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Qualitative research: open-ended and closed-ended questions A set of 5 open-ended questions (one for each day of the week) applying linear functions. Similar to those open-ended questions that appear on state standardized graduation tests. Comes with answer keys and a 4-point scoring rubric that can be applied to all 5 problems. Each answer key includes 2. Math journals are a great way to have your students communicate what they know mathematically about a concept. With the open-ended questions in this resource, your students can show their thinking several different ways. Math journals are a perfect addition to your math block! There are question. ZIP ( MB) An open-ended math question, a math problem where there is more than one solution, approach, and representation, is more than reciting a fact or repeating a procedure. It requires 1st grade students to apply what they have learned while using their problem .
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Example: 4 is a factor for two different numbers. What else might be true about both numbers? Creating a sentence Students are asked to create a mathematical sentence that includes certain numbers and words.

Possible answers: 3 and 4 are more than 2 3 and 4 together are more than 6 34 and 26 are more than 34 and 20 etc. Example: Add two numbers whose sum is close to What can the numbers be? Know your mathematical focus. Develop questions with the right degree of ambiguity vague enough to be interesting and to allow for different responses, but not too vague so as students get frustrated.

Plan for two types of prompts : enabling prompts for students who seem unable to start working extension prompts for students who finish quickly High quality responses from students have the following features: Are systematic e.

If the solutions are finite, all solutions are found. If patterns can be found, then they are evident in the response.

Where a student has challenged themselves and shown complex examples which satisfy the constraints. Make connections to other content areas. Like this: Like Loading Jessica on February 15, at pm. You are welcome! Jen on February 18, at pm. I am happy to have helped even in a small way! Thank you so much Loading Incidentally, my previous blog�. Leave a Reply Cancel reply. Recent Posts Strategy vs. Tactics: Planning, Assessment and Learning 3 Strategy vs.

To do this, we explore saturation, salience, sample size, and domain size in 28 sets of interviews in which respondents were asked to list all the things they could think of in one of 18 topical domains. The domains�like kinds of fruits highly bounded and things that mothers do unbounded �varied greatly in size. The datasets comprise 20�99 interviews each 1, total interviews. Thematic saturation was, as expected, related to domain size.

It was also related to the amount of information contributed by each respondent but, unexpectedly, was reached more quickly when respondents contributed less information.

In contrast, a greater amount of information per person increased the retrieval of salient items. For most domains, item salience appeared to be a more useful concept for thinking about sample size adequacy than finding the point of thematic saturation. Thus, we advance the concept of saturation in salience and emphasize probing to increase the amount of information collected per respondent to increase sample efficiency. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All relevant data are available as an Excel file in the Supporting Information files. Content is solely the responsibility of the authors and does not necessarily represent the official views of the funding agencies. Competing interests: The authors have declared that no competing interests exist.

Open-ended questions are used alone or in combination with other interviewing techniques to explore topics in depth, to understand processes, and to identify potential causes of observed correlations. Open-ended questions may produce lists, short answers, or lengthy narratives, but in all cases, an enduring question is: How many interviews are needed to be sure that the range of salient items in the case of lists and themes in the case of narratives are covered.

Guidelines for collecting lists, short answers, and narratives often recommend continuing interviews until saturation is reached. The concept of theoretical saturation �the point where the main ideas and variations relevant to the formulation of a theory have been identified�was first articulated by Glaser and Strauss [ 1 , 2 ] in the context of how to develop grounded theory. Most of the literature on analyzing qualitative data, however, deals with observable thematic saturation �the point during a series of interviews where few or no new ideas, themes, or codes appear [ 3 � 6 ].

Since the goal of research based on qualitative data is not necessarily to collect all or most ideas and themes but to collect the most important ideas and themes, salience may provide a better guide to sample size adequacy than saturation.

Salience often called cultural or cognitive salience can be measured by the frequency of item occurrence prevalence or the order of mention [ 7 , 8 ]. These two indicators tend to be correlated [ 9 ]. In a set of lists of birds, for example, robins are reported more frequently and appear earlier in responses than are penguins. Salient terms are also more prevalent in everyday language [ 10 � 12 ]. In this article, we estimate the point of complete thematic saturation and the associated sample size and domain size for 28 sets of interviews in which respondents were asked to list all the things they could think of in one of 18 topical domains.

We also examine the impact of the amount of information produced per respondent on saturation and on the number of unique items obtained by comparing results generated by asking respondents to name all the relevant things they can with results obtained from a limited number of responses per question, as with standard open-ended questioning. Finally, we introduce an additional type of saturation based on the relative salience of items and themes� saturation in salience �and we explore whether the most salient items are captured at minimal sample sizes.

