Developmental Patterns of English Modal Verbs in the Writings of Chinese Learners of English: A Corpus-based Approach

Abstract This study aims to examine how Chinese learners develop in their use and misuse of English modal verbs from Grade 7 to 9. Specifically, it examines form-function connections and explores the factors behind learners’ development. The main focus is on the modal verbs can, could, will, would, must, and should. The learner corpus consists of essay samples selected from the writings of 600 participants, stratified across three levels (Grade 7, 8, and 9). The developmental patterns of modal verbs are assessed in terms of completeness, robustness, and proximity to the target language. The analysis of misuse looks at semantic and pragmatic errors. The findings indicate that there are considerable variations in the form-function connections of the six modal verbs at the initial and subsequent stages. The influencing factors include L1 transfer, the influence of textbooks, L2 complexity, universal learning mechanisms, and teachers’ instructions.


Introduction
Modal verbs are often challenging for those teaching and learning English as a second or foreign language (ESL/EFL) due to their formal and functional complexities (Celce-Murcia & Larsen-Freeman, 1999;Decapua, 2008;Manaf, 2007). While the formal and functional problems Lexi Xiaoduo Li ABOUT THE AUTHORS Lexi X. D. Li got her PhD in Applied English Linguistics in The Chinese University of Hong Kong. Her research interests include corpus linguistics, second/foreign language teaching and learning and early childhood education. The present paper is part of the findings of her PhD project. She is active in applied corpus linguistics research, especially corpus-beard research on passive constructions in English textbooks and learner writings.

PUBLIC INTEREST STATEMENT
This paper explores how Chinese learners develop in their use and misuse of English modal verbs from Grade 7 to 9. Modal verbs have been difficult for learners because each modal verb has more than one meaning and it is important to use different modal verbs in appropriate contexts. The examination of the developmental patterns of modal-verb use and misuse is necessary for language teachers and textbook writers to know more about at which stage which aspect(s) of modal verbs is(are) easy or difficult for learners. With this knowledge, English teachers and textbook designers can take preemptive measures to make modal verb teaching and learning more effective.
confronting ESL/EFL learners when it comes to modal verbs have been examined (Hyland & Milton, 1997;Meng & Zhang, 2010), the issues with form-function connections have been left largely untouched. Form-meaning connections (FMCs) in L2 acquisition are a major area of concern in research relating to second-language acquisition (SLA). According to VanPatten et al. (2004), "the acquisition of important subsystems in interlanguage grammars involves almost exclusively the relationship between forms, their meanings, and how the connections between the two are established" (p. 4).
English modal verbs are notoriously multifunctional. A modal verb encodes multiple semantic meanings (Palmer, 1990) and diverse pragmatic meanings (Sinclair, 1990). Therefore, modal verbs feature a complicated, non-linear mapping system between their forms and functions. To help understand why learners have difficulty with the form-function connections of modal verbs, researchers can investigate the use and misuse of the FMCs of English modal verbs by EFL/ESL learners. In addition, the developmental patterns of the use and misuse of FMCs by learners are essential for understanding interlanguage (Selinker, 1972). A better understanding of learner language will no doubt lend insight to modal verb teaching or textbook presentation.
With the development of large-scale learner corpora, SLA researchers are can observe the developmental patterns of interlanguage with a greater degree of validity and reliability. Until now, corpus-based studies (Bai, 2015;Cheng & Qiu, 2007;Jin & Yang, 2013;Meng & Zhang, 2010;Tang, 2012) of the use of English modal verbs by learners has mainly focused on the use or misuse of grammatical items. These studies have rarely addressed developmental issues. Regarding the misuse of modal verbs, research has mainly been concerned with superficial structural errors (Khojasteh & Reinders, 2013). So far, there has been no investigation of semantic and pragmatic errors made when using modal verbs. This study focuses on Chinese secondary learners. Research into the developmental patterns of the use and misuse of modal verbs by learners in mainland China has been rather scarce, especially when it comes to secondary school students.
To fill these research gaps, this paper examines how the use and misuse of the FMCs of modal verbs develop and change over time. The study asks the following questions: (1) How do the use and misuse of modal verb FMCs develop in the writings of learners from Grade 7 to Grade 9?
(2) What factors affect the developmental patterns of modal verb FMCs in the writings of learners?
Deontic modality refers to "the necessity or possibility of acts performed by morally responsible agents" (Lyons, 1977, p. 823). Palmer (1990) further explains that deontic modality is "essentially performative" (p. 69). By using a deontic modal verb, a speaker can give permission (may, can), make a promise or threat (shall), or communicate an obligation (must). For example, "Even though this is my rock, you can use it sometimes" (Palmer, 1990, p. 71). Here, can is used to give permission to the addressee, allowing them to "use it sometimes".
