt	and	practice	a	way	to	manage	their	problem	solv-
ing.

Aspects	of	management	include	(a)	making	sure	that	you	understand	what	a	problem	is	all	about	before	you	
hastily	attempt	a	solution;	(b)	planning;	(c)	monitoring,	or	keeping	track	of	how	well	things	are	going	during	
a	solution;	and	(d)	allocating	resources,	or	deciding	what	to	do,	and	for	how	long,	as	you	work	on	a	problem	
(Schoenfeld,	1987,	p.	109-110).

Schoenfeld	(1987)	contends	that	if	teachers	fail	to	provide	students	a	general	way	to	“self-regulate”	their	prob-
lem	solving	attempts,	students	become	frustrated	when	they	cannot	start	a	problem,	they	continue	a	strategy	that	
never	converges	to	a	solution,	or	they	run	into	a	dead	end	with	their	selected	strategy.	Therefore,	in	addition	to	
learning a collection of processes and concepts, students must learn and practice a management system in order 
to	become	successful	problem	solvers.	In	other	words,	managing	or	thinking	about	problem	solving	differs	from	
solving	problems,	but	it	 is	a	necessity	for	problem	solvers.	Schoenfeld	(1987)	suggests	students	reflect	on	the	
problem-solving	process	by	asking,	“How	well	do	you	keep	track	of	what	you’re	doing	when	(for	example)	you’re	
solving	problems,	and	how	well	(if	at	all)	do	you	use	the	input	from	those	observations	to	guide	your	problem	
solving	actions?”	(p.190).

The seventh characteristic of a community of mathematical inquiry is that it supports students doing math-
ematics	like	mathematicians.	One	essential	ingredient	in	developing	a	community	of	inquiry	environment	in	
high	school	mathematics	is	the	creation	of	a	culture	of	mathematicians	(Clarke,	1997;	Cobb,	Wood,	&	Yackel,	
1991;	Lampert,	1990;	Shoenfeld,	1987,	1996).	In	a	community	of	mathematical	inquiry,	axioms	and	definitions	
remain	open	to	reexamination	by	the	community.	When	members	of	the	community	engaged	in	constructing	a	
proof discover a deficiency, the need for revision is obvious.  Revisions of assertions by community members in 
the	light	of	recently	discovered	inadequacies	allow	mathematics	to	advance	(Lampert,	1990).	Helped	by	a	math-
ematician’s	educated	guessing	about	relationships,	mathematics	develops	in	a	back-and-forth	process	between	
adjusting	assumptions	and	revising	conclusions.	A	conjecture	is	proposed	to	the	mathematical	community	and	
stands	until	a	counterexample	is	discovered	or	the	conjecture	is	proven.	Students	need	practice	in	taking	the	
risk	to	make	conjectures;	they	must	also	understand	that,	because	of	limited	insight,	their	conclusions	might	be	
incorrect	(Lampert,	1990).			

Schoenfeld	(1987)	points	out	three	important	types	of	metacognition	that	mathematicians	need	to	develop:	
a)	A	problem	solver’s	knowledge	about	his/her	own	thought	process,	b)	a	problem	solver’s	self-regulation,	and	c)	



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a	problem	solver’s	beliefs	and	intuitions	(p.	190).	Participation	in	a	community	of	mathematical	inquiry	allows	
each	of	these	qualities	to	be	developed.	Students	become	better	general	problem	solvers	when	these	qualities	
are introduced and their application is practiced. Many attempts at solving mathematical problems depend on 
the	prior	mathematical	knowledge	of	the	students.	When	students	have	a	good	sense	of	what	they	know,	their	
efficiency	as	general	problem	solvers	increases.	In	addition,	students	learn	to	explain	how	they	solved	general	
problems and constantly reflect upon their thought processes in a community of inquiry (Carpenter & Fenne-
ma,	1992).	Simply	put,	a	student	who	self-regulates	during	a	general	problem	solution	is	a	better	problem	solver	
than	one	who	does	not	keep	track	of	what	is	happening.	Student	beliefs,	such	as	the	belief	that	students	must	
memorize rules to be successful in mathematics, should begin to change as their participation in the community 
matures	(Schoenfeld,	1987,	1996).

