Sarah E. Frampton · Judah B. Axe
Accepted: 15 July 2022
© Association for Behavior Analysis International 2022

Abstract

Matrix training consists of arranging targets for instruction to promote fine-grained stimulus control resulting in the establishment of skills without direct training. Recent reviews of the matrix training literature (Curiel et al., 2020a, b.; Kemmerer et al., 2021) highlighted the efficacy and efficiency of the approach with learners with and without disabilities. These reviews noted substantial variations in procedures across studies, suggesting the approach may be flexibly deployed across content areas and teaching procedures. This outcome is positive for practitioners as they may customize matrix training to meet the unique needs of their clients. However, it also necessitates decision making to sort through the variations in the literature. This tutorial was developed to help practitioners weigh various considerations when using matrix training. Tools and resources are provided to illustrate and accelerate adoption into practice settings.

Keywords Efficiency · Matrix training · Recombinative generalization · Skill acquisition · Tutorial

Judah B. Axe
judah.axe@simmons.edu

May Institute, Inc., Randolph, MA, USA
Simmons University, Boston, MA, USA

 

Teaching generative responses under the control of the correct stimuli in a reasonable number of trials, with procedures that are relatively feasible to deploy, is no easy feat. Yet practitioners are asked to do this (and more) daily in service of their clients. One such skill is responding to two-component stimuli or instructions through tacting and listener responding (Skinner, 1957). These skills are addressed in the Verbal Behavior Milestones Assessment and Placement Program (VB-MAPP; Saaybi et al., 2019; Sundberg, 2008). In particular, milestone 9-M under tacts is: “Tacts . . . two-component verb-noun or noun-verb combinations . . . (e.g., washing face, Joe swinging, baby sleeping).” In addition, milestone 9-M under listener responding is: “Follows 50 two-component noun–verb and/ or verb–noun instructions (e.g., Show me the baby sleeping. Push the swing).” These two-component skills may be taught using any prompting/fading procedure, such as time delay and most-to-least prompting. In addition, these skills may be taught efficiently using matrix training, a method of systematically arranging learning targets to promote generative responding (Frampton et al., 2016; Goldstein, 1983).

Matrix training, which is less of a training procedure and more of a planning process, comprises several steps. First, the matrix is designed with two or more dimensions with the components of the target skills isolated on each axis (e.g., actions on one axis and objects on the other axis; Fig. 1, example 1). Within the cells of the matrix are the learning targets consisting of combinations of the components. For example, Frampton et al. (2019) designed twodimensional matrices with colors on one axis and shapes on the other axis. The cells consisted of the varying colorshape combinations (e.g., red star, blue circle, blue star, red circle). Goldstein and Mousetis (1989) created three-dimensional matrices with objects, prepositions, and locations on the axes. The cells consisted of object-preposition-location combinations (e.g., button-under-bed, penny-behindcouch). Matrices may include varying numbers of components on each axis, and additional components increase the number of combinations exponentially (e.g., 3 × 3 = 9; 4 × 4 = 16; 5 × 5 = 25).

Second, cells within the matrix are strategically selected for training. Two common approaches are nonoverlap/diagonal training and overlap/stepwise training (Fig. 2). Third, after training those cells, probes (i.e., tests) of untrained combinations may reveal that additional combinations were learned without being taught. This outcome has been described as recombinative generalization, “producing or responding to novel utterances; when familiar stimuli are recombined in novel ways” (Goldstein & Mousetis, 1989, p. 246). Fourth, effects beyond the training matrix may be observed (Kemmerer et al., 2021). That is, following training within one matrix, responses within an untrained matrix with known components may be established (e.g., Axe & Sainato, 2010; Frampton et al., 2016, 2019; Marya et al., 2021).

