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Content and Pedagogical Knowledge in Colorado 

Teachers’ Mathematics Exams at the Turn of the 20th Century 
 

Robert M. Capraro           Lynn M. Burlbaw 

Texas A and M University 

 

Linda Reichwein Zientek 

Sam Houston State University 

 

Abstract 

 

During the late 1800s and early 1900s, 

several published reports recommended changes 

in teaching mathematics. For this paper, items 

from the Colorado county arithmetic tests (1878 – 

1912) form the basis of investigating teacher’s 

content knowledge and the degree to which 

education was adhering to these 

recommendations. The purpose of this paper is to 

investigate whether or not recommendations 

proposed by the Report of the Committee of Ten 

(NEA, 1893) and the Report of the Committee of 

the Chicago Section (1899) made an impact on 

teacher certification exams in the state of 

Colorado. 

 

Certification of Teachers 

 By portraying teachers as young women 

graduates of the 8
th

 grade who then went on to 

teach children younger than themselves, 

television shows such as “Little House on the 

Prairie” have reinforced a view that 19
th

 and 20
th

 

century teachers lacked sufficient content and 

pedagogical knowledge. The idea that becoming a 

teacher requires little education beyond what was 

taught in the public school remains viable in some 

peoples’ minds. The idea that an 8
th

 grade 

education was sufficient to teach affirms a belief 

that teaching requires little specific knowledge, 

either in content or pedagogy. This idea is further 

substantiated with accelerated programs which 

give a minimal amount of training prior to 

assuming full classroom responsibility (Hallinan 

and Khmelkov, 2001).  

 

Consider the following questions: 

 

A square and a triangle contain an equal 

area, and the base of the triangle is 36.1 

feet and its altitude is 5 feet, what is the 

side of the square?  

 

A man’s expenses are 30 percent of his 

income, and 33 percent of his income 

equals 10 percent of his property which 

is valued at $27,000. What is his 

income?  What are his expenses? 

 

Examination questions such as these may seem 

rather commonplace on a mathematics test.  

 Consider though, a question such as,  

 

“When it is 1 o’clock a.m. on the first 

day of January, 1889, at Bangor, Maine, 

68º 47’ west, what was the time at the 

City of Mexico, 99º 5’ west? 

 

might not routinely appear on a mathematics test. 

These questions (Peavey 1896, 288, 283, 288), 

and others like them, were routine test items for 

people seeking to become elementary level 

teachers in Colorado between 1878 and 1923.  

 A vision of the certification process in 

turn-of-the-century Colorado has been presented 

by Burlbaw (2002, 2003a, 2003b) and Burlbaw 

and Bozeman (2003). These papers show that 

becoming a certified teacher was not automatic 



   

159 

 

and that teachers receiving county certificates 

were continuously studying and completing 

exams.  Some authors (cf. Askey 2005; 2006a; 

2006b;  2007; nd.) have examined contents on 

teachers’ exams in California and Michigan, 

questioning whether current teachers could pass 

those exams. This paper focuses on the nature of 

the content of the arithmetic exams teachers 

completed between 1878 and 1912, examining the 

expected knowledge through a modern lens of 

modern national standards. To contextualize 

teacher preparation (1) Types of Certificates, (2) 

Content of Exams, (3) Criteria for Class (Grade) 

of Certificate, (4) Teacher Performance on 

Exams, and finally (5) Teaching in the 20th 

Century are discussed. 

 

Types of Certificates 

 To teach in the state of Colorado in the 

late 1800s, one had to have one of three types of 

certificates: a county certificate, a state certificate 

or a normal school certificate.  The latter, a life-

time certificate qualified a person to teach at any 

public school in the state and was awarded once a 

person completed a course of study at one of the 

state teachers’ colleges. A second type of 

certificate, the state certificate, also a life-time 

certificate to teach at any school in the state, was 

awarded once a person passed the "state" teachers 

exam. In order to sit for the “state” exam, a 

candidate had to possess a valid “first grade” 

county certificate. Beginning in 1895, a Normal 

Institute Certificate, another lifetime certificate, 

was issued to teachers who attended summer 

institutes, successfully passed the required 

number of courses during the summer sessions 

and paid tuition for university credit (Craig 

1906a).  The Normal Institute Certificate and 

program was an attempt to address the lack of 

preparation of teachers “who had very little or no 

professional training” and provided the 

“minimum of continued professional training 

required by law for those who would continue to 

be certified” (Strayer and Englehardt 1920, 18, 

34). 

 In the period from 1880-1920, county 

certificate exams were not viewed as a way 

around what today would be considered a 

“traditional” normal school certification process 

but the quickest and most efficient mode to 

provide teachers. With a growing population, 

counties needed teachers and a way to ensure 

quality instruction.  In 1891, 86.6 percent of 

school age children under 16 were enrolled in 

school; these 66,750 students were largely taught 

by 1877 county certificate teachers; 419 of whom 

had First Grade Certificates; 719 Second Grade 

Certificates and 739 Third Grade Certificates 

(Coy 1893).  In his first Biennial Report, W. C. 

Lothrop (1872), Superintendent of Public 

Instruction of the Territory of Colorado wrote: 

 

The highest discretion should be used in 

the employment of a teacher for any 

position. It is a popular fallacy to 

suppose that anyone who is a fair scholar 

can teach school, especially doest this 

false notion prevail in regards to schools 

composed of small children. . . Let me 

here suggest to county superintendents 

the necessity for great care in the 

examination of teachers and 

discrimination in issuing certificates of 

qualification. . . .  To county 

superintendents then, we must look for 

the means of preventing the employment 

of incompetent teachers. (7-8) 

 

 The University of Colorado at Boulder 

opened its first education school in 1881; the first 

institution dedicated to teacher preparation, the 

State Normal School, at Greeley, was founded in 

1889 and opened its doors to students in 1890. A 

second normal school at Gunnison (today 



   

160 

 

Western State College) was authorized in 1901 

but did not open its doors until September 12, 

1911. As late as 1919, 48 percent of the teachers 

teaching in one-teacher schools were teaching 

under county certificate authority.  In two-teacher 

and three or more-teacher schools and high 

schools, the percentages were 33 percent, 11 

percent and 9 percent respectively (Bradford 

1921).  In 1925, 41 percent of the teachers in the 

state were county certified while 39% had normal 

school certificates and only 16 percent had state 

certificates (Bradford 1927).  According to a 1911 

report (Updegraff 1911), Colorado was one of 16 

states that used a state-county certification model; 

only in 17 states was the sole certification 

authority the state board of education (138-142).  