Studies Showing Outcomes of Classroom Communities of Inquiry

There is very little empirical literature on mathematical communities of inquiry. The most relevant litera-
ture incorporates community of inquiry characteristics as part of socio-constructivist approaches to teaching 
and learning mathematics. This literature is appropriate as a frame for a study on mathematical communities 
of	 inquiry	 because	 socio-constructivist	 approaches	 to	 teaching	 share	 many	 theoretical	 underpinnings	 as	 well	
as	 pedagogical	 approaches	with	 community	of	 inquiry:	 1)	 “Students	have	 frequent	opportunities	 to	discuss,	
critique,	explain,	and,	when	necessary,	justify	their	interpretations	and	solutions”	(Cobb,	Wood,	Yackel,	&	Per-
lwitz,	1992,	p.485).	2)	“Mathematics	learning	is	a	process	in	which	students	reorganize	their	activity	to	resolve	
situations	that	they	find	problematic”	(Cobb,	Wood,	Yackel,	Nichols,	et	al.,	1991,	p.	4).		3)	Learning	occurs	when	
students	attempt	to	make	meaning	by	actively	negotiating	with	their	peers	(Cobb	et	al.,	1991;	Fawcett,	1938;	
Lampert,	1990;	Schoenfeld,	1987,	1996).	4)	“Social	norms	are	not	static	prescriptions	or	rules	to	be	followed	but	
are	instead	continually	reconstructed	in	the	course	of	classroom	interactions”	(Cobb,	Wood,	Yackel,	Nichols,	et	
al.,	1991,	p.	7).	5)	Students	create	mathematics	as	mathematicians	(Cobb	et	al.,	1991;	Fawcett,	1938;	Lampert,	
1990;	Schoenfeld,	1987,	1996).

In a 1991 study, Cobb, et al. designed instruments to assess mathematics achievement, computational pro-
ficiency, beliefs about success in mathematics, and motivation to study mathematics. Ten second-grade classes 
(project	students)	were	taught	mathematics	using	an	inquiry	approach	while	eight	classes	(non-project	students)	
were	taught	in	a	traditional	manner.	The	instruments	were	administered	at	the	end	of	a	year.		This	study	indi-
cated	that	the	project	students	retained	more	content,	constructed	more	advanced	concepts,	cooperated	with	
their	peers	to	a	greater	extent,	and	did	better	on	challenging	tasks	than	the	non-project	students.	The	amount	of	
time	to	teach	the	concepts	of	the	curriculum	using	the	inquiry	approach	was	not	a	concern	mentioned	in	this	
study.		Students	in	the	project	classes	believed	success	in	mathematics	comes	from	attempting	to	understand	
mathematics	while	communicating	with	their	peers	about	their	thinking.		

In	a	follow-up	study	Cobb,	Wood,	Yackel,	and	Perlwitz	(1992)	tested	former	second	grade	students	from	five	
inquiry	classes	and	six	traditional	classes.	All	students	were	in	traditional	third	grade	classes.	Scores	on	standard	
achievement	tests	together	with	results	from	instruments	designed	to	assess	computational	development,	per-
sonal	goals	in	mathematics,	and	beliefs	about	mathematical	success	were	used	to	compare	students	at	the	end	
of the year. The inquiry students maintained their edge in conceptual understanding and mathematical prob-
lem solving. Beliefs of the inquiry students, that success in mathematics comes from attempting to understand 
concepts	and	from	talking	with	their	classmates	about	their	thinking,	persisted	even	after	a	year	of	traditional	
instruction.