Fig. 1  Example matrices

Fig. 2  Matrix design and training options

The efficacy of matrix training has been demonstrated with varying types of skills and with learners with and without disabilities (see Curiel et al., 2020a, b; Kemmerer et al., 2021). This efficacy may be due to the development of strong stimulus control over unique elements of responses (Goldstein & Mousetis, 1989). Consider the example of a color–shape tact in which the desired outcome is tacting the color under the control of the color element of the stimulus and tacting the shape under the control of the shape element of the stimulus. Matrix training necessitates breaking complex responses into smaller units, consistent with Palmer’s (2012) description of atomic repertoires. The more precisely the behavioral units can be isolated, the more varied the potential combinations. Thus, matrix training requires arranging the desired elements to control responding and considering which combinations to teach and which to probe.

In practice, as time and training resources are often scarce (Cook & Odom, 2013; Odom et al., 2010), the efficiency of interventions is paramount. Matrix training may be desirable as it does not necessitate changes to standard teaching procedures. For example, for the trained cells, researchers have used least-to-most prompting (Wilson et al., 2017), time delayed most-to-least prompting (Pauwels et al., 2015), and video modeling (Kinney et al., 2003). In addition, matrix training has been deployed across curricular areas, such as following action–object instructions in a play context (Wilson et al., 2017), tacting with prepositions (Pauwels et al., 2015), and spelling (Kinney et al., 2003). This diversity of curricular areas and operants suggests flexible use in practice.

The purpose of this tutorial is to assist practitioners in using matrix training. The research-to-practice gap is a substantial barrier to improving outcomes in real world contexts (Carnine, 1997), even in scientist-practitioner models such as ABA. As efficacious procedures are identified in the literature, resources must be developed to support practitioners in customizing them to meet the needs of their clients. This tutorial was designed for master’s-level practitioners who are competent with skills described on the Behavior Analysis Certification Board (BACB) Task List, such as assessment of current skills, discrimination training, selecting interventions based on client-specific variables, and interpreting data to support decision making. We describe several considerations for using matrix training, including (1) selecting the curricular area; (2) designing matrices; (3) developing training conditions; (4) evaluating results; and (5) implementing on a wider scale. For each consideration, we provide clarifying questions designed to prompt more nuanced decision making related to the design of matrix interventions. These considerations and questions are outlined in Fig. 3.

Fig. 3  Flowchart for matrix planning

Considerations When Selecting the Curricular Area

Matrix training has been efficacious across a wide range of skills (Curiel et al., 2020a, b; Kemmerer et al., 2021). The following are four questions to consider (not necessarily in order) when selecting a curricular area to target using matrix training.

Are the Components of the Desired Response Controlled by Unique Environmental Stimuli?

Responses targeted in a matrix must consist of two or more components, each controlled by a unique environmental stimulus. For example, Curiel, Curiel, and Li (2020b) taught adults with disabilities to tact various times on a clock. When tacting time, the hour stimulus controls the hour response, and the minute stimulus controls the minute response. As a nonexample, when tacting a “firefighter,” the responses “fire” and “fighter” are not controlled by distinct elements of a firefighter. Both responses (i.e., “fire” and “fighter”) are evoked as a single unit under the control of the stimulus. Examples of combinations with unique environmental stimuli are object–action (e.g., bear jumping), color–object (e.g., red car), and preposition–object (e.g., under tree). See Fig. 1 for several example matrices.

Can (and should) the Components of the Desired Responses be Flexibly Recombined?

Consider a 3 × 3 matrix with pick up, push, and throw (actions) on one axis and ball, cup, and stapler (objects) on the other axis. Picking up and pushing those three objects poses no problems, but throwing a stapler is dangerous. All combinations in a matrix “need to work” in that sense. For another example, if targeting cutting, all objects on the other axis must be able to be cut (e.g., paper, playdoh). It is critical to ensure each component on one axis can and should interact with all the components on the other axis.

It is also critical to consider whether the target response can be emitted in a similar manner when recombined. Reading in English is a prime example (Mueller et al., 2000). Consider arranging the initial sounds for l and n on one axis and the final sounds of -ow and -et on the other axis. Recombinative generalization with “now” after training “low” is faulty because the client would incorrectly pronounce “now” as “no” (like “low”). It may be that the value of learning recombinations, even if nonsensical, outweighs the relevance to the terminal skill. For example, color and animal combinations may be taught because they are both topics of interest to the client, despite the fact that animals do not come in every shade of the rainbow.