Thus, for over 50 years, county exams, as used in 

many other states, were vital in ensuring quality 

teachers were available for Colorado public 

schools. 

 After an applicant met the criteria for a 

given certification on the state prepared test the 

county superintendent of schools issued the 

certificate. During the 1880s, the county test was 

administered quarterly, usually at the county seat 

although provisions were made for testing in 

another larger city in the county. Although the 

First Biennial Report indicates that tests were 

administered quarterly (Lothrop 1878), the exact 

dates do not appear until the Third Biennial 

Report where the dates are listed as the “last 

Friday of February, May, August and November 

(Cornell 1883). This was a departure from the 

earlier practice of giving the exam on Saturdays. 

L. S. Cornell, State Superintendent of Public 

Instruction, suggested to county superintendents 

that they consider administering the test over two 

days, 6 subjects each day, in a five to six hour 

session (Cornell 1883). By 1898, tests were being 

administered only three times a year (March, 

August, and December) on dates designated by 

the State Superintendent of Public Instruction.   

 County certificates were valid for a 

specific period of time, ranging from six months 

to three years. Three grades or classes of 

certificates were awarded: First, Second and 

Third. The term grade did not refer to school 

grades but hierarchy of certification type. 

Therefore, for this paper we will use the term 

class to refer to the different levels of 

certification. First class certificates entitled the 

holder to teach for three years in any school in 

Colorado. The other certificates for lesser periods 

of time as explained below in the section, 

Criteria for Class (Grade) of Certificate. 

Content of Exams. 

With the exception of removing botany 

and adding civil government in 1895 to the 

subjects tested, the content remained consistent 

over time (Peavey 1896). A review of the 

published exams reveals the following subjects 

were tested each time: arithmetic, reading, U.S. 

history and constitution; physiology/health, 

orthography, school law, natural science, 

grammar, theory and practice, geography, and 

penmanship/writing.    

From 1878 to 1882, the subjects tested were 

arranged into two branches with different passing 

scores required for each branch: 

 

The school law requires examination in 

the “Elements of the Natural Sciences.” 

As this is a new departure it was thought 

best to fix the standard of these 

requirements in these branches a little 

lower than in those branches in which 

applicants had formerly been examined. 

(Lothrop 1878, 20) 

 

Table 1 shows the division of the subjects into 

two branches; the division into branches was 

discontinued in 1885 (Cornell 1887). 



   

161 

 

Table 1: Subjects Tested in Each Branch 

Branch One Branch Two 

Arithmetic 

U. S. History and 

Constitution 

Reading 

Orthography 

Grammar 

Theory and Practice 

Physiology and Laws of 

Health 

School Law 

Botany 

Other Natural Sciences 

Penmanship 

(Lothrop 1878,  19) 

 

The order of the subjects on the exams 

varied by administration but each was tested each 

time. In the early testing years, specific lengths of 

time were allotted for candidates to complete each 

content test. For example the time allotment, for 

arithmetic (10 questions) was 40 minutes; 

physiology and health (5 questions), 25 minutes; 

and theory and practice (10 questions), 35 

minutes (Lothrop 1878).  

In later years, the test was extended to a 

two day session. Three hours in the morning and 

three hours in the afternoon were allotted for 

completing assigned test sections. According to 

Shattuck (1884), “There is work for two days of 

five or six hours each for the average applicant 

and I recommend a two days’ session . . .  

Applicants should have time to do themselves 

justice. . . .   For the first day use Nos. 1 to 6 

inclusive; second day, Nos. 7 to 12 inclusive” 

(29).  For many years, both the county and the 

state certification tests were published in the 

Biennial or Annual Superintendent's report (cf. 

Lothrop 1878; Cornell 1883, 1887; Coy 1893; 

Craig 1908; Grenfell 1900). 

 

Criteria for Class (Grade) of Certificate 

 Because all candidates took the same 

exam, distinction between certificates was 

determined by a teacher’s performance on the 

exam, overall and on individual parts.  Teachers 

had to achieve a certain score on each subject of 

the exam and a specific cumulative average was 

required for each certificate.  The higher 

certificates required both a higher subject and 

overall score.  Table 2 provides a picture of the 

subject scores and test averages needed for each 

of the certificates at various times between 1878 

and 1923.  Beginning with the first quarter 

administration in 1885, subjects were no longer 

differentiated by branch and a single scoring 

standard was used for all certificates. In 1898, the 

minimum criterion score for the third class 

certificate was raised.  

 

Table 2: Required Averages and Scores for Certificates, 1878 – 1923 

  1878; 1882 to 1884 

Administrations 

1879 to 1881 

Administrations 

1885 to 1897 

Administrations
 

1898 and after 

Administrations 

Certificate 

Grade 

Branch
a
 Avg

 b 
Min

 c 
Ave

b
 Min

c
 Ave

b
 Min

c
 Ave

b
 Min

c
 

First One 90 75 90 75 90 70 90 70 

 Two 75 60 75 40     

Second One 75 60 75 60 80 60 80 60 

 Two 60 40 50 40     

Third One 60 50 60 50 70 50 70 60 

 Two 50 40 50 40     

Note.
.  a 

After 1885, subjects were no longer divided into branches; a single minimum score applied to all subjects.  

 
b 
Average = average of all scores received on each subject on the exam.  

 
c 
Minimum = lowest allowable score on any subject on the exam to receive that grade of certificate. 



