# Cyclic shuffle compatibility

Rachel Domagalski
Department of Mathematics, Michigan State University,
East Lansing, MI 48824-1027, USA<EMAIL_ADDRESS>
Jinting Liang
Department of Mathematics, Michigan State University,
East Lansing, MI 48824-1027, USA<EMAIL_ADDRESS>
Quinn Minnich
Department of Mathematics, Michigan State University,
East Lansing, MI 48824-1027, USA<EMAIL_ADDRESS>
Bruce E. Sagan
Department of Mathematics, Michigan State University,
East Lansing, MI 48824-1027, USA<EMAIL_ADDRESS>
Jamie Schmidt
Department of Mathematics, Michigan State University,
East Lansing, MI 48824-1027, USA<EMAIL_ADDRESS>
Alexander Sietsema
Department of Mathematics, Michigan State University,
East Lansing, MI 48824-1027, USA<EMAIL_ADDRESS>
(
Key Words: cyclic permutation, descent, peak, shuffle compatible
AMS subject classification (2010): 05A05 (Primary), 05A19 (Secondary) )

###### Abstract

Consider a permutation $\pi$ to be any finite list of distinct positive
integers. A statistic is a function $\operatorname{St}$ whose domain is all
permutations. Let $\pi\shuffle\sigma$ be the set of shuffles of two disjoint
permutations $\pi$ and $\sigma$. We say that $\operatorname{St}$ is shuffle
compatible if the distribution of $\operatorname{St}$ over $\pi\shuffle\sigma$
depends only on $\operatorname{St}(\pi)$, $\operatorname{St}(\sigma)$, and the
lengths of $\pi$ and $\sigma$. This notion is implicit in Stanley’s work on
$P$-partitions and was first explicitly studied by Gessel and Zhuang. One of
the places where shuffles are useful is in describing the product in the
algebra of quasisymmetric functions. Recently Adin, Gessel, Reiner, and
Roichman defined an algebra of cyclic quasisymmetric functions where a cyclic
version of shuffling comes into play. The purpose of this paper is to define
and study cyclic shuffle compatibility. In particular, we show how one can
lift shuffle compatibility results for (linear) permutations to cyclic ones.
We then apply this result to cyclic descents and cyclic peaks. We also discuss
the problem of finding a cyclic analogue of the major index.

##  1 Introduction

Let ${\mathbb{N}}$ and ${\mathbb{P}}$ be the nonnegative and positive
integers, respectively. If $n\in{\mathbb{N}}$ then let
$[n]=\\{1,2,\ldots,n\\}$. Given a finite $A\subseteq{\mathbb{P}}$ then a
linear permutation of $A$ is a linear arrangement
$\pi=\pi_{1}\pi_{2}\ldots\pi_{n}$ of the elements of $A$. We let

$L(A)=\\{\pi\mid\text{$\pi$ is a linear permutation of $A$}\\}.$

We often drop “linear” if it is understood from context. For example,

$L(\\{1,3,6\\})=\\{136,163,316,361,613,631\\}.$

If $\pi=\pi_{1}\pi_{2}\ldots\pi_{n}$ then $n$ is called the length of $\pi$,
written $\\#\pi=|\pi|=n$. The hash symbol and absolute value sign will also be
used for the cardinality of a set. In fact, we will often treat permutations
as sets if no problem will result and write things such as $x\in\pi$ in place
of the more cumbersome $\pi_{i}=x$ for some $i$.

A (linear) permutation statistic is a function $\operatorname{St}$ whose
domain is all linear permutations. There are four statistics which will
concern us here. A permutation $\pi=\pi_{1}\pi_{2}\ldots\pi_{n}$ has Descent
set