A key conclusion is that saturation may be more meaningfully and more productively conceived of as the point where the most salient ideas have been obtained.

Increasingly, researchers are applying systematic analysis and sampling theory to untangle the problems of saturation and sample size in the enormous variety of studies that rely on qualitative data�including life-histories, discourse analysis, ethnographic decision modeling, focus groups, grounded theory, and more.

For example, Guest et al. Similarly, Hagaman and Wutich [ 20 ] found that they could reliably retrieve the three most salient themes from each of the four sites in the first 16 interviews.

Galvin[ 21 ] and Fugard and Potts[ 22 ] framed the sample size problem for qualitative data in terms of the likelihood that a specific idea or theme will or will not appear in a set of interviews, given the prevalence of those ideas in the population. They used traditional statistical theory to show that small samples retrieve only the most prevalent themes and that larger samples are more sensitive and can retrieve less prevalent themes as well.

This framework can be applied to the expectation of observing or not observing almost anything. Here it would apply to the likelihood of observing a theme in a set of narrative responses, but it applies equally well for situations such as behavioral observations, where specific behaviors are being observed and sampled[ 23 ]. For example, to obtain ideas or themes that would be reported by about one out of five people 0.

Saturation and sample size have also begun to be examined with multivariate models and simulations. Tran et al. They assumed that items were independent and found that sample sizes greater than 50 would add less than one new theme per additional person interviewed.

Similarly, Lowe et al. Lowe et al. In this context, non-independence refers to the fact that some responses are much more likely than others to be repeated across people. Instead of complete saturation, they suggested using a goal such as obtaining a percentage of the total domain that one would like to capture e.

Van Rijnsoever [ 26 ] used simulated datasets to study the accumulation of themes across sample size increments and assessed the effect of different sampling strategies, item prevalence, and domain size on saturation.

As modeling estimates to date have been based on only one or two real-world examples, it is clear that more empirical examples are needed. Here, we use 28 real-world examples to estimate the impact of sample size, domain size, and amount of information per respondent on saturation and on the total number of items obtained.

Using the proportion of people in a sample that mentioned an item as a measure of salience, we find that even small samples may adequately capture the most salient items.

Data were obtained by contacting researchers who published analyses of free lists. Examples with 20 or more interviews were selected so that saturation could be examined incrementally through a range of sample sizes. Fifteen unpublished classroom educational examples were obtained on: soda pops Weller, n. Some interviews were face to face, some were written responses, and some were administered on-line.

Investigators varied in their use of prompts, using nonspecific What other � are there? Brewer [ 29 ] and Gravlee et al. The 28 examples, their topic, source, sample size, the question used in the original data collection, and the three most frequently mentioned items appear in Table 1. All data were collected and analyzed without personal identifying information. For each example, statistical models describe the pattern of obtaining new or unique items with incremental increases in sample size.

Individual lists were first analyzed with Flame [ 31 , 32 ] to provide the Questions In Linear Algebra In list of unique items for each example and the Smith [ 14 ] and Sutrop [ 15 ] item salience scores. Duplicate items due to spelling, case errors, spacing, or variations were combined. To help develop an interviewing stopping rule, a simple model was used to predict the unique number of items contributed by each additional respondent.

Generalized linear models GLM, log-linear models for count data were used to predict the unique number of items added by each respondent incrementing sample size , because number of unique items added by each respondent count data is approximately Poisson distributed.

For each example, models were fit with ordinary least squares linear regression, Poisson, and negative binomial probability distributions. Respondents were assumed to be in random order, in the order in which they occurred in each dataset, although in some cases they were in the order they were interviewed.

Goodness-of-fit was compared across the three models with minimized deviants the Akaike Information Criterion, AIC to find the best-fitting model [ 33 ]. Using the best-fitting model for each example, the point of saturation was estimated as the point where the expected number of new items was one or less.

Sample size and domain size were estimated at the point of saturation, and total domain size was estimated for an infinite sample size from the model for each example as the limit of a geometric series assuming a negative slope. Because the GLM models above used only incremental sample size to predict the total number of unique items domain size and ignored variation in the number of items provided by each person and variation in item salience, an additional analysis was used to estimate domain size while accounting for subject and item heterogeneity.

For that analysis, domain size was estimated with a capture-recapture estimation technique used for estimating the size of hidden populations. Domain size was estimated from the total number of items on individual lists and the number of matching items between pairs of lists with a log-linear analysis.

