How Do You Formulate (Important) Hypotheses?

Abstract

Building on the ideas in Chap. 1, we describe formulating, testing, and revising hypotheses as a continuing cycle of clarifying what you want to study, making predictions about what you might find together with developing your reasons for these predictions, imagining tests of these predictions, revising your predictions and rationales, and so on. Many resources feed this process, including reading what others have found about similar phenomena, talking with colleagues, conducting pilot studies, and writing drafts as you revise your thinking. Although you might think you cannot predict what you will find, it is always possible—with enough reading and conversations and pilot studies—to make some good guesses. And, once you guess what you will find and write out the reasons for these guesses you are on your way to scientific inquiry. As you refine your hypotheses, you can assess their research importance by asking how connected they are to problems your research community really wants to solve.

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Chapter 2
How Do YouFormulate (Important)
Hypotheses?
Part I.Getting Started
We want to begin by addressing a question you might have had as you read the title
of this chapter. You are likely to hear, or read in other sources, that the research
process begins by asking research questions. For reasons we gave in Chap. 1, and
more we will describe in this and later chapters, we emphasize formulating, testing,
and revising hypotheses. However, it is important to know that asking and answer-
ing research questions involve many of the same activities, so we are not describing
a completely different process.
We acknowledge that many researchers do not actually begin by formulating
hypotheses. In other words, researchers rarely get a researchable idea by writing out
a well-formulated hypothesis. Instead, their initial ideas for what they study come
from a variety of sources. Then, after they have the idea for a study, they do lots of
background reading and thinking and talking before they are ready to formulate a
hypothesis. So, for readers who are at the very beginning and do not yet have an idea
for a study, let’s back up. Where do research ideas come from?
There are no formulas or algorithms that spawn a researchable idea. But as you
begin the process, you can ask yourself some questions. Your answers to these ques-
tions can help you move forward.
1. What are you curious about? What are you passionate about? What have you
wondered about as an educator? These are questions that look inward, questions
about yourself.
2. What do you think are the most pressing educational problems? Which problems
are you in the best position to address? What change(s) do you think would help
all students learn more productively? These are questions that look outward,
questions about phenomena you have observed.
3. What are the main areas of research in the eld? What are the big questions that
are being asked? These are questions about the general landscape of the eld.
© The Author(s) 2023
J. Hiebert et al., Doing Research: A New Researcher’s Guide,
Research in Mathematics Education,
https://doi.org/10.1007/978-3-031-19078-0_2

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4. What have you read about in the research literature that caught your attention?
What have you read that prompted you to think about extending the profession’s
knowledge about this? What have you read that made you ask, “I wonder why
this is true?” These are questions about how you can build on what is known in
the eld.
5. What are some research questions or testable hypotheses that have been identi-
ed by other researchers for future research? This, too, is a question about how
you can build on what is known in the eld. Taking up such questions or hypoth-
eses can help by providing some existing scaffolding that others have
constructed.
6. What research is being done by your immediate colleagues or your advisor that
is of interest to you? These are questions about topics for which you will likely
receive local support.
Part II.Paths fromaGeneral Interest
toanInformed Hypothesis
There are many different paths you might take from conceiving an idea for a study,
maybe even a vague idea, to formulating a prediction that leads to an informed
hypothesis that can be tested. We will explore some of the paths we recommend.
We will assume you have completed Exercise 2.1in Part I and have some written
answers to the six questions that preceded it as well as a statement that describes
your topic of interest. This very rst statement could take several different forms: a
description of a problem you want to study, a question you want to address, or a
hypothesis you want to test. We recommend that you begin with one of these three
forms, the one that makes most sense to you. There is an advantage to using all three
and exibly choosing the one that is most meaningful at the time and for a particular
study. You can then move from one to the other as you think more about your
research study and you develop your initial idea. To get a sense of how the process
might unfold, consider the following alternative paths.
Exercise 2.1
Brainstorm some answers for each set of questions. Record them. Then step
back and look at the places of intersection. Did you have similar answers
across several questions? Write out, as clearly as you can, the topic that cap-
tures your primary interest, at least at this point. We will give you a chance to
update your responses as you study this book.
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Beginning withaPrediction If YouHave One
Sometimes, when you notice an educational problem or have a question about an
educational situation or phenomenon, you quickly have an idea that might help
solve the problem or answer the question. Here are three examples.
You are a teacher, and you noticed a problem with the way the textbook pre-
sented two related concepts in two consecutive lessons. Almost as soon as you
noticed the problem, it occurred to you that the two lessons could be taught more
effectively in the reverse order. You predicted better outcomes if the order was
reversed, and you even had a preliminary rationale for why this would be true.
You are a graduate student and you read that students often misunderstand a
particular aspect of graphing linear functions. You predicted that, by listening to
small groups of students working together, you could hear new details that would
help you understand this misconception.
You are a curriculum supervisor and you observed sixth-grade classrooms where
students were learning about decimal fractions. After talking with several experi-
enced teachers, you predicted that beginning with percentages might be a good way
to introduce students to decimal fractions.
We begin with the path of making predictions because we see the other two paths
as leading into this one at some point in the process (see Fig.2.1). Starting with this
path does not mean you did not sense a problem you wanted to solve or a question
you wanted to answer.
Notice that your predictions can come from a variety of sources—your own
experience, reading, and talking with colleagues. Most likely, as you write out your
predictions you also think about the educational problem for which your prediction
is a potential solution. Writing a clear description of the problem will be useful as
you proceed. Notice also that it is easy to change each of your predictions into a
question. When you formulate a prediction, you are actually answering a question,
even though the question might be implicit. Making that implicit question explicit
can generate a rst draft of the research question that accompanies your prediction.
Fig. 2.1 Three Pathways to Formulating Informed Hypotheses
Part II.Paths fromaGeneral Interest toanInformed Hypothesis

20
For example, suppose you are the curriculum supervisor who predicts that teaching
percentages rst would be a good way to introduce decimal fractions. In an obvious
shift in form, you could ask, “In what ways would teaching percentages benet
students’ initial learning of decimal fractions?”
The difference between a question and a prediction is that
a question simply asks what you will find whereas a pre-
diction also says what you expect to find.
There are advantages to starting with the prediction form if you can make an
educated guess about what you will nd. Making a prediction forces you to think
now about several things you will need to think about at some point anyway. It is
better to think about them earlier rather than later. If you state your prediction
clearly and explicitly, you can begin to ask yourself three questions about your pre-
diction: Why do I expect to observe what I am predicting? Why did I make that
prediction? (These two questions essentially ask what your rationale is for your
prediction.) And, how can I test to see if it’s right? This is where the benets of mak-
ing predictions begin.
Asking yourself why you predicted what you did, and then asking yourself why
you answered the rst “why” question as you did, can be a powerful chain of thought
that lays the groundwork for an increasingly accurate prediction and an increasingly
well-reasoned rationale. For example, suppose you are the curriculum supervisor
above who predicted that beginning by teaching percentages would be a good way
to introduce students to decimal fractions. Why did you make this prediction?
Maybe because students are familiar with percentages in everyday life so they could
use what they know to anchor their thinking about hundredths. Why would that be
helpful? Because if students could connect hundredths in percentage form with hun-
dredths in decimal fraction form, they could bring their meaning of percentages into
decimal fractions. But how would that help? If students understood that a decimal
fraction like 0.35 meant 35 of 100, then they could use their understanding of hun-
dredths to explore the meaning of tenths, thousandths, and so on. Why would that
be useful? By continuing to ask yourself why you gave the previous answer, you can
begin building your rationale and, as you build your rationale, you will nd yourself
revisiting your prediction, often making it more precise and explicit. If you were the
curriculum supervisor and continued the reasoning in the previous sentences, you
might elaborate your prediction by specifying the way in which percentages should
be taught in order to have a positive effect on particular aspects of students’ under-
standing of decimal fractions.
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Developing aRationale forYour Predictions
Keeping your initial predictions in mind, you can read what others already know
about the phenomenon. Your reading can now become targeted with a clear purpose.
You can search for chapters or literature reviews related to your
research topic in recent research handbooks and compendia or in
journals. Reading these will help inform your predictions and
provide helpful reference lists of other sources.
By reading and talking with colleagues, you can develop more complete reasons for
your predictions. It is likely that you will also decide to revise your predictions
based on what you learn from your reading. As you develop sound reasons for your
predictions, you are creating your rationales, and your predictions together with
your rationales become your hypotheses. The more you learn about what is already
known about your research topic, the more rened will be your predictions and the
clearer and more complete your rationales. We will use the term more informed
hypotheses to describe this evolution of your hypotheses.
As you develop sound reasons for your predictions, you
are creating your rationales, and your predictions to-
gether with your rationales become your hypotheses.
Developing more informed hypotheses is a good thing because it means: (1) you
understand the reasons for your predictions; (2) you will be able to imagine how you
can test your hypotheses; (3) you can more easily convince your colleagues that
they are important hypotheses—they are hypotheses worth testing; and (4) at the
end of your study, you will be able to more easily interpret the results of your test
and to revise your hypotheses to demonstrate what you have learned by conducting
the study.
Imagining Testing Your Hypotheses
Because we have tied together predictions and rationales to constitute hypotheses,
testing hypotheses means testing predictions and rationales. Testing predictions
means comparing empirical observations, or ndings, with the predictions. Testing
Part II.Paths fromaGeneral Interest toanInformed Hypothesis