Dynamic modality is a category proposed by Palmer (1990) to accommodate subject-oriented modal verbs. These verbs express the ability or volition of the subject, rather than the opinions or attitudes of the speaker. For example, "They can't speak a word of English, of course, not a word, but, you know, they can say what they like" (Palmer, 1990, p. 85). Can in this sentence indicates ability and can be paraphrased as "be able to". The sentence means that they are not able to speak a word of English. Palmer (1990) also puts "neutral modality" under the category of dynamic modality. Neutral modality refers to "what is possible or necessary in the circumstances" (Palmer, 1990, p. 37). The following sentences exemplify this type of modality: "I know the place. You can get all sorts of things here" (Palmer, 1990, p. 37). The modal force of can in the second sentence does not come from the speaker or the subject; instead, it comes from the circumstances. The sentence means that it is possible for you to get all sorts of things (the place I know makes it possible). Appendix 1 provides an overview of the semantic functions of the six modal verbs (can, could, will, would, must, and should) that have been chosen for semantic analysis in this study. All the examples are taken from BNC2014.
One major drawback of Palmer's (1990) descriptive framework is the failure to describe and explain the pragmatic meanings of modal verbs. Even though Palmer (1990) discusses the pragmatic meanings of modal verbs occasionally, he does not describe all the pragmatic functions of modals. Nor does he provide a systematic explanation for the limited examples of pragmatic meanings that he gives. He does not explain how the pragmatic functions can be distinguished from the semantics of modal verbs. This paper proposes that the descriptive and explanatory weakness of Palmer's framework can be overcome by integrating Sinclair's (1990) corpus-based description of modal verb functions and  theory of indirect speech acts. Based on the COHUILD corpus, Sinclair (1990) offers a full inventory of the pragmatic functions of modal verbs, which serves as a descriptive basis for the analysis of the pragmatic functions of modal verbs in this study. Appendix 2 presents the pragmatic functions of the six modal verbs identified by Sinclair (1990). All the examples in this study are taken from Sinclair (1990, p. 511-524).
While it may be easy for native speakers of English to judge whether a modal verb performs a semantic or a pragmatic function in a given sentence, it is difficult for EFL/ESL learners to distinguish between them.  provides a potential solution to this problem in his Indirect Speech Act Theory. An indirect speech act (or primary illocutionary act) is achieved by performing a secondary illocutionary act (or direct speech act). For example, by saying "I have to wash my hair", one is performing the indirect speech act of declining an invitation by asserting that one is going to wash one's hair instead (the secondary illocutionary act).
As for how one speech act can be used to perform another, Searle (1979, p. 169) hypothesizes that it is the "mutually shared background information, both linguistic and nonlinguistic, together with the general powers of rationality and inference on the part of the hearer, that allows the speaker to communicate more than what is said". Specifically, indirect speech acts involve "a theory of speech acts" that relies on "certain general principles of cooperative conversation, and the mutually shared factual background information of the speaker and the hearer, together with an ability on the part of the hearer to make inferences" (ibid).
To illustrate how these factors affect the process of interpreting an indirect speech act, Searle (1979, pp. 34-35) devises a procedure of ten steps for deriving the primary illocutionary meaning from the literal meaning. This study condenses the ten steps into five: Step 1: Understanding the literal meaning of the sentence.
Step 2: Assuming the speaker/writer observes the principles of conversational cooperation.
Step 3: Using the theory of speech acts to predict a relevant response.
Step 4: Knowing that the literal meaning is not relevant given the context (an inference from steps 1 and 3).
Step 5: Assuming that the utterance is relevant and that the speaker/writer probably means more than what he/she says. (inference from steps 2 and 4).
This study uses the above procedure, based on , to help distinguish semantic meanings from the indirect speech-act functions of modal verbs in the data analysis. For clarity, the "indirect speech-act functions" of modal verbs are referred to as the "pragmatic functions" throughout this paper.

Form-meaning connections of modal verbs
There are three form-function possibilities (VanPatten et al., 2004, p. 3): (1) one form encodes one single meaning; (2) one form encodes more than one meaning in one context or different contexts; (3) multiple forms encode the same meaning. Modal verbs fall into the second category due to their multifunctional nature. The meaning of a modal verb is ambiguous until the context is taken into consideration. For example, the meaning of can in the sentence You can come in is "permission". However, the meaning changes in a different context, for instance, You can come in if you can get the door open. In the underlying clause, can indicates "possibility", as in "it is possible for you to come in".
To make matters even more complicated, modal verbs have pragmatic functions (i.e., as indirect speech acts) as well as semantic functions, as in the example, Can you speak louder please? Here, the meaning of can is ability (semantic function). However, the speaker uses "can" to convey "request", thereby going beyond its face-value meaning. Such a complicated, non-linear formfunction mapping system would no doubt pose great difficulties to EFL/ESL learners.
Regarding the framework for the development of FMC acquisition, VanPatten et al. (2004, p. 5) state that there are three processes or stages of acquisition: (1) making the initial connection, (2) subsequently processing the connection, and (3) accessing the connection for use.
The initial FMC can be made when a learner encounters a new form and registers the form as well as its associated meaning. The meaning may be accessed through existing knowledge or the surrounding linguistic context. According to VanPatten et al. (2004), initial FMCs can be evaluated in terms of their completeness, robustness, and proximity to target. The connection may be partial if a learner only ties part of the new form to its meaning or the form to part of its meaning. Even if an FMC is complete, it may be weak. That is, the connection may be weakened if not strengthened by subsequent input. Finally, it is possible for the initial FMC to reflect non-target-like features of a language.