In addition, the age of the client should be considered. For a 20-year-old in a vocational training program, the focus may be exclusively on daily meal preparation actions (e.g., pour milk/oil/batter, scoop flour/sugar/baking powder). As an alternative, for a 3-year-old in early intervention, the team may target silly object–action combinations (e.g., a dog reading, a bunny writing). After drafting a matrix, it is important to review each combination and check for internal consistency and practical relevance.

Does the Learner Demonstrate Prerequisites for the To‑Be‑Combined Skill?

When developing a matrix, consider the client’s skills within the targeted operant and the complexity of the skills to be trained and probed. For example, prior to their study, Axe and Sainato (2010) confirmed that the participants could exhibit listener responding with some pictures but not with the experimental pictures (i.e., prerequisites within the targeted operant). If a client has made limited progress within the targeted operant, there may be barriers to learning, and the skill may be best addressed using a more direct intervention. For example, ensure a client can follow one-step instructions before targeting two-step instructions, the sequence reflected in the VB-MAPP (Sundberg, 2008) and typical child development. If acquiring one-step instructions was lengthy and labor intensive, assume similar barriers when targeting combinations of these skills. We recommend taking time to address these barriers with simpler skills before targeting combinations.

We also recommend ensuring the target response can be successfully occasioned by some form of controlling prompt (Wolery et al., 1992). In essence, does the target response occur presently but under control of different antecedent variables? For example, if targeting two-word tacts, evaluate two-word echoics. If the client requires extensive shaping to emit an acceptable approximation as an echoic, assume similar barriers when targeting tacts. However, if an acceptable approximation can be occasioned as an echoic, all that must occur is transfer of stimulus control to the dictated verbal stimuli. If targeting two-step instructions, evaluate two-step imitation. These considerations are not unique to matrix training but may be of particular importance because the aim of matrix training is developing more refined stimulus control (Frampton et al., 2019). Starting with behaviors that can be reliably occasioned by controlling prompts may narrow the potential sources of error once matrix training is underway.

What Materials Will Be Used?

A final consideration in the area of content selection is related to practicality. When selecting materials for matrix training, consider the exponential increases in the number of targets with each added component. For example, Frampton et al. (2019) created three 3 × 3 matrices for a total of 27 targets. Each card had to be created, laminated for durability, strategically stored, and replaced when lost. The materials were used with six clients, making the time and resource investment worthwhile. However, those same efforts for one client may not be possible depending on the setting and the practitioner’s caseload.

Whenever possible, we recommend using existing materials. Using materials from the individual’s typical learning environment may promote generalization (Stokes & Baer, 1977) and save time on material creation. Curiel and Curiel (2021) used bills and coins to teach listener responding with sums of money. As not every dollar/coin combination was evaluated, the investigation required a simple array of four bills and four coins. Marya et al. (2021) used animal figurines and accessory items from the clinical space to illustrate object–action targets. This was likely faster than finding or creating pictures of each animal engaging in each action. Using three-dimensional stimuli for illustrating actions may also be advantageous as not all actions can be easily represented in a two-dimensional format.

Videos may be another means to effectively illustrate actions. As organization of videos is important, Kohler and Malott (2014) embedded 162 videos into PowerPoint slides for ease of locating and to preclude needing to scroll through video files on a tablet or other device. A consideration for selecting actors is that if they are known, the client may tact them by name (e.g., “Steve eats cake”). However, if another client does not know Steve, rerecording the videos with known actors may be more time efficient than teaching the new tacts of the actors. Finally, we recommend using easily adaptable materials whenever possible. Consider the ease of writing varying digital times on a whiteboard in comparison to the time needed to create the 720 hour–minute combinations on index cards or PowerPoint slides.

