Finger Patterns as a Tool for Teaching and Learning About Number Relations Exceeding 10 in the Many Hands Activity

Abstract

In this study, we investigate the learning opportunities offered in the enactment of a finger pattern activity with numbers exceeding 10 that shows how smaller units can be composed into larger units. Research on early arithmetic learning shows the importance of students understanding numbers as composed units and making use of arithmetic strategies that are based on unitizing rather than single-unit counting. The Many Hands activity was enacted in an intervention program focusing on 6-year-olds’ learning of structuring numbers and number relations during one school year, conducted in collaboration with teachers. The activity, with numbers exceeding 10, was enacted at the end of the program. Video observations of 19 teaching episodes in which the activity was used were analyzed using the variation theory of learning. The analysis focused on identifying which aspects of numbers were made visible for students to discern and how finger patterns became a tool for structuring numbers and number relations. Five aspects were made visible in the enactments of the Many Hands activity: (i) small numbers as composed units; (ii) units within units; (iii) units within units and new, larger units; (iv) relationships between units in the number system; and (v) place value. In 12 of the 19 episodes, the teacher or the students used their fingers to show and see the structure of numbers in relation to the identified aspects.

1. Introduction

Children across cultures naturally use their fingers to count objects and show their age and when performing simple calculations. The advantages of finger use in early mathematics teaching and learning have been highlighted in recent years (e.g., Björklund, 2024; Crollen et al., 2011; Poletti et al., 2022). It has been suggested that finger patterns, when used in a static way, can help in visualizing part–whole relationships in numbers—making it possible to “see” numbers within other numbers (Björklund, 2024; Björklund et al., 2019). This contrasts with the dynamic use of fingers, as seen in activities such as counting on them or using them to keep track of individual units (Carpenter & Moser, 1982), which is considered a less productive strategy and by some a dead end (Ellemor-Collins & Wright, 2009). Having a good understanding of the structure of numbers and operations in terms of part–whole relationships has been proposed as crucial for the development of arithmetic skills (Cheng, 2012; Hunting, 2003; Kullberg et al., 2024; Venkat et al., 2019). Interventions where fingers are used to support young students in structuring numbers as part–whole relationships have been shown to enhance learning outcomes compared to control groups (Kullberg et al., 2020, 2024). Activities that support young students in learning decomposition strategies for solving arithmetic problems have also been studied (e.g., Cheng, 2012; Ekdahl, 2019; Ollivier et al., 2020). However, we still know little about the role of finger patterns in such activities and what specific learning opportunities different activities afford to students.

In this study, we analyze 19 video recordings of teaching sessions from an intervention program focused on 6-year-olds’ learning to “see” numbers as structural phenomena and to make use of part–whole relationships in arithmetic problem solving. The program was implemented over the course of one school year. The Many Hands activity, based on finger patterns, was one of the activities in the program, aimed at supporting the composition of numbers and number relations beyond ten. The aim of this study is to identify and discuss the learning opportunities provided by the enacted activity and how the use of finger patterns further supported students’ understanding of numbers and number relations exceeding ten. Our research questions were as follows:

• What aspects of numbers are made visible in the enactment of the Many Hands activity?
• How do students and teachers use finger patterns to support the discernment of the identified aspects?

1.1. Finger Use in Mathematics Teaching and Learning

A growing body of empirical research has recently emphasized the potential of finger use for learning as the fingers can act as representations of numbers (Moeller et al., 2011; Poletti et al., 2025). Finger use in mathematical learning can provide assistance in understanding foundational mathematical constructs, such as one-to-one correspondence and part–whole relationships (Soylu et al., 2018). Finger patterns are useful for seeing ten as equal to two fives (the sub-base-five system) and five as a benchmark, which can support arithmetic problem solving (Neuman, 1987). Considering this relationship between 5 and 10 allows for the flexible use of numbers (e.g., a hand +2 fingers is 7) and emphasizes the whole (10). Moreover, it has been argued that finger patterns can assist in helping learners discover necessary relationships between and within numbers (Brissiaud, 1992) and also in larger number ranges exceeding 10 (Sensevy et al., 2015). Björklund et al. (2019) studied 5-year-olds’ use of finger patterns when solving an arithmetic task and showed that children used their fingers in different ways and that despite not being taught formal mathematics, they used finger patterns to represent a part–whole relationship. This means that young students perceive a finger pattern as constituting smaller units that can be decomposed and added to other finger patterns, thereby “seeing numbers within numbers” and making use of the internal structure of a pattern to figure out, for example, a missing addend.