$\operatorname{Des}\pi=\\{i\mid\pi_{i}>\pi_{i+1}\\},$

descent number

$\operatorname{des}\pi=\\#\operatorname{Des}\pi,$

Peak set

$\operatorname{Pk}\pi=\\{i\mid\pi_{i-1}<\pi_{i}>\pi_{i+1}\\},$

and peak number

$\operatorname{pk}\pi=\\#\operatorname{Pk}\pi.$

To illustrate, if $\pi=4218596$ then $\operatorname{Des}\pi=\\{1,2,4,6\\}$,
$\operatorname{des}\pi=4$, $\operatorname{Pk}\pi=\\{4,6\\}$, and
$\operatorname{pk}\pi=2$. We will be particularly concerned with a particular
type of statistic which is determined by the descent set in the following
sense. Statistic $\operatorname{St}$ is a descent statistic if
$\operatorname{Des}\pi=\operatorname{Des}\sigma$ and $\\#\pi=\\#\sigma$
implies $\operatorname{St}\pi=\operatorname{St}\sigma$. Note that
$\operatorname{Des}$ itself, $\operatorname{des}$, $\operatorname{Pk}$, and
$\operatorname{pk}$ are all descent statistics. A statistic which is not a
descent statistic would be $\operatorname{St}\pi=\pi_{1}$ for nonempty $\pi$
and $\operatorname{St}\emptyset=0$. We will often need to evaluate a statistic
$\operatorname{St}$ on a set of permutations $\Pi$. So we define the
distribution of $\operatorname{St}$ over $\Pi$ to be

$\operatorname{St}\Pi=\\{\\{\operatorname{St}\pi\mid\pi\in\Pi\\}\\}.$

Note that this is a set with multiplicity since $\operatorname{St}$ may
evaluate to the same thing on various members of $\Pi$. We will sometimes use
exponents to denote multiplicities where, as usual, no exponent means
multiplicity $1$. For example

$\operatorname{des}\\{123,132,312,321\\}=\\{\\{0,1,1,2\\}\\}=\\{\\{0,1^{2},2\\}\\}.$

We will also need to combine functions on permutations and statistics. If
$\Pi$ and $\Pi^{\prime}$ are sets of permutations and $\operatorname{St}$ is a
statistic, then a function $f:\Pi\rightarrow\Pi^{\prime}$ is
$\operatorname{St}$-preserving if

$\operatorname{St}f(\pi)=\operatorname{St}\pi$

for all $\pi\in\Pi$.

If $\pi\in L(A)$ and $\sigma\in L(B)$ where $A\cap B=\emptyset$ then these
permutations have shuffle set

$\pi\shuffle\sigma=\\{\tau\in L(A\uplus B)\mid\text{$\pi,\sigma$ are subwords
of $\tau$}\\}$

where a subwords of a permutation $\tau$ is a subsequence of (not necessarily
consecutive) elements of $\tau$. For example,

$25\shuffle 73=\\{2573,2753,2735,7253,7235,7325\\}.$ (1)

Whenever we write the shuffle of two permutations, we will tacitly assume that
they are from disjoint sets.

An important use of the shuffle set is in computing the product of two
fundamental quasisymmetric functions in the algebra $\operatorname{QSym}$ of
quasisymmetric functions. More information about $\operatorname{QSym}$ can be
found in the texts of Luoto, Mykytiuk, and van Willigenburg [LMvW13], Sagan
[Sag20], or Stanley [Sta99]. In order to prove that this product formula is
well defined, one needs shuffle compatibility. Roughly speaking, a statistic
$\operatorname{St}$ is shuffle compatible if the distribution of
$\operatorname{St}$ over $\pi\shuffle\sigma$ depends only on
$\operatorname{St}\pi$, $\operatorname{St}\sigma$, and the lengths $\\#\pi$
and $\\#\sigma$. To be precise, $\operatorname{St}$ is shuffle compatible if
for all quadruples $\pi,\pi^{\prime},\sigma,\sigma^{\prime}$ with
$\operatorname{St}\pi=\operatorname{St}\pi^{\prime}$,
$\operatorname{St}\sigma=\operatorname{St}\sigma^{\prime}$,
$\\#\pi=\\#\pi^{\prime}$, and $\\#\sigma=\\#\sigma^{\prime}$ we have

$\operatorname{St}(\pi\shuffle\sigma)=\operatorname{St}(\pi^{\prime}\shuffle\sigma^{\prime}).$

For example, from equation (1) it is easy to see that

$\operatorname{des}(25\shuffle 73)=\\{\\{1^{3},2^{3}\\}\\}$

The reader can check that this is also the distribution
$\operatorname{des}(12\shuffle 43)$ which is because $\operatorname{des}$ is
shuffle compatible. Shuffle compatibility is implicit in Stanley’s theory of
$P$-partitions [Sta72] and he proved a result implying the shuffle
compatibility of $\operatorname{des}$. Gessel and Zhuang [GZ18] were the first
to define the concept explicitly and prove many shuffle compatibility results,
including those for the other three statistics defined previously.

######  Theorem 1.1.

The statistics

$\operatorname{Des},\operatorname{des},\operatorname{Pk},\operatorname{pk}$

are all shuffle compatible. ∎

Other work on shuffle compatibility has been done by Baker-Jarvis and Sagan
[BJS20], by Oğuz [Oğu], and by Grinberg [Gri18].

