paper uses the _monotonicity sensitivity-axiom_ (this is equivalent to the _transfer-sensitivity axiom_ ) as an additional axiom which can be extremely relevant to measure the progress of educational systems.

The properties of _subgroup consistency_ and _additive decomposability_ are quite useful to understand how much of the overall learning poverty can attributed to the learning poverty of a particular group, or in the nested case, to each one of its components, given that learning poverty can be expressed as a collection of population subgroups. This property requires that the overall learning poverty can be expressed as a population-weighted average of subgroup learning poverty levels, or by the different components of the nested measure.

The learning poverty headcount ratio respects the focus axiom, since it is clearly not sensitive to changes in learning above the deprivation cut-offs. But it violates the stronger version of the monotonicity axiom and monotonicity-sensitivity axiom.

Even so, why would the axiomatic properties often discussed in the poverty and inequality literature also matter in education? In the poverty and inequality literature, the properties are often supported in terms of hypothetical income transfers that have no direct analog in education, since one cannot physically transfer a unit of learning from one individual to another.

Since school-age children should be learning in school settings, given the multiple externalities that those settings offer, the axioms are relevant to comparing educational systems and school-age populations, both in regular times and after exogenous shocks such as the COVID-19 pandemic. The case for monotonicity is simple and builds on the fact that a headcount measure does not distinguish between different levels of learning deprivation among those below the chosen threshold.

There are three main arguments to seek a measure that would satisfy the _monotonicity axiom_ in education. First, it seems plausible that children or educational systems with lower scores among the poor are worse off, other things equal, a difficulty that then an aggregation should reflect. Being worse off means that the lower level of learning is associated with an inefficiency and ineffectiveness in the educational system which fails to provide good quality education to all, as stated in the SDGs. Second, the world is in a learning crisis, with 53 percent of children at age 10 unable to read and understand a simple age-appropriate text.<sup>28</sup> In certain regions such as Africa and country groupings such as low-income countries, the learning poverty headcount ratio approaches 90 percent. In that context, it is critical to use a measure that differentiates levels of learning poverty among countries with similar headcount ratios. Third, financial requirements might differ depending on the level of deprivation in a country's school-age population.<sup>29</sup> The relevance of the monotonicity-sensitive axiom in the context of school-age populations is even greater. First, as previously mentioned, learning is a cumulative and progressive process that requires time, and a good measure should be able to capture foundational changes, which take place at lower levels of the scale. Such sensitivity can help policy makers prioritize foundational learning and prevent the understating of the progress that their educational system might be achieving when investing in foundational skills. Second, there is overwhelming evidence that <mark>teaching at a level too high for students’ proficiency has</mark> a detrimental effect on how much they learn, so a good measure should be able to take student’s initial conditions into consideration.<sup>30</sup> Third, a distribution-sensitive measure captures an important dimension of the complexity of the problem—how unequal the distribution of learning among

> 28 World Bank 2019.

> 29 As Denny (2002) suggests, in allocating funds to alleviate the low learning of different subpopulations with the same sample incidence of low learning, it is clearly helpful to know whether the severity of the problem differs across groups, since more resources would typically be needed.

> 30 Banerjee et al. 2016.

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