Poletti et al. (2022) showed that kindergarteners who used their fingers to solve addition problems showed greater arithmetic performance in year 2 compared to children who did not use their fingers. In grade 2, these students did not use their fingers anymore. Hence, finger patterns serve as representations of number relations during a period when they develop an understanding of numbers. However, students who still used their fingers in grade 2 to solve arithmetic tasks showed poorer arithmetic performance. This may be due to how the students used their fingers, using finger patterns to solve tasks or counting in single units on their fingers to solve tasks.

It has been shown that a limited (4 weeks) finger training intervention can improve young students’ quantitative skills (Jay & Betenson, 2017). Jay and Betenson (2017) showed that a teacher-led or research-led intervention that combined finger training and number games was more effective in developing improvements in numeration scores compared to only teacher-led finger training or only teacher-led games. The authors suggested that young students’ development of knowledge about numbers is best supported by the “experience of a combination of representations of number, fingers plus verbal, symbolic and non-symbolic representation, than by a particular set in isolation” (p. 6). Previous studies on teaching covering a longer period have focused, for example, on the use of finger patterns in arithmetic problem solving, as a tool for structuring numbers when solving arithmetic problems in lower number ranges, e.g., 1–15 (Kullberg et al., 2020, 2024). These studies have shown that when finger patterns were used to structure number relations in learning activities about partitioning numbers and solving addition and subtraction problems during one school year, students’ learning improved more than in the control group and that the difference between the intervention and control group was also sustained one year later. It was also shown that the students in the intervention group, to a larger extent, used finger patterns to solve addition and subtraction tasks. In sum, there is support for finger use being beneficial for student learning in teaching interventions. There are, however, few studies about teaching where finger patterns are used in a number range exceeding 10, which is the focus of this study.

1.2. The Variation Theory of Learning

The variation theory of learning is used as a theoretical framework in the study presented in this paper (Marton, 2015). The theory was developed by Marton from insights gained from Phenomenographic research about learners’ ways of experiencing the same phenomena, for instance, how children experience numbers (Ahlberg, 1997; Neuman, 1987). In variation theory, learning is seen as experiencing aspects of phenomena not previously discerned, and these aspects contribute to seeing a phenomenon in a new way. Hence, some aspects are necessary to discern to master an object of learning. Finger patterns afford a sensual experience of numbers and a simultaneous experience of numbers’ part and whole relationships. For instance, it is possible to simultaneously experience the whole ten as four and six or two and eight.

As variation is seen as a prerequisite for learning, what is varied and what remains invariant are of importance. If an aspect of an object of learning is varied against an invariant background, it is more likely to be noticed. Simultaneously discerning several aspects of an object of learning, or several representations of an aspect, may affect the learners’ ways of experiencing that object of learning, compared to if aspects are handled one at a time or a single representation of that aspect was used. The analysis conducted in this study focuses on what aspects of numbers can possibly be discerned in the enactment of the activity. How students and teachers used finger patterns as representations and to support the discernment of aspects is also reported.

2. Materials and Methods

2.1. Participants and Design

During one school year, two researchers and twelve teachers from five schools in a medium-sized Swedish community collaborated on a project called SATSA. The SATSA project focused on teaching numbers with a structural approach to 6-year-olds (Swedish preschool class). The teaching activities that were planned, enacted, and reflected on with colleagues and researchers built on a structural approach to early numbers. The activities focused on cardinality, numbers as composite units (units > 1), and relationships within numbers (part-whole). These aspects of numbers had been identified in previous research (e.g., Baroody, 2016; Neuman, 1987; Polotskaia & Savard, 2018), from earlier projects conducted by our research team (e.g., Björklund et al., 2021; Ekdahl, 2019), and from individual task-based interviews conducted at the beginning of the preschool class. Finger patterns for representing numbers and number relations were emphasized in most activities in the intervention program. Also, dot patterns, bead strings, and part–part–whole diagrams (triads) were used to represent numbers and additive part–whole relationships. In the fifth theme, the numbers in the teaching activities exceeded 10. In

Table 1. The five themes in the intervention program.

Theme	Focus of Activities
1	Cardinality, five as benchmark, numbers as composite units
2	Relationships within numbers (part–whole) and between numbers
3	Relationships within numbers, five and ten as benchmarks, numbers as composite units
4	Relationships within numbers (part–whole relationships)
5	Number relations, numbers as composite units, decimal and semi-decimal structures

, a short description of the five themes in the project is presented.

The teachers and the researchers met once a month and had short follow-up digital meetings between these meetings. Five themes, including 2–4 activities, were to be enacted several times in their classes and reflected on with colleagues and researchers. The teachers video-recorded some of their teaching episodes and uploaded them to a server.