Considerations When Designing Matrices

Recent reviews have highlighted the diversity in matrix sizes and components across the literature (Curiel et al., 2020a, b; Kemmerer et al., 2021). Two-dimensional matrices are most common, ranging in size from 2 × 3 to 12 × 12. These ranges suggest that there are many choices and that no format or size has been established as the best practice. The following are three questions to consider when designing matrices.

How Many Dimensions Will Be Included?

The number of dimensions should be based on how many active variables one is aiming to manipulate. If a skill (e.g., agent-verb tacts) is already established at strength, continued manipulation of these combinations may not be necessary in subsequent matrices. The established components could be grouped as a single component of the matrix and manipulated with respect to a new variable of interest (e.g., adverbs). Thus, rather than creating a three-dimensional matrix (e.g., agent–verb–adverb), a two-dimensional matrix could be formed (e.g., agent + verb–adverb) with fewer permutations (see Fig. 4). This may permit closer evaluation of the most critical, new aspect of the response and keep the complexity of the matrix from accelerating too quickly. We recommend using the least needed dimensions to evaluate the development of strong stimulus control for the included components.

Fig. 4  Options for three dimensional matrices

Although designing two-dimensional matrices is fairly straightforward, three-dimensional matrices involve rapid increases in combinations with each additional component (e.g., 3 × 3 × 3 = 27; 3 × 3 × 4 = 36).

Matrices need not have the same number of components on each axis, such as a matrix with three objects, five prepositions, and six locations (Goldstein & Mousetis, 1989). Although not evaluated in research, matrices could include many (e.g., five) dimensions to capture more sophisticated skills, such as agent–verb–adverb–preposition–location (e.g., the cat walked quickly to the fence; the dog jumped eagerly on the couch). As the matrix size increases, consider the sequence of the trained responses, and check for alignment with established grammatical structures. Practitioners interested in teaching skills of this level of complexity may consider packaged interventions aimed at teaching grammatical skills in a generative manner, such as Language for Learning (Engelmann & Osborn, 2008).

How Many Components Per Dimension Will be Included?

When determining the number of components per dimension, consider whether the purpose of the intervention is content- or cusp-oriented. If the purpose is to efficiently teach content of educational or functional relevance to the client, all combinations are important to teach. For example, a client has not mastered telling time if they can respond only to quarter increments (e.g., 1:15; 2:30; 3:45). A practitioner may choose to target only certain increments to build an initial foundation of success (see Curiel & Curiel, 2021; Curiel et al., 2020b). But eventually the client should be able to respond to any combination to truly master the content.

If the purpose is to establish a behavioral cusp, it may be necessary to evaluate only a subset of components. With a cusp orientation, the taught combinations are less critical than the end goal of establishing fluent recombinative generalization. For example, Frampton et al. (2016) and Marya et al. (2021) evaluated a subset of agent–action tacts in 3 × 3 matrices. The evaluations continued until the participants could emit recombined responses from the trained matrices, as well as combined responses to targets from an untrained matrix. Performance of this nature is suggestive of an atomic repertoire (Palmer, 2012), such that any permutation of known components may be established. This approach precludes the need to teach every combination (Goldstein et al., 1987). As the client masters new components (e.g., more objects and actions), they may be recombined with all previously learned components. In this case, teaching should continue until evidence of this fluid and flexible repertoire of recombination is demonstrated across a variety of new targets.

Will the Effects of Instruction be Assessed Beyond the Training Matrix?

Several studies have assessed the effects of matrix training across additional exemplars and operants (Kemmerer et al., 2021). Researchers have used a generalization matrix in which known components are evaluated in combinations with trained or other known components (Fig. 5). For example, Axe and Sainato (2010) evaluated combinations of trained preliteracy skills and known pictures, letters, and numbers. Frampton et al. (2016, 2019) evaluated combinations of known components with other known components similar to those trained in the initial matrix. Performance on generalization matrices indicates the effects of the intervention on skills of the same operant and level of complexity without additional training. Components in generalization matrices must be known or trained.