Once an initial FMC is forged, there are several possibilities when it comes to subsequent processing. These depend on the extent of exposure to different forms in the initial FMC. An initially incomplete FMC may become more complete or continue to be incomplete. A weak FMC may be strengthened due to subsequent repeated exposures. Moreover, it is possible that a learner may be forced to adjust a non-target-like FMC to a target-like one if he/she subsequently gets substantial target-like form-meaning mappings in the input.
The final stage of the acquisition of FMCs, according to VanPatten et al. (2004), is the access for use, which applies to both comprehension and production. Each time a form is accessed for comprehension, the form and its associated contextual meaning become strengthened. The same holds for production. Regarding access for use, De Bot (1996) emphasized that repeated access is more beneficial for acquisition than simple encounters in the input. This paper adopts VanPatten et al.'s (2004) framework for evaluating the development of formmeaning acquisition for modal verbs in the written products of EFL students in China. As the last stage (accessing the connection for use) is a psychological process, it is not directly observable in learner performance data. Therefore, this study changed the three stages of the process into two stages: the initial stage and the subsequent stage. VanPatten et al. (2004) also provide a discussion of a range of factors that may affect FMCs in L2 acquisition, including the first language, universal processing mechanisms, L2 proficiency, input frequency, and the nature of L2 form. Regarding L1 influence, learners may apply L1 parsing procedures to L2 processing, which may result in a block for the subsequent processing of L2 forms. In addition, if the L1 encodes meanings differently from the L2 (e.g., due to a different conceptualization of democracy), learners may link L2 forms to L1 concepts. As a result, the FMC may be based more on L1 than the target language.
Universal processing mechanisms can support or prevent the establishment of FMCs. VanPatten (1996VanPatten ( , 2003 claimed that L2 learners tend to make FMCs for content words, which are more critical for comprehension. Another universal processing mechanism that may lead to non-targetlike FMCs is described by Andersen's (1984Andersen's ( , 1990 one-to-one principle, which states that one form is connected to a single meaning, at least initially. This one-to-one connection may delay or suppress further associations between the form and its additional meanings. L2 proficiency may influence a learner's ability to process grammatical and lexical forms. Learners with low proficiency levels tend to pay more attention to lexical FMCs than grammatical ones because the former are more useful for communication. Input frequency is an important factor that affects the establishment and subsequent strengthening of FMCs. Its importance for lexical learning has been recognized by N. Ellis (2002) and Horst et al. (1998), and its importance for morphology and multiword units has been recognized by N. Ellis (1996Ellis ( , 2001 and Myles et al. (1998). Nevertheless, the frequency effects may fail when learners are not ready (Gass & Mackey, 2002), when the form is not salient, or when it must be learned explicitly because of its complexity (N. Ellis, 2002).
As well as frequency, complexity and salience are two influential characteristics of L2 forms in the input. In L2 acquisition, complex structures present more difficulty than simple structures (DeKeyser, 1998;Hulstijn, 1995). Complexity may arise when (a) the form-meaning relationship is either one-to-many or many-to-one (Bensoussan & Laufer, 1984;Goldschneider & DeKeyser, 2001 (b) the FMC is not transparent (DeKeyser, 1995;Ellis & Beaton, 1993;Laufer, 1997); (c) the formmeaning relationship does not adhere to expected rules (Hulstijn & De Graaff, 1994;Goldschneider & DeKeyser, 2001 (d) the L2 form can be confused with other forms due to superficial similarities (Laufer, 1997), and (e) the L2 form is not easy to pronounce or spell (Ellis & Beaton, 1993;Laufer, 1997). It is important to note that these factors are generally researched in isolation, so it is not clear which effect is the main one, or how the effects interact (Vidal, 2003).
Although salience is an important factor, it is difficult to define it without the argument becoming circular: salient forms are noticeable, and learners tend to notice forms that are salient. Furthermore, a form may be salient when it is frequent or when it is salient only in a particular context if it encodes crucial information for communication. Salience can also be related to acoustic properties. Goldschneider and DeKeyser (2001) attempted a clearer definition of the acoustic salience of inflectional morphemes.
FMC research is important for SLA. Understanding how and why it works is crucial for understanding interlinguistic development as a whole. As well as its significance for SLA research, studying FMCs is also important for foreign language teaching and learning. First, in a foreign language learning context, inputs can be manipulated for the benefit of learners. This means that inputs, whether in the form of textbooks, syllabuses, or teachers' presentations, can be carefully designed to include more complete, target-like form-meaning mappings of grammatical features and lexical words. Given the incremental nature of the acquisition process of FMCs, those important or difficult features of languages can be made to occur more frequently in the input to strengthen the form-function connections. Second, observing the patterns of the development of FMCs in foreign language learners can provide insights into how FMCs evolve in the interlanguage in a foreign-language context. They can also show how textbook writers and teachers can adjust the design of their teaching and presentations to promote better outputs.

The acquisition of modal verb form-meaning connections by Chinese learners of english
This section reviews studies of the use of modal verbs by Chinese learners of English. Several issues are addressed, including the linguistic aspects of modal verbs (i.e., forms, functions), data collection, and the various assessments of the reasons for the patterns used by Chinese learners of English.