2.2. The Many Hands Activity

The Many Hands activity (in theme 5) originates from a French curriculum, called Arithmetic and Comprehension at Elementary School (ACE), in grade 1 (Sensevy et al., 2015). The purpose of the activity is to demonstrate how functional it can be to group numbers into tens. Students should gain the experience of understanding how effective the method of grouping in tens is for quickly and efficiently determining the sum of a larger number and for understanding the decimal number system, with units and tens.

The Many Hands activity was adapted to fit the Swedish context. For instance, in the French version of the Many Hands activity, the teacher displayed 24 images of finger patterns on a board, whereas in the SATSA intervention program, the number of finger patterns was lower. In the SATSA program’s Teacher Guide, the Many Hands activity is described as follows: Students are presented with a horizontal row of finger patterns (e.g., 1, 2, 3, 3, 1, 2). The teacher initiates the activity by asking how the same number of fingers can be represented using the fewest hands possible. Students are encouraged to compose and regroup the patterns into units of five. When a group of five is identified, a whole hand (finger pattern 5) is placed in a new horizontal row below, and lines are drawn between the original finger patterns and the grouped unit. This process continues until all possible groups of five are formed. In the next step, a discussion takes place on how to compose two units of five (two whole hands) into units of ten (finger pattern 10). Subsequently, the class explores how two units of five can be composed into units of ten (finger pattern 10), with lines and sometimes with symbols. The activity concludes with a discussion on the total number of fingers represented. The number and range of finger patterns may be adjusted based on students’ engagement and reasoning. Two formats of the activity are used: a teacher-led version with students sitting in front of the board (see Figure 1a, Figure 2a, Figure 3a and Figure 4a), and a pair work version (see Figure 1b, Figure 2b, Figure 3b and Figure 4b). Both formats follow the same structure.

2.3. Data

The analysis presented in this study is based on the transcriptions of 19 video-recorded episodes (in total 120 min, with the shortest lasting 1 min and 38 s and the longest spanning 14 min and 38 s). This analysis encompasses all video recordings of the Many Hands activity uploaded by eight teachers. Each recording consisted of one activity involving the hands, eight recordings of board work (labeled BW1–BW8), and eleven recordings from paired work activities (labeled PW1–PW11). Although the teaching guide suggested a total number of fingers and hands to be used, teachers selected finger patterns. As a result, the number of finger patterns varied across the recorded films (see

Table 2. The number of video recordings showing board and pair work, the total number of finger patterns used in the activities, and the number of hands used.

Number of Hands	Total Number Of Fingers	Board Work	Pair Work
5	12/13	2	1
6	15	0	2
7	16/17/21	3	1
8	20/23	3	0
9	22	0	2
10	24	0	5

).

Eight teachers and their students participated in the Many Hands activity (in groups of 2 to 6 students). In total, 40 students were involved in the video-recorded films uploaded by the teachers. The children’s guardians and participating teachers gave their written consent to participate, according to the ethical guidelines of the Swedish Research Council (2024). The children’s names were changed to pseudonyms.

The board work and pair work versions were analyzed separately. In pair work, the students were more active in composing, regrouping, exchanging finger patterns to fives or tens, and making notes, compared to those in board work. However, in board work, the students composed and exchanged finger patterns to units of fives and tens and drew lines and arrows on the board when given the opportunity.

2.4. The Process of Analysis

To reveal what aspects of numbers are made visible in the enactment of the activity, we used principles from the variation theory of learning (Marton, 2015). Following the assumptions from the theory, variation is seen as a prerequisite for learning. Furthermore, simultaneously discerning aspects of an object of learning may affect the learners’ ways of experiencing that object of learning in a more powerful way (Marton, 2015). Hence, what is possible to discern simultaneously in the teaching process has an impact on what students have the opportunity to learn. Teaching can also involve the variation that is made possible to experience within an aspect, e.g., when the teacher simultaneously shows how smaller units can be composed into larger units, which in turn can be composed into new, larger units.

In this study, we analyzed 19 teaching episodes using films and transcripts. The analysis focuses on the aspects of numbers that are possible to discern in the enactment of the Many Hands activity. In each episode, we identified which aspects of numbers were made visible and how students and teachers used finger patterns to support the discernment of these aspects. Some aspects were similar to those found in structural approaches to teaching numbers (see p. 4), while others emerged through careful video analysis. Following the principles of variation theory (Marton, 2015), for each aspect, we identified what varied and what was kept invariant. We then summarized which aspects were identified in each film and whether and how the students and teachers used finger patterns to support the discernment of the identified aspects.