   

162 

 

For example, in 1895, a teacher whose 

scores on the subject exams were at least 70 but 

whose overall average on the 12 subject exams 

was in excess of 90, would receive a First class 

certificate. If each subject exam was at least 70 

but overall average was less than 90 or if on one 

subject the score was below 70, the teacher would 

be awarded a Second class certificate. In addition 

to achieving a high score on the test, First class 

certificate teachers had to demonstrate success in 

teaching before taking the test that qualified them 

for the First class certificate (Lothrop 1878; 

Cornell 1883; Craig 1908; Wixson 1912). Second 

class certificates were issued to those scoring 

within a certain band, below that of a First class 

certificate, on the exam and qualified a teacher to 

teach for 12 to 18 months at any school in the 

county of issuance with no transferability to 

another county.  

The Third class certificate, depending on 

the time issued, was valid for either six or nine 

months and also not transferable to another 

county. Holders of a Third class certificates 

scored within the lowest band of acceptable 

scores. A teacher could only receive a third class 

certificate twice; after that a teacher could not be 

certified to teach without attaining a score high 

enough to warrant a First or Second class 

certificate. Beginning in 1901, the State 

Superintendent of Public Instruction office 

published a listing of those who had received a 

Third class certificate that was distributed to all 

county superintendents (cf. Grenfell 1900, 1902; 

Craig 1906b, 1908; Wixson, 1912).  

 

Teacher Performance on Exams 

Information on how teachers performed 

on the various exams can be found in two types of 

reports. The first, and most general, is tabular 

information found in the Biennial Reports. These 

tables report on how many teachers received 

First, Second, and Third class certificates in each 

county and how many teachers were teaching on 

each certificate type among other things. The 

second source of information, a richer and more 

detailed record, is also found in the Biennial 

Reports. This source is a listing of the test items 

which appeared on each of the offered exams. 

The items from the arithmetic tests (1878 – 1912) 

form the basis for this paper’s analysis of teacher 

content knowledge. The practice of publishing the 

exams was not limited to Colorado.  The 

Superintendent of Public Instruction in Michigan 

(1900) also published the exams as did E. L. 

Kellogg and Company which published test items 

from the New York State Uniform Examination 

(Kellogg 1889, 1890). 

Influences on Mathematics Education 

In the period between 1890 and 1900, 

several national reports were issued and changes 

within Colorado’s educational administration 

were occurring. The 1893 Report of the 

Committee of Ten and the 1899 Report of the 

Committee of the Chicago Section made 

recommendations regarding what teachers should 

know and how students should be taught (as cited 

in Bidwell and Clason 1970/2002).  In the 1893 

report, the committee divided the special reports 

into four subheadings: (a) arithmetic, (b) concrete 

geometry, (c) algebra, and (d) formal geometry. 

The committee recommended that arithmetic be 

both abridged and enriched with an elimination of 

the majority of commercial arithmetic problems. 

Recommendations were given on the importance 

of accuracy and speed scholars should be able to 

perform the four fundamental operations. 

Practical applications should be extended to real-

world applications that are interesting to students. 

The 1899 Report of the Committee of the 

Chicago Section advocated different presentation 

styles and believed more would be gained by 



   

163 

 

presenting problems in more than one way. 

Arithmetic problems should be confined to (a) the 

fundamental processes, (b) factorizations, (c) 

prime numbers, (d) denominate numbers, (e) 

operations in fraction, (7) money, (8) decimals, 

(9) simple interest, and (10) method of analysis 

for simple and compound proportion (Bidwell 

and Clason 1970/2002). The Committee believed 

high school teachers should have sufficient 

mathematics knowledge through calculus as well 

as the theory of equations. In addition to content 

knowledge, the Committee advocated 

pedagogical content knowledge acquired under 

the supervision of an experienced mathematics 

teacher. While the 1899 report provided insights 

into mathematics such as how it should be taught, 

what should be taught, and when in school 

specific content should be taught it did not 

provide a clear framework for analyzing the 

exams of the day. “Various method of teaching 

mathematics are in vogue. The good teacher will 

not tie himself to any one method, but, on 

occasion, make use of the good features of every 

one.  . . . The committee recommends no single 

method above all others,  . . . the aim should 

always be to cultivate independent thinking to the 

part of the pupil. A method that . . . permits rote 

work or mechanical manipulations , is radically 

wrong.” (196).  From these excerpts it seems the 

use of current NCTM standards as an analytic 

framework is logical and justified even though, as 

we know, mathematics education took a very 

different turn with the rise of behaviorism at the 

turn of the century. 

 

Purpose of Paper 

The purpose is to report on our 

investigation into whether either of two 

conditions had an effect on the content of the 

certification exams in the state of Colorado at the 

turn of the 20
th

 century; hence, influencing what 

and how it was taught:  

1. Did the recommendations of the 

1893 Report of the Committee of 

Ten, the 1899 Report of the 

Committee of the Chicago Section 

make an impact on teacher 

certification?  

2. Did the effect of the leadership of 

Helen Grenfell, State Superintendent 

of Public Instruction between 1899 

and 1905, who was known for her 

support for pedagogical knowledge, 

have an impact on teacher 

certification exams?  

 

The topics discussed at the turn of the 21
st
 

century were some of the same topics discussed at 

the turn of the 20
th

 century. Therefore, the NCTM 

content and process strands were used as a lens to 

classify and contextualize the problems and to 

examine how they changed in regard to content 

and pedagogy at the turn of the 20
th

 century.  

Helen Grenfell - a former classroom 

teacher - was elected in 1899 to the position of 

State Superintendent of Public Instruction. 

Grenfell’s support for pedagogical knowledge 

appears to align with the pedagogical 

recommendations of the 1899 Report. Grenfell 

served from 1899 to 1905 and was the former 

president of the Colorado Education Association.  

In 1903, during her tenure as State 

Superintendent, she was elected the first female 

Vice-President of the National Education 

Association. Grenfell was an ardent supporter of 

improving public education and entered the 

public school system of Colorado as a teacher in 

Eire, Colorado, in 1880. In 1882 she received a 

Master’s Degree at State Normal School in 

Albany, New York. She later taught in 

Blackhawk, Colorado, eventually being appointed 



   

164 

 

County Superintendent in 1896. Grenfell 

continued her studies in education and 

campaigned for school improvement throughout 

her life (Grenfell 1939). 