Recently Adin, Gessel, Reiner, and Roichman [AGRR20] introduced a cyclic
version of quasisymmetric functions with a corresponding cyclic shuffle
operation. A linear permutation $\pi=\pi_{1}\pi_{2}\ldots\pi_{n}$ has a
corresponding cyclic permutation $[\pi]$ which is the set of all its
rotations, namely

$[\pi]=\\{\pi_{1}\pi_{2}\ldots,\pi_{n},\ \pi_{2}\ldots,\pi_{n}\pi_{1},\
\ldots,\ \pi_{n}\pi_{1}\ldots\pi_{n-1}\\}.$

For example

$[3725]=\\{3725,7253,2537,5372\\}$

so that

$[3725]=[7253]=[2537]=[5372].$

The reader should bear in mind the difference between the notation $[n]$ for
an interval of integers and $[\pi]$ for a cyclic permutation. We let

$C(A)=\\{[\pi]\mid\text{$[\pi]$ is a cyclic permutation of $A$}\\}.$

To illustrate

$C(\\{2,3,5,7\\})=\\{[2357],[2375],[2537],[2735],[2573],[2753]\\}.$

We can lift the linear permutation statistics we have introduced to the cyclic
realm as follows. Define the cyclic descent set and cyclic descent number for
a linear permutation $\pi=\pi_{1}\pi_{2}\ldots\pi_{n}$ to be

$\operatorname{cDes}\pi=\\{i\in[n]\mid\text{$\pi_{i}>\pi_{i+1}$ where $i$ is
taken modulo $n$}\\}$

and

$\operatorname{cdes}\pi=\\#\operatorname{cDes}\pi,$

respectively. Returning to $\pi=4218596$ we have
$\operatorname{cDes}\pi=\\{1,2,4,6,7\\}$ and $\operatorname{cdes}\pi=5$. Now a
cyclic permutation $[\pi]$ will have cyclic descent set

$\operatorname{cDes}[\pi]=\\{\\{\operatorname{cDes}\sigma\mid\sigma\in[\pi]\\}\\}.$

Notice that this is a multiset since two different $\sigma$ can have the same
cyclic descent set. Also define the cyclic descent number

$\operatorname{cdes}[\pi]=\operatorname{cdes}\pi.$

Note that this is well defined since all elements of $[\pi]$ have the same
number of cyclic descents. To illustrate

$\operatorname{cDes}[3725]=\\{\\{\operatorname{cDes}3725,\operatorname{cDes}7253,\operatorname{cDes}2537,\operatorname{cDes}5372\\}\\}=\\{\\{\
\\{1,3\\}^{2},\ \\{2,4\\}^{2}\ \\}\\}$

and $\operatorname{cdes}[3725]=2$. An important point later will be that if
$S$ is one of the sets in $\operatorname{cDes}[\pi]$ then all of the other
member sets can be obtained by adding $i$ to the elements of $S$ modulo
$n=\\#\pi$ as $i$ runs over $[n]$.

We deal with peaks in a similar manner, defining the cyclic peak set and
cyclic peak number in the linear case by

$\operatorname{cPk}\pi=\\{i\in[n]\mid\text{$\pi_{i-1}<\pi_{i}>\pi_{i+1}$ where
$i$ is taken modulo $n$}\\}$

and

$\operatorname{cpk}\pi=\\#\operatorname{cpk}\pi,$

respectively. For $\pi=4218593$ we have $\operatorname{cPk}\pi=\\{1,4,6\\}$
and $\operatorname{cpk}\pi=3$. The extension to cyclic permutations is as
expected

$\operatorname{cPk}[\pi]=\\{\\{\operatorname{cPk}\sigma\mid\sigma\in[\pi]\\}\\}$

and

$\operatorname{cpk}[\pi]=\operatorname{cpk}\pi.$

In general, a cyclic permutation statistic is any function
$\operatorname{cSt}$ whose domain is cyclic permutations. And
$\operatorname{cSt}$ is a cyclic descent statistic if
$\operatorname{cDes}[\pi]=\operatorname{cDes}[\sigma]$ and $\\#\pi=\\#\sigma$
implies $\operatorname{cSt}[\pi]=\operatorname{cSt}[\sigma]$. All four of our
cyclic statistics are cyclic descent statistics. Functions preserving cyclic
permutation statistics are defined in the obvious way.