3. Results

The analysis of the episodes of teaching revealed that the following aspects were made visible in the enactment: (i) small numbers as composed units; (ii) units within units; (iii) units within units and new, larger units; (iv) relationships between units in the number system; and (v) place value. In the Results Section, we first present a summary of 19 episodes of teaching, the aspects made visible in each episode, and in which episodes finger patterns (teachers’ or students’) draw attention to these aspects (see

Table 3. Aspects made visible in each episode (board work and pair work) and episodes in which teachers or students use finger patterns to draw attention to aspects.

	(i) Small Numbers as Composite Units <5	(ii) Units Within Units = 5	(iii) Units Within Units and New, Larger Units 5 ≤ 10	(iv) Relationships Between Different Units in the Number System	(v) Place Value	Teachers’/Students’ Finger Patterns, Drawing Attention to Aspects 1
Board Work
BW1	√	√	√	√	√	√
BW2	√	√	√	√	√	√
BW3	√	√	√	√
BW4	√	√	√	√
BW5	√	√	√			√
BW6	√	√	√	√		√
BW7	√	√	√	√		√
BW8	√	√	√	√		√
Pair Work
PW1	√	√				√
PW2		√	√			√
PW3	√	√	√		√
PW4	√	√	√	√		√
PW5	√	√	√	√		√
PW6	√	√	√	√		√
PW7	√	√	√	√
PW8	√	√	√	√
PW9		√	√			√
PW10	√	√	√	√
PW11	√	√	√	√

¹ We marked episodes where finger patterns were used by teachers or students, without considering the number of times they were used or their relationship to specific aspects.

). Thereafter, the enactment of the Many Hands activity is presented, step by step.

The aspects made visible across the 19 teaching episodes were largely consistent. No major differences were identified between the two formats of the activity, board work and pair work. However, the aspect of place value was notably different from the others and occurred less frequently. In 12 of the 19 episodes, finger patterns were used by the teacher or the students to draw attention to one or more of the identified aspects.

Figure 1, Figure 2, Figure 3 and Figure 4 represent instances of board work (BW3) and pair work (PW6). These figures serve to illustrate the enactment of the activity. Additional excerpts from various episodes are used to elucidate how the aspects of numbers are made visible in teaching when the number range exceeds 10. Furthermore, these excerpts illustrate how finger patterns can become a tool for structuring numbers and number relations.

3.1. Small Numbers as Composed Units

At the starting point of the Many Hands activity, the teacher presents several different finger patterns < 5. Figure 1 shows an example of board work where 8 finger patterns represent a total number of 20 fingers (Figure 1a) and in pair work, 10 finger patterns represent a total number of 24 fingers (Figure 1b).

Figure 1. (a,b) The starting point of the Many Hands activity.

Figure 1. (a,b) The starting point of the Many Hands activity.

The starting point of the Many Hands activity, which presents several different finger patterns (<5), supports students in identifying the number of fingers (cardinality). This made it possible for students to discern the aspect, small numbers as composite units, and how the same small number can be represented using different finger patterns and hands.

For instance, the composite number 2 can be represented by the ring finger and little finger on the right hand or by the thumb and index finger on the left hand (see the second and fourth finger patterns in Figure 1b). Most often, the teacher, after having asked “How can we show the same number of fingers with as few hands as possible”, pointed to one finger pattern at a time and asked the students to say how many they saw. In 3 (of 19) episodes, the teacher at the same time wrote numerals below the row of finger patterns. Irrespective of the number of finger patterns (<5) announced, the students could discern small numbers (<5) as composed units without having to count each finger as a single unit. Hence, the activity started with something that all students were familiar with (finger patterns within five) from previous activities in the intervention program.

In some episodes, finger patterns were used to support the discernment of the aspect small numbers as composite units. In episodes BW5 and BW8, the students were asked to observe the presented finger patterns and simultaneously show each finger pattern with their fingers. The teachers focused on the number of fingers (cardinality), not the exact finger patterns in the images. Consequently, the students were able to represent the numbers represented with finger patterns in different ways, compared to the images, which made it possible to discern that the same number can be represented with different finger patterns on one hand.

3.2. Units Within Units

The knowledge of smaller numbers as composite units supported learners in reasoning about strategies for reducing the number of hands and finding units of fives (whole hands) and identifying units within the unit. When finger patterns were sequentially presented in some episodes (e.g., 4, 1, 2, 3, 3), it became less challenging for students to compose groups of five. However, the introduction of additional finger patterns and an expanded number range facilitated the exploration of different solutions and fostered more extensive discussions. Figure 2a,b illustrate step 2, the process of grouping in fives (same episodes as those in Figure 1a, board work, and Figure 1b, pair work).