This review concentrates on the forms and functions of modal verbs, leaving out the syntactic features of modal verbs in the surrounding context. (For more on the syntactic use of modal verbs, see, Kang, 2018;Liang, 2008;Zhang, 2013) Some studies examine only the forms of modal verbs (Cheng & Qiu, 2007;Liu, 2016;Xiao, 2017). As far as the semantic functions of modal verbs are concerned, Tang (2012) has examined the use of deontic and epistemic modalities. Additionally, several studies have looked at the distributions of different meanings of modal verbs (Ji & Lu, 2008;Kang, 2018;Xie, 2009;Zhang, 2013). Finally, some researchers have considered the pragmatic aspects of modal verbs. For example, Meng and Zhang (2010) examined modal verbs used as boosters and hedges in discourse. Nevertheless, no study has yet investigated the use of modal verbs by learners in terms of form-function mappings and the developmental patterns of FMCs. This study will fill this gap.
There has been some progress in the research into the semantic and pragmatic functions of modal verbs. For example, Xie (2009) found that the "obligative" meaning of must occurred more frequently than the epistemic meaning. On the different meanings of must, can, and could, Zhang (2013) claimed that the meaning of must as "obligation" and the meaning of can and could as "ability" tended to be overused in learner's written work. Ji and Lu (2008) obtained similar findings on must and can. On the pragmatic functions of modal verbs, studies by Hyland and Milton (1997) and Meng and Zhang (2010) found that Chinese learners of English are very likely to overuse boosters (e.g., must and will) and underuse hedges (e.g., would and could), resulting in unintended pragmatic consequences. Unfortunately, no study has yet examined how semantic and pragmatic uses develop in learners. This study will contribute to this research area by observing how learners use the different functions of English modal verbs over time.
In terms of data collection, most studies (e.g., Cheng & Qiu, 2007;Liang, 2008;Xie, 2009) gather mixed data from learners of different proficiency levels at different times. Some studies Bai, 2015;Jin & Yang, 2013;Wang & S, 2015) are cross-sectional. For instance, Jin and Yang (2013) distinguish four levels of learners, including two levels for non-English majors and two levels for English majors. Furthermore, most studies have focused on tertiary level education. Secondary level education is rarely addressed, and even less so the younger years in secondary education (i.e., Grade 7, 8, and 9) This study will fill this gap by investigating the modal verb use of secondaryschool students in Grade 7, 8 and 9.
Finally, previous studies have claimed that there are several reasons for the use of modal verbs: L1 transfer, cultural differences, lack of L2 knowledge, and teaching materials. L1 transfer is mainly responsible for the overuse of certain modal verbs (Cheng & Qiu, 2007;Jin & Yang, 2013). A lack of L2 knowledge is considered a possible reason for the underuse of could and would (Cheng & Qiu, 2007;Jin & Yang, 2013;Liang, 2008;Liu, 2016). For instance, Liang (2008) argues that learners tend to avoid using could and would because they have not mastered how to signal hedging and convey hypothetical meanings with modal verbs. Moreover, teaching materials such as textbooks contribute to the overuse or underuse of modal verbs or modal functions (Bai, 2015;Ji & Lu, 2008;Wang & S, 2015;Zhang, 2013). These causes were identified using cross-sectional data. The potential causes for the developmental patterns of modal verb use remain unknown due to a lack of longitudinal data. The present study examines how learners use modal verbs from a developmental perspective and explores the reasons for their developmental patterns.

Corpora and data analysis
The researcher constructed the learner corpus. It was designed to be cross-sectional. The participants were secondary-school students from Guangdong province. Generally, they started learning English at the age of 6 upon entering primary school, and they received approximately two to four hours of English instruction every week in primary school. When they entered secondary school, their English instruction increased to between six and eight hours each week. They were encouraged to read English books and watch English TV shows or movies after school.
To track the developmental route of the students' modal verb use, the data were sampled from Grade 7, 8, and 9 during the first term of 2019 (a period of four months). The corpus was composed of the students' written essays. The corpus followed the design criteria for a learner corpus proposed by Granger (2002, p. 9). The control variables in this project pertained to the context in which the data were collected, the learners, and the task settings (Table 1). Regarding the learning context, English is taught as a foreign language in a classroom setting. The learners are from Grade 7 to 9 in a secondary school in Guangdong Province. They use the textbook series that is used in this study. Their mother tongue is Chinese (Mandarin or Cantonese). Their in-class essays and exam essays are timed, and reference books and dictionaries are not allowed. Their after-class essays are untimed, and reference tools can be used. The essay types are narrative or expository and cover a wide range of topics, such as "My Hero" or "Should Students Do Chores at Home?" The essays were accessed through a secondary school portal to the platform where scanned exam papers from Grade 7, Grade 8, and Grade 9 were stored. There were on average 200 students in each grade, and they took four English tests each semester. This means that there were around 800 essay samples for each grade. The researcher randomly selected 500 essays from each grade, resulting in 1500 samples in total. All the essays were typed into txt files. They were stored in three separate computer folders. The total number of tokens in the three corpora was 140,099, with 2,324 in the Grade-7 corpus, 24,336 in the Grade-8 corpus and 113,439 in the Grade-9 corpus.