The definitions for shuffles follow the same pattern already established.
Given $[\pi]\in C(A)$ and $[\sigma]\in C(B)$ where $A\cap B=\emptyset$ we
define their cyclic shuffle set to be

$[\pi]\shuffle[\sigma]=\\{[\tau]\in C(A\uplus B)\mid\text{$[\pi],[\sigma]$ are
circular subwords of $[\tau]$}\\}.$

Alternatively, there are rotations $\pi^{\prime}$ and $\sigma^{\prime}$ of
$\pi$ and $\sigma$, respectively, which are both linear subwords of $\tau$. To
illustrate,

$[13]\shuffle[24]=\\{[1324],[1342],[1234],[1432],[1243],[1423]\\}.$ (2)

Call a cyclic permutation statistic $\operatorname{cSt}$ cyclic shuffle
compatible if for all quadruples $[\pi]$, $[\pi^{\prime}]$, $[\sigma]$,
$[\sigma^{\prime}]$ with
$\operatorname{cSt}[\pi]=\operatorname{cSt}[\pi^{\prime}]$,
$\operatorname{cSt}[\sigma]=\operatorname{cSt}[\sigma^{\prime}]$,
$\\#\pi=\\#\pi^{\prime}$, and $\\#\sigma=\\#\sigma^{\prime}$ we have

$\operatorname{cSt}([\pi]\shuffle[\sigma])=\operatorname{cSt}([\pi^{\prime}]\shuffle[\sigma^{\prime}]).$

The aim of the present work is to study this concept. In particular, we will
prove the following theorem.

######  Theorem 1.2.

The statistics

$\operatorname{cDes},\operatorname{cdes},\operatorname{cPk},\operatorname{cpk}$

are all cyclic shuffle compatible.

The rest of this paper is structured as follows. Rather than proving each of
the cases of Theorem 1.2 in an ad hoc manner, we will develop a method for
lifting linear shuffle compatibility results to cyclic ones. This will be done
in the next section. Then in Section 3 we will apply our Lifting Lemma to each
of the four statistics in turn. Finally, we will end with a section of
comments and a future direction for research.

##  2 The Lifting Lemma

In this section we will provide a general method for proving cyclic shuffle
compatibility results as corollaries of linear ones. We will first set some
notation. In the notations like $C(A)$ and $L(A)$ we will often drop the
parentheses if they would cause double delimiters. For example $L[n]$ is all
the linear permutations of $[n]$.

If $A\subset{\mathbb{P}}$ and $n\in{\mathbb{N}}$ then let

$A+n=\\{a+n\mid a\in A\\}.$

In particular,

$[m]+n=\\{n+1,n+2,\ldots,n+m\\}.$

We will also need to add integers to sets modulo some $m\in{\mathbb{P}}$. So
if $A\subseteq[m]$ then define

$A+n\ (\operatorname{mod}m)=\\{a+n\ (\operatorname{mod}m)\mid a\in A\\}$

where the representatives are chosen to be in $[m]$. For example if
$A=\\{2,4,5\\}$ and $m=6$ then

$A+3=\\{5,7,8\\}$

and

$A+3\ (\operatorname{mod}6)=\\{1,2,5\\}.$

We will also need the notion of standardization. Suppose
$A,B\subset{\mathbb{P}}$ with $\\#A=\\#B=n$ and
$\pi=\pi_{1}\pi_{2}\ldots\pi_{n}\in L(A)$. The standardization of $\pi$ to $B$
is

$\operatorname{std}_{B}\pi=f(\pi_{1})f(\pi_{2})\ldots f(\pi_{n})\in L(B)$

where $f:A\rightarrow B$ is the unique order-preserving bijection between $A$
and $B$. To illustrate, if $A=\\{2,4,6,7\\}$ and $B=\\{1,3,8,9\\}$ then
$\operatorname{std}_{B}(4762)=3981$. If $B=[n]$ then we write just
$\operatorname{std}\pi$ for $\operatorname{std}_{[n]}\pi$ and call this the
standardization of $\pi$. Standardization for cyclic permutations is defined
in the analogous manner. For example, $\operatorname{std}[79254]=[45132]$.

We first prove a result about cyclic descent statistics which is a cyclic
analogue of one in the linear case [BJS20].