Figure 2. (a,b) Step 2 in the Many Hands activity.

Figure 2. (a,b) Step 2 in the Many Hands activity.

The students proposed various methods for composing the announced numbers, occasionally rearranging the order of the finger patterns. The aspect units within units was made visible when the teacher asked the learner to either compose two finger patterns for one unit of five or find the missing finger pattern when five (whole hand) and one number were known. The teacher or the students moved the two identified finger patterns to a new row, exchanged them with a whole hand (unit of five), and drew lines between the two parts and the five (extra images of finger pattern 5 were used, as shown in Figure 2b). In Figure 2a and Figure 2b, respectively, two units of five are shown that were composed during board work (2 and 3; 3 and 2) and pair work (4 and 1; 3 and 2). Small numbers as composed units and different units within the same unit (five) were made visible simultaneously. It was only in episode BW8 that a student suggested that three finger patterns (2, 1, 2) could be composed into a 5-unit. In episode BW1, the discussion of finding fives and exchanging two smaller units to one whole hand was enacted as follows:

T: How could we do that with fewer hands (points to: 1, 4, 1, 2, 3, 3, on the top of the board)?

No responses

T: How many fingers do we have on one hand (holds up her hand)?

Eve: Five

T. So, the maximum number of fingers you can have on a hand is five.

Eve: Maybe three and two (walks up to the board and points to finger pattern 2 and finger pattern 3).

T: Exactly, two and three together are five (shows with her fingers). We compose them (draws two lines from the finger patterns 2 and 3) into one hand (places one whole hand below).

Eve: Five (shows 2 and 3 on her fingers).

Another way to engage the students in composing smaller units into units of fives was identified when teachers noticed students having chosen one finger pattern, hesitating which number to combine, to obtain a 5-unit (whole hand). The teacher could say the following: “What do we need to get a full hand?”. Thus, this was a different way to discern the units within units (the same unit of five) compared to finding two units that constitute a five. In episode PW5, ten finger patterns were placed in a row (a total of 24 fingers), and two students had composed 3 and 2, 3 and 2, and 4 and 1 to make fives (whole hands). The finger patterns 4, 2, 2, and 1 were left on the top row. The teacher asked the following:

T: Do you think we can do some more? (…)

Betty selects finger patterns 4 and 2.

T: Show with your fingers.

Betty holds up four fingers on one hand and two fingers on the other hand.

T: Put your fingers on the floor. Four (circling the student’s four fingers) four, five, six …

Betty: Six (puts back the finger pattern 2 and chooses another finger pattern 2 (inaudible).

T: What will it be?

Betty: Five.

T: Bring out your fingers again. Four and two, what is four and two? Four there and two there (puts a pen between her four fingers on the left hand and the left thumb and right thumb)

Betty: (hesitates) six.

T: What do we need to get a whole five? What do we need to add to four to make it a full hand?

Betty: (takes the finger pattern 1 and places it next to finger pattern 4). But I’m not sure.

T: If you have four, show with your fingers (the student shows 4) and add one, what do you get?

Betty: (watches her hand) five.

The aspect units within units was made visible when the teacher asked the students to compose two numbers into a whole and when Betty chose finger patterns 4 and 2. The teacher underpinned the aspect (units within units) by encouraging Betty to represent the units (4 and 2) with her fingers twice and pointing out the 4 and 2 and 6 on her fingers. In the end, Betty looked at her fingers, identifying the two smaller units within 5 (4 and 1). In this situation, Betty’s finger patterns supported her in discerning the aspect units within units. In episode BW1, as well as in other episodes, teachers encouraged students to represent smaller units within the five-unit with their fingers. In some episodes, when the video recordings allowed for it, we identified students who used finger patterns for structuring the number relations more spontaneously. In almost half of the episodes, the teachers used their finger patterns, emphasizing units within the unit of five (see Table 3).

3.3. Units Within Units and New, Larger Units (Finger Patterns 5 ≤ 10)

Having identified units of fives (whole hands) and the finger patterns left over, the teacher discussed with the students how to compose units of fives into larger units (two groups of five equal to ten). Figure 3a,b (from the same episodes as those in Figure 1 and Figure 2) illustrate the process in step 3 of composing fives into tens. They show how the teacher and students draw arrows and lines between two groups of five and tens.

Figure 3. (a,b) Step 3 in the Many Hands activity.

Figure 3. (a,b) Step 3 in the Many Hands activity.

