Alberto Capurro
email: alberto.capurro@newcastle.ac.uk

Jack Thornton

Bruno Cessac

Lyle Armstrong

Evelyne Sernagor

On the role of Nav1.7 sodium channels in chronic pain: an experimental and computational study

Chronic pain is a global healthcare problem with a huge societal impact. Its management remains generally unsatisfactory, with no single treatment clinically approved in most cases. In this study we use an in vitro model of erythromelalgia consisting of dorsal root ganglion neurons derived from human induced pluripotent stem cells obtained from a patient (carrying the mutation F1449V) and a control subject. We combine neurophysiology and computational modelling to focus on the Nav1.7 voltage gated sodium channel, which acts as an amplifier of the receptor potential in nociceptive neurons and plays a critical role in erythromelalgia due to gain of function mutations causing the channel to open with smaller depolarisations.

Using extracellular recordings, we found that the scorpion toxin OD1 (a Nav1.7 channel opener) increases dorsal root ganglion cell excitability in cultures obtained from the control donor, evidenced by an increase in spontaneous discharges, firing rate and spike amplitude. In addition, we confirmed previous reports of voltage clamp experiments concerning an increase in spontaneous discharge in the patient cell cultures and the analgesic effects of the Nav1.7 blocker PF-05089771. Our findings are explained with a conductance-based model of the dorsal root ganglion neuron, exploring its behaviour for different values of half activation voltage and inactivation removal rate of the Nav1.7 current. Erythromelalgia was simulated through a decrease of the Nav1.7 half activation voltage, turning previously subthreshold stimuli to pain-inducing, and successfully counteracted with the channel blocker. The painful effects of OD1 were simulated through a quicker removal of Nav1.7 inactivation that reproduced the effects of the toxin not only on the spike frequency but also on its amplitude.

New & Noteworthy

We investigate Nav1.7 channel gating mechanisms in human iPSC-derived dorsal root ganglion cell cultures using multielectrode array recordings. The scorpion toxin OD1 increases firing frequency and spike amplitudes whilst the analgesic PF-05089771 decreases or even abolishes spontaneous

Introduction

Chronic pain is a global healthcare problem, particularly affecting elderly people, women and persons with lower socio-economic status [START_REF] Van Hecke | Chronic pain epidemiology and its clinical relevance[END_REF]. It is one of the most common reasons for physician consultation in developed countries, interfering with quality of life and causing large socio-economic impacts that include significant loss of working hours and the need of clinical care. Current therapies have limitations in their effectiveness and side effects, creating an urgent need to develop more precise and effective treatments for pain management [START_REF] Khouzam | Chronic pain and its management in primary care[END_REF]. In this study we focus on the voltage dependent properties of the sodium channel Nav1.7 that is implicated in inherited erythromelalgia, a vascular peripheral pathology causing attacks of severe chronic pain [START_REF] Mcdonnell | Inherited erythromelalgia due to mutations in SCN9A: natural history, clinical phenotype and somatosensory profile[END_REF].

Pain evoked spiking activity starts in peripheral terminals of dorsal root ganglion (DRG) neurons. The central extensions of these neurons form the Aδ and C fibres which establish glutamatergic synapses onto second order neurons within the spinal cord. They can be stimulated by mechanical, thermal or chemical stimuli as well as by inflammatory mediators. Distributed throughout the body (skin, viscera, muscles, joints, meninges), they carry noxious sensory information into the central nervous system [START_REF] Serpell | Anatomy, physiology and pharmacology of pain[END_REF]. The recent discovery of nociceptive Schwann cells has changed the notion of bare nerve terminals being the starting point of pain sensation, and introduced the concept of a glioneural end organ in the skin that transmits nociceptive information to the nerve, resembling the specialized receptor cells found in other sensory systems [START_REF] Abdo | Specialized cutaneous Schwann cells initiate pain sense[END_REF].

DRG neurons express several types of voltage gated sodium channels with different properties (Lera [START_REF] Ruiz | Voltage-Gated Sodium Channels: Structure, Function, Pharmacology, and Clinical Indications[END_REF]. The opening of Nav1.7 channels requires smaller depolarisations than other types of sodium channels, being more easily activated by the graded generator potentials. For this 3 reason it is considered to be a threshold channel that acts as an amplifier of the receptor potential, increasing the probability of triggering a spike via the activation of other, higher threshold sodium channels such as Nav1.8. This amplification property makes Nav1.7 a major contributor to pain signalling in humans [START_REF] Dib-Hajj | The Na(V)1.7 sodium channel: from molecule to man[END_REF].

Erythromelalgia is caused by gain of function mutations of the gene SCN9A, which encodes for the Nav1.7 channels pain [START_REF] Mcdonnell | Inherited erythromelalgia due to mutations in SCN9A: natural history, clinical phenotype and somatosensory profile[END_REF]. Different types of mutations have in common the need for less depolarization to open the channel than in the wild type, resulting in a decrease of the rheobase (i.e., minimal amplitude of an infinite depolarizing current to evoke a spike) [START_REF] Cao | Pharmacological reversal of a pain phenotype in iPSC-derived sensory neurons and patients with inherited erythromelalgia[END_REF]. DRG neurons fire spontaneously in erythromelelgia patients, although normally these cells are silent unless receiving strong peripheral pain stimulus.

In this study, we have used extracellular recordings of spontaneous activity in DRG neuron cultures derived from human induced pluripotent stem cells (hiPSCs) obtained from an erythromelalgia patient and a control subject to study the voltage dependent gating properties of Nav1.7 channels.

To start, the effects on the neuronal firing of two different pharmacological compounds -a pain eliciting scorpion toxin (OD1, [START_REF] Motin | Modulation of human Nav1.7 channel gating by synthetic α-scorpion toxin OD1 and its analogs[END_REF] and an analgesic drug (PF-05089771, [START_REF] Cao | Pharmacological reversal of a pain phenotype in iPSC-derived sensory neurons and patients with inherited erythromelalgia[END_REF]-were assessed in these cultures using multi electrode array (MEA) recordings. We then present a simple conductance-based computational model of the DRG neuron to explain our findings in terms of the Nav1.7 gating process, focusing on the channel activation (opening) and repriming (i.e., recovery from inactivation).

Materials and Methods

Cells and MEA recordings

For this study we used hiPSCs from a control subject (cell line AD3) and an erythromelalgia patient (cell line RCi002-A, carrying the mutation F1449V) made available at the European Bank for induced pluripotent stem cells (EBiSC). The cells were differentiated into sensory neurons using a small 4 molecule based protocol described previously (Cao et al., 2016, and references therein). Once they reached the stage of DRG neurons, the cells were re-plated in 24 wells MEA plates (MEA700, Multichannel Systems, Reutlingen, Germany) where the spontaneous activity was recorded after 10 weeks of maturation. Each well contains 12 circular electrodes (100 µm diameter, 700 µm electrode pitch), making a total of 288 electrodes per plate. The large distance between electrodes makes it very unlikely that the activity originating from an individual neuron will be detected by adjacent electrodes. On the other hand, due to the relatively large cell density in these cultures, which organize in structures called neural rosettes (e.g., Wilson and Stice, 2006), it is quite likely that one electrode can detect the activity of more than one neuron (up to four in our data sets, although most channels had only one or two).

We compared the spontaneous activity of DRG neurons obtained from a control subject (plated in 18 wells, making a total of 216 electrodes in MEA 1) with an erythromelalgia patient (plated in 24 wells, making a total of 288 electrodes in MEA 2). In addition we used a third MEA plated with diseased cells stimulated with higher concentration of potassium (from 4.16 mM in the growing medium to 6 mM) to enhance spontaneous activity (plated in 24 wells, making a total of 288 electrodes in MEA 3). This allowed us to find more active neurons following the addition of PF-05089771 to make paired comparisons before and after this treatment that tends to suppress the firing in many channels. Each well constitutes an independent cell culture.

The cultures were treated with OD1 (100 nM) in the control subject (MEA 1) and with PF-05089771 (100 nM) in the patient (MEAs 2 and 3). The activity was recorded for 5 minutes immediately before the application of each substance and compared with a recording of the same duration performed 5 minutes after the onset of drug exposure. Wells contained 200 µl of growing medium and 5 µl drops were added to apply the treatments. The experimental doses were selected to be in a near saturation range to ensure a strong effect, based on dose response curves published previously (Cao et al., 2016 for PF-05089771;Motin et al., 2016 for OD1). Both compounds were purchased from Tocris (Bio-Techne, Abingdon, UK).

Spike sorting and pairing of units

Recordings were performed with the Multiwell-MEA-System and the software Multi Channel Experimenter (Multi Channel Systems, Reutlingen, Germany). Each channel was band-pass filtered (100 to 3500 Hz) and acquired with a sampling rate of 20 kHz.