Methodology 

 A historical synthesis and data reduction of 

mathematics teacher certification exams was 

conducted on Colorado tests from 1878-1912. In 

the Biennial Report to the governor on the status 

of education in Colorado, the State 

Superintendents of Public Instruction published 

the items used on each of the tests during the 

previous biennium. These tests, published in the 

Biennial State Superintendent Bulletins, were 

located in the Denver Public Library and 

Colorado State Archives and were photocopied. 

The certification exam items for the arithmetic 

section from 1878 – 1912 were coded.  

Each question was coded by (a) year, (b) 

process strand: problem solving, reasoning and 

proof, communication, connections, or 

representation, (c) content strand: number and 

operations, algebra, geometry, measurement, and 

data analysis or probability, (d) according to real-

world problems: business, life, or neither, and (e) 

as pedagogy problems. If a problem was coded as 

business or commercial, it was examined to 

determine if reform suggestions from the Report 

of the Committee of Ten (National Education 

Association 1893) were implemented.  

The NCTM process strands were used as 

markers to determine changes in the Colorado 

educational system to match to recommendations 

set forth by mathematical organizations and 

educational entities of that time period. Further, 

problem difficulty, depth of content knowledge 

required to successfully answer the question, and 

the scope of topics covered was classified. 

Therefore, the use of process strands served as a 

proxy for influences on the state educational 

system. 

 

Problem Coding 

The criteria for coding an item as problem 

solving were: (1) several obvious solution 

strategies and (2) higher order thinking skills 

required. Reasoning and proof problems were 

problems in which formulas could be applied to 

the problem or an application of basic order of 

operations could result in the correct answer. 

Communication problems asked the prospective 

teacher to explain or demonstrate a concept. 

Connection problems asked the prospective 

teacher to convert or reduce a number to another 

form. Representation problems asked the 

prospective teacher to illustrate or show 

concretely the problem solution.  

Data were coded as “prior to” and “1893 

and after” to determine the effects of the Report 

of the Committee of Ten. Because of the 

complexity of some items, multiple codings were 

required. A problem-solving item that required 

proficiency with numbers as well as conceptual 

understandings of geometry would, therefore, be 

coded into three categories: problem-solving, 

geometry, and numbers and operations. In 

contrast, items may have made incidental use of 

two content strands such as geometry or 

measurement but these skill concepts were not 

instrumental in solving the item; hence, items 

were only coded according to the skills necessary 

to complete the problem. For example, an item 

may have used units of measure but a successful 

answer did not require manipulation or 

transformation between groups of units for 

successfully completing the question. In this case, 

the item was not coded as measurement.  

Two raters knowledgeable of the coding 

scheme coded all items. After coding, overall 

agreement was .85. The codes were later 



   

165 

 

reviewed by two other raters not involved in the 

study who examined the codes for accuracy of 

embodiment of the intent of the process and 

content strands. Agreement between the two 

external reviewers was .98 and .91 with the 

original coders.  

Because interest lies in the comparison of 

the proportion of items in any given year 

corresponding to the codes, arithmetic means 

were reported to provide insights into educational 

reform issues of the time. To determine if the 

report of 1893 had an impact on the items 

appearing on the teacher certification exams, 

correlations were calculated between process and 

content strands and real-world applications and 

year. Kruskal-Wallis test for independent samples 

with Bonferroni corrections were conducted 

between types of application problems (i.e., 

business or no real-world applications) 

administered before and after 1893. Year was 

dummy coded as “0” prior to 1893 or “1” 1893 or 

later.  

To determine if the instillation of the first 

woman superintendent impacted teacher 

certification, correlations were calculated between 

years 1900 and 1905 in algebra and pedagogy, 

and Kruskal-Wallis tests were conducted (a) 

between the time period 1900 to 1905 and algebra 

and pedagogy and (b) between the time period 

prior to and after 1900 and the NCTM process 

strands. Relationships between pedagogy and 

real-world application problems and content and 

process strands were qualitatively analyzed. The 

relationship between algebra problems and 

business problems were also qualitatively 

analyzed. Examples of problems and coding are 

contained in Table 3. 

 

Results 

Spearman Rho correlations indicated a 

statistically significant relationship between the 

year the exam was administered and business (



  

= .131, p =.002) and commercial problems (



  = -

.085, p =.043) meaning that as years increased so 

did business problems but commercial problems 

decreased. There was a statistically significant 

relationship between problem solving and 

reasoning (



  = -.602, p < .001) which is 

indicative of increases in problem solving 

resulted in a decrease in reasoning items. A 

statistically significant relationship existed 

between problem solving and algebra (



  = .237, 

p < .001), geometry (



  = -.097, p = .021), and 

measurement (



  = -.096, p = .023). Regression 

results indicated the year did not depend upon 

content and process strands but year did depend 

upon real-world application problems (R
2 

= .054, 

p <
 
.001). 

 

Influences of the 1983 Report of the Committee 

of Ten.  

A statistically significant relationship 

existed between business problems prior to and 

after 1893 (



  = -.099; p = .019). A Kruskal-

Wallis test for independent samples was 

conducted between types of application problems 

(i.e., business or no real-world applications) 

administered before and after 1893. Statistically 

significant differences existed at p <.025 (alpha 

set by Bonferroni correction) between time before 

1893 and business (
2 

(1, n = 561) = 5.75, p = 

.016) and no real world applications problems (
2 

(1, n = 561) = 11.20, p = .001). As shown in 

Table 4, investigations of arithmetic means by 

year indicate there was an increase in business 

applications and a decrease in application 

problems not related to real-world problems after 

1893. Figure 1 illustrates how both the percent of 

business problems and the percent commercial 

non-arithmetic business problems steadily 

increased over time. 



   

166 

 

 

Table 3 Sample Problems and Codes 

Code Year Item 

 

R and P B but 

not CA 

 

1901 

 

 

What will be the expense of plastering a room 18 feet long, 15 feet wide, 

and 8 feet high, at 30 cents a square yard, allowing 150 square feet for 

doors, windows, etc.? 