22
rationales means using these comparisons to evaluate the adequacy or soundness of
the rationales.
Imagining how you might test your hypotheses does not mean working out the
details for exactly how you would test them. Rather, it means thinking ahead about
how you could do this. Recall the descriptor of scientic inquiry: “experience care-
fully planned in advance” (Fisher, 1935). Asking whether predictions are testable
and whether rationales can be evaluated is simply planning in advance.
You might read that testing hypotheses means simply assessing whether predic-
tions are correct or incorrect. In our view, it is more useful to think of testing as a
means of gathering enough information to compare your ndings with your predic-
tions, revise your rationales, and propose more accurate predictions. So, asking
yourself whether hypotheses can be tested means asking whether information could
be collected to assess the accuracy of your predictions and whether the information
will show you how to revise your rationales to sharpen your predictions.
Cycles ofBuilding Rationales andPlanning toTest Your Predictions
Scientic reasoning is a dialogue between the possible and the actual, an interplay between
hypotheses and the logical expectations they give rise to: there is a restless to-and-fro
motion of thought, the formulation and rectication of hypotheses (Medawar, 1982, p.72).
As you ask yourself about how you could test your predictions, you will inevitably
revise your rationales and sharpen your predictions. Your hypotheses will become
more informed, more targeted, and more explicit. They will make clearer to you and
others what, exactly, you plan to study.
When will you know that your hypotheses are clear and precise enough? Because
of the way we dene hypotheses, this question asks about both rationales and pre-
dictions. If a rationale you are building lets you make a number of quite different
predictions that are equally plausible rather than a single, primary prediction, then
your hypothesis needs further renement by building a more complete and precise
rationale. Also, if you cannot briey describe to your colleagues a believable way to
test your prediction, then you need to phrase it more clearly and precisely.
Each time you strengthen your rationales, you might need to adjust your predic-
tions. And, each time you clarify your predictions, you might need to adjust your
rationales. The cycle of going back and forth to keep your predictions and rationales
tightly aligned has many payoffs down the road. Every decision you make from this
point on will be in the interests of providing a transparent and convincing test of
your hypotheses and explaining how the results of your test dictate specic revi-
sions to your hypotheses. As you make these decisions (described in the succeeding
chapters), you will probably return to clarify your hypotheses even further. But, you
will be in a much better position, at each point, if you begin with well-informed
hypotheses.
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23
Beginning by Asking Questions toClarify Your Interests
Instead of starting with predictions, a second path you might take devotes more time
at the beginning to asking questions as you zero in on what you want to study. Some
researchers suggest you start this way (e.g., Gournelos etal., 2019). Specically,
with this second path, the rst statement you write to express your research interest
would be a question. For example, you might ask, “Why do ninth-grade students
change the way they think about linear equations after studying quadratic equa-
tions?” or “How do rst graders solve simple arithmetic problems before they have
been taught to add and subtract?”
The rst phrasing of your question might be quite general or vague. As you think
about your question and what you really want to know, you are likely to ask follow-
up questions. These questions will almost always be more specic than your rst
question. The questions will also express more clearly what you want to know. So,
the question “How do rst graders solve simple arithmetic problems before they
have been taught to add and subtract” might evolve into “Before rst graders have
been taught to solve arithmetic problems, what strategies do they use to solve arith-
metic problems with sums and products below 20?” As you read and learn about
what others already know about your questions, you will continually revise your
questions toward clearer and more explicit and more precise versions that zero in on
what you really want to know. The question above might become, “Before they are
taught to solve arithmetic problems, what strategies do beginning rst graders use
to solve arithmetic problems with sums and products below 20 if they are read story
problems and given physical counters to help them keep track of the quantities?”
Imagining Answers toYour Questions
If you monitor your own thinking as you ask questions, you are likely to begin form-
ing some guesses about answers, even to the early versions of the questions. What
do students learn about quadratic functions that inuences changes in their propor-
tional reasoning when dealing with linear functions? It could be that if you analyze
the moments during instruction on quadratic equations that are extensions of the
proportional reasoning involved in solving linear equations, there are times when
students receive further experience reasoning proportionally. You might predict that
these are the experiences that have a “backward transfer” effect (Hohensee, 2014).
These initial guesses about answers to your questions are your rst predictions.
The rst predicted answers are likely to be hunches or fuzzy, vague guesses. This
simply means you do not know very much yet about the question you are asking.
Your rst predictions, no matter how unfocused or tentative, represent the most you
know at the time about the question you are asking. They help you gauge where you
are in your thinking.
Part II.Paths fromaGeneral Interest toanInformed Hypothesis

24
Shifting totheHypothesis Formulation andTesting Path
Research questions can play an important role in the research process. They provide
a succinct way of capturing your research interests and communicating them to
others. When colleagues want to know about your work, they will often ask “What
are your research questions?” It is good to have a ready answer.
However, research questions have limitations. They do not capture the three
images of scientic inquiry presented in Chap. 1. Due, in part, to this less expansive
depiction of the process, research questions do not take you very far. They do not
provide a guide that leads you through the phases of conducting a study.
Consequently, when you can imagine an answer to your research question, we
recommend that you move onto the hypothesis formulation and testing path.
Imagining an answer to your question means you can make plausible predictions.
You can now begin clarifying the reasons for your predictions and transform your
early predictions into hypotheses (predictions along with rationales). We recom-
mend you do this as soon as you have guesses about the answers to your questions
because formulating, testing, and revising hypotheses offers a tool that puts you
squarely on the path of scientic inquiry. It is a tool that can guide you through the
entire process of conducting a research study.
This does not mean you are nished asking questions. Predictions are often cre-
ated as answers to questions. So, we encourage you to continue asking questions to
clarify what you want to know. But your target shifts from only asking questions to
also proposing predictions for the answers and developing reasons the answers will
be accurate predictions. It is by predicting answers, and explaining why you made
those predictions, that you become engaged in scientic inquiry.
Cycles ofRening Questions andPredicting Answers
An example might provide a sense of how this process plays out. Suppose you are
reading about Vygotsky’s (1987) zone of proximal development (ZPD), and you
realize this concept might help you understand why your high school students had
trouble learning exponential functions. Maybe they were outside this zone when
you tried to teach exponential functions. In order to recognize students who would
benet from instruction, you might ask, “How can I identify students who are within
the ZPD around exponential functions?” What would you predict? Maybe students
in this ZPD are those who already had knowledge of related functions. You could
write out some reasons for this prediction, like “students who understand linear and
quadratic functions are more likely to extend their knowledge to exponential func-
tions.” But what kind of data would you need to test this? What would count as
“understanding”? Are linear and quadratic the functions you should assess? Even if
they are, how could you tell whether students who scored well on tests of linear and
quadratic functions were within the ZPD of exponential functions? How, in the end,
would you measure what it means to be in this ZPD? So, asking a series of
2 How Do YouFormulate (Important) Hypotheses?

25
reasonable questions raised some red ags about the way your initial question was
phrased, and you decide to revise it.
You set the stage for revising your question by dening ZPD as the zone within
which students can solve an exponential function problem by making only one addi-
tional conceptual connection between what they already know and exponential
functions. Your revised question is, “Based on students’ knowledge of linear and
quadratic functions, which students are within the ZPD of exponential functions?”
This time you know what kind of data you need: the number of conceptual connec-
tions students need to bridge from their knowledge of related functions to exponen-
tial functions. How can you collect these data? Would you need to see into the
minds of the students? Or, are there ways to test the number of conceptual connec-
tions someone makes to move from one topic to another? Do methods exist for
gathering these data? You decide this is not realistic, so you now have a choice:
revise the question further or move your research in a different direction.
Notice that we do not use the term research question for all these early versions
of questions that begin clarifying for yourself what you want to study. These early
versions are too vague and general to be called research questions. In this book, we
save the term research question for a question that comes near the end of the work
and captures exactly what you want to study. By the time you are ready to specify a
research question, you will be thinking about your study in terms of hypotheses and
tests. When your hypotheses are in nal form and include clear predictions about
what you will nd, it will be easy to state the research questions that accompany
your predictions.
To reiterate one of the key points of this chapter: hypotheses carry much more
information than research questions. Using our denition, hypotheses include pre-
dictions about what the answer might be to the question plus reasons for why you
think so. Unlike research questions, hypotheses capture all three images of scientic
inquiry presented in Chap. 1 (planning, observing and explaining, and revising
one’s thinking). Your hypotheses represent the most you know, at the moment, about
your research topic. The same cannot be said for research questions.
Beginning withaResearch Problem
When you wrote answers to the six questions at the end of Part I of this chapter, you
might have identied a research interest by stating it as a problem. This is the third
path you might take to begin your research. Perhaps your description of your prob-
lem might look something like this: “When I tried to teach my middle school stu-
dents by presenting them with a challenging problem without showing them how to
solve similar problems, they didn’t exert much effort trying to nd a solution but
instead waited for me to show them how to solve the problem.” You do not have a
specic question in mind, and you do not have an idea for why the problem exists,
so you do not have a prediction about how to solve it. Writing a statement of this
problem as clearly as possible could be the rst step in your research journey.
Part II.Paths fromaGeneral Interest toanInformed Hypothesis