The raw voltage traces were first plotted and inspected using a zoom tool to discard artefacts and confirm the existence of spikes. The voltage time series were then fed into the Matlab toolbox Waveclus (Quian [START_REF] Quiroga | Unsupervised Spike Detection and Sorting with Wavelets and Superparamagnetic Clustering[END_REF] to perform spike sorting in all active channels. In the toolbox, the continuous data were filtered again with a non-causal band pass filter between 300 and 3000 Hz and the firing times were detected with an amplitude threshold. We used a dual threshold (i.e., picking deflections in both up and down directions) set to 5 median absolute deviations of the filtered voltage signal with a refractory period of 2 ms to avoid double annotations due to fast voltage oscillations in the vicinity of the threshold. Spikes were aligned to their maximum, after interpolating the waveforms to locate the peak time more accurately. The toolbox uses a wavelet based method for feature extraction, and the grouping of spikes into clusters is done with superparamagnetic clustering [START_REF] Blatt | Superparamagnetic Clustering of Data[END_REF], a stochastic algorithm that does not assume any particular distribution of the data. In few cases the dual threshold created mistakes in the detection, so we decided to use a single threshold, kept for the units that corresponded to the same neuron before and after a given treatment. We consider this spike sorting strategy as supervised, always following the criterion of the biologist as ground true. The firing times and voltage cut outs of all units were stored to document the parameters and quality of the spike sorting in each channel.

Units that were active both before and after each pharmacological treatment were identified. The only set-in-stone criterion for deciding if they are the same cell is the coordinates of the current sources, using high density MEAs [START_REF] Hilgen | Unsupervised Spike Sorting for Large-Scale, High-Density Multielectrode Arrays[END_REF]. As our data were recorded with low density 6 MEAs, we cannot provide this level of certainty, but assumed that they corresponded to the same neuron if the wave shape remained similar across recordings performed within few minutes of each other. Possible ambiguities were minimized by the fact that most channels yielded only one or two active neurons. These pairs were used to investigate the changes in spike frequency and amplitude caused by the pharmacological treatments (Figs 1 and2). If we were reasonably convinced that a given unit corresponded to the same neuron before and after treatment, we designated the two recordings as a pair. In the case of the control cultures, few neurons had spontaneous discharges and most cells started to fire only after the OD1 treatment. Conversely, in the case of the patient cultures, several neurons were active before, but few remained active after the application of PF-05089771. In that case, we made an additional recording after increasing the spontaneous activity with potassium, as mentioned above. This manipulation allowed to get more active neurons in control conditions, and then more neurons remained active after the treatment to configure pairs, although we did not pool the data recorded with and without higher potassium concentration.

For statistical comparisons between two conditions we used the Wilcoxon signed rank test (for paired values) or the Mann-Whitney U test (for unpaired values). We also used cross correlation histograms of the spike times recorded from the same channel to assess whether neurons tend to fire at fixed delay from another.

Numerical simulations

We performed numerical simulations of the membrane potential and sodium channels gate parameters of a single DRG neuron using the software NEURON (www.neuron.yale.edu), e.g., [START_REF] Choi | Physiological interactions between Nav1.7 and Nav1.8 sodium channels: a computer simulation study[END_REF]. The integration method was the IDA algorithm (e.g., [START_REF] Carnevale | Neuron simulation environment[END_REF] with a fixed time step of 0.025 ms (i.e., 40 time steps per ms). Our purpose is to explain the changes in frequency and amplitude induced by the drug treatments covered in the previous section. In this context we adopted a parsimonious approach to keep the model as simple as possible, using Hodgkin and Huxley (HH) type of Na + and K + currents [START_REF] Ermentrout | Mathematical Foundations of Neuroscience[END_REF] with parameters (Table 1) adapted to be plausible for mammal cells [START_REF] Krouchev | From Squid to Mammals with the HH Model through the Nav Channels' Half-Activation-Voltage Parameter[END_REF] plus a single Nav1.7 type current added using the Channel Builder GUI [START_REF] Mcdougal | Twenty years of ModelDB and beyond: building essential modeling tools for the future of neuroscience[END_REF].

The two different types of sodium channels that we included differed only in their parameters (Table 1), the form of the equations was the same (S1 Appendix). The first, with larger conductance ( 0 1). These changes allowed to create a rapidly activating but slowly repriming (slow recovery from inactivation) current, resembling Nav1.7. To simulate erythromelalgia we varied the half activation voltage of Nav1.7 from -55 to -60 mV. To simulate PF-05089771 effect, we varied the maximum conductance of Nav1.7 from 0.1 to 0.07 S/cm 2 (Figs 3 and4) andto simulate OD1 effects (Figs 5, 6 and7) we increased 10 times the parameter A h that multiplies the inactivation removal rate a h . Model parameters are provided in Table 1, the equations that are most relevant for our results appear in section Results and all model equations are listed in S1 Appendix.

Step depolarisations of 0.04 nA lasting 60 ms were used to evaluate the effect of the half activation voltage (d h ) and the inactivation removal rate (a h ) on the firing frequency of the spike (Figs 3 and5).

To estimate more reliably the spike frequency and amplitude by averaging over many spikes we used depolarization steps lasting 5 s (Figs 4 and6).

In order to assess for bifurcations in the dynamical system, we performed simulations of the

Results

Experimental recordings

We compared the spontaneous activity of DRG neurons obtained from a control subject (cell line AD3, n = 18 cultures) with an erythromelalgia patient (cell line RCi002-A, n = 24 cultures). The percentage of electrodes showing activity was 1.85 % in the control and 5.9 % in the disease cultures. After spike sorting we found 30 spontaneously active units in the patient and only 7 in the control. The firing frequency and amplitude of these units were not significantly larger in the disease cultures, implying that although much more units are active, they do not necessarily fire faster or have spikes of larger amplitude. It is worth mentioning that the fastest neurons and the tallest spikes were found in the disease cultures, but the distributions were not significantly different because slow units with small amplitude were present in both groups. The larger number of spontaneously active neurons found in the disease cultures is consistent with the results of a previous study [START_REF] Cao | Pharmacological reversal of a pain phenotype in iPSC-derived sensory neurons and patients with inherited erythromelalgia[END_REF].

The cultures from the control subject were treated with OD1 (100 nM), resulting in the percentage of active channels increasing from 1.85 to 6.02 % and a concomitant increase in the number of active units, from 7 to 24. An example of the voltage data before and after treatment is presented in Fig 1-A, note the large increase in spontaneous activity. The same picture was observed in an erythromelalgia culture conducted just as a verification (not shown).

Pairing all the units that were active before and after OD1 application in the control cells (n = 7), we found that both the spike frequency and amplitude increased significantly (Fig 1-B). Although the increase in amplitude was not dramatic (20 % in average), it was consistently found in every pair of units. The extent of the increase in firing frequency was more pronounced (70 %). The unpaired units that were initially silent and started to fire after the OD1 were not significantly different in rate and amplitude from those already active before the treatment.

The cells from the patient were treated with PF05089771 (100 nM). The percentage of channels with spiking activity fell from 5.9 to 2.08 %. An example of the raw voltage trace before and after treatment is presented in Fig 2-A, note that the spiking activity is decreased, although this particular recording was done in the presence of higher K + concentration.

Pairing the units that were active before and after the treatment (n=6), we found that the spike frequency decreased significantly but the amplitude did not change (Fig 2-B). This result was confirmed with a larger number of paired units (n=16) in a MEA plate previously exposed to high potassium concentration (Fig 2-C). In both cases the unpaired units that stopped firing as a result of the PF05089771 application were not significantly different in rate and amplitude to the ones that remained active after treatment. In order to assess for possible functional connectivity, we calculated cross correlation histograms between the firing times of the neurons that were recorded from the same electrode. No histogram peaks at fixed latency were found, which is compatible with the established notion that DRG neurons do not form synaptic contacts between them.

Numerical simulations

In this subsection we present the results obtained with the computational model of the single DRG neuron explained in section Methods, including the equations directly related with the quantities plotted in the figures. The equations of all currents are provided in S1 Appendix.

Our DRG neuron was simulated as a cylindrical soma of 30 µm diameter and height. The ionic conductances are based on a HH model with mammalian-plausible parameters plus an additional Na + current with the features of a Nav1.7 (Table 1). The disease and health conditions were simulated through changes in the Nav1.7 half-activation voltage parameter (d m in equation 1). In 10 erythromelalgia patients, this parameter is shifted up to 15 mV in hyperpolarizing direction [START_REF] Cao | Pharmacological reversal of a pain phenotype in iPSC-derived sensory neurons and patients with inherited erythromelalgia[END_REF], causing a decrease in the level of depolarization needed to activate the Nav1.7 current.

General parameter set K HH parameters L (µm) 30

Opening (a m ) The rate of Nav1.7 channel opening a m is given by a m ൌ Am ሺkm ሺ௩ିdm ሻሻ ଵି షkm ሺೡషdm ሻ

(1

)
where v is the membrane potential, d m is the half-activation voltage, and k m and A m are constants (Table 1). The opening and closing dynamics of the current is given by the differential equation

ௗ ௗ௧ ൌ a m ሺ1 െ ݉ሻ െ b m ݉ ( 2
)
where m is the opened state of the channel, 1-m the closed state and b m the closing rate.

The conductance is a cubic function of m, gNav1.7 ൌ gmax ݉ ଷ ݄

(3

)
where h is the inactivation process and g max the maximum conductance.