 

R and P 

B and CA 

1897 

 

Find the difference between the true discount and the bank discount of 

$1,000, for 1 year, 3 months, 15 days at 7%. 

 

Problem 

Solving  

1902 

 

A, B and C are to share $1,200 in the proportion of 3, 4, and 5, 

respectively. B dies. How should the whole sum be divided between A and 

C? 

 

Pedagogy 1878 Explain to a child how to add three digit numbers. 

 

Pedagogy 1905 Tell how you would develop one of the processes of finding the area of a 

circle, triangle, parallelogram or trapezoid to a pupil. 

 

Pedagogy 1900 In division of decimals how would you lead a child to understand where in 

the quotient the decimal point should be placed? Illustrate. 

 

Pedagogy 1902 Divide 0.75 by 17 5/8 by 4/5 of 0.035, giving the answer the form of a 

decimal number. Show by discussion and by examples how to teach pupils 

to place the decimal point correctly in the quotient. 

 

Pedagogy 1905 

 

Tell how you would develop one of the processes of finding the area of a 

circle, triangle, parallelogram or trapezoid 

 

Connections 

and not-

business 

problem 

1897 

 

What is the difference between figures and numbers? What is a complex 

decimal? What is a perfect power? What is the difference between an 

arithmetical and geometrical progression? What is a scalene triangle? 

 

 

Communication 1898 What is the mental effect of the study of arithmetic? 

 

Note. Items prior to 1900 asked prospective teachers to explain concepts, whereas items after 1900 asked prospective teachers 

to apply reasoning and proof skills and required them to make connections between various mathematical concepts.  

 

Codes: R and P= Reasoning and Proof; B= Business problem; CA=Commercial Arithmetic.  

 



   

167 

 

Table 4: Reported Mean Testing of Items 

Y
ear 

N
u
m

b
er 

o
f 

E
x
am

s 

R
easo

n
in

g
 an

d
 

P
ro

o
f 

P
ro

b
lem

 

S
o
lv

in
g
 

C
o
m

m
u
n
icatio

n
 

C
o
n
n
ectio

n
s 

R
ep

resen
tatio

n
s 

B
u
sin

ess 

L
ife P

ro
b
lem

s 

N
o
 

B
u
sin

ess 

o
r L

ife 

C
o
m

m
ercial 

P
ed

ag
o
g
y

 

1878 2 .550 .050 .300 .000 .050 .300 .200 .500 .100 .050 

1881 1 .600 .300 .000 .100 .100 .300 .200 .500 .300 .000 

1883 1 .600 .200 .100 .100 .000 .300 .100 .600 .200 .000 

1888 2 .850 .050 .100 .000 .000 .200 .100 .700 .150 .000 

1889 1 .800 .200  .000 .000 .000 .100 .000 .900 .000 .000 

1890 2 .750 .050 .150 .000 .050 .400 .250 .350 .100 .000 

1891 4 .650 .075 .175 .075 .025 .333 .325 .425 .125 .000 

1892 4 .675 .125 .075 .125 .000 .425 .250 .433 .125 .000 

1895 4 .775 .150  .000 .050 .025 .425 .375 .200 .150 .000 

1896 3 .433 .233 .233 .067 .033 .267 .267 .467 .000 .000 

1897 3 .700 .100 .067 .100 .033 .400 .167 .433 .167 .000 

1898 3 .667 .033 .233 .033 .033 .267 .300 .433 .133 .000 

1899 3 .700 .133 .067 .067 .033 .467 .267 .267 .000 .000 

1900 3 .833 .167 .050 .000 .033 .500 .200 .300 .167 .067 

1901 3 .733 .167 .100 .033 .000 .400 .233 .333 .133 .133 

1902 5 .600 .220 .120 .080 .020 .360 .240 .400 .060 .040 

1903 1 .600 .200 .200 .000 .000 .300 .400 .300 .100 .100 

1904 3 .667 .200 .100 .033 .000 .500 .233 .267 .100 .033 

1905 3 .667 .200 .100 .112 .000 .433 .167 .400 .033 .033 

1906 2 .700 .150 .100 .050 .000 .500 .250 .250 .100 .000 

1908 1 .500 .200 .100 .100 .100 .600 .000 .300 .000 .000 

1910* 1 .636 .182 .090 .000 .000 .545 .182 .182 .090 .090 

1912 1 .700 .300 .000 .000 .000 .400 .500 .100 .000 .000 

Note. Each exam had 10 problems except the 1910 exam; * The 1910 exam had 11 problems 

 



   

168 

 

 

Figure 1. The Increase in Business Problems and the Percent of Business Problems Categorized as 

Not Commercial and Commercial Arithmetic. 

 

 

Algebra problems usually related to 

business problems (80%) with the majority of 

these consisting of investment problems. 

Probability and statistics problems were almost 

non-existent - only two problems on the entire set 

of exams. Both of these problems had prospective 

teachers compute the arithmetic mean.  Forty-two 

percent (n = 237) of the questions related to the 

content strand numbers and operations. The 

problems related to numbers and operations were 

complex by today’s standards and tested 

knowledge on multiple concepts such as fractions, 

percentages, and order of operations. A problem 

from a test administered in 1904 (Grenfell 1904, 

115) is given below 

 

4

3

5

2

16

3

8

3
69

3

2

9

2
8

3

1

5

1
2

6

5
1

2

1
22

8

3



















of

 

 



   

169 

 

Twenty-three of the 56 exams (41%) 

contained definition problems. There does not 

appear to be a relationship between year and the 

number of definition problems; however, there 

does appear to be relationship between year and 

the type of definition problems proposed. Prior to 

1893, nine of the 10 exams (90%) contained 

definitions relating to mathematical terms typical 

in a mathematics classroom such as define an 

abstract number or a prime number [1888 test]. 