26
As you think more about this problem, it will feel natural to ask questions about
it. For example, why did some students show more initiative than others? What
could I have done to get them started? How could I have encouraged the students to
keep trying without giving away the solution? You are now on the path of asking
questions—not research questions yet, but questions that are helping you focus your
interest.
As you continue to think about these questions, reect on your own experience,
and read what others know about this problem, you will likely develop some guesses
about the answers to the questions. They might be somewhat vague answers, and
you might not have lots of condence they are correct, but they are guesses that you
can turn into predictions. Now you are on the hypothesis-formulation-and-testing
path. This means you are on the path of asking yourself why you believe the predic-
tions are correct, developing rationales for the predictions, asking what kinds of
empirical observations would test your predictions, and rening your rationales and
predictions as you read the literature and talk with colleagues.
A simple diagram that summarizes the three paths we have described is shown in
Fig.2.1. Each row of arrows represents one pathway for formulating an informed
hypothesis. The dotted arrows in the rst two rows represent parts of the pathways
that a researcher may have implicitly travelled through already (without an intent to
form a prediction) but that ultimately inform the researcher’s development of a
question or prediction.
Part III.One Researcher’s Experience Launching
aScientic Inquiry
Martha was in her third year of her doctoral program and beginning to identify a
topic for her dissertation. Based on (a) her experience as a high school mathematics
teacher and a curriculum supervisor, (b) the reading she has done to this point, and
(c) her conversations with her colleagues, she has developed an interest in what
kinds of professional development experiences (let’s call them learning opportuni-
ties [LOs] for teachers) are most effective. Where does she go from here?
A natural thing for Martha to do at this point is to ask herself some additional
questions, questions that specify further what she wants to learn: What kinds of LOs
do most teachers experience? How do these experiences change teachers’ practices
and beliefs? Are some LOs more effective than others? What makes them more
effective?
Exercise 2.2
Before you continue reading, please write down some suggestions for Martha
about where she should start.
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27
To focus her questions and decide what she really wants to know, she continues
reading but now targets her reading toward everything she can nd that suggests
possible answers to these questions. She also talks with her colleagues to get more
ideas about possible answers to these or related questions. Over several weeks or
months, she nds herself being drawn to questions about what makes LOs effective,
especially for helping teachers teach more conceptually. She zeroes in on the ques-
tion, “What makes LOs for teachers effective for improving their teaching for
conceptual understanding?”
This question is more focused than her rst questions, but it is still too general
for Martha to dene a research study. How does she know it is too general? She uses
two criteria. First, she notices that the predictions she makes about the answers to
the question are all over the place; they are not constrained by the reasons she has
assembled for her predictions. One prediction is that LOs are more effective when
they help teachers learn content. Martha makes this guess because previous research
suggests that effective LOs for teachers include attention to content. But this ratio-
nale allows lots of different predictions. For example, LOs are more effective when
they focus on the content teachers will teach; LOs are more effective when they
focus on content beyond what teachers will teach so teachers see how their instruc-
tion ts with what their students will encounter later; andLOs are more effective
when they are tailored to the level of content knowledge participants have when
they begin the LOs. The rationale she can provide at this point does not point to a
particular prediction.
A second measure Martha uses to decide her question is too general is that the
predictions she can make regarding the answers seem very difcult to test. How
could she test, for example, whether LOs should focus on content beyond what
teachers will teach? What does “content beyond what teachers teach” mean? How
could you tell whether teachers use their new knowledge of later content to inform
their teaching?
Before anticipating what Martha’s next question might be, it is important to
pause and recognize how predicting the answers to her questions moved Martha
into a new phase in the research process. As she makes predictions, works out the
reasons for them, and imagines how she might test them, she is immersed in scien-
tic inquiry. This intellectual work is the main engine that drives the research pro-
cess. Also notice that revisions in the questions asked, the predictions made, and the
rationales built represent the updated thinking (Chap. 1) that occurs as Martha con-
tinues to dene her study.
Based on all these considerations and her continued reading, Martha revises the
question again. The question now reads, “Do LOs that engage middle school math-
ematics teachers in studying mathematics content help teachers teach this same con-
tent with more of a conceptual emphasis?” Although she feels like the question is
more specic, she realizes that the answer to the question is either “yes” or “no.”
This, by itself, is a red ag. Answers of “yes” or “no” would not contribute much to
understanding the relationships between these LOsfor teachers and changes in their
teaching. Recall from Chap. 1 that understanding how things work, explaining why
things work, is the goal of scientic inquiry.
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28
Martha continues by trying to understand why she believes the answer is “yes.”
When she tries to write out reasons for predicting “yes,” she realizes that her predic-
tion depends on a variety of factors. If teachers already have deep knowledge of the
content, the LOs might not affect them as much as other teachers. If the LOs do not
help teachers develop their own conceptual understanding, they are not likely to
change their teaching. By trying to build the rationale for her prediction—thus for-
mulating a hypothesis—Martha realizes that the question still is not precise and
clear enough.
Martha uses what she learned when developing the rationale and rephrases the
question as follows: “Under what conditions do LOs that engage middle school
mathematics teachers in studying mathematics content help teachers teach this same
content with more of a conceptual emphasis?” Through several additional cycles of
thinking through the rationale for her predictions and how she might test them,
Martha species her question even further: “Under what conditions do middle
school teachers who lack conceptual knowledge of linear functions benet from
LOs that engage them in conceptual learning of linear functions as assessed by
changes in their teaching toward a more conceptual emphasis on linear functions?”
Each version of Martha’s question has become more specic. This has occurred
as she has (a) identied a starting condition for the teachers—they lack conceptual
knowledge of linear functions, (b) specied the mathematics content as linear func-
tions, and (c) included a condition or purpose of the LO—it is aimed at conceptual
learning.
Because of the way Martha’s question is now phrased, her predictions will
require thinking about the conditions that could inuence what teachers learn from
the LOs and how this learning could affect their teaching. She might predict that if
teachers engaged in LOs that extended over multiple sessions, they would develop
deeper understanding which would, in turn, prompt changes in their teaching. Or
she might predict that if the LOs included examples of how their conceptual learn-
ing could translate into different instructional activities for their students, teachers
would be more likely to change their teaching. Reasons for these predictions would
likely come from research about the effects of professional development on teach-
ers’ practice.
As Martha thinks about testing her predictions, she realizes it will probably be
easier to measure the conditions under which teachers are learning than the changes
in the conceptual emphasis in their instruction. She makes a note to continue search-
ing the literature for ways to measure the “conceptualness” of teaching.
As she renes her predictions and expresses her reasons for the predictions, she
formulates a hypothesis (in this case several hypotheses) that will guide her research.
As she makes predictions and develops the rationales for these predictions, she will
probably continue revising her question. She might decide, for example, that she is
not interested in studying the condition of different numbers of LO sessions and so
decides to remove this condition from consideration by including in her question
something like “. . . over ve2-hour sessions . . .”
At this point, Martha has developed a research question, articulated a number of
predictions, and developed rationales for them. Her current question is: “Under
2 How Do YouFormulate (Important) Hypotheses?