In board work (see Figure 3a), eight finger patterns (2 and 3; 3 and 2, 1 and 4; 1 and 4) were composed into four 5-units (four 5s) to make larger units (four 5s equal two 10s). In pair work (see Figure 3b), ten finger patterns (3 and 2; 3 and 2; 1 and 4; 1 and 4, 2 and 2) were composed intofour whole hands and two finger patterns of 2 and reduced from 5-units to two finger patterns of 10 with 4 left over. This way of composing units made it possible for the students to discern units within units and new, larger units.

In episode PW11, when two students composed four 5-units, the teacher asked them how to compose the fives in some way. The student wrote the numerals 5, 5, 5, and 5 below the whole hands. However, the teacher said the following: “If we returned to how we started, you said four and one (points to the finger pattern 1 and 4, and then points towards finger pattern 5), that’s five” and the next (points to finger pattern 2 and 3, points towards next whole hand) that’s five (etc.), then how can you make tens?” In this episode, the teacher directed the students’ attention to the aspect units within units and new, larger units (finger patterns 5 –> 10). In episode PW11, while the students were focused on just adding fives, using numerals, the teacher wanted them to discern how smaller units are composed into five-units and that two groups of five equals ten. In another episode (BW5), when one learner composed the last two finger patterns into a unit of five, the teacher asked the following:

T: Can we compose them in another way?

Cai: Twenty.

T: Five plus five plus five plus five (draws plus sign between the finger patterns (units of fives). Five plus five is ten; fifteen plus five is twenty. Can you do it in any other way? You can show five, five, five, five (showing her hand, repeatedly), or can you do it in another way?

Dave shows two hands.

T: That’s right, ten. Then, units of five in two groups, that’s two hands, two hands. One hand five, another five makes ten (holding up her two hands). Just like Cai said (…) ten and ten is twenty.

In this enactment (BW5), the teacher confirmed that the learner added four 5s to get 20 (5 + 5 + 5 + 5 = 20) but also asked for another way to achieve this, emphasizing that two units of 5 can be composed to create a new larger unit, a 10-unit (5 + 5 = 10), and that the same number (20) can also be seen as two groups of 10 (10 + 10 = 20). By keeping the 20 constant and varying the size of the units (5s and 10s), the decimal and semi-decimal structures were made visible.

In step 3, students and teachers also used finger patterns to support the discernment of units within units and new, larger units. For instance, in episode BW5, the teacher emphasized the decimal and semi-decimal structures, highlighting the student (Dave) who demonstrated how to compose tens using a finger pattern. In the same episode, the teacher illustrated the relationships between units of five and units of ten by showing the differences with her finger patterns—repeating one full hand four times and comparing it with the finger pattern for ten, shown twice. Thereby, the teacher drew attention to the aspects units within units and new, larger units and relationships between different units in the number system (see next section also).

3.4. Relationships Between Different Units in the Number System

In the final step of the Many Hands activity, connections between different units were identified in the enactments. Some of these connections were also identified in step 3 (see, for instance, episode BW5). Looking at the board work and the paper from the pair work conducted in the final step, the documentation of the Many Hands activity often included numerals and signs (+ and =), written by the students or the teacher. The aspects small numbers as composite units; units within units; units within units and new, larger units; and the relationships between different units in the number system were made visible in the enactment, as well as a new aspect not previously identified: place value. Figure 4a,b show the same episodes of teaching as those in Figure 1, Figure 2 and Figure 3.

Figure 4. (a,b) The final step in the Many Hands activity.

Figure 4. (a,b) The final step in the Many Hands activity.

The horizontal rows of finger patterns (three rows in Figure 4a,b) show the additive relations of numbers of announced finger patterns, units of fives (whole hands) on the second row, and ten units (group of tens) in the third row. In 15 (of 19) episodes, teachers and students also wrote numerals and signs (+ and =) to emphasize the addition of units. For instance, the teacher in episode BW3, after having composed and documented the original eight finger patterns to four finger patterns of five and new and larger units (two tens), discussed with the learner how two units of fives equal ten (see Figure 4a).

T: How many (pointing to two whole hands to the right in the second row)?

Emy: Five plus five is ten,

T: (places the finger pattern 10 below the two fives). Now we have reduced the number of finger patterns from eight hands (pointing to the top row) to how many hands?

Billy: Four

T: (…) Now we have four hands (points to the second row). How much is two hands? (circling the left 10 finger pattern in the third row).

Emy: Ten.

T: (writes 10 below the finger patterns). What sign do we use when composing? (…) Ten and ten (shows also with finger pattern)?

Emy: Twenty.

In this enactment, the teacher draws the students’ attention to the decimal and semi-decimal structures of numbers (relationships between different units in the number system) and the original numbers of small numbers as composed units, by comparing the groups of fives and the two tens and the original numbers. Also, introducing how numbers (10 and 20) can be represented differently (with numerals and finger patterns) makes it visually possible to discern these aspects simultaneously.