After 1893, definitions evolved from classical 

mathematics terms to terms used in the real-

world. On the 13 exams containing definitions 

after 1893, seven of the exams (54%) contained 

definitions typical to a traditional mathematics 

course. Almost half - six of the remaining exams 

(46%) - asked definitions of terms such as bond, 

ad valorem, duty [1902 test] or define promissory 

or times notes [1908 test].  

Influence of Changes in Leadership  

 Non-parametric tests (i.e. Kruskal-Wallis) 

indicated statistically significant relationships 

between the number of algebra (
2 

(1, n = 561) = 

6.75, p = .009) and pedagogy (
2 

(1, n = 561) = 

6.10, p = .014) problems before, after, and during 

the time period from 1900 to 1905. The years 

from 1900 to 1905 were influenced by 

superintendent Helen Grenfell and the publication 

of the Report of the Chicago Section of the 

American Mathematical Society. Prior to 1900, 

few pedagogical items appeared. However, after 

1900 not only did the number of pedagogical 

problems increase, but the nature of the 

pedagogical problems appeared to have evolved 

from basic understanding to in-depth pedagogical 

knowledge. Items prior to 1900 asked prospective 

teachers to explain concepts; whereas items after 

1900 had prospective teachers apply reasoning 

and proof skills and required them to make 

connections between various mathematical 

concepts. For example, a problem in 1878 asked a 

prospective teacher to explain to a child how to 

add three digit numbers whereas a problem in 

1905 asks the prospective teacher to tell how they 

would develop one of the processes of finding the 

area of a circle, triangle, parallelogram or 

trapezoid to a pupil (Grenfell, 1906). No 

statistically significant relationship existed 

between the time period before and after 1900 in 

regards to process strands.  

 

Discussion 

 As educators, we must examine the 

history of education. Investigating our past gives 

us a glimpse of who we were and helps us 

determine how to be more effective in the future. 

Studying the impact of recommendations by 

major organizations and changes in leadership 

helps us identify effective methods of instilling 

changes in our system that will lead to improved 

teaching and an increase in student achievement.  

Analysis of Colorado teacher exams 

indicated educators attempted to follow many of 

the recommendations set forth by mathematical 

organizations and educational entities at the turn 

of the 20th century. The Committee of Ten (1893) 

was unanimous in implementing a radical change 

in the teaching of arithmetic. They recommended 

omitting subjects “which perplex and exhaust the 

student without affording any really valuable 

mental discipline, and enriched by a greater 

number of exercises in simple calculation, and in 

the solution of concrete problems” (Bidwell and 

Clason, 1970/2002, 23) and suggested omitting or 

curtailing topics such as compound proportion, 

cube root and a large part of commercial 

arithmetic. The committee recommended that 

algebra, with options for bookkeeping or 

commercial arithmetic, be taught in grades ten 

and eleven (Willis, Schubert, Bullough, Kridel 

and Holton 1994).  

 



   

170 

 

Business Problems  

 Business problems were coded as 

commercial arithmetic or not commercial 

arithmetic. According to our analysis of the 

Colorado teaching exams, there was a modest 

attempt to curtail commercial arithmetic 

problems. Review of the arithmetic means of 

commercial arithmetic problems in Table 3 shows 

a modest decline in the number of commercial 

problems. As was noted by the committee, there 

was the consequence of people desiring a system 

suited to be more practical to commercial and 

business life, which could explain the only 

modest decline in commercial arithmetic items 

and the slight increase in business life items. The 

committee did recommend that if commercial 

arithmetic items were to prevail, pupils should 

master definitions regarding the business terms or 

else inexperienced students would struggle with 

questions they could not comprehend. A statically 

significant relationship existed between business 

and commercial problems and the year, as 

illustrated in Figure 1, and to business and real-

world application problems before and after the 

1893 Report of the Committee of Ten with an 

increase in the number of business problems. The 

educational system did take heed of the 

committees warning regarding students’ 

understanding of terms with a noticeable increase 

in the number of definition problems pertaining to 

business on the teacher certification exams after 

1893.  

   

Problem Solving and Reasoning and Proof 

  Problem solving is the medium in which 

children can develop mathematical ideas. 

Reasoning problems represent the shift from 

memorization to doing mathematics and is “the 

logical thinking that helps us solve problems and 

decide if and why our answers make sense” (Van 

de Walle 2001, 7). Van de Walle expanded his 

thoughts on problem solving to afford that “[i]f 

problem solving is the focus of mathematics, 

reasoning is the logical thinking that helps us 

solve problems and decide if and why answers 

make sense” (8).  

There was an inverse relationship between 

problem solving and reasoning and proof 

indicating that as the number of problem solving 

questions increased, the number of reasoning 

problems decreased. Most of the questions 

focused on reasoning and proof without giving 

prospective teachers the opportunity to apply their 

problem solving skills. Despite the lack of 

problem solving items, each exam did contain 

items which required prospective teachers to use 

problem solving skills and there were multiple 

items requiring reasoning and proof skills. The 

tests just lacked the opportunity for teachers to 

simultaneously apply these skills. The problem 

solving strand was positively correlated to 

algebra, geometry, and measurement, which is 

indicative of being able to apply higher order 

thinking skills. 

 

Pedagogy Problems 

  From 1900 until 1905, there was a 

statistically significant increase in mathematical 

pedagogy problems and a change in the 

complexity of these problems. The 1899 election 

of a former classroom teacher, Helen Grenfell, to 

the position of State Superintendent of Public 

Instruction and the 1899 Report of the Chicago 

Section of the American Mathematical Society 

appears to have influenced the increase in 

pedagogical items. While there was a gradual 

decrease in pedgagogy problems after 1905, 

educators did not fall back into the pattern of 

completely eliminating pedagogy problems. 

Today, NCTM and teacher educations support the 

need to understand pedagogy and prepare teacher 

exams that reflect this mode of thought.  



   

171 

 

The 1899 Report of the Chicago Section of 

the American Mathematical Society advocated the 

different presentation styles and believed more 

would be gained by presenting problems in more 

than one way (Bidwell and Clason 1970/2002). 