29
what conditions do middle school teachers who lack conceptual knowledge of linear
functions benet from ve 2-hour LO sessions that engage them in conceptual
learning of linear functions as assessed by changes in their teaching toward a more
conceptual emphasis on linear functions?” Her hypothesis is:
• Prediction: Participating teachers will show changes in their teaching with a
greater emphasis on conceptual understanding, with larger changes on linear
function topics directly addressed in the LOs than on other topics.
• Brief Description of Rationale: (1) Past research has shown correlations
between teachers’ specic mathematics knowledge of a topic and the quality of
their teaching of that topic. This does not mean an increase in knowledge causes
higher quality teaching but it allows for that possibility. (2) Transfer is usually
difcult for teachers, but the examples developed during the LO sessions will
help them use what they learned to teach for conceptual understanding. This is
because the examples developed during the LO sessions are much like those that
will be used by the teachers. So larger changes will be found when teachers are
teaching the linear function topics addressed in the LOs.
Notice it is more straightforward to imagine how Martha could test this prediction
because it is more precise than previous predictions. Notice also that by asking how
to test a particular prediction, Martha will be faced with a decision about whether
testing this prediction will tell her something she wants to learn. If not, she can
return to the research question and consider how to specify it further and, perhaps,
constrain further the conditions that could affect the data.
As Martha formulates her hypotheses and goes through multiple cycles of ren-
ing her question(s), articulating her predictions, and developing her rationales, she
is constantly building the theoretical framework for her study. Because the theoreti-
cal framework is the topic for Chap. 3, we will pause here and pick up Martha’s
story in the next chapter. Spoiler alert: Martha’s experience contains some surpris-
ing twists and turns.
Before leaving Martha, however, we point out two aspects of the process in
which she has been engaged. First, it can be useful to think about the process as
identifying (1) the variables targeted in her predictions, (2) the mechanisms she
believes explain the relationships among the variables, and (3) the denitions of all
the terms that are special to her educational problem. By variables, we mean things
that can be measured and, when measured, can take on different values. In Martha’s
case, the variables are the conceptualness of teaching and the content topics
addressed in the LOs. The mechanisms are cognitive processes that enable teachers
to see the relevance of what they learn in PD to their own teaching and that enable
the transfer of learning from one setting to another. Denitions are the precise
descriptions of how the important ideas relevant to the research are conceptualized.
In Martha’s case, denitions must be provided for terms like conceptual understand-
ing, linear functions, LOs, each of the topics related to linear functions, instruc-
tional setting, and knowledge transfer.
A second aspect of the process is a practice that Martha acquired as part of her
graduate program, a practice that can go unnoticed. Martha writes out, in full
Part III.One Researcher’s Experience Launching aScientic Inquiry

30
sentences, her thinking as she wrestles with her research question, her predictions
of the answers, and the rationales for her predictions. Writing is a tool for organiz-
ing thinking and we recommend you use it throughout the scientic inquiry process.
We say more about this at the end of the chapter.
Here are the questions Martha wrote as she developed a clearer sense of what
question she wanted to answer and what answer she predicted. The list shows the
increasing renement that occurred as she continued to read, think, talk, and write.
Early questions: What kinds of LOs do most teachers experience? How do these
experiences change teachers’ practices and beliefs? Are some LOs more effective
than others? What makes them more effective?
First focused question: What makes LOs for teachers effective for improving
their teaching for conceptual understanding?
Question after trying to predict the answer and imagining how to test the predic-
tion: Do LOs that engage middle school mathematics teachers in studying mathe-
matics content help teachers teach this same content with more of a conceptual
emphasis?
Question after developing an initial rationale for her prediction: Under what con-
ditions do LOs that engage middle school mathematics teachers in studying math-
ematics content help teachers teach this same content with more of a conceptual
emphasis?
Question after developing a more precise prediction and richer rationale: Under
what conditions do middle school teachers who lack conceptual knowledge of lin-
ear functions benet from ve 2-hour LO sessions that engage them in conceptual
learning of linear functions as assessed by changes in their teaching toward a more
conceptual emphasis on linear functions?
Part IV.AnIllustrative Dialogue
The story of Martha described the major steps she took to rene her thinking.
However, there is a lot of work that went on behind the scenes that wasn’t part of the
story. For example, Martha had conversations with fellow students and professors
that sharpened her thinking. What do these conversations look like? Because they
are such an important part of the inquiry process, it will be helpful to “listen in” on
the kinds of conversations that students might have with their advisors.
Here is a dialogue between a beginning student, Sam (S), and their advisor, Dr.
Avery (A). They are meeting to discuss data Sam collected for a course project. The
dialogue below is happening very early on in Sam’s conceptualization of the study,
prior even to systematic reading of the literature.
2 How Do YouFormulate (Important) Hypotheses?

31
S: Thanks for meeting with me today.
As you know, I was able to collect some
data for a course project a few weeks
ago, but I’m having trouble analyzing
the data, so I need your help. Let me try
to explain the problem. As you know, I
wanted to understand what middle-
school teachers do to promote girls’
achievement in a mathematics class. I
conducted four observations in each of
three teachers’ classrooms. I also inter-
viewed each teacher once about the four
lessons I observed, and I interviewed
two girls from each of the teachers’
classes. Obviously, I have a ton of data.
But when I look at all these data, I don’t
really know what I learned about my
topic. When I was observing the teach-
ers, I thought I might have observed
some ways the teachers were promoting
girls’ achievement, but then I wasn’t
sure how to interpret my data. I didn’t
know if the things I was observing were
actually promoting girls’ achievement.
A: What were some of your
observations?
S: Well, in a couple of my classroom
observations, teachers called on girls to
give an answer, even when the girls
didn’t have their hands up. I thought that
this might be a way that teachers were
promoting the girls’ achievement. But
then the girls didn’t say anything about
that when I interviewed them and also
the teachers didn’t do it in every class.
So, it’s hard to know what effect, if any,
this might have had on their learning or
their motivation to learn. I didn’t want to
ask the girls during the interview spe-
cically about the teacher calling on
them, and without the girls bringing it
up themselves, I didn’t know if it had
any effect.
A: Well, why didn’t you want to ask the
girls about being called on?
S: Because I wanted to leave it as open
as possible; I didn’t want to inuence
what they were going to say. I didn’t
want to put words in their mouths. I
wanted to know what they thought the
teacher was doing that promoted their
mathematical achievement and so I only
asked the girls general questions, like
“Do you think the teacher does things to
promote girls’ mathematical achieve-
ment?” and “Can you describe specic
experiences you have had that you
believe do and do not promote your
mathematical achievement?”
A: So then, how did they answer those
general questions?
S: Well, with very general answers,
such as that the teacher knows their
names, offers review sessions, grades
their homework fairly, gives them
opportunities to earn extra credit, lets
them ask questions, and always answers
their questions. Nothing specic that
helps me know what teaching actions
specically target girls’ mathematics
achievement.
A: OK. Any ideas about what you
might do next?
S: Well, I remember that when I was
planning this data collection for my
course, you suggested I might want to
be more targeted and specic about
what I was looking for. I can see now
that more targeted questions would have
made my data more interpretable in
terms of connecting teaching actions to
the mathematical achievement of girls.
But I just didn’t want to inuence what
the girls would say.
Part IV.AnIllustrative Dialogue

32
A: Yes, I remember when you were
planning your course project, you
wanted to keep it open. You didn’t want
to miss out on discovering something
new and interesting. What do you think
now about this issue?
S: Well, I still don’t want to put words
in their mouths. I want to know what
they think. But I see that if I ask really
open questions, I have no guarantee they
will talk about what I want them to talk
about. I guess I still like the idea of an
open study, but I see that it’s a risky
approach. Leaving the questions too
open meant I didn’t constrain their
responses and there were too many ways
they could interpret and answer the
questions. And there are too many ways
I could interpret their responses.
∗∗∗∗∗∗
By this point in the dialogue, Sam has realized that open data (i.e., data not test-
ing a specic prediction) is difcult to interpret. In the next part, Dr. Avery explains
why collecting open data was not helping Sam achieve goals for her study that had
motivated collecting open data in the rst place.
******
A: Yes, I totally agree. Even for an expe-
rienced researcher, it can be difcult to
make sense of this kind of open, messy
data. However, if you design a study
with a more specic focus, you can cre-
ate questions for participants that are
more targeted because you will be inter-
ested in their answers to these specic
questions. Let’s reect back on your
data collection. What can you learn
from it for the future?
S: When I think about it now, I realize
that I didn’t think about the distinction
between all the different constructs at
play in my study, and I didn’t choose
which one I was focusing on. One con-
struct was the teaching moves that
teachers think could be promoting
achievement. Another is what teachers
deliberately do to promote girls’ mathe-
matics achievement, if anything.
Another was the teaching moves that
actually do support girls’ mathematics
achievement. Another was what teach-
ers were doing that supported girls’
mathematics achievement versus the
mathematics achievement of all stu-
dents. Another was students’ perception
of what their teacher was doing to pro-
mote girls’ mathematics achievement. I
now see that any one of these constructs
could have been the focus of a study and
that I didn’t really decide which of these
was the focus of my course project prior
to collecting data.
A: So, since you told me that the topic
of this course project is probably what
you’ll eventually want to study for your
dissertation, which of these constructs
are you most interested in?
S: I think I’m more interested in the
teacher moves that teachers deliberately
do to promote girls’ achievement. But
I’m still worried about asking teachers
directly and getting too specic about
what they do because I don’t want to
bias what they will say. And I chose
2 How Do YouFormulate (Important) Hypotheses?