######  Lemma 2.1.

Let $\operatorname{cSt}$ be a cyclic descent statistic. For any four cyclic
permutations $[\pi]$, $[\pi^{\prime}]$, $[\sigma]$, $[\sigma^{\prime}]$ such
that

$\operatorname{std}[\pi]=\operatorname{std}[\pi^{\prime}]\quad\mbox{and}\quad\operatorname{std}[\sigma]=\operatorname{std}[\sigma^{\prime}]$

we have

$\operatorname{cSt}([\pi]\shuffle[\sigma])=\operatorname{cSt}([\pi^{\prime}]\shuffle[\sigma^{\prime}]).$

###### Proof.

Since $\operatorname{cSt}$ is a cyclic descent statistic, its values only
depend on the relative order of adjacent elements. So it suffices to prove the
case when

$[\pi]\uplus[\sigma]=[\pi^{\prime}]\uplus[\sigma^{\prime}]=[m+n]$

where $m=\\#\pi=\\#\pi^{\prime}$ and $n=\\#\sigma=\\#\sigma^{\prime}$. For
simplicity, let $A=[m]$ and $B=[n]+m$.

We claim that it suffices to find, for any $\pi$ and $\sigma$ as in the
previous paragraph, a $\operatorname{cSt}$-preserving bijection

$[\pi]\shuffle[\sigma]\rightarrow\operatorname{std}_{A}[\pi]\shuffle\operatorname{std}_{B}[\sigma].$

For from this map and the hypothesis of the lemma we have

$\operatorname{cSt}([\pi]\shuffle[\sigma])=\operatorname{cSt}(\operatorname{std}_{A}[\pi]\shuffle\operatorname{std}_{B}[\sigma])=\operatorname{cSt}(\operatorname{std}_{A}[\pi^{\prime}]\shuffle\operatorname{std}_{B}[\sigma^{\prime}])=\operatorname{cSt}([\pi^{\prime}]\shuffle[\sigma^{\prime}]).$

We will show the existence of this bijection by induction on the size of the
set of what we will call out-of-order pairs

$O=\\{(i,j)\in\pi\times\sigma\mid i>j\\}.$

If $\\#O=0$ then $[\pi]\in C(A)$ and $[\sigma]\in C(B)$. It follows that
$[\pi]=\operatorname{std}_{A}[\pi]$ and
$[\sigma]=\operatorname{st}_{B}[\sigma]$ so the identity map will do.

For the induction step, let $\\#O>0$. Then there must be a pair $(i,i-1)\in
O$. Let $\pi^{\prime\prime}=(i-1,i)\pi$ and
$\sigma^{\prime\prime}=(i-1,i)\sigma$ where $(i-1,i)$ is the transposition
which exchanges $i-1$ and $i$. We will be done if we can construct a
$\operatorname{cSt}$ preserving bijection

$T_{i}:[\pi]\shuffle[\sigma]\rightarrow[\pi^{\prime\prime}]\shuffle[\sigma^{\prime\prime}].$

This is because $\pi^{\prime\prime},\sigma^{\prime\prime}$ have fewer out-of-
order pairs and so, by induction, there is a $\operatorname{cSt}$-preserving
bijection
$[\pi^{\prime\prime}]\shuffle[\sigma^{\prime\prime}]\rightarrow\operatorname{std}_{A}[\pi^{\prime\prime}]\shuffle\operatorname{std}_{B}[\sigma^{\prime\prime}]$
which, when composed with $T_{i}$, will finish the construction.

Define $T_{i}$ by

$T_{i}[\tau]=\begin{cases}[(i-1,i)\tau]&\text{if $i-1,i$ are not cyclically
adjacent in $[\tau]$,}\\\ [\tau]&\text{else.}\end{cases}$

We must first check that $T_{i}$ is well defined in that
$T_{i}[\tau]\in[\pi^{\prime\prime}]\shuffle[\sigma^{\prime\prime}]$. This is
true if $i-1$ and $i$ are not cyclically adjacent since $i-1$ and $i$ have
been swapped in all three cyclic permutations involved. If they are adjacent
then the relative order of the elements of $[\tau]$ corresponding to $[\pi]$
and $[\pi^{\prime\prime}]$ are the same, and similarly for $[\sigma]$ and
$[\sigma^{\prime\prime}]$. So leaving $[\tau]$ fixed again gives a shuffle in
the range.