Fig. 5 Variations of matrix designs

Additional types of generalization are possible in matrix training. After training multiple combinations, testing for stimulus generalization is important (LaFrance & Tarbox, 2020; Stokes & Baer, 1977), such as across instructors and materials (Goldstein & Mousetis, 1989) and across novel peers and settings (Hatzenbuhler et al., 2019). If training is conducted in a tightly controlled environment (i.e., structured teaching session at a table or desk), it is important to demonstrate the effects with caregivers and peers in real-world, meaningful contexts. In addition, Goldstein and Mousetis (1989) taught tacts and tested for the emergence of listener responses, and vice versa. This type of testing in the opposite modality is reasonable given findings from the bidirectional naming literature (Horne & Lowe, 1996; Miguel, 2016). Achieving these types of generalization will further enhance the efficiency of matrix training.

Considerations for Selecting Training and Testing Arrangements

We recommend conceptualizing “matrix training” as “matrix planning” because the effects are based on planning the sequence of training and probing, rather than the training procedures (e.g., time delay, most-to-least prompting). We share two considerations for deciding which targets will be trained and which will be probed.

What Variation of Matrix Training will be Used?

The decision of which variation of matrix training to use (i.e., what to train and what to probe) should be linked to the status of the component skills as known or unknown. The prevailing recommendation is that with known components, nonoverlap training (i.e., “diagonal training”) may be sufficient (Curiel et al., 2020a, b; Kemmerer et al., 2021; Pauwels et al., 2015). In nonoverlap training, the targets along the diagonal of a matrix are trained (see Fig. 2). Overall, these targets include one component on each axis trained with one component on the other axis (or axes). As each component is trained with only one target, the trained combinations do not overlap across multiple components on other axes.

Inclusion of known components may be the most conservative approach. Using known components ensures some prerequisite skills for learning the combinations. Progression from simple (i.e., component) to complex (i.e., combination) skills aligns with typical developmental pathways. For example, children may consistently tact with single words before tacting with combinations of words (Brown, 1973). The VB-MAPP (Sundberg, 2008) reflects this progression, such as actions in Listener Milestone 8 and noun–verb actions in Milestone 9.

On the other hand, when unknown components are used or mixed with known targets, overlap training may be required (Curiel et al., 2020a, b; Kemmerer et al., 2021), in which a minimum of two targets are trained per component (see Fig. 2). Trained combinations are those on the diagonal and those to the right of each diagonal combination to create a stair-step pattern (also known as “stepwise training”). As each component is taught across at least two targets, there is overlap within and across components.

Using unknown components may be more efficient than using known components. Two types of skills are taught in matrix training: (1) the components and (2) the combinations (Goldstein, 1983). For example, Axe and Sainato (2010) used unknown components, and when they taught each diagonal skill (e.g., “underline the pepper”), they essentially taught three skills: the listener response of “underline” (action), the listener response of “pepper” (object), and the combination of performing the action with the object. On the other hand, Frampton et al. (2016) used known components as the participants could tact “dog” and “jumping” but not the combination, “dog jumping.” With known components, one skill is trained: the combination. With unknown components, three (or more) skills are trained: the component from each axis and the combination.

Learning a combination may entail learning an autoclitic frame (i.e., grammatical structure, word-order rule; Skinner, 1957), as it is correct to say, “dog jumping” but not “dogging jump” or “jump dogging.” This learning is heightened with three-dimensional matrices, such as the autoclitics involved in object–preposition–location (Goldstein & Mousetis, 1989) or agent–action–object (Kohler & Malott, 2014). When the components are known, training only one cell may be needed to learn the autoclitic frame/combination (Goldstein et al., 1987). A final consideration is that although nonoverlap training with unknown components may be most efficient, training unknown components in combination may take more trials to criterion than training known components in combination or training unknown components in isolation (Bergmann et al., 2022).

How will Training be Sequenced?