33
qualitative methods and an exploratory
design because I thought it would allow
for a more open approach, an approach
that helps me see what’s going on and
that doesn’t bias or predetermine the
results.
A: Well, it seems to me you are conat-
ing three issues. One issue is how to
conduct an unbiased study. Another
issue is how specic to make your study.
And the third issue is whether or not to
choose an exploratory or qualitative
study design. Those three issues are not
the same. For example, designing a
study that’s more open or more explor-
atory is not how researchers make stud-
ies fair and unbiased. In fact, it would be
quite easy to create an open study that is
biased. For example, you could ask very
open questions and then interpret the
responses in a way that unintentionally,
and even unknowingly, aligns with what
you were hoping the ndings would say.
Actually, you could argue that by adding
more specicity and narrowing your
focus, you’re creating constraints that
prevent bias. The same goes for an
exploratory or qualitative study; they
can be biased or unbiased. So, let’s talk
about what is meant by getting more
specic. Within your new focus on what
teachers deliberately do, there are many
things that would be interesting to look
at, such as teacher moves that address
math anxiety, moves that allow girls to
answer questions more frequently,
moves that are specically tted to stu-
dent thinking about specic
mathematical content, and so on. What
are one or two things that are most inter-
esting to you? One way to answer this
question is by thinking back to where
your interest in this topic began.
******
In the preceding part of the dialogue, Dr. Avery explained how the goals Sam had
for their study were not being met with open data. In the next part, Sam begins to
articulate a prediction, which Sam and Dr. Avery then sharpen.
******
S: Actually, I became interested in this
topic because of an experience I had in
college when I was in a class of mostly
girls. During whole class discussions,
we were supposed to critically evaluate
each other’s mathematical thinking, but
we were too polite to do that. Instead,
we just praised each other’s work. But it
was so different in our small groups. It
seemed easier to critique each other’s
thinking and to push each other to better
solutions in small groups. I began won-
dering how to get girls to be more criti-
cal of each other’s thinking in a whole
class discussion in order to push every-
one’s thinking.
A: Okay, this is great information. Why
not use this idea to zoom-in on a more
manageable and interpretable study?
You could look specically at how
teachers support girls in critically evalu-
ating each other’s thinking during whole
class discussions. That would be a much
more targeted and specic topic. Do you
have predictions about what teachers
could do in that situation, keeping in
mind that you are looking specically at
girls’ mathematical achievement, not
students in general?
S: Well, what I noticed was that small
groups provided more social and emo-
Part IV.AnIllustrative Dialogue

34
tional support for girls, whereas the
whole class discussion did not provide
that same support. The girls felt more
comfortable critiquing each other’s
thinking in small groups. So, I guess I
predict that when the social and emo-
tional supports that are present in
small groups are extended to the
whole class discussion, girls would be
more willing to evaluate each other’s
mathematical thinking critically dur-
ing whole class discussion. I guess
ultimately, I’d like to know how the
whole class discussion could be used to
enhance, rather than undermine, the
social and emotional support that is
present in the small groups.
A: Okay, then where would you start?
Would you start with a study of what the
teachers say they will do during whole
class discussion and then observe if that
happens during whole class discussion?
S: But part of my prediction also
involves the small groups. So, I’d also
like to include small groups in my study
if possible. If I focus on whole groups, I
won’t be exploring what I am interested
in. My interest is broader than just the
whole class discussion.
A: That makes sense, but there are
many different things you could look at
as part of your prediction, more than
you can do in one study. For instance, if
your prediction is that when the social
and emotional supports that are pres-
ent in small groups are extended to
whole class discussions, girls would be
more willing to evaluate each other’s
mathematical thinking critically dur-
ing whole class discussions, then you
could ask the following questions: What
are the social and emotional supports
that are present in small groups?; In
which small groups do they exist?; Is it
groups that are made up only of girls?;
Does every small group do this, and for
groups that do this, when do these sup-
ports get created?; What kinds of small
group activities that teachers ask them
to work on are associated with these
supports?; Do the same social and emo-
tional supports that apply to small
groups even apply to whole group
discussion?
S: All your questions make me realize
that my prediction about extending
social and emotional supports to whole
class discussions rst requires me to
have a better understanding of the social
and emotional supports that exist in
small groups. In fact, I rst need to nd
out whether those supports commonly
exist in small groups or is that just my
experience working in small groups. So,
I think I will rst have to gure out what
small groups do to support each other
and then, in a later study, I could ask a
teacher to implement those supports
during whole class discussions and nd
out how you can do that. Yeah, now I’m
seeing that.
******
2 How Do YouFormulate (Important) Hypotheses?

35
The previous part of the dialogue illustrates how continuing to ask questions
about one’s initial prediction is a good way to make it more and more precise (and
researchable). In the next part, we see how developing a precise prediction has the
added benet of setting the researcher up for future studies.
******
A: Yes, I agree that for your rst study,
you should probably look at small
groups. In other words, you should
focus on only a part of your prediction
for now, namely the part that says there
are social and emotional supports in
small groups that support girls in cri-
tiquing each other’s thinking. That
begins to sharpen the focus of your pre-
diction, but you’ll want to continue to
rene it. For example, right now, the
question that this prediction leads to is a
question with a yes or no answer, but
what you’ve said so far suggests to me
that you are looking for more than that.
S: Yes, I want to know more than just
whether there are supports. I’d like to
know what kinds. That’s why I wanted
to do a qualitative study.
A: Okay, this aligns more with my
thinking about research as being predic-
tion driven. It’s about collecting data
that would help you revise your existing
predictions into better ones. What I
mean is that you would focus on collect-
ing data that would allow you to rene
your prediction, make it more nuanced,
and go beyond what is already known.
Does that make sense, and if so, what
would that look like for your prediction?
S: Oh yes, I like that. I guess that would
mean that, based on the data I collect for
this next study, I could develop a more
rened prediction that, for example,
more specically identies and differ-
entiates between different kinds of
social and emotional supports that are
present in small groups, or maybe that
identies the kinds of small groups that
they occur in, or that predicts when and
how frequently or infrequently they
occur, or about the features of the small
group tasks in which they occur, etc. I
now realize that, although I chose quali-
tative research to make my study be
more open, really the reason qualitative
research ts my purposes is because it
will allow me to explore ne-grained
aspects of social and emotional supports
that may exist for girls in small groups.
A: Yes, exactly! And then, based on the
data you collect, you can include in your
revised prediction those new ne-
grained aspects. Furthermore, you will
have a story to tell about your study in
your written report, namely the story
about your evolving prediction. In other
words, your written report can largely
tell how you lled out and rened your
prediction as you learned more from
carrying out the study. And even though
you might not use them right away, you
are also going to be able to develop new
predictions that you would not have
even thought of about social and emo-
tional supports in small groups and your
aim of extending them to whole-class
discussions, had you not done this study.
That will set you up to follow up on
those new predictions in future studies.
For example, you might have more
rened ideas after you collect the data
about the goals for critiquing student
thinking in small groups versus the
goals for critiquing student thinking
Part IV.AnIllustrative Dialogue

36
during whole class discussion. You
might even begin to think that some of
the social and emotional supports you
observe are not even replicable or even
applicable to or appropriate for whole-
class discussions, because the supports
play different roles in different contexts.
So, to summarize what I’m saying, what
you look at in this study, even though it
will be very focused, sets you up for a
research program that will allow you to
more fully investigate your broader
interest in this topic, where each new
study builds on your prior body of work.
That’s why it is so important to be
explicit about the best place to start this
research, so that you can build on it.
S: I see what you are saying. We started
this conversation talking about my
course project data. What I think I
should have done was gure out explic-
itly what I needed to learn with that
study with the intention of then taking
what I learned and using it as the basis
for the next study. I didn’t do that, and
so I didn’t collect data that pushed for-
ward my thinking in ways that would
guide my next study. It would be as if I
was starting over with my next study.
******
Sam and Dr. Avery have just explored how specifying a prediction reveals addi-
tional complexities that could become fodder for developing a systematic research
program. Next, we watch Sam beginning to recognize the level of specicity
required for a prediction to be testable.
******
A: One thing that would have really
helped would have been if you had had
a specic prediction going into your
data collection for your course project.
S: Well, I didn’t really have much of an
explicit prediction in mind when I
designed my methods.
A: Think back, you must have had some
kind of prediction, even if it was
implicit.
S: Well, yes, I guess I was predicting
that teachers would enact moves that
supported girls’ mathematical achieve-
ment. And I observed classrooms to
identify those teacher moves, I inter-
viewed teachers to ask them about the
moves I observed, and I interviewed stu-
dents to see if they mentioned those
moves as promoting their mathematical
achievement. The goal of my course
project was to identify teacher moves
that support girls’ mathematical
achievement. And my specic research
question was: What teacher moves sup-
port girls’ mathematical achievement?
A: So, really you were asking the
teacher and students to show and tell
you what those moves are and the effects
of those moves, as a result putting the
onus on your participants to provide the
answers to your research question for
you. I have an idea, let’s try a thought
experiment. You come up with data col-
lection methods for testing the predic-
tion that there are social and emotional
supports in small groups that support
girls in critiquing each other’s think-
ing that still puts the onus on the partici-
pants. And then I’ll see if I can think of
data collection methods that would not
put the onus on the participants.
2 How Do YouFormulate (Important) Hypotheses?