Finally, we need to verify that $T_{i}$ is $\operatorname{cSt}$ preserving.
Since $\operatorname{cSt}$ is a descent statistic, it suffices to show that
$T_{i}$ is $\operatorname{cDes}$ preserving. Certainly this is true if
$[\tau]$ is fixed. And if it is not, then $i-1,i$ are not cyclically adjacent
in $[\tau]$. But switching $i-1$ and $i$ could only change a cyclic descent
into a cyclic ascent or vice-versa if these two elements were adjacent. So in
this case $\operatorname{cDes}[(i-1,i)\tau]=\operatorname{cDes}[\tau]$ and we
are done. ∎

As an example of the map $T_{i}$, consider the shuffle set in (2). Here we can
take $i=3$ since $3\in 13$ and $2\in 24$. For $[\tau]=[1324]$ we have $2$ and
$3$ cyclically adjacent so

$T_{3}[1324]=[1324]\in[12]\shuffle[34]$

as desired. On the other hand, in $[1342]$ the $2$ and $3$ are not cyclically
adjacent so

$T_{3}[1342]=[1243].$

Note that $[1243]\in[12]\shuffle[34]$ and

$\operatorname{cDes}[1243]=\\{\\{\ \\{1,2\\},\ \\{2,3\\},\ \\{3,4\\},\
\\{1,4\\}\ \\}\\}=\operatorname{cDes}[1342].$

We can use the previous lemma to drastically cut down on the number of cases
which need to be checked to obtain cyclic shuffle compatibility. In
particular, the component permutations in the shuffles to be considered can be
on consecutive intervals of integers. And one can keep one component of the
shuffle constant while the other varies.

######  Corollary 2.2.

Suppose that $\operatorname{cSt}$ is cyclic descent statistic. The following
are equivalent.

  1. (a)

The statistic $\operatorname{cSt}$ is cyclic shuffle compatible.

  2. (b)

If $\operatorname{cSt}([\pi])=\operatorname{cSt}([\pi^{\prime}])$ where
$[\pi],[\pi^{\prime}]\in C[m]$ and $[\sigma]\in C([n]+m)$ for some
$m,n\in{\mathbb{N}}$ then

$\operatorname{cSt}([\pi]\shuffle[\sigma])=\operatorname{cSt}([\pi^{\prime}]\shuffle[\sigma]).$

  3. (c)

If $\operatorname{cSt}([\sigma])=\operatorname{cSt}([\sigma^{\prime}])$ where
$[\sigma],[\sigma^{\prime}]\in C([n]+m)$ and $[\pi]\in C[m]$ for some
$m,n\in{\mathbb{N}}$ then

$\operatorname{cSt}([\pi]\shuffle[\sigma])=\operatorname{cSt}([\pi]\shuffle[\sigma^{\prime}]).$

###### Proof.

We will prove the equivalence of (a) and (b) since the equivalence of (a) and
(c) is similar. Clearly (a) implies (b). For the converse, let
$\pi,\pi^{\prime},\sigma,\sigma^{\prime}$ be any four permutations satisfying
the hypothesis of the cyclic shuffle compatible definition and let
$m=\\#\pi=\\#\pi^{\prime}$, $n=\\#\sigma=\\#\sigma^{\prime}$. Also let
$A=[m]$, $B=[n]+m$, $A^{\prime}=[m]+n$, and $B^{\prime}=[n]$. Then, using
Lemma 2.1 and (b) alternately

$\displaystyle\operatorname{cSt}([\pi]\shuffle[\sigma])$
$\displaystyle=\operatorname{cSt}(\operatorname{std}_{A}[\pi]\shuffle\operatorname{std}_{B}[\sigma])$
$\displaystyle=\operatorname{cSt}(\operatorname{std}_{A}[\pi^{\prime}]\shuffle\operatorname{std}_{B}[\sigma])$
$\displaystyle=\operatorname{cSt}(\operatorname{std}_{A^{\prime}}[\pi^{\prime}]\shuffle\operatorname{std}_{B^{\prime}}[\sigma])$
$\displaystyle=\operatorname{cSt}(\operatorname{std}_{A^{\prime}}[\pi^{\prime}]\shuffle\operatorname{std}_{B^{\prime}}[\sigma^{\prime}])$
$\displaystyle=\operatorname{cSt}([\pi^{\prime}]\shuffle[\sigma^{\prime}])$

which is what we wished to prove. ∎

In order to prove the Lifting Lemma, we will need to define two functions. For
$i\in[n]$ we construct the splitting map $S_{i}:C[n]\rightarrow L[n]$ as
follows. If $[\pi]\in C[n]$ then let $S_{i}[\pi]$ be the unique linear
permutation in $[\pi]$ which starts with $i$. For example,

$S_{3}[45132]=32451.$

Also define the maximum removal map $M:C[n]\rightarrow L[n-1]$ by first
applying $S_{n}$ to $[\pi]$ and then removing the initial $n$. To illustrate

$M[45132]=1324.$

Note that $M$ is a bijection. We similarly define $M:C([n]+m)\rightarrow
L([n-1]+m)$ using $m+n$ in place of $n$, as in