An	early	pretest-posttest,	quasi-experimental	study	conducted	by	Fawcett	(1938)	showed	the	viability	of	the	in-
quiry	approach.	Several	geometry	classrooms	totaling	seventy-five	students	were	taught	by	the	traditional	method	
in	two	schools;	these	students	became	the	control	group.	In	the	experimental	group,	a	class	of	25	students	deter-
mined	the	pace	and	sequence	of	their	learning,	constructed	their	own	definitions,	and	questioned	assumptions.	
Pretest	scores	were	the	same	for	all	groups,	but	the	experimental	group	scored	significantly	better	on	the	posttest	



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53

despite	the	fact	that	the	experimental	group	only	covered	half	of	the	curriculum.	Two	of	Fawcett’s	(1938)	conclu-
sions	were	that	“following	these	procedures	improves	reflective	thinking”	and	that	“the	usual	formal	course	in	
demonstrative	geometry	does	not	improve	the	reflective	thinking	of	pupils”	(p.	119).

Lampert	(1990)	conducted	a	classic	qualitative,	first-person	study	indicating	how	a	middle	school	classroom	
could	be	set-up	in	a	community	of	inquiry.	She	found	many	examples	of	how	students	tried	to	use	irrational	
strategies	to	have	their	assertions	accepted	by	other	students.	Lampert	used	the	work	of	Polya	(1954)	and	Laka-
tos	(1976)	to	identify	how	to	establish	a	culture	of	mathematicians	in	the	classroom	and	bring	students	to	the	
realization	of	what	it	means	to	do	mathematics.	Her	students	studied	exponents	like	mathematicians	by	making	
conjectures	and	dialoguing	about	their	thoughts.	In	her	role	as	the	teacher,	Lampert	would	at	different	times	tell	
students	whether	their	behavior	was	appropriate	or	inappropriate,	model	appropriate	ways	of	behaving,	or	do	
mathematics	with	her	students.			

The list of seven characteristics of a community of mathematical inquiry based on current literature estab-
lishes	a	starting	point	for	determining	if	a	community	of	inquiry	exists.	Since	additional	characteristics	could	
certainly	be	extracted	from	the	literature,	there	is	no	claim	that	this	list	is	exhaustive.	Each	of	the	characteristics	
can be observed and measured by the frequency of its occurrence on the individual or class level. Literature re-
lated	to	high	school	community	of	mathematical	inquiry	was	limited	to	Fawcett’s	1938	classic	quantitative	study	
about	geometry	students.	Lampert’s	1990	classic,	first-person,	qualitative	study	generalized	student	attempts	to	
thwart	dialogue	in	a	middle	school	community	of	inquiry	that	would	be	applicable	to	the	high	school	level.	The	
Cobb et al. (1991, 1992) studies demonstrated the effectiveness of a community of inquiry approach for math-
ematics at the second grade level.

References
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vances	in	Research	on	Teaching,	(2),	pp.	1-48.
Burbules,	N.	(1993).	Dialogue	in	Teaching.	New	York:	Teachers	College	Press.
Carpenter,	T.P.	&	Fennema,	E.	(1992).	Cognitively	guided	instruction:	Building	on	the	knowledge	of	students	

and	teachers.	International	Journal	of	Educational	Research,	17,	457-470.
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problem-centered	second	grade	mathematical	project.		Journal	for	Research	in	Mathematics	Eduaction	(22),	
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Cobb,	P.,	Wood,	T.,	Yackel,	E.	&	Perlwitz,	M.	(1992).	A	follow-up	assessment	of	a	second-grade	problem-centered	
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Cobb,	P.,	Yackel,	E.,	&	Wood,	T.	(1992).	A	constructivist	alternative	to	the	representational	view	of	mind	in	
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Address correspondences to:

 Raymond Siegrist
 Suny College at Oneonta
	 Box	146	Fitzelle
	 Oneonta,	NY	13820
 siegrir@oneonta.edu