37
S: Hmm, well. .. I guess I could simply
interview girls who participated in small
groups and ask them “are there social
and emotional supports that you use in
small groups that support your group in
critiquing each other’s thinking and if
so, what are they?” In that case, I would
be putting the onus on them to be aware
of the social dynamics of small groups
and to have thought about these con-
structs as much as I have. Okay now can
you continue the thought experiment?
What might the data collection methods
look like if I didn’t put the onus on the
participants?
A: First, I would pick a setting in which
it was only girls at this point to reduce
the number of variables. Then, person-
ally I would want to observe a lot of
groups of girls interacting in groups
around tasks. I would be looking for
instances when the conversation about
students’ ideas was shut down and
instances when the conversation about
students’ ideas involved critiquing of
ideas and building on each other’s think-
ing. I would also look at what happened
just before and during those instances,
such as: did the student continue to talk
after their thinking was critiqued, did
other students do anything to encourage
the student to build on their own think-
ing (i.e., constructive criticism) or how
did they support or shut down continued
participation. In fact, now that I think
about it, “critiquing each other’s think-
ing” can be dened in a number of dif-
ferent ways. I could mean just
commenting on someone’s thinking,
judging correctness and incorrectness,
constructive criticism that moves the
thinking forward, etc. If you put the
onus on the participants to answer your
research question, you are stuck with
their denition, and they won’t have
thought about this very much, if at all.
S: I think that what you are also saying
is that my denitions would affect my
data collection. If I think that critiquing
each other’s thinking means that the
group moves their thinking forward
toward more valid and complete mathe-
matical solutions, then I’m going to
focus on different moves than if I dene
it another way, such as just making a
comment on each other’s thinking and
making each other feel comfortable
enough to keep participating. In fact, am
I going to look at individual instances of
critiquing or look at entire sequences in
which the critiquing leads to a goal?
This seems like a unit of analysis ques-
tion, and I would need to develop a more
nuanced prediction that would make
explicit what that unit of analysis is.
A: I agree, your denition of “critiquing
each other’s thinking” could entirely
change what you are predicting. One
prediction could be based on dening
critiquing as a one-shot event in which
someone makes one comment on
another person’s thinking. In this case
the prediction would be that there are
social and emotional supports in
small groups that support girls in
making an evaluative comment on
another student’s thinking. Another
prediction could be based on dening
critiquing as a back-and-forth process in
which the thinking gets built on and
rened. In that case, the prediction
would be something like that there are
social and emotional supports in
small groups that support girls in cri-
tiquing each other’s thinking in ways
that do not shut down the conversa-
tion but that lead to sustained conver-
Part IV.AnIllustrative Dialogue

38
sations that move each other toward
more valid and complete solutions.
S: Well, I think I am more interested in
the second prediction because it is more
compatible with my long-term interests,
which are that I’m interested in extend-
ing small group supports to whole class
discussions. The second prediction is
more appropriate for eventually looking
at girls in whole class discussion. During
whole class discussion, the teacher tries
to get a sustained conversation going
that moves the students’ thinking for-
ward. So, if I learn about small group
supports that lead to sustained conver-
sations that move each other toward
more valid and complete solutions,
those supports might transfer to whole
class discussions.
******
In the previous part of the dialogue, Dr. Avery and Sam showed how narrowing
down a prediction to one that is testable requires making numerous important deci-
sions, including how to dene the constructs referred to in the prediction. In the nal
part of the dialogue, Dr. Avery and Sam begin to outline the reading Sam will have
to do to develop a rationale for the specic prediction.
******
A: Do you see how your prediction and
denitions are getting more and more
specic? You now need to read exten-
sively to further rene your prediction.
S: Well, I should probably read about
micro dynamics of small group interac-
tions, anything about interactions in
small groups, and what is already known
about small group interactions that sup-
port sustained conversations that move
students’ thinking toward more valid
and complete solutions. I guess I could
also look at research on whole-class dis-
cussion methods that support sustained
conversations that move the class to
more mathematically valid and com-
plete solutions, because it might give me
ideas for what to look for in the small
groups. I might also need to focus on
research about how learners develop
understandings about a particular sub-
ject matter so that I know what “more
valid and complete solutions” look like.
I also need to read about social and
emotional supports but focus on how
they support students cognitively, rather
than in other ways.
A: Sounds good, let’s get together after
you have processed some of this litera-
ture and we can talk about rening your
prediction based on what you read and
also the methods that will best suit test-
ing that prediction.
S: Great! Thanks for meeting with me.
I feel like I have a much better set of
tools that push my own thinking for-
ward and allow me to target something
specic that will lead to more interpre-
table data.
2 How Do YouFormulate (Important) Hypotheses?

39
Part V.Is It Always Possible toFormulate Hypotheses?
In Chap. 1, we noted you are likely to read that research does not require formulat-
ing hypotheses. Some sources describe doing research without making predictions
and developing rationales for these predictions. Some researchers say you cannot
always make predictions—you do not know enough about the situation. In fact,
some argue for the value of not making predictions (e.g., Glaser & Holton, 2004;
Merton, 1968; Nemirovsky, 2011). These are important points of view, so we will
devote this section to discussing them.
Can YouAlways Predict What YouWill Find?
One reason some researchers say you do not need to make predictions is that it can
be difcult to imagine what you will nd. This argument comes up most often for
descriptive studies. Suppose you want to describe the nature of a situation you do
not know much about. Can you still make a prediction about what you will nd? We
believe that, although you do not know exactly what you will nd, you probably
have a hunch or, at a minimum, a very fuzzy idea. It would be unusual to ask a ques-
tion about a situation you want to know about without at least a fuzzy inkling of
what you might nd. The original question just would not occur to you. We acknowl-
edge you might have only a vague idea of what you will nd and you might not have
much condence in your prediction. However, we expect if you monitor your own
thinking you will discover you have developed a suspicion along the way, regardless
how vague the suspicion might be. Through the cyclic process we discussed above,
that suspicion or hunch gradually evolves and turns into a prediction.
The Benets ofMaking Predictions Even When They Are Wrong:
AnExample fromthe1970s
One of us was a graduate student at the University of Wisconsin in the late 1970s,
assigned as a research assistant to a project that was investigating young children’s
thinking about simple arithmetic. A new curriculum was being written, and the
developers wanted to know how to introduce the earliest concepts and skills to kin-
dergarten and rst-grade children. The directors of the project did not know what to
expect because, at the time, there was little research on ve- and six-year-olds’ pre-
instruction strategies for adding and subtracting.
After consulting what literature was available, talking with teachers, analyzing
the nature of different types of addition and subtraction problems, and debating with
each other, the research team formulated some hypotheses about children’s perfor-
mance. Following the usual assumptions at the time and recognizing the new
Part V.Is It Always Possible toFormulate Hypotheses?

40
curriculum would introduce the concepts, the researchers predicted that, before
instruction, most children would not be able to solve the problems. Based on the
rationale that some young children did not yet recognize the simple form for written
problems (e.g., 5+3=___), the researchers predicted that the best chance for suc-
cess would be to read problems as stories (e.g., Jesse had 5 apples and then found 3
more. How many does she have now?). They reasoned that, even though children
would have difculty on all the problems, some story problems would be easier
because the semantic structure is easier to follow. For example, they predicted the
above story about adding 3 apples to 5 would be easier than a problem like, “Jesse
had some apples in the refrigerator. She put in 2 more and now has 6. How many
were in the refrigerator at the beginning?” Based on the rationale that children
would need to count to solve the problems and that it can be difcult to keep track
of the numbers, they predicted children would be more successful if they were given
counters. Finally, accepting the common reasoning that larger numbers are more
difcult than smaller numbers, they predicted children would be more successful if
all the numbers in a problem were below 10.
Although these predictions were not very precise and the rationales were not
strongly convincing, these hypotheses prompted the researchers to design the study
to test their predictions. This meant they would collect data by presenting a variety
of problems under a variety of conditions. Because the goal was to describe chil-
dren’s thinking, problems were presented to students in individual interviews.
Problems with different semantic structures were included, counters were available
for some problems but not others, and some problems had sums to 9 whereas others
had sums to 20 or more.
The punchline of this story is that gathering data under these conditions, prompted
by the predictions, made all the difference in what the researchers learned. Contrary
to predictions, children could solve addition and subtraction problems before
instruction. Counters were important because almost all the solution strategies were
based on counting which meant that memory was an issue because many strategies
require counting in two ways simultaneously. For example, subtracting 4 from 7
was usually solved by counting down from 7 while counting up from 1 to 4 to keep
track of counting down. Because children acted out the stories with their counters,
the semantic structure of the story was also important. Stories that were easier to
read and write were also easier to solve.
To make a very long story very short, other researchers were, at about the same
time, reporting similar results about children’s pre-instruction arithmetic capabili-
ties. A clear pattern emerged regarding the relative difculty of different problem
types (semantic structures) and the strategies children used to solve each type. As
the data were replicated, the researchers recognized that kindergarten and rst-
grade teachers could make good use of this information when they introduced sim-
ple arithmetic. This is how Cognitively Guided Instruction (CGI) was born
(Carpenter etal., 1989; Fennema etal., 1996).
To reiterate, the point of this example is that the study conducted to describe
children’s thinking would have looked quite different if the researchers had made no
2 How Do YouFormulate (Important) Hypotheses?