Combinations may be trained all at once (i.e., simultaneously), a few at a time (i.e., sequentially), or some combination of the two. Hatzenbuhler et al. (2019) trained all four play actions simultaneously. Curiel and Curiel (2021) trained targets 1 and 2 simultaneously, then 3 and 4 simultaneously, then 1–4 in mixed training, then 5 and 6, then 1–6 in mixed training. This type of sequence was most efficacious when teaching algebra to college students (Mayfield & Chase, 2002). As an alternative, several studies trained sequentially by dividing matrices into submatrices (e.g., Axe & Sainato, 2010; Jimenez-Gomez et al., 2019). For example, Axe and Sainato (2010) taught submatrices 1 and 2 simultaneously and 3 and 4 sequentially; each submatrix entailed training and probing for recombinative generalization. Decisions related to training sequences may depend on the size of the matrix. If using a small matrix (e.g., 2 × 2, 3 × 3), simultaneous training may be optimal as it promotes conditional discriminations (Grow et al., 2011, 2014). However, with a larger matrix, simultaneous training may be cumbersome, and submatrices and/or sequential training may be indicated.

Considerations for Evaluating Results

Kemmerer et al. (2021) noted variations in which probes for untrained targets interacted with the training conditions. In some studies, probes were conducted only after completing all training conditions (i.e., a posttest; Curiel & Curiel, 2021; Frampton et al., 2016, 2019; Jimenez-Gomez et al., 2019; Marya et al., 2021; Naio et al., 2006). In other studies, probes were conducted over the course of training (e.g., Axe & Sainato, 2010; Curiel et al., 2016, 2018). Decisions of when and what to probe may be influenced by several factors, detailed in the following three considerations.

When will Probes be Conducted in Relation to Training?

Administering one posttest for a large matrix may be likened to the “train and hope” strategy Stokes and Baer (1977) cautioned against. In other words, this is using summative assessment rather than formative assessment (Fuchs et al., 1993). The number of trials needed for mastery may depend on the duration between pretest and posttest (Fuller & Fienup, 2018). Long time lapses may weaken stimulus control for the trained targets, which may undermine the success of the intervention. Long time lapses also delay remedial procedures if optimal results are not obtained. On the other hand, if there is no recombinative generalization in a small matrix or submatrix (see Fig. 5), additional targets may be trained to achieve recombinative generalization before progressing to later submatrices. In an ideal situation, less training will be required across submatrices (see Trey and Rex’s performance in Axe & Sainato, 2010, for an example) consistent with the concept of learning set (Saunders & Spradlin, 1993).

The frequency of probes may depend on the matrix size as the larger the number of components, the greater number of probed combinations. With large matrices, a random selection of untrained combinations may be probed. A sample of instances of recombinative generalization should be representative of others. Thus, targets 1B, 2C, and 3A may be probed in Session 1, and targets 1C, 2A, and 3B probed in Session 2 (see Fig. 5). This approach saves time spent on probing, while allowing an ongoing assessment of the target outcome.

What Contingencies for Responding will be Used During Probe Conditions?

Researchers have to tightly control the procedures to be able to make firm conclusions that the independent variable, and nothing else, produced the change in the dependent variable. Therefore, many matrix training studies utilized extinction during the probes (e.g., Curiel & Curiel, 2021; Curiel et al., 2020a, b; Frampton et al., 2016, 2019; Kohler & Malott, 2014; Marya et al., 2021; Solano et al., 2021). According to Stokes and Baer (1977), to claim results are attributable to generalization, one must demonstrate that, “no extratraining manipulations are needed for extratraining changes” (p. 350). Reinforcement may be considered a form of manipulation, thus applications of the training contingencies to untrained targets undermines the analysis of these responses as a product of generalization.

However, use of lean or extinction schedules of reinforcement may also weaken responding. For example, in the study by Frampton et al. (2019), two participants (George and Tony) demonstrated mastery-level recombinative generalization during initial posttraining probes. As the probes continued under extinction, rates of responding decreased, all the way to 0% in some instances. Had reinforcement been provided in the probes, these participants may not have required remedial training sessions, saving instructional time.