In where the stimulus does no elicit pain (i.e., no spiking). The pink traces of the middle row represent mild erythromelalgia (d m = -58 mV) where the same stimulus is painful (spiking in the left panel) and can be successfully treated by blocking 20% of the Nav1.7 channels, i.e., decreasing the maximum conductance (g max in Equation 3) from 0.1 to 0.08 S/cm 2 (no spiking in the right panel). In the red traces of the lower row we simulate severe erythromelalgia (d m = -60 mV), showing faster spiking rate (left) and spontaneous spikes occurring before and after the stimulus. In this case, blocking 20% of the Nav1.7 channels decreases the firing but is not able to stop it (middle panel). The pain eliciting firing can be stopped by blocking a larger percentage of Nav1.7 channels (30% with g max = 0.07 S/cm 2 , right panel).

In In a second series of simulations we explored the inactivation properties of Nav1.7 channels (parameter sets displayed in Table 1). The scorpion toxin OD1 has been reported to increase the rate at which Nav1.7 inactivation is removed after being established [START_REF] Motin | Modulation of human Nav1.7 channel gating by synthetic α-scorpion toxin OD1 and its analogs[END_REF]. In order to emulate this effect in the model, we increased 10 times the removal rate of Nav1.7 inactivation (a h in Equation 4).

The inactivation removal rate is given by ah ൌ Ah ݁ khሺ௩ିdhሻ

(4)

where v is the membrane potential, k h and A h are constants (Table 1) and d h is the half-inactivation removal parameter. The inactivation process dynamic is given by the temporal derivative of h ௗ ௗ௧ ൌ ah ሺ1 െ ݄ሻ െ bh ݄

(5)

where 1-h is the inactivated state and h is the non-inactivated state that allows ionic conduction.

To increase a h we increased the multiplicative constant A h by 10 times (Equation 4). This change caused the model neuron to fire in response to a depolarizing pulse of 0.04 nA that was not effective before (upper panels in during a spike, in a simulation before (blue plots of left panel) and after OD1 (red plots of right panel). Both panels display the Nav1.7 current with full thick lines. Note that in the simulation after OD1, Nav1.7 has larger amplitude, and the values of h are higher, with a faster return to 1 after the negative peak. These large changes of h were not observed in the HH type high threshold sodium current, although the current amplitude was slightly increased (Fig 7 -B).

Discussion

In this study, we performed to our knowledge the first test of the scorpion toxin OD1 in hiPSCderived DRG neurons. This toxin enhances the recovery from fast inactivation of the threshold current Nav1.7 [START_REF] Motin | Modulation of human Nav1.7 channel gating by synthetic α-scorpion toxin OD1 and its analogs[END_REF]. Our extracellular recordings also confirmed previous findings of [START_REF] Cao | Pharmacological reversal of a pain phenotype in iPSC-derived sensory neurons and patients with inherited erythromelalgia[END_REF] with patch clamp regarding the increased spontaneous firing of erythromelalgia DRG cells and the analgesic effects of PF-05089771. This drug was shown to be clinically effective to control heat induced pain attacks by stabilizing the voltage sensor domain of the Nav1.7 channel in a non-conducting conformation. The effects of both compounds were explained using a conductancebased computational model, comparable with [START_REF] Choi | Physiological interactions between Nav1.7 and Nav1.8 sodium channels: a computer simulation study[END_REF] but simpler and particularly focalized on the issue of pain signalling. In this section we first discuss our main findings and emphasize the usefulness of extracellular recordings to observe changes in ionic currents. To close the discussion, we provide a short remark about the importance of the interaction between different sodium currents in erythromelalgia.

Effects of erythromelalgia, PF-05089771 and OD1 [START_REF] Cao | Pharmacological reversal of a pain phenotype in iPSC-derived sensory neurons and patients with inherited erythromelalgia[END_REF] performed voltage clamp experiments in whole-cell configuration in DRG cells derived from erythromelalgia patients and control subjects. Despite the heterogeneity of the samples, they found a significantly higher proportion of spontaneously firing cells in patients compared to those from control donors. Moreover, the patient's cells showed a lower rheobase and reached higher firing rates in response to stimulation with current steps of increasing amplitudes.

These findings point to the existence of elevated excitability in erythromelalgia cells, a fact that we confirmed in the present study by showing higher prevalence of spontaneous activity in extracellular MEA recordings (section Results).

Spontaneous firing in our cultures is due only to intrinsic cell excitability, as there are no reports about the formation of synapses between DRG cells in vitro. Accordingly, we did not find evidence of functional connections in cross correlation histograms between neuronal firing times. Furthermore, in dissociated cell cultures the development of a synaptic network is usually accompanied by the emergence of population bursts [START_REF] Maeda | The mechanisms of generation and propagation of synchronized bursting in developing networks of cortical neurons[END_REF] which we did not observe here. DRG glia cells are not present in the cultures, so their reported contribution to abnormal neuronal activity in in vitro studies done on intact DRG [START_REF] Belzer | Nitric oxide as a messenger between neurons and satellite glial cells in dorsal root ganglia[END_REF] can also be ruled out.

Regarding the effects of Nav1.7 channel blockers, [START_REF] Cao | Pharmacological reversal of a pain phenotype in iPSC-derived sensory neurons and patients with inherited erythromelalgia[END_REF] reported a dose dependent reduction in spontaneous firing and an increase in the action potential rheobase. We were able to reproduce the first finding on spontaneous firing with PF-05089771 (Fig 2 ), supporting the fact that the excitability of the cells plummets after the treatment, in accordance with the reported clinical efficacy of the drug [START_REF] Cao | Pharmacological reversal of a pain phenotype in iPSC-derived sensory neurons and patients with inherited erythromelalgia[END_REF]. We did not, however, observe significant changes in spike amplitudes before and after drug treatment.

The effect of OD1 (and other synthetic toxin analogs) on the gating properties of the Nav1.7 sodium channels was studied by [START_REF] Motin | Modulation of human Nav1.7 channel gating by synthetic α-scorpion toxin OD1 and its analogs[END_REF] (2016) also found that the current-voltage relationships of Nav1.7 are shifted to more negative potentials in the presence of OD1, but this effect has a small magnitude (-3 mV), in accordance with earlier observations in Xenopus laevis oocytes with Nav1.7 channels expression [START_REF] Maertens | Potent modulation of the voltage-gated sodium channel Nav1.7 by OD1, a toxin from the scorpion Odonthobuthus doriae[END_REF].

Our findings corroborate those of [START_REF] Motin | Modulation of human Nav1.7 channel gating by synthetic α-scorpion toxin OD1 and its analogs[END_REF]. Indeed, we observed an increase in the proportion of electrodes showing spontaneous firing in the presence of OD1, as well as an increase in firing rate in cells that did fire before adding the toxin. In addition to the firing frequency rise, we found a consistent increase in the spike amplitude after OD1, in contrast with the PF-05089771 treatment which affected only the firing rate. These larger extracellular spikes are in accordance with the increase in peak current observed in the presence of OD1 in voltage clamp experiments [START_REF] Maertens | Potent modulation of the voltage-gated sodium channel Nav1.7 by OD1, a toxin from the scorpion Odonthobuthus doriae[END_REF][START_REF] Motin | Modulation of human Nav1.7 channel gating by synthetic α-scorpion toxin OD1 and its analogs[END_REF]. Since the amplitude of the current generated by the opening of a single channel was not modified by OD1 [START_REF] Motin | Modulation of human Nav1.7 channel gating by synthetic α-scorpion toxin OD1 and its analogs[END_REF], the size increase of the extracellular spikes must be due to the dynamics of the gating process. In the next section, we address the numerical simulations to offer an explanation for the changes in spike rate and amplitude described above.

Numerical simulations

The computational model that we present here is able to provide satisfactory explanations at a qualitative level for the main experimental findings of our study and for previous studies on erythromelalgia and the effects of pharmacological manipulations on Nav1.7 channels. We did not attempt to perform a complete exploration of the parameter space but all values are biologically plausible. In the absence of stimulation the membrane potential tends to its fixed point which is the resting potential. By injecting an amount of current also plausible for a patch clamp experiment (0.037 nA for d m = -58 mV and g max = 0.1 S/cm 2 ), the system reaches a Hopf bifurcation (e.g., [START_REF] Ermentrout | Mathematical Foundations of Neuroscience[END_REF], the critical point corresponding to the spike threshold. At a critical point, the stability of the system switches and a periodic solution of repetitive spiking arises, eliciting pain. In the dark blue area of no firing of Fig 4 the membrane potential is at a fixed point (resting potential). The periodic solution occurs in the areas having other colors (including lighter tones of blue). The Hopf bifurcation happens at the limit between the dark blue and the other colors, roughly resembling a diagonal trajectory in the heat map. The descending transition from 1 to 2 and 4 exemplifies the trajectory towards pain in erythromelalgia and the horizontal transitions from 2 to 3 and from 4 to 5 and 6 depict the analgesic path triggered by PF-05089771.

While the decrease of Nav1.7 maximum conductance accounted for the analgesic effects of PF-05089771, OD1 effects on the firing rate were reproduced by enhancing the constant governing the removal rate of the inactivation a h (Fig 5). The interpretation of these results is quite straight forward, having fewer available Nav1.7 receptors reduces the firing by interfering with the amplification of subthreshold potentials performed by the channel, while the faster inactivation removal increases the current size and duration (Fig 7), leading to increased spiking and excitability. This is in line with the proved analgesic effects of Nav1.7 blockers [START_REF] Cao | Pharmacological reversal of a pain phenotype in iPSC-derived sensory neurons and patients with inherited erythromelalgia[END_REF] and with the intense pain known to be experienced after a scorpion sting (e.g., [START_REF] Garfunkel | Pediatric Clinical Advisor[END_REF].