The 1899 Report stated that "The good teacher 

will not tie himself to any one method, but on 

occasion, will make use of the good features of 

every one. . . A method which encourages, or 

even permits, rote work, or mechanical 

manipulations is radically wrong” (Bidwell and 

Clason, 196). The committee supported different 

presentations with the goal of cultivating 

independent thinking.  

The pedagogy problems on the Colorado 

teacher exams usually related to the 

communication process strand. The pedagogy 

problems after 1900 differed from the items prior 

to 1878 in regards to complexity. Since the exam 

in 1878 is assumed to be representative of the 

population, one can concur that most exams 

contained problems similar to the ones exhibited 

in our sample. Whereas, the 1878 exam had 

prospective teachers explain to a child an 

arithmetic problem; the exams after 1900, 

evolved into more complex descriptions regarding 

proving how formulas were derived. While 

explaining problems is important, showing that 

prospective teachers understand the underlying 

concepts is an important focus in current teacher 

preparation programs. Our results shows that the 

turn of the 20
th

 century did begin this 

transformation in Colorado by changing from 

mainly assessing prospective teachers’ procedural 

skills to assessing both procedural skills and 

conceptual knowledge.   

Content Strands. Of the content strands, 

probability and statistics was almost nonexistent 

and when represented consisted of computing 

arithmetic means. This is not surprising since the 

first major introduction of probability and 

statistics in the western world came in the 1960s 

as part of the new mathematics curriculum 

(Truran, 2001). The large percentage of 

operations and number items was also not 

surprising because the understanding of numbers 

and operations is instrumental in understanding 

other content strands. The lack of problems 

pertaining to algebra (24%) and the large number 

relating to business and finance was a surprise 

since algebra has been heralded as an important 

subject for everyone to learn (Usiskin, 1995) and 

is an important topic in today’s curriculum.  

Conclusion 

Today’s major goal of producing qualified 

teachers whose students achieve a high degree of 

success are similar to those voiced in the late 

1800s and early 1900s. The 1893 Report of the 

Committee of Ten had three recommendations for 

better trained teachers: a) utilize agencies already 

in existence to practicing teachers by professional 

development opportunities and financial support 

to pursue these endeavors, b) colleges and 

universities should give stated courses of 

instruction in elementary and secondary subjects, 

and c) a mentoring system in which the best 

teacher in each department gives part of his/her 

time towards “. . .helping the other teachers by 

inspecting and criticizing their work, and showing 

them, both by precept and example, how to do it 

better” (Bidwell and Clason 1970/2002, 54). 

The committee recommended that 

secondary subjects should be taught the same to 

every pupil no matter what their probable 

destination or when their education might cease. 

The goal of secondary schools was not to prepare 

boys and girls for college but was “to prepare for 

the duties of life that small proportion of all the 

children in the country . . . who show themselves 

able to profit by an education prolonged to the 

eighteenth year” (Bidwell and Clason 1970/2002, 

51).  



   

172 

 

The rigor of the 1904 mathematics 

problem presented above exemplifies the 

complexity of items for those certifying to teach  

and are seemingly more demanding procedurally 

than those items appearing on present day first 

through eighth grade teacher certification exams. 

Although some might continue to argue that if 

one possesses the ability to solve problems one 

probably possesses sound process and conceptual 

knowledge, ; this is not reflected in contemporary 

practice. Today, preparation of teachers focuses 

on mathematical processes and conceptual 

development rather than limiting preparation to 

algorithmic procedures and processes. However, 

this study identified differences in the level of 

rigor on procedural problems. Maybe, as 

educators we should revisit the mode of thought 

prior to and around 1900 and create assessments 

that contain more rigorous and demanding 

mathematical problems as well as problems 

emphasizing conceptual knowledge.  

This study indicates that educational 

entities in charge of determining who would teach 

in the classroom adhered to the recommendations 

by leading mathematical organizations. This 

study also showed how teacher testing changed in 

Colorado around the turn of the 20
th

 century, and 

aids in dispelling the myth that 19
th

 and 20
th

 

century teachers lacked sufficient content or 

pedagogical knowledge. Input from experts, 

accompanied by changes in leadership within the 

school system, cause revision in teacher testing, 

which in turn, had an impact on classroom 

instruction. In conclusion, this study suggests that 

reports produced by people who were experts in 

their field of study were considered valuable to 

the educational community and had an impact 

upon teacher certification in the state of Colorado.  

 

 

 

References 

 

Askey, R. (Fall 2005).  Mathematics for teaching: 

Then and Now. Teacher Educator, 23. 

Retrieved September 28, 2005 from 

http://www.aft.org/pubs-reports/ 

american_educator/issues /fall2005/Askey.pdf  

Askey, R. (Spring 2006a). Mathematics for 

teaching: Then and now. Teacher Educator, 

np. Retrieved September 28, 2006 from 

http://www.aft.org/pubs-reports/ american 

_educator/issues /fall2005/Askey.pdf 

Askey, R. (2006b). “Mathematical content 

knowledge of teachers – a view from 1875,” 

Wisconsin Teacher of Mathematics, 56 (2), 

31-35. 

Askey, R. (nd). Learning from Assessment. 

Mimeograph paper obtained from the author. 

Askey, R. (2007). Learning from Assessment. 

Chapter 9 in Title: Assessing mathematical 

proficiency / edited by Alan H. Schoenfeld. 

Publisher: Cambridge ; New York : 

Cambridge University Press, 2007.  Pages 

125-136. 

Bidwell, J. K., and R. G. Clason, (Eds.). 

(1970/2002). Readings in the history of 

mathematics education. Reston, VA: National 

Council of Teachers of Mathematics. 

Bradford, Mary C. C. (1921). Twenty-third 

biennial report of the superintendent of public 

instruction of the state of Colorado for the 

official biennium ending November 30, 1920. 

Denver: Eames Brothers. 

Bradford, Mary C. C. (1927). Twenty-fifth 

biennial report of the superintendent of public 

instruction of the state of Colorado for the 

official biennium ending November 30, 1926. 

Denver: Bradford-Robinson Ptg. Co. 