$M[67354]=3546.$

Finally, we say that $y$ is between $x$ and $z$ in $[\pi]$ if this is true of
any linear permutation in $[\pi]$ where $x$ occurs to the left of $z$. If
$[\pi]\in C[m]$, $i\in[m]$, and $[\sigma]\in C([n]+m)$ define

$[\pi]\shuffle_{i}[\sigma]=\\{[\tau]\in[\pi]\shuffle[\sigma]\mid\text{only
elements of $\sigma$ are between $m+n$ and $i$ in $[\tau]$}\\}.$

For example,

$[12]\shuffle[34]=\\{[1234],[1243],[1324],[1423],[1342],[1432]\\}$

with

$[12]\shuffle_{1}[34]=\\{[1234],[1243],[1324]\\}$

and

$[12]\shuffle_{2}[34]=\\{[1423],[1342],[1432]\\}.$

Note that $[\pi]\shuffle[\sigma]$ is the disjoint union of the
$[\pi]\shuffle_{i}[\sigma]$ for $i\in[m]$ since we can decompose the full
shuffle set by finding the first element $i\in[m]$ that one encounters
cyclically after $m+n$.

######  Lemma 2.3 (Lifting Lemma).

Let $\operatorname{cSt}$ be a cyclic descent statistic and $\operatorname{St}$
be a shuffle compatible linear descent statistic such that the following
conditions hold.

  1. (a)

For any $[\tau],[\tau^{\prime}]$ of the same length

$\operatorname{St}(M[\tau])=\operatorname{St}(M[\tau^{\prime}])\quad\mbox{implies}\quad\operatorname{cSt}[\tau]=\operatorname{cSt}[\tau^{\prime}].$

  2. (b)

Given any $[\pi],[\pi^{\prime}]\in C[m]$ such that
$\operatorname{cSt}[\pi]=\operatorname{cSt}[\pi^{\prime}]$, there exists a
bijection $f:[m]\rightarrow[m]$ such that for all $i$ and $j=f(i)$

$\operatorname{St}(S_{i}[\pi])=\operatorname{St}(S_{j}[\pi^{\prime}]).$

Then $\operatorname{cSt}$ is cyclic shuffle compatible.

###### Proof.

By Corollary 2.2, it suffices to show that if $[\pi],[\pi^{\prime}]$ are as
given in (b) and $\sigma\in C([n]+m)$ then
$\operatorname{cSt}([\pi]\shuffle[\sigma])=\operatorname{cSt}([\pi^{\prime}]\shuffle[\sigma])$.
The remarks preceding the lemma show that this reduces to proving

$\operatorname{cSt}([\pi]\shuffle_{i}[\sigma])=\operatorname{cSt}([\pi^{\prime}]\shuffle_{j}[\sigma])$
(3)

for all $i\in[m]$ and $j=f(i)$.

It follows that
$\operatorname{St}(S_{i}[\pi])=\operatorname{St}(S_{j}[\pi^{\prime}])$ by (b).
So, since $\operatorname{St}$ is shuffle compatible, we have
$\operatorname{St}(S_{i}[\pi]\shuffle\sigma^{\prime})=\operatorname{St}(S_{j}[\pi^{\prime}]\shuffle\sigma^{\prime})$
where $\sigma^{\prime}=M[\sigma]$. Thus there is an
$\operatorname{St}$-preserving bijection

$\theta:S_{i}[\pi]\shuffle\sigma^{\prime}\rightarrow
S_{j}[\pi^{\prime}]\shuffle\sigma^{\prime}.$

This gives rise to a map

$\theta^{\prime}:[\pi]\shuffle_{i}[\sigma]\stackrel{{\scriptstyle
M}}{{\rightarrow}}S_{i}[\pi]\shuffle\sigma^{\prime}\stackrel{{\scriptstyle\theta}}{{\rightarrow}}S_{j}[\pi^{\prime}]\shuffle\sigma^{\prime}\stackrel{{\scriptstyle
M^{-1}}}{{\rightarrow}}[\pi^{\prime}]\shuffle_{j}[\sigma]$

Since this is a bijection, to show (3) it suffices to prove that
$\theta^{\prime}$ is $\operatorname{cSt}$-preserving. So take
$[\tau]\in[\pi]\shuffle_{i}[\sigma]$ and
$[\tau^{\prime}]=\theta^{\prime}[\tau]$. Then $M[\tau^{\prime}]=\theta\circ
M[\tau]$. Since $\theta$ is $\operatorname{St}$ preserving, we have
$\operatorname{St}M[\tau^{\prime}]=\operatorname{St}M[\tau]$. But then
hypothesis (a) implies that
$\operatorname{cSt}[\tau]=\operatorname{cSt}[\tau^{\prime}]$ which is what we
wished to prove. ∎

##  3 Applications

The hypotheses in the Lifting Lemma may seem strange at first glance. But they
can be quite easy to verify, making it a useful tool. We will see four
instances of this by proving Theorem 1.2 using its aid.