In the case of spike amplitude, the explanation of the results requires a deeper elaboration. Starting from the fact that spike depolarization is sustained by sodium currents, we can see that the maximum amplitude that a spike can potentially reach in the absence of inactivation is given by the sodium equilibrium potential. The inactivation process will also limit the maximum value reached by the current and the peak intracellular voltage, because the current starts to inactivate before the opening process is completed. If this reasoning is correct, an increase of the inactivation removal rate of Nav1.7 will create a larger spike, because more channels will be free from inactivation during the critical period of the spike rising phase.

This last interpretation is exemplified in the numerical simulations of Figs 6 and7. The consequences of the faster inactivation removal rate of Nav1.7 after OD1 are noticeable in the higher values of the process h, as well as in its faster return to 1 following the negative peak. The closer to 0 h reaches, the more channels are inactivated, so a larger h value implies more open channels potentially available, which is reflected in a larger Nav1.7 current (Fig 7-A). The high threshold Na + current also grows slightly following enhancement of Nav1.7, without large changes of h. This increase is less pronounced than for Nav1.7, but still contributes to increase the spike size because the total HH type current is bigger. We must keep in mind that the extracellular spike is a filtered version of the intracellular spike and we cannot necessarily compare the amplitude differences linearly. In addition, Nav1.7 and Nav1.8 interact in a complex manner to regulate DRG neuronal excitability, as has been

shown in a previous modelling study of the rat HRV neurons [START_REF] Choi | Physiological interactions between Nav1.7 and Nav1.8 sodium channels: a computer simulation study[END_REF]. Despite these caveats, we can assume that in the situation depicted in our simulation both currents are adding up to enhance the intracellular spike (Fig 6). This effect was captured in our extracellular recordings (Fig 1).

Model limitations

We are interested in understanding the reason for the observed changes in spike rate and amplitude created by the two compounds tested using a model simulation with biologically plausible parameters. In this context, we validated our interpretation of biological facts with a numerical simulation. In order to keep the model simple and with a clear biological meaning and physical unit for all variables, we opted for a modified HH scheme, which also has the virtue of being familiar to neurobiologists. We included only one potassium current (non-inactivating, HH type), to avoid unnecessary complexity. The only added current to the original HH scheme is the Nav1.7, because it is strictly needed to implement the biological mechanisms under consideration. A detailed quantitative modelling would require Ca

2+

currents and several types of Na + and K + currents that have been found in mammalian DRG cells (e.g., [START_REF] Rush | Multiple sodium channels and their roles in electrogenesis within dorsal root ganglion neurons[END_REF][START_REF] Scroggs | Multiple Ca2+ currents elicited by action potential waveforms in acutely isolated adult rat dorsal root ganglion neurons[END_REF][START_REF] Choi | Physiological interactions between Nav1.7 and Nav1.8 sodium channels: a computer simulation study[END_REF][START_REF] Newberry | Development of a spontaneously active dorsal root ganglia assay using multiwell multielectrode arrays[END_REF][START_REF] Mandge | A biophysically detailed computational model of urinary bladder small DRG neuron soma[END_REF]Zemel et al., 2018), and is beyond the scope of this study. Although more refined kinetic models of the Nav1.7 sodium and other channels have been proposed (e.g., [START_REF] Sigg | Modeling ion channels: Past, present, and future[END_REF], we think that we are adopting here a level of simplification in the modelling process that is fitted for our purpose.

Extracellular recordings and ionic currents

Although extracellular recordings are typically used to detect firing times, they also have the theoretical ability to report about what are considered to be intracellular features of the action potential [START_REF] Gold | On the Origin of the Extracellular Action Potential Waveform: A Modeling Study[END_REF]. We want to highlight this possibility of using extracellular recordings to detect changes in ionic currents. If researchers first evaluate them together with voltage clamp data for validation purposes, they can then benefit from the opportunity to observe more neurons simultaneously. A small amplitude change of the DRG neuron spike might not play an important physiological role in pain signalling, which is mainly conveyed by the spike rate, but we provided here a proof of principle regarding the footprint of a current in the extracellular spike.

The fact that the extracellular spike did not show large changes in waveform shape with the treatments (only scaling up moderately with OD1) facilitated our analysis, allowing reasonable assumptions regarding neuron identities before and after the treatments, but this might not be the case for other drugs or experimental systems (Hilgen, personal communication). It has been reported that an intracellular spike broadening caused by K + current modulators can increase Ca ++ influx at the synaptic level in mouse hippocampal cultures (Vivekananda et al., 2017). In this context it makes sense to speculate about the possibility that, if a larger spike could elicit more neurotransmitter release, then the spike amplitude would also contribute to pain signalling by eliciting a discharge of higher frequency in the ascending spinal neurons.

Importance of the interaction between sodium currents

The interaction between sodium currents can have profound functional consequences, and the clinical signs of erythromelalgia provide a striking example [START_REF] Rush | A single sodium channel mutation produces hyper or hypoexcitability in different types of neurons[END_REF]. As explained above, in pain pathways Nav1.7 increases neuronal excitability mainly by amplifying the depolarization caused by peripheral stimuli, allowing to trigger Nav1.8. This facilitates firing and pain, an effect that is exaggerated in erythromelalgia patients. A key property for this firing to occur (as implemented in our model) is that Nav1.8 does not inactivate until the voltage is quite high (e.g., around -35 mV). In the sympathetic nervous system the situation is different because there are no Nav1.8 channels and spikes are sustained by other sodium currents that inactivate at lower voltage [START_REF] Rush | A single sodium channel mutation produces hyper or hypoexcitability in different types of neurons[END_REF].

Then, the excessive depolarization created by a mutated Nav1.7 in erythromelalgia can be sufficient to trigger inactivation, terminating the firing. The excessively active mutated Nav1.7 increases firing in HRV nociceptive neurons but blocks firing in the sympathetic nerves that mediate the contraction of peripheral blood vessels. This results in excessive vasodilatation, creating the typical sign of red extremities that originated part of the name to the disease (erythromelalgia could be translated as "red neuralgia of the extremities"). The importance of the interaction between sodium currents is highlighted by this example, because the same mutation produces opposite effects in the firing of pain afferents and autonomic motor neurons. nA before (blue plots in left panels) and after (red plots in right panels) a 10 fold increase in the inactivation removal rate a h to simulate OD1 treatment. The value of d m was varied from 57.8 mV (upper panels) to 58 mV (lower panels), while A h was varied from 0.92 (left panels) to 9.2 (right panels). Nav1.7 inactivation (parameter A h = 0.92 in blue plot and 9.2 in red plot, with d m = -58 mV). The red spikes are slightly taller than the blue spikes (7 % average amplitude increase in a 5 s simulation).

  more depolarized value of half activation voltage (d = -40 mV) resembles a HH type high threshold Na + channel (like Nav1.8 in mammals). The second type of Na + channel, with a half activation voltage shifted in the hyperpolarizing direction (d = -55 to -60 mV) and a smaller value of maximum conductance (0.1 S/cm 2 ), represented a Nav1.7. It had faster opening/closing and activating/inactivating rates and a more depolarized value of half inactivation voltage removal than the HH type channel (Table

8

  maximum and minimum values of the voltage oscillation as a function of the amount of external current injected (i.e., a one dimensional bifurcation diagram, e.g., Fig 3 in Doi et al., 2001, for d m = - 58 mV and g max = 0.1 S/cm 2 . All figures of the article were created with Matlab (R2018b, The MathWorks, Inc., Natick, Massachusetts, United States) and finalized with Adobe illustrator (Adobe Inc.).

  Fig 3 we show the effects of varying d m (Equation1) in the model neuron by displaying the response to a depolarizing current step of 0.04 nA lasting 60 ms. This level of current is slightly above the firing threshold at d m = -58 mV and g max = 0.1 S/cm 2 , which is 0.037 nA. At this threshold the one dimensional bifurcation diagram (not shown) presents a discontinuity that corresponds to a Hopf bifurcation (see section Discussion). The upper blue trace represents a normal subject (d m= -55 mV)

  Fig 4 we provide a qualitative frame to understand the situations described in the previous figure. Using a heat map, the spike rate of the neuron model (stimulated with 0.04 nA depolarizing current) is represented with a color bar for different values of maximum conductance g max (Equation 3, abscissas) and half activation voltage parameter d m (Equation 1, ordinates) of the Nav1.7 current. The white circles with numbers on the heat map indicate the coordinates (g max and d m ) used for the simulations depicted in Fig 3. The numbers inside the circles of Fig 4 correspond to the numbers in the panels of Fig 3. Erythromelalgia mutations cause pain through a downward shift toward the red values of the heat map (e.g., trajectories 1 to 2 and 2 to 4). Conversely, the treatment with Nav1.7 blockers (e.g., PF05089771) alleviates pain through a shift to the right, towards the blue values (e.g., trajectories 2 to 3 and 4 to 5 and 6).