To investigate the frequency effect of the input on learner use of modal verbs, this study compiled the textbook series that the participants were using at the time of the data collection into three corpora, corresponding to Grade 7, Grade 8, and Grade 9. The compilation of the textbook corpora involved three steps. First, PDF versions of the textbooks were obtained from an online textbook supplier. These PDF files were then processed with ABBYY FineReader Pro (version 12.0), an optical character recognition (OCR) software. The highlighting tool in ABBYY FineReader Pro was used to select all parts of the pages containing words, excluding pictures. The pages with words were then converted and saved as txt files, which were manually checked for misspellings and missing words. These txt files comprised the textbook corpus, containing 1,495,222 running words. The Grade-7 corpus contained 458,660 tokens; the Grade-8 corpus contained 684,378 tokens; and the Grade-9 corpus contained 352,184 tokens.
The study used WordSmith Tools (Scott, 2008, version 5.0) to analyze the data. Three analysis tools from the WordSmith Tools program were used: Concord, WordList, and Keywords. The Concord function retrieved all the concordance lines of the six modal verbs from the learner and textbook corpora for Grade 7, Grade 8, and Grade 9. All the concordances were saved in excel sheets for semantic and pragmatic analysis. After the researcher had calculated the raw and relative frequency of each function of each modal verb, the distributional profiles of the formfunction connections in the learner corpora for Grade 7, Grade 8, and Grade 9 were compared with the three corresponding textbook corpora.
To answer the two research questions, this study analyzed the distributional features of modal semantic and pragmatic functions in the learner and textbook corpora, as well as the learner errors of these functions in the three learner corpora. The semantic analysis followed the coding scheme (see Appendix 1) based on Palmer (1990). The pragmatic analysis (see Appendix 2) was based on Sinclair (1990) and . Sinclair (1990) offers an inventory of the pragmatic functions of modal verbs, and Searle (1975) provides a basis for a step-wise procedure (see, Section 2.1) to help distinguish a semantic function from a pragmatic function in a given sentence.
The starting point of the design of the coding scheme was the assumption that the primary source of the multiplicity of modal semantics was the modal verb's dependency on the meanings of the surrounding lexical resources in that context (Papafragou, 1998). Therefore, the coding scheme was characterized by the specification of contextual cues that were used to pin down each meaning of a modal verb. The cues include "the subject", "the main verb", and the "source of the modality". For example, for the "permission" meaning of can, the "subject" is usually animate under the authority of the speaker or a social law; "the main verb" denotes a physical or mental activity; and the source of the possibility of the action is the speaker's authority or the social law. In the sentence erm you can have that bag if you want (BNC2014, S23A 3930), the speaker (probably the owner of the bag) has the authority to let others have the bag. Therefore, the meaning of can in this sentence is "permission". The error analysis was based on the aforementioned coding scheme. In the process of identifying errors, the present study distinguished indeterminate cases from semantic/pragmatic errors. The cases where the meaning of a modal verb could not be determined because the context was not clear were labeled "indeterminate". The cases where the context was clear but the modal verb was inappropriately used were labeled "semantic/pragmatic error". Sentence a is a typical example of indeterminate cases, as there is no way of knowing what the writer means by "small writings" and "huge books". It is also not possible to pin down what exactly "them" and "it" refer to. In sentence b, the writer is clearly advising against using online food delivery services due to the potential risk of releasing personal information on public platforms. Therefore, can is not semantically appropriate given the context, as it conveys "permission" (i.e., they are not permitted to do), rather than "suggestion" (i.e., it is best not to do). Sentence c exemplifies the use of a modal verb that is pragmatically odd. The sentence comes from an essay that requires learners to discuss what qualities make a good teacher. The writer of the sentence intends to say that one of the qualities of a good teacher is that they love teaching. The use of must is inappropriate because it means "obliging" and, as a student, he/she is not in a position to place an obligation on a teacher.
To ensure the reliability of the data analysis of the semantic and pragmatic functions of modal verbs and learners' errors, all data were coded by the researcher and another rater, who is an English teacher with four years of teaching experience and a score of 8 on the IELTS. The researcher trained the second rater on the semantics and pragmatics of modal verbs, using the work of Palmer (1990), Searle (1975), and Sinclair (1990). The analysis of the learner data required certain linguistic knowledge of the semantics and pragmatics of Chinese modal verbs. Whenever a disagreement regarding Chinese linguistics arose, the two raters consulted another individual with a Ph.D in Chinese linguistics, who is a currently a lecturer at a Chinese university. The results of this study of semantics, pragmatics, and error analysis are based on the negotiations between the two raters and the Chinese linguistics consultant. Cohen's κ was used to determine if there was an agreement between the two raters' judgements regarding the semantic and pragmatic functions and the learner errors. Based on the interpretations of Kappa listed by Landis and Koch (1977), there was substantial agreement between the two raters' judgements, κ = .756 (95% CI, .746 to .766), p < .0005.