In pair work (episode PW10), the students composed two groups of ten and two single units (finger pattern 2). Like the pair work shown in Figure 4b, they added the numbers and recorded the sum. This method enables the identification of four units of five when compared to two finger patterns of ten, along with their symbolic representations (10, 10, 2 = 22), simultaneously. This enactment, connecting and comparing different units, is illustrated in Figure 5a,b.

T: [Points to the two tens and the finger patterns 2].

Frank: Ten, ten, and two.

T: How could it all be added up?

Frank: Twenty-two. Ten plus ten is twenty, plus two.

T: Could you show what you just said in another way? You will get a new paper.

[Frank writes the additive expression with symbols (See Figure 5b)].

Thus, adding the units (four fives and two) and (two finger patterns of ten and one finger pattern of two) and being encouraged by the teacher to write the expressions with numerals and signs (10 + 10 + 2 = 22) to obtain a sum made it possible to discern how finger patterns made up larger units (fives and tens) when the number range exceeded ten. Hence, the documentation of this process shows the relationships between smaller composed units, units of fives, and units of tens simultaneously. Looking at board work vertically (see, for instance, Figure 4a), it becomes visible that, by composing two finger patterns to create one unit of five and two fives (whole hands), and drawing arrows or lines between the small, composed units (first row) and the finger pattern five (second row), allows for the visualization of the relationship between smaller units and units within units. And to compose units into new, defined larger units by drawing lines and arrows between two fives and one ten (third row), the students might discern the connections between smaller units (<5) and units of fives, as well as units of ten, simultaneously.

In 4 out of 19 episodes, the original announcement of finger patterns (first row) is compared with the number of hands in the final row, following up from the question posed at the start: “How can we show the same number of fingers with as few hands as possible?” Episode BW4 illustrates this enactment.

T: If I say this, which is the easiest to figure out, if I say this four plus one plus one plus four plus two plus three, plus three plus two plus three equal 23, or I say: five plus five, plus five, plus five plus three equals …?

T: Have we done anything different? Have we taken more fingers?

Students: Yes… no

T: Do we have more, … look here… (…) We have made full hands. (…)

T: If we say ten plus ten plus three, what will it be?

Harry: Twenty-three.

T: What is the easiest way to figure out if you look at the pictures?

Harry: This one (points to the bottom row)

T: Yes, there are only three numbers we need to remember in our heads. How many fingers do you have on your hands?

Helen: If you put your fingers together with a friend …

In this enactment (BW4), the teacher contrasted many hands with fewer hands by displaying the number of fingers desired in different ways, focusing on the relationships between units of different sizes and how to create them. This is achieved through a demonstration on the board, where the teacher moved forward with the same sum (23), enumerated all the numbers (4 + 1 + 1 + 4 + 2 + 3 + 3 + 2 + 3), composed 5s (5 + 5 + 5 + 5 + 3) and 10s (10 + 10 + 3), and encouraged students to visualize these connections and relationships simultaneously. Thereby, the students had the opportunity to discern relationships between different units in the number system.

Another aspect that is made visible in the Many Hands activity is place value. In 3 of the 19 episodes (see Table 3), units of tens and units of ones were underpinned by the teacher’s enactment. For instance, in episode BW2, six finger patterns (1, 2, 4, 3, 3, 2, 1) were composed into three whole hands and one single and in the next step, into one 10-unit, one 5-unit, and one single finger. Between the finger patterns, plus signs were written. In the final step, the teacher wrote 10 + 5 + 1 = underneath the finger patterns. She said, “Ten plus five plus one, how much is ten plus five … plus one?”, and asked a student how to write sixteen. She then said the following: “Which fingers show six (pointing to 6 in 16)?”. The student pointed to the finger patterns five and one (5 + 1). Then, the teacher asked what the 1 in 16 meant. The same student pointed to finger pattern 10. In this enactment, the teacher emphasized that the values of numerals in the expressions are connected to the representation of finger patterns of tens (decimal structure) and finger patterns of single units. Finger patterns 5 and 1 (altogether 6) constitute the units and finger pattern 10 the value of one ten.

In this final step of the activity, the use of finger patterns supported the discernment of aspects. For instance, at the end of episode BW4, when the teacher referred to the number of fingers on their hands, one student (Helen) proposed that they could collaborate with the classmates showing more than 10 on their fingers. Also, the teachers underpinned the composed units, illustrating or mapping with their finger patterns what is made visible on the board, whole hands (units of 5), and finger pattern 10 (units of 10). This mapping was identified more often in board work than in pair work.