  Fig 5) and elicited a faster firing rate if the neuron was already active in the control condition (lower panels in Fig 5). The faster removal of Nav1.7 inactivation (A h varied from 0.92 to 9.2) also increased the model spike amplitude to a moderate amount in the simulation after OD1 (red plot in Fig 6), as observed in the extracellular recordings (Fig 1-B). In Fig 7-A, we show plots of m (thin full lines) and h (dashed lines)

  using voltage clamp whole cell and single channel recordings in Chinese hamster ovary (CHO) cells expressing human Nav1.7 channels. They found that the decay phase of the Nav1.7 channel opening slows down in a concentration-dependent manner in the presence of these toxins. Using a paired-pulse protocol in the voltage clamp configuration, the authors demonstrated an enhancement of Nav1.7 recovery from fast inactivation. Single channel recordings showed that the mean open time of the channels was not substantially changed by the toxin, but the channels exhibited a prolonged flickering behavior between open and closed states, which enabled a more efficient inactivation removal. These experiments pointed to a voltage-sensor trapping interpretation, in which the toxin prevents a conformational change in the domain IV voltage sensor, as the main cause for the observed prolongation in the current duration. Motin et al.

Fig 1 .

 1 Fig 1. Effects of OD1 in nociceptive neurons. (A) Voltage data of the most active channel in MEA 1

Fig 3 .

 3 Fig 3. Erythromelalgia and PF05089771 effects: simulation of membrane voltage. Response of the

Fig 4 .

 4 Fig 4. Erythromelalgia and PF05089771 effects: heat map of the spiking activity. The neuron model

Fig 6 .

 6 Fig 6. Simulation of OD1 effects on the spike amplitude. Spikes with different removal rates of

Fig 7 .

 7 Fig 7. Simulation of OD1 effects on Na + currents and their gating variables. Values of m (thin full

Table 1 .

 1 Default parameter values used for the simulations. Particular cases are indicated in the corresponding figures.

	d (µm)			30	A (ms -1	)	0.1
	cm (µF/cm 2 )	1	k (ms -1 )	0.1
	gNa_hh (S/cm 2 )	0.3	d (mV)		-55
	gNav1.7_hh (S/cm 2 )	0.1	Closing (b m )
	gK_hh (S/cm 2 )	0.15	A (ms -1	)	0.125
				3.00E-
	gL_hh (S/cm 2 )	05	k (ms -1 )	-0.0125
	eL_hh (mV)	-65	d (mV)		-65
	eK (mV)	-90
	eNa (mV)	60
	Initial V (mV)	-75
	dt (ms)	0.025
	Temp (Celsius)	37
	Na HH parameters		Nav1.7 parameters
	Opening (a m )		Opening (a m )
	A (ms -1	)	1	A (ms -1	)	13.78
	k (ms -1 )	0.1	k (ms -1 )	0.1
	d (mV)			-40	d (mV)		-58
	Closing (b m )		Closing (b m )
	A (ms -1	)	4	A (ms -1	)	55.11
	k(ms -1 )		-0.055	k (ms -1 )	-0.055
	d (mV)			-65	d (mV)		-65
	Inactivation removal (a h )		Inactivation removal (a h )
	A (ms -1	)	0.07	A (ms -1	)	0.92
	k (ms -1 )	-0.05	k (ms -1 )	-0.05
	d (mV)			-65	d (mV)		-40
	Inactivation establishment (b h )		Inactivation establishment (b h )
	A (ms -1	)	1	A (ms -1	)	8.76
	k (ms -1 )	-0.1	k (ms -1 )	-0.1
	d (mV_	-35	d (mV)		-35

  Supporting information S1 Appendix. Equations Appendix. Equations are from the HH model formulation provided by the NEURON software and the Channel Builder GUI (https://www.neuron.yale.edu/neuron/static/docs/chanlbild/main.html). Parameter values are listed in Table 1 and provided with the corresponding figures.

Acknowledgements

Study funded by MRC BH171892.


Bozena Maj-Tatsis

Konstantinos Tatsis
email: ktatsis@uoi.gr

Task characteristics that promote mathematical reasoning among young students: An exploratory case study

Keywords: Task design, mathematical reasoning, task characteristics

In the paper we examine possible factors that enhance young students' mathematical reasoning. Particularly, we present the characteristics of seven mathematical tasks, which were chosen based on an a priori analysis, in order to elicit reasoning among a pair of young students. Additionally, we examine the reasoning processes that were manifested during the students' interactions while solving the tasks. Our results show how particular task characteristics affect reasoning processes.

Introduction

Algebra constitutes one of the most significant elements of mathematics, and children, even at a small age, have access to mathematical ideas such as mathematisation, connections, argumentation, number sense and mental computation, algebraic reasoning, spatial and geometric thinking, data and probability sense [START_REF] Perry | Young children's access to powerful mathematical ideas[END_REF]. This has led to the "Early Algebra" movement, which has been the background of a number of studies (see, e.g., [START_REF] Kieran | Early algebra: Research into its nature, its learning, its teaching[END_REF]. The main process involved in algebraic thinkingand in all mathematical activitiesis undoubtedly generalisation [START_REF] Mason | Developing thinking in algebra[END_REF], which therefore has been the focus of the majority of studies in early algebra. However, other significant processes are also involved in algebraic thinking: [START_REF] Blanton | Developing essential understandings of algebraic thinking for teaching mathematics in grades 3-5[END_REF] (as cited in Kieran et al., 2016, p. 12, our emphasis) refer to "the processes of generalizing, representing, justifying, and reasoning with mathematical structure and relationships". One of the questions that may then arise is whether it is possible to design and propose mathematical tasks that enhance reasoning among young students. Having this question in mind we designed a case study aiming to examine how particular task characteristics relate to the reasoning processes of two eight-year-old students. Thus, our main research question is what are the characteristics of the tasks which promote mathematical reasoning. Particularly, we aim to examine how specific factors of the tasks are related to reasoning processes.

Algebraic reasoning and task design

Algebraic thinking is defined as "a process in which students generalize mathematical ideas from a set of particular instances, establish those generalizations through the discourse of argumentation, and express them in increasingly formal and age-appropriate ways" (Kaput & Blanton, 2005, p. 99). This is in line with "Early Algebra", which can be characterised by the following shift:

... from a traditional content-centered characterization of algebra to that of the mathematical reasoning processes and representations that would seem appropriate for young children, as well as to the nature of the early algebra activities that might promote the development of these processes and representations (Kieran et al., 2016, p. 5).

The above statement leads us to the importance of engaging young students in activities that promote reasoning. But how should such activities look like? This question might prove misleading, since there are various factors that contribute to a learning situation: besides the task itself, we may refer to the teacher and the students participating and interacting. This list is not exhaustive; one could also consider the wider social and classroom context, the available resources and artefacts, the histories of the students and their interactions, etc.

In our research paper, we focus on the teacher (as a task designer), as well as on the students (their reasoning activities as they manifest during their interactions). The importance of task design in enhancing students' engagement and promoting mathematical reasoning has been stressed in research focused on reasoning [START_REF] Francisco | Conditions for promoting reasoning in problem solving: Insights from a longitudinal study[END_REF] as well as assessment (van den Heuvel-Panhuizen, 2005). These studies have provided comprehensive suggestions on task characteristics, including the expected teacher and students' roles. They can be summarised as follows:

 "the problems must be accessible, inviting and worthwhile to solve" and "the students must have the opportunity to give their own answers in their own words" (van den Heuvel-Panhuizen, 2005, p. 3); moreover, pictures play an important role as context-bearers and they serve multiple functions (van den Heuvel-Panhuizen, 2005);  "the development of a sense of ownership of the mathematical activity enhances the building of personally meaningful mathematical understanding and students' confidence in their abilities" and working collaboratively and discussing "helps the students become flexible in problem comprehension and adaptive rather [than] routine experts at solving problems" (Francisco & Maher, 2005, p. 371).

Context of the study and methodology

Based on the type of our research questions we have adopted an exploratory case study approach [START_REF] Yin | Case Study Research Design and Methods[END_REF]. The participants were two eight-year-old female students studying together in the 2nd grade of a public primary school in an urban area in Poland. They both had good marks in mathematics and volunteered to participate in the research. The session took place in one of the students' house, lasted for two hours and was videotaped, after the parents' consent was obtained. Both authors of the paper were present in the session and both were familiar to the students, who worked together for solving the tasks. The role of the first author of the paper consisted of the following: challenge pupils, support pupils, evaluate their progress, think mathematically in order to pose challenging and interesting mathematical questions, supply and recall information and promote pupils' reflection [START_REF] Da Ponte | Investigating in mathematics and in learning to teach mathematics[END_REF].

A priori analysis of the tasks

Task design was based on the assumptions mentioned in the previous section. Seven tasks were chosen from two sources [START_REF] Lankiewicz | Matematyka[END_REF][START_REF] Treffers | Grade 1 (and 2). Calculation up to 20[END_REF], based on: their potential to stimulate students' interest, be solvable, or at least approachable, in more than one way and without the use of tricks, illustrate important mathematical ideas, serve as first steps towards mathematical explorations and be extensible and generalisable [START_REF] Schoenfeld | Reflections on doing and teaching mathematics[END_REF]. Table 1 below presents all tasks, except Tasks 3 and 5, which will be presented and analysed later. Task 2: Fish How many fish can you colour by three colours so that every fish is different? You have three colours: green, orange and blue.