Fortunately, such tight control is not needed in practice, and desired behaviors should be reinforced. A consideration for selecting contingencies during probes is examining the schedules of reinforcement in the “natural environment” (Stokes & Baer, 1977). If the target skills are intended to occur in a lean-reinforcement contexts (e.g., taking a spelling test), use of extinction or a lean schedule of reinforcement may be appropriate. As an alternative, if the target skills are intended to occur in the context of an enriched play activity, asking a parent to withhold reinforcement following the first occurrence of a skill they have been working on for weeks may be unacceptable. An approach for transitioning from reinforcement (during training) to extinction (during probes) is schedule thinning (Solano et al., 2021).

How will Results be Analyzed?

Analyzing results requires defining the target behaviors, measurement systems, and visual analysis procedures (e.g., AB design, multiple baseline design). When determining parameters for mastery, decide if the purpose of the matrix is to establish content (probe all combinations) or a cusp (probe a sample of combinations). In addition, consider the likelihood of chance-responding. If evaluating an auditory–visual conditional discrimination skill with a field of three, a client may be correct by chance in one out of three opportunities. Thus, multiple evaluations may be needed to increase confidence correct selections were not due to chance. Chance responding is less of a concern with tacting.

When evaluating the effects of matrix training, it may be useful to mix probes for trained and untrained targets to permit the strongest analysis of obtained results. Figure 6 details potential results and suggested remedial or future steps. If both the trained and untrained targets occur at low levels, conditions within the probe session may be to blame (i.e., overall extinction effect, low motivation). The training could be repeated or enhanced, or the probe conditions could be modified to support performance (e.g., Frampton et al., 2019). Should the trained targets occur at strength and the untrained targets at low levels, additional targets may be trained (e.g., Pauwels et al., 2015). If correct trained and untrained responding occurs across targets in an initial matrix but not in a generalization matrix, training across matrices may be effective (e.g., Frampton et al., 2016; Marya et al., 2021). If correct responding is observed across matrices, more targets, including across operant classes, may be trained and probed (e.g., Axe & Sainato, 2010; Curiel et al., 2016; Goldstein & Mousetis, 1989).

Fig. 6  Troubleshooting steps across outcomes. Steps progress from left to right

Considerations for Implementation on a Wider Scale

Matrix training has the potential to enhance the overall efficiency of clinical programming on both a small and large scale. As matrix training lies in the planning, time and energy may be saved by pooling resources and developing shared matrix-based protocols. This consideration should not diminish the obligation to meet each client’s unique needs. Rather, dissemination on a wide scale should encourage personalization where appropriate while hardwiring the effective mechanisms, such as disseminating predeveloped matrices that require only the insertion of client-specific targets (see Fig. 5, Supplemental Materials 1 and 2). In the end, the time to switch out an object (e.g., bird) for a favorite character (e.g., Donatello) within a predeveloped matrix would require less time and effort than developing a new matrix from scratch. The following two considerations may assist in leveraging the power of shared program banks to accelerate the use of matrix training.

How will Matrix Programs be Stored and Shared?

The efficiency of matrix training may be enhanced if deployed within an organization’s shared protocol or data collection systems. In the simplest form, template data sheets (see Supplemental Material 1) may be developed and shared that connect to template matrices (Fig. 5). One could use the answers from this tutorial to generate their matrix and adjust the templates based on the client. The data sheets should be designed to highlight critical steps, such as identifying the targets to be trained and the flow of procedures (e.g., probe Matrix 1 then Generalization Matrix). Embedded prompts and cues may support the design and implementation of matrices. These data sheets may be supplemented with template protocols (see Supplemental Material) with key procedures noted and left blank to tailor to clients’ unique needs. Template protocols may reduce the time spent writing and developing documents to allow rapid transition into intervention.