Results and discussion
This section presents the major findings regarding the development of modal verb FMCs in the writings of learners from Grade 7 to Grade 9. It also discusses the possible factors that influence the development. The presentation of the results and the discussion answers the two research questions raised at the beginning of this paper. Table 2 provides an overview of the form-function connections of the six modal verbs (can, could, will, would, must, and should) in the learner corpora for Grade 7, Grade 8, and Grade 9. Based on the table, this section examines which meanings the learners favor or avoid and how their preferences change as their proficiency increases from Grade 7 to Grade 9. Furthermore, the possibility that textbooks influence the choice of modal functions is investigated by comparing the distributions and patterns of change in the cross-sectional learner data for Grade 7, Grade 8, and Grade 9 (Table 3).

The development of the FMC of MODAL Semantics
Among the different meanings of can, "epistemic possibility" does not occur as a possibility in the textbooks for the three grades. The patterns of use of different functions of can in the learner corpora are similar to those in the textbook series, except for the distributional patterns in Grade 8 and Grade 9. The Grade 8 textbook gives relatively more weight to "ability" (26.8%) and less attention to "dynamic possibility" (68.9%) than the Grade 8 learner corpus does. Moreover, in Grade 9, "epistemic possibility" does not appear in the learner essays, but the function occurs in the Grade 9 textbook. The differences between the learner output and the textbook input might be due to the influence of the essay topics. The less frequent occurrence of can as "ability", and the absence of its epistemic meaning, do not necessarily point to a poor understanding of its functions. Learners may simply respond to the requirements of the essay prompts.
Regarding could, the difference between learner use and textbook presentation lies in Grade 7. In the Grade 7 textbook, there is only one token of could expressing "dynamic possibility", and the 7 th grade learners use could exclusively to convey "permission". Unfortunately, there are two few tokens of could (n = 6) in the Grade 7 learner essays to claim that the learner's have a preference for the "permission" function.
The above findings on the use of different meanings of can and could are not consistent with previous studies. Ji and Lu (2008) and Zhang (2013) found that learners tended to overuse the "ability" meaning of can and could. However, this paper has found that learners showed an overwhelming preference for the "dynamic possibility" meaning of can and the "permission" meaning of could. There may be two reasons for this inconsistency. First, it is likely that the learners in this study and the learners in Ji and Lu (2008) and Zhang (2013) were influenced by different textbooks, as the findings in this paper indicate that textbooks are a major factor when it comes to learners' usage of can, especially when it comes to the meanings related to "dynamic possibility". Second, essay topics might be another factor. That is, learners may have used can and could differently because they were responding to different essay prompts.
The distributions of the two meanings of will in the Grade 8 and Grade 9 essays were roughly similar to those in the textbooks for those year groups. However, in Grade 7, "volition" was absent from the textbook, whereas 70.5% of the examples of will used by learners were in the sense of "volition". The fact that learners picked up this meaning without being exposed to it in the input might be due to this term being used as an explicit instruction by their teachers, or they may have been exposed to it outside the curriculum.
In the Grade 7 textbook, would does not appear. Although would occurred in the writings of students in Grade 7, there were only two tokens. Therefore, the near absence of would in the learners' essays seems correlated with its absence in the teaching materials. The most noticeable difference between the textbooks and the learners' essays is that the Grade 9 textbook gives relatively equal weight to "volition" (58.8%) and "prediction" (41.2%), whereas learners tended to use "prediction" (87.1%) more often than "volition" (12.9%). This difference might be ascribed to the constraints of the essay topics, which often required students to prediction the future (e.g., What will you do in ten years?).
The semantic functions of will and would have not been researched in previous studies. Previous studies of these two modal verbs have tended to focus on their pragmatic functions. This paper is one of the first to investigate the semantic meanings of these two volitional modals. It is hoped that the present study will provide some insights into learners' use of will and would for future studies.
When learners use the modal verbs related to "obligation/necessity" (must and should), they exclusively opt for the meaning of "obligation". This is the pattern throughout the three proficiency levels. What is different about the use of must and should in the teaching materials from their usage by learners is the presence of "epistemic necessity" in Grade 8 for must (31.3%) and should (5.7%) and in Grade 9 for must (60%). Besides the influence of the essay topics, this pattern may have been influenced by the Confucian cultural ethos, which lionizes the virtue of fulfilling one's social roles (Buchtel, et al., 2015). This is reflected in the learners' use of must. In the data, the learners tended to use must to express their obligations to study hard (e.g., I must study hard and make progress every day, Learner Corpus, Grade 7, No. 14), or to help do chores at home (e.g., We must sweep the floor, Learner Corpus, Grade 9, No. 44), or to protect the environment (e.g., We must take actions to protect our planet, Learner Corpus, Grade 9, No. 55).
The findings regarding the different meanings of must and should are in line with those from previous studies. Xie (2009) andZhang (2013) also found that learners strongly favor the meaning of "obligation" when it comes to must and should. As explained previously, learners' preference for "obligation" may be due to the fact that it is a social construct of Chinese culture heavily influenced by Confucianism. Chinese learners of English were brought up with a strong obligation to fulfill their social roles as devoted sons or daughters, diligent students, responsible individuals, and more.