4. Discussion

It has been shown that finger patterns can support young learners in understanding numbers and number relations (e.g., Brissiaud, 1992; Neuman, 1987; Kullberg et al., 2020). In this study, we add to this by showing how finger patterns are beneficial for learning, even in larger number ranges when numbers exceed 10. Finger patterns in the Many Hands activity became a means not only for composing and decomposing units within 10, using learners’ fingers and images of different finger patterns, but also for composing numbers larger than 10 when composing 5s into several 10s using images of finger patterns. In this sense, learners’ finger patterns become extended into finger pattern images that can represent larger numbers. We suggest that this way of using fingers to show and represent parts and wholes of larger numbers is beneficial for young learners’ understanding of numbers and number relationships (cf. Hopkins et al., 2022).

We showed how several aspects of numbers were possible to perceive in the enactment of the activity, including (i) small numbers as composed units; (ii) units within units; (iii) units within units and new, larger units; (iv) relationships between units in the number system (decimal and semi-decimal structures); and (v) place value. It is possible that other aspects not identified in this study can also be enacted in the activity. These aspects focus on a relational understanding of numbers and arithmetic operations, which has been suggested earlier (Björklund et al., 2019, 2021; Polotskaia & Savard, 2018) and as a reaction to an approach based on counting in single units to gain knowledge of numbers and early arithmetic skills (Fuson & Secada, 1986). These aspects are likely necessary for learners to discern, allowing them to be able to develop a sustainable way of handling numbers and early arithmetic, but from our experience, they are most often not explicitly taught. The aspect place value was made visible in only three episodes (see Table 3). While place value may not be a primary learning goal for six-year-olds, some teachers nonetheless visualized this aspect, which also provided learning opportunities for foundational knowledge about numbers.

The Many Hands activity illustrates the potential of finger patterns to support students in composing and recognizing units of ten, as well as identifying patterns that do not constitute a complete ten. In this study, teachers emphasized place value by connecting numerals (e.g., 16) to images of finger patterns representing tens and single units (see Figure 4a, Section 3.4) or, as shown at the end of episode BW3 (see Figure 4a in Section 3.4), by emphasizing the place value in twenty (two ten-units). Consistent with the original French version of the game (Sensevy et al., 2015), the activity could be extended by increasing the number of hands involved. This would allow students to become familiar with multiple 10-units and to appreciate the efficiency of grouping larger quantities into 10s when determining larger sums.

Children’s bodily interactions, including the use of hands and fingers, play a critical role in numerical development (Soylu et al., 2018), particularly in exploring 10-units and smaller units within the number system. Given their awareness of having ten fingers, students may naturally engage in adding several tens (see episode BW5 in Section 3.3). The combination of bodily interactions and visual finger pattern representations may thus enhance students’ understanding of the decimal number system and serve as a foundational pathway to learning about place value.

The analyzed enactments of the Many Hands activity revealed that none of the eight teachers in the analyzed episodes asked the students to count the fingers in the images as single units. Instead, they were encouraged to discern small numbers as composite units and as units within units (part–whole relationships). The students represented these units using their finger patterns, encouraged by the teachers (cf. not by modeling numbers on fingers, Carpenter & Moser, 1982).

We suggest that the Many Hands activity, presented in two formats—board work and pair work—has the potential to become a highly meaningful activity when learning about numbers and number relations. The analyses of video recordings revealed a high level of student involvement and engagement in both versions of the activity. This high level of engagement was also evident while planning meetings and reflected in the teachers’ observations. The students’ active participation in an activity that presents numerous aspects related to numbers and number relations underscores the potential of the activity. However, it is not just the activity itself that matters. Even though the activity is designed based on aspects of numbers and number relations, its learning potential depends on how the teacher underpins these aspects through variation in the enactment. Some limitations of this study need to be considered. The teachers themselves decided what teaching episodes they uploaded to the server. Their selections of recordings and how long the video recordings were may have influenced which aspects were identified in the analysis. The results reveal that some teachers used finger patterns more frequently and encouraged students to represent patterns using their fingers. For instance, in episode BW9, the teacher constantly encouraged the students to demonstrate numbers and number relations with their fingers.

Finger patterns, used as a tool for structuring numbers and number relations, seem to be powerful (Björklund, 2024; Kullberg et al., 2020). The finger patterns used are built on students’ knowledge of units being composed of smaller units and emphasize the relationships between different units in the number system—a method of understanding numbers that is possible to generalize. The findings support the use of finger patterns in early childhood and primary education, when used as tools for structuring numbers, not only within the 1–10 number range but also as a tool for developing arithmetic skills.