Burlbaw, Lynn M. (2002, September). Teacher 

testing – could you pass the test? Paper 

presented at Twenty-fourth Mid-America 

Conference on History, Fayetteville, AR. 

Burlbaw, Lynn M. (2003a, September). In context 

of the times: Historical, geographic, and 

political content in elementary certification 

exams at the turn of the century Colorado 

teachers’ exams. Paper presented at Twenty-



   

173 

 

fifth Annual Mid-America Conference on 

History, Memphis, TN. 

Burlbaw, Lynn M. (2003b, October). In Context 

of the times: Pedagogical exams for turn of 

the century Colorado teachers. Paper 

presented at Midwest History of Education 

Society meeting, Chicago, IL. 

Burlbaw, Lynn M., and Bozeman, Dane. (2003, 

October). A Scientifically literate profession – 

the knowledge base of county certified 

elementary teachers in the state of Colorado, 

1880-1912. Paper presented at Midwest 

History of Education Society meeting, 

Chicago, IL. 

Cornell, Leonidas S. (1883). Third Biennial 

Report of the Superintendent of Public 

Instruction of the State of Colorado for the 

two years ending August 31, 1881 and August 

31, 1882. Denver: Times Public Printer.  

Cornell, Leonidas  S. (1887). Fifth Biennial 

Report of the Superintendent of Public 

Instruction of the State of Colorado for the 

two years ending August 31, 1885 and August 

31, 1886. Denver: The Collier and Cleaveland 

Lith. Co.  

Coy, Nathan B. (1893). Eighth Biennial Report of 

the Superintendent of Public Instruction of the 

State of Colorado, December, 1892. Denver: 

Smith Brooks Printing. 

Craig, Katherine L. (1906a). Report of the State 

Superintendent of Public Instruction of the 

State of Colorado for the years 1905-1906.  

Denver: Smith-Brooks Printing, Co.  

Craig, Katherine L. (1906b). Third Grade 

Certificates issued Aug. 16 and 17.  Denver, 

CO: Smith-Brooks Press. 

Craig, Katherine L. (1908). Report of the State 

Superintendent of Public Instruction of the 

State of Colorado for the years 1907-1908.  

Denver: Smith-Brooks Printing, Co.  

Grenfell, E. I. (1939). A brief sketch of the life 

and works of Helen Thatcher Loring Grenfell. 

Denver, CO: Smith-Brooks Printing Co. 

Grenfell, Helen L. (1900).  Twelfth Biennial 

Report of the Superintendent of Public 

Instruction of the State of Colorado to the 

Governor.  Denver: Smith Brooks Printing. 

Grenfell, Helen L. (1902). Third Grade 

Certificates issued December 20 and 21, 

1901.  Denver, CO: Smith-Brooks Press. 

Grenfell, Helen L. (1904). Fourteenth Biennial 

Report of the Superintendent of Public 

Instruction of the State of Colorado to the 

Governor.  Denver: Smith Brooks Printing. 

Grenfell, Helen L. (1906). Fifteenth Biennial 

Report of the Superintendent of Public 

Instruction of the State of Colorado to the 

Governor.  Denver: Smith Brooks Printing. 

Hallinan, M., and Khmelkov, V. (2001). Recent 

developments in teacher education in the 

United States of America. Journal of 

Education for Teaching, 27, 175-185. 

Kellogg, E. L. (March, 1889). New York uniform 

state examination questions. The Teachers 

Institute and Practical Teacher,11(7), 221-

223. 

Kellogg, E. L. (June, 1890). Second and third 

grades. The Teachers Institute and Practical 

Teacher, 12(10), 320-322. 

Lothrop, W. C. (1872). First biennial report of 

the superintendent of public instruction of the 

territory of Colorado for the school years 

ending September 30, 1870, and September 

30, 1871. Central City, CO: D. C. Collier. 

Lothrop, W. C. (1878). First Biennial Report of 

the Superintendent of Public Instruction of the 

State of Colorado for the two years ending 

August 31, 1878. Denver: Tribune Steam 

Printing House. 

Michigan (1900). Annual report of the 

Superintendent of Public Instruction of the 

State of Michigan: with accompanying 

documents, for the year 1900. Lansing: The 

Superintendent. 

National Council of Teachers of Mathematics. 

(2000). Principles and standards for school 

mathematics. Reston, VA: NCTM.  

National Education Association. (1893). Report of 

the Committee of Ten on Secondary Schools.  

Washington, DC: U. S Government Printing 

Office. 

Peavey, Angenette J. (1896). Tenth Biennial 

Report of the Superintendent of Public 



   

174 

 

Instruction of the State of Colorado. Denver, 

CO: Smith-Brooks Printing. 

Shattuck, Joseph C. (1884). Fourth Biennial 

Report of the Superintendent of Public 

Instruction of the State of Colorado for the 

years ending August 31, 1883, and August 31, 

1884. Denver: Times Printing Company.  

Strayer, George, and Englehardt, Nate. (1920).  

The classroom teacher at work in American 

schools.  New York:  American Book 

Company. 

Truran, J. (2001). Postscript: Researching 

stochastic understanding-The place of a 

developing research field in PME. 

Educational Studies in Mathematics, 45, 9-13. 

Updegraff, H. (1911). Teachers’ certificates 

issued under general state laws and 

regulations. United States Bureau of 

Education, Bulletin 1911, No. 18.  

Washington: GPO. 

Usiskin, Z. (1995). Why is algebra important to 

learn? American Educator, 32, 30-37. 

Van de Walle, J. A. (2001). Elementary and 

middle school mathematics: Teaching 

developmentally (4th ed.). New York: 

Addison Wesley Longman. 

Willis, George, William H. Schubert, Robert V. 

Bullough, Jr., Craig Kridel, and John T. 

Holton. (1994). The American curriculum: A 

documentary history. Westport, CT: Praeger. 

Wixson, Helen M. (1912). Eighteenth Biennial 

Report of the State Superintendent of Public 

Instruction for the two years ending 1912. 

Denver: Smith-Brooks Printing Co. 