The development of the FMC of modal pragmatics
This section presents the results of the distributions of the pragmatic functions of the six modal verbs in the learner corpora for Grade 7, Grade 8, and Grade 9. The presentation highlights the general trends in terms of the change in the range of functions and the distributional profiles of these functions from lower to higher levels of proficiency. The pattern of the use of pragmatic functions by learners is compared with that in the textbook series. Will and must were not used by learners to perform pragmatic functions. This correlates with the fact that the pragmatic functions of these two modal verbs are rarely seen in the textbooks. There are only two tokens of will as an "appeal for help" and no tokens of must are found as a speech act. In terms of the range of functions, there is an expansion in the functions of can from Grade 7 to 8 and a decrease from Grade 8 to 9. Could and should remain the same from Grade 7 to 9. As for would, Grade 7 and Grade 8 show one function, and Grade 9 shows a second function. These patterns of change in learner essays are different from those in the textbooks, where there is a collective increase in the functional range from Grade 7 to 8, and a unanimous decline from Grade 8 to 9. This might be explained by the influence of the essay topics, which require learners to use different functions from those presented in the teaching materials.
As for the individual pragmatic functions of each modal verb, learners tended to favor one or two functions over the others (Table 4). For can, "request" and "suggestion" were much more popular than the other functions in Grade 7 ("request" = 58.5%; "suggestion" = 41.5%) and Grade 9 ("request" = 18.2%; "suggestion" = 81.2%). In Grade 8, "suggestion" was dominant (94.3%). Could performed the function of making requests and should conveyed "suggestion" throughout the three grades. Similar to could and should, would was mainly used to perform one function (i.e., expressing wishes). The only difference was that one more function (i.e., "request") appeared in Grade 9. The distributional patterns of the pragmatic use in learner essays were similar to those in the textbooks, especially for could, will, would, must, and should ( Table 5). The only exception was can. In contrast to the patterns in the learner data, "offer", "request", and "suggestion" were the main functions in the Grade 7 and Grade 8 textbooks. In Grade 9, only "request" appeared.
The above findings on the pragmatic functions of modal verbs are partially consistent with the literature. Learners in the present study seldom used the pragmatic meanings of would and could. This is consistent with the findings of Hyland and Milton (1997) and Meng and Zhang (2010). However, in the studies of Hyland and Milton (1997) and Meng and Zhang (2010), learners tended to overuse boosters such as must and will, which was not the case in the present study. This inconsistency might be due to the different frameworks used for the pragmatic analysis of modal verbs. The two previous studies focused on linguistic devices that expressed certainty (boosters) and doubt (hedges). Will and must were classified as boosters, and would and could were classified as hedges. By contrast, the present study adopted the framework of Sinclair (1990) and .

The development of learner errors of modal FMC
The error analysis identified two major sources of errors in the present learner data: interlingual and intralingual. Interlingual errors are caused by negative L1 transfer or interference. James (1998) explains that it is the co-existence of differences and similarities between a learner's mother tongue and the target language that leads to transfer errors.
A typical example of this type of error is the use of must. The similarity between must and its Chinese equivalent "必须bi xu" is that they both encode "obligation" or "necessity", whereas the difference is that the Chinese version of "must" lacks the authoritative overtone that goes with English version of "must". Failing to understand this difference can result in a pragmatic breakdown. Take the example, We can't use food delivery. We must go outside and do exercise (Learner Corpus,Grade 9,No. 56). In this example, it would be more appropriate if must were replaced with should. In Chinese, "必须" (bi xu) or "一定" (yi ding) can be used to give suggestions or make obligations, but the modal force is much weaker than must in English. The inappropriate use of must found in this study is consistent with the findings of Hinkel (1995) and Hu, Brown, and Brown (1982).
Besides being responsible for pragmatic errors, L1 interference can also cause semantic errors, as in the example Burgers and ice cream can get fat (Learner Corpus, Grade 8, No. 40). In Chinese, this sentence makes perfect sense, whereas in English, it is semantically confusing. What can and its Chinese equivalent (会hui; 能 neng) have in common is that they both convey dynamic possibility. Most of the time they are used in the same way and have no difference in meaning. For example: You can get anything you want in a supermarket.

你能在超市买到所有你想要的。
There are times when they are used in different ways. The English sentence where can is present has a different informational structure to the Chinese sentence where "会hui" is used. While Chinese is characterized by its topic-comment structure, English uses a subject-predicate structure. The sentence "Burgers and ice cream can get fat" is a typical topic-comment structure that uses an L2 form. The misuse of can is caused by its placement in the Chinese topic-comment structure makes the message semantically odd.
The second category is intralingual errors. In the learner data, incomplete rule application was the main source of errors. According to James (1998, p. 185-186), incomplete rule application refers to situations where learners are confronted by multiple rules, and they do not apply all the rules that are relevant. The basic strategy behind this is simplification. That is, to reduce the cognitive load, learners tend to apply only the rules they are familiar with and ignore the rules they are not. For example, I wish I can be my teacher (Learner Corpus,Grade 7,No. 91). The context for this sentence in the original essay is that the writer was introducing his/her favorite teacher. In the sentence, the learner was aware of the need for a modal verb before the copular verb be but he/she ignored another rule that was at play, namely that "wish" requires could instead of can. Table 6 presents the tallies of the two types errors in relation to the two different causes. It is obvious that L1 transfer was the major source of inappropriateness. L2 interference was less influential. When it comes to individual modal verbs, can seemed to cause semantic errors, resulting from both interlingual and intralingual errors. The following sentences exemplify the typical errors and their causes: (a) I wish you can write the article with me. (Learner Corpus, Grade 9, No. 8) (b) Keeping safe can make us study better. (Learner Corpus,Grade 7,No. 92) In sentence a, can should be replaced by could. The error is caused by interference from the target language, resulting in an incomplete application of L2 rules. The semantic confusion in sentence b comes from the use of an L1 topic-comment information structure expressed by L2 forms. The learner is trying to express the idea that personal safety is a condition of studying better. In English, it calls for a conditional if, as in If we are safe, we can study better.