Matrix training may also be hardwired into electronic program banks such as CentralReach ©. Programs can be designed to hardwire matrix elements into curricular areas that fit with matrix training (e.g., time, money, play, following instructions, tacting). Several suggestions for accomplishing this are included in Supplemental Material 2. Trained targets may be sequenced as initially on the diagonal, then with overlapping targets, and then with the remainder of the untrained targets. This can be done in CentralReach © with each target as a child branch within a matrix folder or within a task analysis with each step serving as a target. Depending on the method of target building (as child branches or a task analysis), various auto-progression features may be used to move from intervention to probes or manually move targets between phases.

How will Matrix Materials be Stored and Shared?

Within organizations, it may be useful to develop shared material banks. Videos may be created and stored across various actor–action combinations. Stimulus cards, PowerPoints ©, or Boom Cards can be created with all the needed color–shape, hour–minute, dollar–cent combinations. Depending on the organization’s size, storage may be purely electronic within a secure file-sharing system or uploaded as a resource within CentralReach ©. Transitioning to online libraries allows behavior analysts in California to benefit from the efforts of their colleagues in Georgia. Regardless of the methods used, clear organization and labeling of stimuli are necessary to ensure quick and easy location of the desired materials. In addition, sufficient exemplars should be created to support flexible programming driven by client preferences and culture. If a client’s favorite colors are teal and violet but all the color–shape stimuli are primary colors, using the available materials misses a chance to infuse a program with preferred materials. If the available holiday materials reflect only one faith tradition, practitioners must take extra steps to ensure the client’s cultural practices are represented. Creating and sharing larger libraries of materials will reduce the response effort on individual practitioners and promote more inclusive practices on a large scale. Overall, these efforts to create and organize materials will streamline “matrix planning” and matrix training.

Conclusions

Matrix training is a system of planning two-component (or more) responses to teach and probe in which probed responses may emerge based on the concept of recombinative generalization. Carefully arranging learning targets into a matrix and following the considerations we have outlined will translate into efficient instruction and learning. As illustrated throughout this tutorial, the efficacy of the matrix approach lies in the planning and design. By attending to these considerations and related questions, we hope practitioners will feel increased confidence deploying this approach to benefit their clients.

To summarize the key points, matrix training is not appropriate for all skills; rather, the skills need to be twocomponent (or more) responses where each component is controlled by a stimulus or part of a stimulus. When arranging matrices, the combinations “need to work” in that each component may be combined with all components on the other axis. As emitting two-component responses can be challenging, one-component responses should be strong in a client’s repertoire. Two guiding principles for selecting materials for matrix training are practicality and programming for stimuli in the natural environment. The number of dimensions of a matrix (e.g., two-dimensional, three-dimensional) should be based on the client’s repertoire and goals. The number of components on each axis may be based on teaching content (teach all components and combinations) or a cusp (can sample some components and combinations).

There are two main arrangements of what to teach and what to probe: nonoverlap/diagonal and overlap/stepwise. A common recommendation is that if components are known, the nonoverlap/diagonal method may be used; if the components are unknown, the overlap/stepwise method may be needed. Matrix training with unknown components may be most efficient as both the components and combinations are trained, though this is not always the case. Simultaneous training may be most efficient but also challenging for learners; sequential training, perhaps with submatrices, may improve outcomes. In addition, the components and combinations learned in one matrix may be learned in additional, generalization matrices.

We recommend probing untrained targets throughout the course of a matrix to determine if recombinative generalization is occurring or if remedial strategies, such as training additional combinations, are needed. Even though untrained responses are often probed in extinction in research, reinforcing untrained/probed responses is recommended in practice. Finally, implement matrix training on a wide scale by using template data sheets stored in electronic program banks.

In conclusion, a teaching approach that requires substantial planning time is not necessarily more efficient than other approaches. We hope the tools and resources provided in this tutorial assist practitioners in implementing matrix training effectively and efficiently.

Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/s40617–022-00733-5.

Acknowledgments We thank Dr. Bill Heward and Dr. Gretchen

Dittrich for encouraging us to write this article.

Declarations

Conflicts of Interest We have no conflicts of interest relevant to this article to disclose.

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