Task 7: Caterpillar You choose two numbers less than 100. Each succesive number is the sum of the previous two. For example:

Try to make a 5-part caterpillar with 100 as the last number.

Task 6: The parrot Waku-Waku This is a talking parrot; his name is Waku-Waku. Unfortunately, he can say only one word: five. You want to show your friends how clever parrot you have. Ask him some questions, for which the answer is "five".

Table 1: Five of the given tasks

Our a priori analysis of the tasks initiated by identifying the mathematical concepts and processes involved in the tasks' solutions as anticipated by us and, wherever applicable, by the original task designers. The mathematical fields contained in the tasks correspond to those included in the Polish curriculum for the particular age group. All tasks contained a written description, but in our a priori analysis we also analysed the role of the accompanying images, according to van den Heuvel-Panhuizen's (2005) pictures' functions: motivator (M: making the task more attractive), situation describer (S: describing the context of the task), information provider (I: providing necessary information), action indicator (A: an action is elicited that has the potential of a strategy that leads to a solution), model supplier (MS: the picture structuring possibilities that can be used to solve the problem), and solution and solution-strategy communicator (SC: the picture contains the solution and aspects of the applied strategies). The analysis of the tasks' contexts was based on de Lange's (1999) assessment framework: zero-order (0): fake context, not considered for the solution (to this category we also inserted mathematical contexts, e.g. Task 5, which will be analysed later); firstorder (1): relevant context, needed for the solution; second-order (2): relevant context, mathematisation needed for the solution; third-order (3): the context serves for constructing or reinventing mathematical concepts. Finally, each task was categorised as open or closed. As shown in

A posteriori analysis of the tasks

In our a posteriori analysis, we focused on the interactions that took place among the students and the researcher, during the research session. For this analysis we have adopted Lannin, Ellis and Elliot's (2011) manifestations of essential understandings related to mathematical reasoning: developing conjectures, generalizing to identify commonalities, generalizing by application, conjecturing and generalizing using terms, symbols, and representations, investigating why, justifying based on already-understood ideas, refuting a statement as false, justifying and refuting the validity of arguments and validating justifications. To these we added: reformulating conjectures or justifications, monitoring each other (a form of collaborative reasoning), concluding and explaining (a form of informal justification). After identifying the above understandings, we made a cross-analysis of all tasks, in order to identify commonalities and differences among them.

Results

As we see in Table 3, the reasoning processes which persevered were: developing conjectures and reformulating conjectures or justifications. Next, we present the analyses of Tasks 3 and 5; these tasks are chosen because they demonstrate the range of reasoning processes that we have encountered. In Figure 1, on the left, we read: "Design different boxes in which you can put 12 Christmas baubles in such a way that every bauble is in one compartment." The mathematical concepts come from geometry (shapes, dimensions, area) as well as arithmetic (numbers and multiplication); we anticipated that students will shift from the geometrical representation of the problem to an arithmetic one. The image of the grid can be categorised as a situation describer and a model supplier. The studentssignified by A and M, while R stands for the researcher (the first author of the paper)started by drawing a 4×6 rectangle on the grid and placing their drawn baubles in every second compartment. Initially, it was hard for them to imagine how the baubles can be arranged. Later, they discovered that the box they designed can contain more baubles than they thought, and they justified it. However, they were not able to use this knowledge for the design of boxes of different sizes; they were more focused on the shape, the colours and the sizes of the baubles. In the following excerpts our notes are in brackets and (...) signifies an omitted part of the discussion. The number(s) of the reasoning processes (as they appear in Table 3) are put in brackets at the end of the respective lines. During the whole interaction only two aspects of mathematical reasoning were manifested: justification and reformulating the justification. The students eventually discovered different arrangements of the baubles (in different boxes) but they could not find any relations between them, neither geometric nor arithmetic.

Reasoning processes /

Task 5: Magic squares

In Figure 1, on the right, we read: "Complete the empty fields so that the sum of the numbers in rows and columns is 20"; and then: "Try to create a similar table for your friend or somebody from your family". The main mathematical concept is addition and its properties: identity, commutative and associative. The images of the task are information providers. An extensive analysis of the students' interactions [START_REF] Maj-Tatsis | Investigations in magic squares: a case study with two eightyear-old girls[END_REF] has led to the following understandings related to mathematical reasoning: developing conjectures, generalizing by application, refuting a statement as false, justifying and refuting the validity of arguments, validating justifications, reformulating conjectures, monitoring each other and explaining. Thus, this task has proved fruitful for mathematical reasoning among our students. In the following excerpt the students have completed working on the first magic square and are discussing on the second. It is interesting that student M started filling the square by number 9 (which means that then one has to put 0 in the third square) which was the extreme case in the first square, and it took them some time to accept it. The differences between this task and the Christmas baubles task lie not only in the extent of reasoning processes, but also in their depth. Although in this task the students were able to articulate, justify and validate mathematical conjectures in order to support their mathematical reasoning, in the baubles task they did not manage to cross the boundaries between arithmetic and geometry; this has resulted in limited opportunities for extensive mathematical reasoning.

Cross-analysis of the tasks

We have seen the greatest extent of reasoning processes in Tasks 5 and 7 (followed by Tasks 2 and 4), while Tasks 1, 3 and 6 proved the poorest concerning the manifested reasoning processes by the students. The common characteristic of Tasks 1 and 6 is that they both required the identification of commonalities and explanations; furthermore, the students' interactions have shown that there was no explicit need for justification or generalisation. Tasks 5 and 7 were similar concerning the mathematical concepts involved (addition and its properties); additionally, they were both open, but with a finite number of solutions (unlike Task 6). At the same time, the only task with a single solution (Task 2) allowed for a considerable extent of mathematical reasoning; this was probably due to the many ways of solution. Task 3 was the one that demonstrated the influence of context: the students were engaged in a non-mathematical investigation concerning the size and the type of baubles, which in turn minimised the reasoning processes and hindered a thorough investigation of the task. It might be also the influence of the geometrical nature of the task that had created obstacles in its generalisation.

Discussion

The fact that reasoning processes were manifested in all tasks goes in line with relevant studies of the "Early Algebra" movement and strengthens the claim that students at a young age are able to reason mathematically. The variety of our tasks has proved that young students, even with a minimum help, can develop, justify, validate and refute mathematical conjectures.

Our main research question was what are the characteristics of the tasks which promote mathematical reasoning. Firstly, we have not seen any influence of the images that accompanied the tasks; however, we have seen that context might move students away from reasoning (Task 3). Some tasks have proven more fruitful than others concerning the reasoning processes they evoked.

It seems that open tasks do not pose any difficulties to the students, when they have a finite number of solutions (Tasks 5, 7). The same is the case with closed tasks that can be solved in many different ways (Task 2). On the contrary, our students experienced difficulties when faced with a task that required a shift from geometrical structures to numerical ones (Task 3). At the same time, they manifested a variety of reasoning processes during another geometrical task (Task 4), which did not require an explicit shift to numerical properties. Based on all the above, although our exploratory study has not led to conclusive results, we believe that it may assist the design of mathematical tasks aiming to promote reasoning among young students.

Task 1 :

 1 ChairsHow you can divide the chairs into 2 sets of 4 chairs in each? Task 4: Tiles Uncle Ted tiles three terraces. Here are the tiles he uses.  Help uncle Ted in his work and draw your propositions for the tiling. Check different possibilities. Write under every proposition the number of used tiles.  Try to tile the following terraces by using the tiles of uncle Ted.  Count and write how many tiles you used for every terrace. Do you notice any changes? Which ones?

Figure 1

 1 Figure 1: Task 3 and Task 5, containing students' work Task 3: Christmas baubles

R:

  Could we put more baubles to this box? [the box on the right edge of the grid]. For example, could we put one more bauble? A: I guess not. R: If I had one more bauble and I would tell you: girls, put that bauble to your box… M: I would pack it in such a square. (…) R: (…) then how many could we pack here? A [immediately]: More, 24. R: How do you know that 24? You didn't count. A: Because there are 12 baubles and 12 empty places. [R5] R: And how do you know that there are 12 empty places? A: Because here in the 4 squares there are 4 places and in 2 of them there are baubles and those 2 are left. And in the same way you can count to the bottom so it will be 24. [R9] M: Or 4×6 [showing the dimensions of the rectangle]. (…) [R9] R: Are those boxes different? [showing the four boxes drawn on the grid] A: Uhm. R: By what? [A and M thinking] A: Here are bigger baubles [the second and third box] and here smaller [the first box]. M: And these are medium ones [the fourth box].

M:

  Here for example 8 [in the centre] and put 1 here [above 8] [R1] R: Aha (…) What if here we put 7 [in the centre], then what will be here? [above] when it was 7 then it was 2. And if we decrease it more then it will be 3. [R1, R12] R: What if we put 5? M: Then 4 (…) R: What's the biggest number we can put in the centre? (…) M: For me 9, because 10+11 would be already 21.[R1, R7]

Table 2 ,

 2 all tasks but one were open.