The errors with will were all semantic in nature, resulting from L2 interference. For example, I wish I will be rich (Learner Corpus, Grade 9, No. 6). Here, will should be replaced by would or could. The error is caused by an incomplete application of rules. As for would, sometimes it was mistakenly used in some constructions such as "had better" due to L2 interference (i.e., learners confuse "had better" with "would better").

Summary of FMC Development of Learner Use and Misuse of Modal Verbs
The developmental stages of form-function connection indicate an uneven development when it comes to modal verbs. The following evaluates the developmental patterns according to the twostage framework proposed by VanPatten et al. (2004), namely, making the initial connection, processing the connection. The FMCs in the initial stage are evaluated in terms of their completeness, robustness, and proximity to the target language, and the subsequent stage is assessed based on whether the connections forged in the initial stage are weakened or strengthened by the three aspects mentioned above. It is expected that the observation and evaluation of the development of modal verb acquisition will contribute to the understanding of FMC evolution in the interlanguage. This should have implications for teaching and learning modal verbs. Table 7 summarizes the developmental paths of the six modal verbs from Grade 7 to Grade 9.
Can starts out as very multi-functional in Grade 7. More functions are added in Grade 8, but the number of functions drops in Grade 9. Could connects with one function in Grade 7, expands to two functions in Grade 8, and develops into three functions in Grade 9. Will starts out as one-to-two connections in Grade 7 and remains the same in Grades 8 and 9. Would begins with one-to-one mappings in Grade 7 and expands to one-to-two connections in Grades 8 and 9. Finally, must and should have a single function throughout the three grades. Andersen (1984, p. 79) states that the initial stage of interlanguage may be characterized by a transparent mapping of one form to one meaning. This universal learning mechanism (i.e., a oneto-one principle) can explain the one-to-one form-function connection of could, must, would, and should in the initial stage (i.e., Grade 7), but it does not shed any light on the possible reason for the multiple mappings for some modal verbs (such as can and will) at the early developmental stage. The reason for this phenomenon might be traced to input frequency effects, as the textbook series presents can and will as multi-functional in Grade 7.
Another reason may be L1 positive transfer. According to VanPatten et al. (2004, p. 11), If an L2 has a system for semantic concepts that differs from the learner's L1, then mappings might be made only between L2 forms and the meanings existing in the L1. The flipside of this claim is that if L1 and L2 share the same system for certain semantic fields, it is possible that the form-meaning connections will be established completely. This might explain why Chinese learners of English successfully establish one-to-many connections from the very beginning, especially for can and will, as Chinese and English share the same system of modal concepts (i.e., "ability", "permission", "dynamic possibility", "epistemic possibility", "volition", "prediction").
In the initial stage, the form-meaning connections for most of the modal verbs are incomplete and weak, except in the case of will. Although most of the connections are target-like, the error-free situation might only be a façade, as the proximity to the target-language features may be due to avoidance (e.g., could and would). In the subsequent stage, for most modal verbs, the connections remain incomplete and weak. Only some form-meaning connections get strengthened (e.g., the "dynamic" meaning of can and the "prediction" meaning of will). This particular picture of modal verb form-function connections in the interlanguage may be the result of multiple factors, including input frequency effects, the semantic complexity of L2 forms, L1 transfer, and universal learning mechanisms.

Conclusion and future directions
This paper has discussed how Chinese learners of English use and misuse modal verbs and how the developmental patterns of the use and misuse of modal verbs evolve from Grade 7 to Grade 9. The results reveal considerable variations in the development of the form-meaning connections of modal verbs at initial and subsequent stages. In addition, this study has found that the potential factors that influence the developmental patterns may be L1 transfer, textbook influence, L2 complexity, universal learning mechanisms, and teachers' instructions.
The findings carry important pedagogical implications. First, the insights into the use and misuse of modal verb FMCs by learners at different developmental stages may help the writers of textbooks to develop teaching materials that are developmentally appropriate for target learners. Second, a better understanding of the possible causes of the use and misuse patterns of modal verbs should enhance the effectiveness of instruction, as teachers can take preemptive measures to address potential learning difficulties.
This study contributes to the understanding of the developmental patterns of SLA. Specifically, this project advances the research into the use and misuse of the semantic and pragmatic functions of modal verbs by learners, as well as the developmental patterns of the formmeaning connections of modal verbs. Also, this paper demonstrates the importance of learner corpora in the study of interlanguage processes. Future research could investigate the developmental patterns of other grammatical structures (e.g., articles or phrasal verbs) by adopting the methodology used in this study. Moreover, it would be best if future studies could interview teachers to explore how their methods of instruction influence language learning.