For example, population size can be estimated from the responses of two people as the product of their number of responses divided by the number of matching items assumed to be due to chance. A log-linear solution generalizes this logic from a 2 x 2 table to a 2 K table [ 34 ]. An implementation in R with GLM uses a log-linear form to estimate population size based on recapture rates Rcapture [ 35 , 36 ]. In this application, it is assumed that the population does not change between interviews closed population and models are fit with: 1 no variation across people or items M 0 ; 2 variation only across respondents M t ; 3 variation only across items M h ; and 4 variation due to an interaction between people and items M ht.

Variation among items heterogeneity is a test for a difference in the probabilities of item occurrence and, in this case, is equivalent to a test for a difference in item salience among the items. Due to the large number of combinations needed to estimate these models, Rcapture software estimates are provided for all four models only up to a sample of size For larger sample sizes all examples in this study had sample sizes of 20 or larger , only model 1 with no effects for people or items the binomial model and model 3 with item effects item salience differences were tested.

Therefore, models were fit at size 10, to test all four models and then at the total available sample size. Descriptive information for the examples appears in Table 2. The first four columns list the name of the example, the sample size in the original study, the mean list length with the range of the list length across respondents , and the total number of unique items obtained.

The free-list counts showed a characteristic descending curve where an initial person listed new themes and each additional person repeated some themes already reported and added new items, but fewer and fewer new items were added with incremental increases in sample size. All examples were fit using the GLM log-link and identity-link with normal, Poisson, and negative binomial distributions. The negative binomial model resulted in a better fit than the Poisson or identity-link models for most full-listing examples, providing the best fit to the downward sloping curve with a long tail.

Of the 28 examples, only three were not best fit by negative binomial log-link models: the best-fitting model for two examples was the Poisson log-link model GoodTeam1 and GoodTeam2Player and one was best fit by the negative binomial identity-link model CultInd1.

Sample size was a significant predictor of the number of new items for 21 of the 28 examples. Using the best-fitting GLM models we estimated the predicted sample size for reaching saturation.

Saturation was defined as the point where less than one new item would be expected for each additional person interviewed. Table 2 , column five, reports the sample size where saturation was reached N SAT.

For the Fruit domain, saturation occurred at a sample size of Saturation was reached at sample sizes of 15�, with a median sample size of Only five examples Holiday1, Fruits, Birds, Flowers, and Drugs reached saturation within the original study sample size and most examples did not reach saturation even after four or five dozen interviews.

Some domains were well bounded and were elicited with small sample sizes. Some were not. In fact, most of the distributions exhibited a very long tail�where many items were mentioned by only one or two people.

Fig 1 shows the predicted curves for all examples for sample sizes of 1 to Although the expected number of unique ideas or themes obtained for successive respondents tends to decrease as the sample size increases, this occurs rapidly in some domains and slowly or not at all in other domains.

Fruits, Holiday1, and Illness-G are domains with the three bottom-most curves and the steepest descent, indicating that saturation was reached rapidly and with small sample sizes.

The three top-most curves are the Moms-F2F, Industries1, and Industries2 domains, which reached saturation at very large sample sizes or essentially did not reach saturation. Because saturation appeared to be related to domain size and some investigators state that a percentage of the domain might be a better standard [ 25 ], domain size was also estimated.

First, total domain size was estimated with the GLM models obtained above. Thus, the model predicted that approximately 51 holidays would be obtained by the time saturation was reached. The total domain size was estimated using a geometric series, summing the estimated number of unique items obtained cumulatively across people in an infinitely large sample.

So for the Holiday1 domain, although the total domain size was estimated to be 57, the model predicted that saturation occurred when the sample size reached 17, and at that point 51 holidays should be retrieved. Model predictions were close to the empirical data, as 62 holidays were obtained with a sample of Larger sample sizes were needed to reach saturation in larger domains; the largest domains were MomsF2F, Industries1, and Industries2 each estimated to have about 1, items and more than interviews needed to approach saturation.

Second, total domain size was estimated using a capture-recapture log-linear model with a parameter for item heterogeneity [ 35 , 36 ]. Literature Circles. Microsoft OneDrive. Movie Guides. Novel Study. PowerPoint Presentations. Professional Documents. Scaffolded Notes. Science Centers. Study Guides. Task Cards. Teacher Manuals. Test Prep. Thematic Unit Plans. Unit Plans.

Video Files. Whole Courses. Word Walls. All Resource Types. Challenge your students with open-ended math questions for journals and do-nows with this yearlong bundle for first and second grade. Homework , Printables. Show more details.

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