41
predictions. They would have had no reason to choose the particular problems and
present them under different conditions. The fact that some of the predictions were
completely wrong is not the point. The predictions created the conditions under
which the predictions were tested which, in turn, created learning opportunities for
the researchers that would not have existed without the predictions. The lesson is
that even research that aims to simply describe a phenomenon can benet from
hypotheses. As signaled in Chap. 1, this also serves as another example of “failing
productively.”
Suggestions forWhat toDo When YouDo Not Have Predictions
There likely are exceptions to our claim about being able to make a prediction about
what you will nd. For example, there could be rare cases where researchers truly
have no idea what they will nd and can come up with no predictions and even no
hunches. And, no research has been reported on related phenomena that would offer
some guidance. If you nd yourself in this position, we suggest one of three
approaches: revise your question, conduct a pilot study, or choose another question.
Because there are many advantages to making predictions explicit and then writ-
ing out the reasons for these predictions, one approach is to adjust your question just
enough to allow you to make a prediction. Perhaps you can build on descriptions
that other researchers have provided for related situations and consider how you can
extend this work. Building on previous descriptions will enable you to make predic-
tions about the situation you want to describe.
A second approach is to conduct a small pilot study or, better, a series of small
pilot studies to develop some preliminary ideas of what you might nd. If you can
identify a small sample of participants who are similar to those in your study, you
can try out at least some of your research plans to help make and rene your predic-
tions. As we detail later, you can also use pilot studies to check whether key aspects
of your methods (e.g., tasks, interview questions, data collection methods) work as
you expect.
A third approach is to return to your list of interests and choose one that has been
studied previously. Sometimes this is the wisest choice. It is very difcult for begin-
ning researchers to conduct research in brand-new areas where no hunches or pre-
dictions are possible. In addition, the contributions of this research can be limited.
Recall the earlier story about one of us “failing productively” by completing a dis-
sertation in a somewhat new area. If, after an exhaustive search, you nd that no one
has investigated the phenomenon in which you are interested or even related phe-
nomena, it can be best to move in a different direction. You will read recommenda-
tions in other sources to nd a “gap” in the research and develop a study to “ll the
gap.” This can be helpful advice if the gap is very small. However, if the gap is large,
too large to predict what you might nd, the study will present severe challenges. It
will be more productive to extend work that has already been done than to launch
into an entirely new area.
Part V.Is It Always Possible toFormulate Hypotheses?

42
Should YouAlways Try toPredict What YouWill Find?
In short, our answer to the question in the heading is “yes.” But this calls for further
explanation.
Suppose you want to observe a second-grade classroom in order to investigate
how students talk about adding and subtracting whole numbers. You might think, “I
don’t want to bias my thinking; I want to be completely open to what I see in the
classroom.” Sam shared a similar point of view at the beginning of the dialogue: “I
wanted to leave it as open as possible; I didn’t want to inuence what they were
going to say.” Some researchers say that beginning your research study by making
predictions is inappropriate precisely because it will bias your observations and
results. The argument is that by bringing a set of preconceptions, you will conrm
what you expected to nd and be blind to other observations and outcomes. The
following quote illustrates this view: “The rst step in gaining theoretical sensitivity
is to enter the research setting with as few predetermined ideas as possible—espe-
cially logically deducted, a priori hypotheses. In this posture, the analyst is able to
remain sensitive to the data by being able to record events and detect happenings
without rst having them ltered through and squared with pre-existing hypotheses
and biases” (Glaser, 1978, pp.2–3).
We take a different point of view. In fact, we believe there are several compelling
reasons for making your predictions explicit.
Making Your Predictions Explicit Increases Your Chances
ofProductive Observations
Because your predictions are an extension of what is already known, they prepare
you to identify more nuanced relationships that can advance our understanding of a
phenomenon. For example, rather than simply noticing, in a general sense, that
students talking about addition and subtraction leads them to better understandings,
you might, based on your prediction, make the specic observation that talking
about addition and subtraction in a particular way helps students to think more
deeply about a particular concept related to addition and subtraction. Going into a
study without predictions can bring less sensitivity rather than more to the study of
a phenomenon. Drawing on knowledge about related phenomena by reading the
literature and conducting pilot studies allows you to be much more sensitive and
your observations to be more productive.
Making Your Predictions Explicit Allows YoutoGuard Against Biases
Some genres and methods of educational research are, in fact, rooted in philosophi-
cal traditions (e.g., Husserl, 1929/1973) that explicitly call for researchers to tempo-
rarily “bracket” or set aside existing theory as well as their prior knowledge and
2 How Do YouFormulate (Important) Hypotheses?

43
experience to better enter into the experience of the participants in the research.
However, this does not mean ignoring one’s own knowledge and experience or turn-
ing a blind eye to what has been learned by others. Much more than the simplistic
image of emptying one’s mind of preconceptions and implicit biases (arguably an
impossible feat to begin with), the goal is to be as reective as possible about one’s
prior knowledge and conceptions and as transparent as possible about how they may
guide observations and shape interpretations (Levitt etal., 2018).
We believe it is better to be honest about the predictions you are almost sure to
have because then you can deliberately plan to minimize the chances they will inu-
ence what you nd and how you interpret your results. For starters, it is important
to recognize that acknowledging you have some guesses about what you will nd
does not make them more inuential. Because you are likely to have them anyway,
we recommend being explicit about what they are. It is easier to deal with biases
that are explicit than those that lurk in the background and are not acknowledged.
What do we mean by “deal with biases”? Some journals require you to include a
statement about your “positionality” with respect to the participants in your study
and the observations you are making to gather data. Formulating clear hypotheses
is, in our view, a direct response to this request. The reasons for your predictions are
your explicit statements about your positionality. Often there are methodological
strategies you can use to protect the study from undue inuences of bias. In other
words, making your vague predictions explicit can help you design your study so
you minimize the bias of your ndings.
Making Your Predictions Explicit Can Help YouSee What YouDid
Not Predict
Making your predictions explicit does not need to blind you to what is different than
expected. It does not need to force you to see only what you want to see. Instead, it
can actually increase your sensitivity to noticing features of the situation that are
surprising, features you did not predict. Results can stand out when you did not
expect to see them.
In contrast, not bringing your biases to consciousness might subtly shift your
attention away from these unexpected results in ways that you are not aware of. This
path can lead to claiming no biases and no unexpected ndings without being con-
scious of them. You cannot observe everything, and some things inevitably will be
overlooked. If you have predicted what you will see, you can design your study so
that the unexpected results become more salient rather than less.
Returning to the example of observing a second-grade classroom, we note that
the eld already knows a great deal about how students talk about addition and
subtraction. Being cognizant of what others have observed allows you to enter the
classroom with some clear predictions about what will happen. The rationales for
these predictions are based on all the related knowledge you have before stepping
into the classroom, and the predictions and rationales help you to better deal with
Part V.Is It Always Possible toFormulate Hypotheses?

44
what you see. This is partly because you are likely to be surprised by the things you
did not anticipate. There is almost always something that will surprise you because
your predictions will almost always be incomplete or too general. This sensitivity to
the unanticipated—the sense of surprise that sparks your curiosity—is an indication
of your openness to the phenomenon you are studying.
Making Your Predictions Explicit Allows YoutoPlan inAdvance
Recall from Chap. 1 the descriptor of scientic inquiry: “Experience carefully
planned in advance.” If you make no predictions about what might happen, it is very
difcult, if not impossible, to plan your study in advance. Again, you cannot observe
everything, so you must make decisions about what you will observe. What kind of
data will you plan to collect? Why would you collect these data instead of others? If
you have no idea what to expect, on what basis will you make these consequential
decisions? Even if your predictions are vague and your rationales for the predictions
are a bit shaky, at least they provide a direction for your plan. They allow you to
explain why you are planning this study and collecting these data. They allow you
to “carefully plan in advance.”
Making Your Predictions Explicit Allows YoutoPut Your Rationales
inHarm’s Way
Rationales are developed to justify the predictions. Rationales represent your best
reasoning about the research problem you are studying. How can you tell whether
your reasoning is sound? You can try it out with colleagues. However, the best way
to test it is to put it in “harm’s way” (Cobb, Confrey, diSessa, Lehrer, & Schauble,
2003 p.10). And the best approach to putting your reasoning in harm’s way is to test
the predictions it generates. Regardless if you are conducting a qualitative or quan-
titative study, rationales can be improved only if they generate testable predictions.
This is possible only if predictions are explicit and precise. As we described earlier,
rationales are evaluated for their soundness and rened in light of the specic dif-
ferences between predictions and empirical observations.
Making Your Predictions Explicit Forces YoutoOrganize andExtend Your
(and theField’s) Thinking
By writing out your predictions (even hunches or fuzzy guesses) and by reecting
on why you have these predictions and making these reasons explicit for yourself,
you are advancing your thinking about the questions you really want to answer. This
means you are making progress toward formulating your research questions and
2 How Do YouFormulate (Important) Hypotheses?