	Task	Field	Concepts	Image	Context	Type
	1	set theory	categorisation	I	1	open
	2	combinatorics	combinations	I	0	closed
	3	geometry & arithmetic	dimensions, shapes, areas, multiplication & properties	S & MS	2	open
	4	geometry & combinatorics	areas, tiling, combinations	I & A	2	open
	5	arithmetic	addition & properties	I	0	open
	6	arithmetic & algebra	numbers, operations, equations	M	0	open
	7	arithmetic	addition, subtraction & properties	I	0	open

Table 2 : Task characteristics

 2

Table 3 : Tasks and reasoning processes

 3

		Tasks	1	2	3	4	5	6	7
	R1	developing conjectures							
	R2	identifying commonalities						
	R3	generalizing by application					
	R4	investigating why							
	R5	justifying based on already-understood ideas				
	R6	refuting a statement as false							
	R7	justifying and refuting the validity of arguments					
	R8	validating justifications					
	R9	reformulating conjectures or justifications							
	R10 monitoring each other							
	R11 concluding							
	R12 explaining						


Nanae Matsuo
email: matsuo@faculty.chiba-u.jp

Nagisa Nakawa
email: nagisa@kanto-gakuin.ac.jp

Preschool children's understanding of length and area measurement in Japan

Keywords: Preschool children, mathematical activity, length measurement, area measurement

This study assesses preschool children's understanding of length and area measurement through the activities of a mathematics program in Japan. Japanese preschool children do not formally learn mathematics; thus, we designed mathematical measurement activities for young, less-experienced preschool teachers to implement for nine five-to six-year-old children. We conducted structured clinical interviews with the children individually, both before and after the measurement activities, and qualitatively analyzed the results by comparing their answers and connecting them to the activity contents. Results indicate that children learn to understand direct comparison through activities; however, it is more difficult to establish understanding of measurement by non-universal units.

Research on measurement in preschool

Much of the current research focuses on elementary mathematics education for preschool children and younger elementary school students (e.g., [START_REF] Brandt | Everyday pedagogical practices in mathematical play situations in German "kindergarten[END_REF]; however, measurement has not been sufficiently included in these studies [START_REF] Sarama | Evaluation of a learning trajectory for length in the early years[END_REF][START_REF] Smith | Learning, teaching, and using measurement: Introduction to the issue[END_REF], even though it is an important real-world area of mathematics used in everyday life. In addition, [START_REF] Hachey | Early childhood mathematics education: The critical issue is change[END_REF] states that mathematical conceptual change in young children is as effective as a substantial change in early childhood mathematical teaching practice, as a large body of developmental research advocates that young children are born mathematicians, and therefore, early childhood mathematics education is vital.

It is known that the development of children's understanding is related to foundational or key ideas of measurement such as comparison, unit iteration, number assignment, and proportionality [START_REF] Lehrer | Developing an understanding of measurement in the elementary grades[END_REF][START_REF] Wilson | Foundational ideas in teaching about measure[END_REF], and several kinds of teaching methods for measurement have been developed [START_REF] Battista | Understanding the development of students' thinking about length[END_REF][START_REF] Kamii | Measurement of length: How can we teach it better?[END_REF]. Although [START_REF] Piaget | Quantification, conservation, and nativism[END_REF] assumed that children up to seven years old are unable to consider more than one stimulus dimension in their judgments, subsequent research has demonstrated that preschoolers can consider two dimensions; for example, they can consider the width and length of rectangles to estimate their area (Wilkening, 1979). Further, [START_REF] Ebersbach | Achieving a new dimension: Children integrate three stimulus dimensions in volume estimations[END_REF] addressed the question of whether children can also take three stimulus dimensions into account and showed that preschoolers have the cognitive competencies required for multidimensional reasoning. There have also been numerous other important studies on preschool children's understanding of measurement. [START_REF] Sarama | Evaluation of a learning trajectory for length in the early years[END_REF] proposed a learning trajectory for length and area in the early years and evaluated it. llner and Benz (2013) concluded that four-to six-year-old children could compare directly and indirectly, but cannot measure using a non-standardized unit. [START_REF] Tzekaki | Teaching intervention for developing generalization in early childhood: the case of measurement[END_REF] indicated that a seven-month intervention helped preschoolers improve their abilities to reflect on their own activities and express their ideas regarding measurement. Skoumpourdi (2015) verified that preschoolers do have some strategies for length measurement, but confuse the concept of length with perimeter and area. Finally, [START_REF] Kotsopoulos | The effects of different pedagogical approaches on the learning of length measurement in kindergarten[END_REF] showed that a free exploration approach in the context of a play-based learning environment is more effective in teaching length measurement than guided instruction and center-based learning. This article examines whether preschool children can improve their understanding of length and area measurement, especially direct and indirect measurement, as well as measurement by non-universal units. It does so with an elementary mathematics play-based program, which can even be implemented by young, less-experienced teachers who have not been trained in mathematics instruction, thereby fostering children's understanding of measurement. These play-based activities are intended as a further contribution to the existing research. The article also seeks to explain the specific reasons the program is effective. First, we compare pre-and post-interview results for five-to six-year-old children. Next, we intentionally choose two children with different degrees of change between their pre-and post-interview results. Finally, we discuss the reason the change occurred, scrutinizing each child's actions and attitude during the activities, as well as his/her post-interview.

The early childhood mathematics program in Japan

Elsewhere, we have proposed a framework for constructing an early childhood mathematical curriculum [START_REF] Matsuo | Framework for an early mathematical preschool curriculum in Japan[END_REF]. The theoretical and methodological underpinnings of our program are social constructivism [START_REF] Ernest | Social constructivism and the psychology of mathematics education[END_REF], mathematical guided-play [START_REF] Weisberg | Guided play: Where curricular goals meet a playful pedagogy[END_REF], and the Structure of the Observed Learning Outcome (SOLO) taxonomy [START_REF] Biggs | Evaluating the quality of learning: SOLO taxonomy[END_REF]. This program presents a scope for the content of preschoolers' activities, while the learning process of those activities are explained from the viewpoint of sequence, based on the relationship between age and modes of representation and focusing on activities for five-to six-year-old children to create a program tailored to these concerns. Based on this framework, we have proposed a mathematical education program to enable a smooth transition from preschool to elementary school mathematical education [START_REF] Matsuo | Shugakuzen sansu kyoiku puroguram no teian: Hirosa kurabe/zukei no hamekomi katsudou ni tsuite. [The preschool mathematical education program: Focus on the domains of area measurement and embedding of geometric figures[END_REF]. Less-experienced preschool teachers should be able to use these activities to better incorporate mathematical content into children's play and recognize activities that will make children think and act mathematically. This practical program was designed for preschools and elementary schools.

Methodology

Research participants

The preschool kindergarten in an urban area of Kanagawa Prefecture of Japan participated in our pilot study. This article focuses on nine five-to six-year-old children in the oldest classes. The children are all Japanese, coming from middle-and upper-middle-class families in Japan. There were six boys and three girls. The school's focus was on physical activities, as well as entrance to candidate schools for the International Baccalaureate (IB). Children started to learn Japanese letters and English to prepare themselves for primary education in September 2017.

The oldest classes of the preschool had two teachers: Teacher A, a male teacher with three years of experience teaching preschool, and Teacher B, a female teacher with two years of experience teaching preschool. Teacher A taught the first three sessions of measurement activities in the first cycle of the program and Teacher B taught the latter three sessions in the second cycle. For the analysis, we intentionally chose two children, T1 and T7, based on the results of pre-and post-interviews, to intensively examine their outcomes and learning processes, as well as to determine the relations between the interview results, their actions, and the detailed observations of the teachers during the activities. T1 was actively engaged in the activities, but provided incorrect answers in the post-interview. T7 was not actively engaged in the activities, but made correct observations and provided correct answers in the post-interview. To compare the two children's understanding, we discuss the level of understanding of direct and indirect measurement, as well as measurement with non-universal units, in connection with the young teachers' instruction.

Data collection and analysis

In the project, we developed six play-based activities related to numbers, measurement, and shapes. Among these activities, we carefully chose those that were better tailored to relate to numbers, shapes, and measurements, things that are necessary for our curriculum to focus on and develop. To evaluate the improvement of children's mathematical skills after these activities, we developed preand post-interview questions. Pre-interviews were conducted in June 2017 and all six activities were implemented twice; therefore, twelve activities were implemented after the pre-interview. After all activities had been completed, a post-interview was conducted with each child in February 2018. Sixteen questions were developed for the pre-/post-interviews, corresponding with the planned activities and previous research by the author [START_REF] Matsuo | Framework for an early mathematical preschool curriculum in Japan[END_REF], as well as the SOLO taxonomy. Four of the sixteen questions were in the area of measurement, as shown in Table 1. Prepare four different pencils, each of a different length, as shown in the diagram, and ask, "Which do you think is the longest pencil of the four? And tell me the order of the length, from longest to shortest."

Methodologically, the authors conducted structured clinical interviews [START_REF] Goldin | Observing mathematical problem solving through task-based interviews[END_REF]. To alleviate children's anxiety, the teachers conducted the interview themselves, while one of the authors sat next to the child to record. The assigned teachers rehearsed the interviews to be consistent with mathematical words and expressions. During the implementation, each child came to the room randomly and the teacher sat in front of the child. The pre-interviews took fifteen to twenty-five minutes, and the post-interviews took ten to fifteen minutes. During the interview, the children could use a ruler, transparent paper, grid paper, and clips to answer questions 12 to 15. The results were qualitatively analyzed.