45
your nal hypotheses. Making more progress in your own thinking before you con-
duct your study increases the chances your study will be of higher quality and will
be exactly the study you intended. Making predictions, developing rationales, and
imagining tests are tools you can use to push your thinking forward before you even
collect data.
Suppose you wonder how preservice teachers in your university’s teacher prepa-
ration program will solve particular kinds of math problems. You are interested in
this question because you have noticed several PSTs solve them in unexpected
ways. As you ask the question you want to answer, you make predictions about what
you expect to see. When you reect on why you made these predictions, you realize
that some PSTs might use particular solution strategies because they were taught to
use some of them in an earlier course, and they might believe you expect them to
solve the problems in these ways. By being explicit about why you are making par-
ticular predictions, you realize that you might be answering a different question
than you intend (“How much do PSTs remember from previous courses?” or even
“To what extent do PSTs believe different instructors have similar expectations?”).
Now you can either change your question or change the design of your study (i.e.,
the sample of students you will use) or both. You are advancing your thinking by
being explicit about your predictions and why you are making them.
The Costs ofNot Making Predictions
Avoiding making predictions, for whatever reason, comes with signicant costs. It
prevents you from learning very much about your research topic. It would require
not reading related research, not talking with your colleagues, and not conducting
pilot studies because, if you do, you are likely to nd a prediction creeping into your
thinking. Not doing these things would forego the benets of advancing your think-
ing before you collect data. It would amount to conducting the study with as little
forethought as possible.
Part VI.How Do YouFormulate Important Hypotheses?
We provided a partial answer in Chap. 1 to the question of a hypothesis’ importance
when we encouraged considering the ultimate goal to which a study’s ndings
might contribute. You might want to reread Part III of Chap. 1 where we offered our
opinions about the purposes of doing research. We also recommend reading the
March 2019 editorial in the Journal for Research in Mathematics Education (Cai
etal., 2019b) in which we address what constitutes important educational research.
As we argued in Chap. 1 and in the March 2019 editorial, a worthy ultimate goal
for educational research is to improve the learning opportunities for all students.
Part VI.How Do YouFormulate Important Hypotheses?

46
However, arguments can be made for other ultimate goals as well. To gauge the
importance of your hypotheses, think about how clearly you can connect them to a
goal the educational community considers important. In addition, given the descrip-
tors of scientic inquiry proposed in Chap. 1, think about how testing your hypoth-
eses will help you (and the community) understand what you are studying. Will you
have a better explanation for the phenomenon after your study than before?
One potentially useful way to start nding an important area of
mathematics education in which to conduct research is to consult with
teachers about a problem of practice that affects their students’ learning
opportunities. If you can connect that problem to research that helps you
develop a prediction, you may have a promising candidate for a good
research problem.
Although we address the question of importance again, and in more detail, in
Chap. 5, it is useful to know here that you can determine the signicance or impor-
tance of your hypotheses when you formulate them. The importance need not
depend on the data you collect or the results you report. The importance can come
from the fact that, based on the results of your study, you will be able to offer revised
hypotheses that help the eld better understand an important issue. In large part, it
is these revised hypotheses rather than the data that determine a study’s importance.
A critical caveat to this discussion is that few hypotheses are self-evidently
important. They are important only if you make the case for their importance. Even
if you follow closely the guidelines we suggest for formulating an important hypoth-
esis, you must develop an argument that convinces others. This argument will be
presented in the research paper you write.
Few hypotheses are self-evidently important. They are im-
portant only if you make the case for their importance.
Consider Martha’s hypothesis presented earlier. When we left Martha, she pre-
dicted that “Participating teachers will show changes in their teaching with a greater
emphasis on conceptual understanding with larger changes on linear function topics
directly addressed in the LOs than on other topics.” For researchers and educators
not intimately familiar with this area of research, it is not apparent why someone
should spend a year or more conducting a dissertation to test this prediction. Her
rationale, summarized earlier, begins to describe why this could be an important
hypothesis. But it is by writing a clear argument that explains her rationale to read-
ers that she will convince them of its importance.
2 How Do YouFormulate (Important) Hypotheses?

47
How Martha lls in her rationale so she can create a clear written argument for
its importance is taken up in Chap. 3. As we indicated, Martha’s work in this regard
led her to make some interesting decisions, in part due to her own assessment of
what was important.
Part VII.Beginning toWrite theResearch Paper
forYour Study
It is common to think that researchers conduct a study and then, after the data are
collected and analyzed, begin writing the paper about the study. We recommend an
alternative, especially for beginning researchers. We believe it is better to write
drafts of the paper at the same time you are planning and conducting your study. The
paper will gradually evolve as you work through successive phases of the scientic
inquiry process. Consequently, we will call this paper your evolving research paper.
We believe it is better to write drafts of the paper at the
same time you are planning and conducting your study.
You will use your evolving research paper to communicate your study, but you
can also use writing as a tool for thinking and organizing your thinking while plan-
ning and conducting the study. Used as a tool for thinking, you can write drafts of
your ideas to check on the clarity of your thinking, and then you can step back and
reect on how to clarify it further. Be sure to avoid jargon and general terms that are
not well dened. Ask yourself whether someone not in your eld, maybe a sibling,
a parent, or a friend, would be able to understand what you mean. You are likely to
write multiple drafts with lots of scribbling, crossing out, and revising.
Used as a tool for communicating, writing the best version of what you know
before moving to the next phase will help you record your decisions and the reasons
for them before you forget important details. This best-version-for-now paper also
provides the basis for your thinking about the next phase of your scientic inquiry.
At this point in the process, you will be writing your (research) questions, the
answers you predict, and the rationales for your predictions. The predictions you
make should be direct answers to your research questions and should ow logically
from (or be directly supported by) the rationales you present. In addition, you will
have a written statement of the study’s purpose or, said another way, an argument
for the importance of the hypotheses you will be testing. It is in the early sections of
your paper that you will convince your audience about the importance of your
hypotheses.
In our experience, presenting research questions is a more common form of stat-
ing the goal of a research study than presenting well-formulated hypotheses. Authors
Part VII.Beginning toWrite theResearch Paper forYour Study

48
sometimes present a hypothesis, often as a simple prediction of what they might
nd. The hypothesis is then forgotten and not used to guide the analysis or interpre-
tations of the ndings. In other words, authors seldom use hypotheses to do the kind
of work we describe. This means that many research articles you read will not treat
hypotheses as we suggest. We believe these are missed opportunities to present
research in a more compelling and informative way. We intend to provide enough
guidance in the remaining chapters for you to feel comfortable organizing your
evolving research paper around formulating, testing, and revising hypotheses.
While we were editing one of the leading research journals in mathematics edu-
cation (JRME), we conducted a study of reviewers’ critiques of papers submitted to
the journal. Two of the ve most common concerns were: (1) the research questions
were unclear, and (2) the answers to the questions did not make a substantial contri-
bution to the eld. These are likely to be major concerns for the reviewers of all
research journals. We hope the knowledge and skills you have acquired working
through this chapter will allow you to write the opening to your evolving research
paper in a way that addresses these concerns. Much of the chapter should help make
your research questions clear, and the prior section on formulating “important
hypotheses” will help you convey the contribution of your study.
Part VIII.TheHeart ofScientic Inquiry
In this chapter, we have described the process of formulating hypotheses. This pro-
cess is at the heart of scientic inquiry. It is where doing research begins. Conducting
research always involves formulating, testing, and revising hypotheses. This is true
regardless of your research questions and whether you are using qualitative, quanti-
tative, or mixed methods. Without engaging in this process in a deliberate, intense,
relentless way, your study will reveal less than it could. By engaging in this process,
you are maximizing what you, and others, can learn from conducting your study.
In the next chapter, we build on the ideas we have developed in the rst two
chapters to describe the purpose and nature of theoretical frameworks. The term
Exercise 2.3
Look back at your answers to the sets of questions before part II of this
chapter.
(a) Think about how you would argue for the importance of your current
interest.
(b) Write your interest in the form of (1) a research problem, (2) a research
question, and (3) a prediction with the beginnings of a rationale. You will
update these as you read the remaining chapters.
2 How Do YouFormulate (Important) Hypotheses?

49
theoretical framework, along with closely related terms like conceptual framework,
can be somewhat mysterious for beginning researchers and can seem like a requirement
for writing a paper rather than an aid for conducting research. We will show how
theoretical frameworks grow from formulating hypotheses—from developing ratio-
nales for the predicted answers to your research questions. We will propose some
practical suggestions for building theoretical frameworks and show how useful they
can be. In addition, we will continue Martha’s story from the point at which we
paused earlier—developing her theoretical framework.
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Part VIII.TheHeart ofScientic Inquir y