Description of measurement activities in the intervention

Teacher A taught the children the first course in how to measure the length of something carefully using clips as non-universal units. They measured assorted items by connecting clips (e.g., measuring the edge of the desk, the mat, the height of the teacher). Teacher B taught the second course, in which the children tried to connect eight clips in order to use them to measure shorter objects than the ones measured in the first course. After the children used the connected clips to measure a building block or a box, the teacher asked the children to find objects with lengths equivalent to eight clips. Next, they looked for objects with lengths equivalent to sixteen clips. They lined up sixteen red magnets and set out to measure them using the clips.

In the area measurement activity, the children employed indirect comparison and measured using non-universal units, such as length measurement. The activity was implemented like a code-breaking game, and when the code was solved, figures whose areas were objects for measurement were supposed to be arranged in descending order. A teacher directed the children to count the square grids covered in rectangular paper to measure the total area. First, the children copied the target onto plain paper, put this paper on grid paper, and counted the number of squares as non-universal units. The sizes of the three kinds of rectangles (thirty-two, forty, and forty-eight squares) were relatively large, and the children had been actively counting quite carefully. Finally, the paper with a number written on it was matched with a corresponding letter, which enabled them to solve the code. Letters with meaning were arranged in order so that the numbers became larger.

Results and Discussion

Table 2 shows that there is not a significant difference of the correctness toward questions between length and area measurement. Looking at the transition of the results from the pre-to post-interview, problems whose answers are incorrect in both pre-and post-interview cannot be seen except for T1. In the case of Q12 and Q14, children got the correct answers in the pre-and post-interview, excluding T7. Since the edges of some part of the figures in these problems are aligned, it is possible to judge the difference in size from the difference in other parts by visual observation. Children were thought to be mature, developable, and able to maintain natural demeanor. Regarding Q13 and Q15, about half of the children answered correctly in the pre-interview, but they had incorrect answers in the post-interview. T2 had correct answers for everything both pre-and post-interviews. T9 also had correct answers, except for Q15, in the pre-interview. T8, who was absent from the length and area measurement activity of the second course, was excluded from analysis.

Comprehensive analysis of eight children's understanding of measurement

Table 2 indicates that every child understood direct measurement in Q12 and Q14 in the post-interview, though in Q13 and Q15, a few children did not understand indirect measurement nor measurement by non-universal units. An erroneous view prevails that the area measurement is harder to understand than length measurement, as the number of dimensions is higher. As [START_REF] Ebersbach | Achieving a new dimension: Children integrate three stimulus dimensions in volume estimations[END_REF] stated, not only length and area but also volume can be estimated by a young child, and the rate of correct answers does not change much between children and adults. It is possible for preschoolers to tackle activities that are not limited to length, area, and volume measurement, as we inferred from the results of our survey; however, even though children were working well on the measurement activities, most of them gave incorrect answers for area measurement. T3, T4, and T7 answered incorrectly in the pre-interview but correctly in the post-interview for Q13 and Q15. Although they answered visually for Q13 and Q15, they were correct. We infer the program implemented had a positive effect, especially because the measurement using non-universal units is not commonly done at home or in preschools. While there were scenes in which they could not be regarded as proactively working on the program activities, in many cases, it is mentioned that they were working on collaborative and individual work well and could observe other subjects' behavior.

Child

No. Gender Birthday Q12 Q13 Q14 Q15 Pre Post Pre Post Pre Post Pre Post T1 F MAY 1 1 0 0 1 1 1 0 T2 M JULY 1 1 1 1 1 1 1 1 T3 M NOVEMBER 1 1 0 1 1 1 0 1 T4 F NOVEMBER 1 1 0 1 1 1 0 1 T5 F AUGUST 1 1 1 0 1 1 1 0 T6 M JUNE 1 1 0 1 1 1 1 0 T7 M DECEMBER 0 1 0 1 0 1 0 1 T9 M JULY 1 1 1 1 1 1 0 1
Table 2: Pre-and post-interview results for measurement of length and area * 1 means correct, 0 means incorrect Conversely, T1, T5, and T6 had incorrect answers in the post-interview even if they were correct in the pre-interview, revealing that the influence exerted by the activities of the mathematics program are strongly related to this fact. Let us consider why they did not have correct answers for Q15, even if they were trying to answer based on measurements in non-universal units using tools and enjoyed it in the post-interview. It is probably because children could not distinguish between the measurement of length and area and did not associate the numerical value measured in non-universal units with the results of the comparison of length and area, or because they could not understand the meaning of the work and retain this understanding even if they acquired skills related to measurement. This can be judged from the state of the post-interview.

Qualitative analysis for two children who had different results and processes of playing

We will now compare and examine the results of two children, T7 and T1, their post-interview responses, and the teacher's findings regarding them in Table 3.

T1 was engaged in acting as a group leader in length measurement. After the clip-connecting activity, she was looking for various items to measure, like building blocks, etc. She was highly interested in the length-measurement activity. Further, in the area-measurement activity, she actively worked on counting. She did the measurement work again, but her answer was not correct. She did not seem to have an opportunity to objectively review the meaning of the work after she had concentrated on it. It can be surmised that because the meaning of the work was unclear, she confused length and area. The result of this study is consistent with previous studies (e.g., Skoumpourdi, 2015). The length and the area are different in dimension, and in daily life, for a child, the length comparison is familiar and more understandable than the area comparison. Conversely, T7 did not do much work and wandered around. He did not work as actively as the other children, though he observed the other children's activities well. During area measurement, everyone was absorbed in drawing figures and counting the squares, while he also worked and looked around.

It is highly likely that his reply was affected by this behavior. As a result, T7 who did not actively work, gave the correct answer by visual observation. It can be inferred that this is due to the child's rich sense of length and area, fostered during the activities [START_REF] Schoenfeld | Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics (reprint)[END_REF]. Through these activities, it seems that the sense of long/short, wide/narrow is more sharpened. Preschool children may not have this vocabulary, but instead use "big" or "small" to express differences in quantity. Since children who observed better than doing their own work seemed to grasp the size correctly by visual observation in the post-interview, rather than just doing work, by sharing the work process of early mathematical activities, it can be said that the sense of quantity has been enhanced.

Further, even when it seems that a child is not actively working on a problem, or appears not to be thinking too much, we must accept the fact that the child considers silently and may still answer correctly. According to [START_REF] Shinohara | Kyojyu genron [Teaching principle[END_REF], since observation is an educationally effective activity, like experiments, it is important for children to observe others' behavior and imitate it. In the case of young children, it is often difficult for them to express in words; therefore, it is important for the teacher to observe the situation and guess from the usual activities or the activities with other children.

From the countereffect by the program, it appears that by depending on teachers' questioning, summary, etc., children concentrate on the work without thinking about its meaning.

Q12 She replied visually, pointing with her finger.

Q13

She took the fine-grid tracing paper, and after a little measuring, she answered, "Oh, I understood it." Q14 She replied visually.

Q15

She laughed and took the clips first, and asked, "Can I measure them sideways?" She joined the clips, measured the length of the rectangle sideways, adjusting the length of the joined clips. Next, using the tracing sheet (with a grid), she aligned the edges and measured the length of the rectangle.

Post-test state

She said, "I was looking forward to it today," and "It's fun to find different lengths in measuring activities."

Teachers' findings

She is not good at detailed work. She is good at drawing. She loves playing football and moving her body. She would rather move her body than sit and work. She is interested in the alphabet. Q12 He pointed in a sequential order visually. Q13 He pointed in the descending order visually and quickly without using tools. Q14 He pointed in the descending order visually and quickly. Q15 He pointed in the descending order visually and quickly.

Post-test state

He worked silently. He did not use any tools at all in the measurement, and answered quickly and visually. He enjoyed the math activities and especially enjoyed using the clips to measure the room. He did not seem to be an noteworthy type.

Teachers' findings

He is good at fine work and studying. Currently, he is learning division and fractions progressively. He is shy and not good at speaking in front of people.

T1 T7

Conclusion

Children in this study showed different outcomes of learning in measurement. The children continued to understand direct comparison in all activities; however, it was difficult to establish understanding of measurement by non-universal units. The intervention activities seemed to affect their learning outcomes, although we only focused on how two children's actions related to their interview results. Play-based activities offered by the young teachers influenced children's understanding in both positive and negative ways. This suggests that play-based activities are effective, but should be meaningful and substantial mathematical activities, not superficial. This study only analyzed a small number of children's understanding of measurement, and therefore cannot be generalized further. Future research should analyze a larger number of children.

Q13

  Show four different pencils, each of slightly different length as shown in the diagram, and ask, "Which do you think is the longest pencil of the four? And tell me the order of the length, from longest to shortest." Q14 Prepare four different rectangles as shown in the diagram, and ask, "What do you think is the biggest rectangle of the four? And tell me the order of the area, from biggest to smallest." Q15 Prepare four different rectangles as shown in the diagram, and ask, "What do you think is the biggest rectangle of the four? And tell me the order of the area, from biggest to smallest."

Table 1 : Description of the interview questions and the problem contexts Q12

 1

Table 3 : Results of T1 and T7 for Q12-15, post-interview results and teachers' findings

 3

Acknowledgment

This research was supported by a Grant-in-Aid for Science Research (B) (15H02911) in Japan.