######  Theorem 3.1.

The statistic $\operatorname{cDes}$ is cyclic shuffle compatible.

###### Proof.

We will verify the hypotheses of Lemma 2.3 using
$\operatorname{St}=\operatorname{Des}$. For (a), suppose that
$\\#\tau=\\#\tau^{\prime}=n$ and

$\operatorname{Des}(M[\tau])=\operatorname{Des}(M[\tau^{\prime}]).$ (4)

Let us assume that $\tau,\tau^{\prime}$ were chosen from their cyclic
equivalence class so that $\tau_{1}=\max\tau$ and similarly for
$\tau^{\prime}$. Then $M[\tau]=\tau_{2}\tau_{3}\ldots\tau_{n}$. And by the
choice of $\tau_{1}$ we see that

$\operatorname{Des}\tau=\\{1\\}\uplus(\operatorname{Des}M[\tau]+1).$ (5)

Since the same statement holds with $\tau^{\prime}$ in place of $\tau$ and (4)
holds, we have $\operatorname{Des}\tau=\operatorname{Des}\tau^{\prime}$. But
$\operatorname{Des}\tau$ is one set in the multiset
$\operatorname{cDes}[\tau]$, and the others are gotten by adding each
$i\in[n]$ to all elements of $\operatorname{Des}\tau$ modulo $n$. The same
being true of $\operatorname{cDes}[\tau^{\prime}]$ shows that
$\operatorname{cDes}[\tau]=\operatorname{cDes}[\tau^{\prime}]$ as desired.

For (b), we are given $[\pi],[\pi^{\prime}]\in C[m]$ with
$\operatorname{cDes}[\pi]=\operatorname{cDes}[\pi^{\prime}]$ and must
construct the necessary bijection. From this assumption, we can choose $\pi$
and $\pi^{\prime}$ such that

$\operatorname{cDes}\pi=\operatorname{cDes}\pi^{\prime}=A.$ (6)

for some set $A$. Define $f:[m]\rightarrow[m]$ by
$f(\pi_{k})=\pi_{k}^{\prime}$ for all $k\in[m]$. Let $i=\pi_{k}$ and
$j=\pi_{k}^{\prime}$. To show that
$\operatorname{Des}(S_{i}[\pi])=\operatorname{Des}(S_{j}[\pi^{\prime}])$ there
are two cases depending on whether $k-1\in A$ or not, where $k-1$ is taken
modulo $m$.

If $k-1\not\in A$ then $\pi_{k},\pi_{k}^{\prime}$ are not the second elements
in cyclic descents of their respective permutations. From this and equation
(6) it follows that

$\operatorname{Des}(S_{i}[\pi])=A-k+1\
(\operatorname{mod}m)=\operatorname{Des}(S_{j}[\pi^{\prime}]).$

If $k-1\in A$ then the same equalities hold with $A$ replaced by $A^{\prime}$
which is $A$ with $k-1$ removed. The completes the proof of (b) and of the
theorem. ∎

######  Theorem 3.2.

The statistic $\operatorname{cdes}$ is cyclic shuffle compatible.

###### Proof.

We proceed as in the previous proof with
$\operatorname{St}=\operatorname{des}$. For (a) we assume
$\operatorname{des}(M[\tau])=\operatorname{des}(M[\tau^{\prime}])$. But from
equation (5) we see that
$\operatorname{des}\tau=\operatorname{des}(M[\tau])-1$ and the same is true
for $\tau^{\prime}$. Combining this with our initial assumption gives
$\operatorname{des}\tau=\operatorname{des}\tau^{\prime}$. But
$\tau,\tau^{\prime}$ begin with their largest element so that

$\operatorname{cdes}[\tau]=\operatorname{des}\tau=\operatorname{des}\tau^{\prime}=\operatorname{cdes}[\tau^{\prime}]$

which is what we wished to show.

To prove (b), we are given
$\operatorname{cdes}[\pi]=\operatorname{cdes}[\pi^{\prime}]$. As in the
preceding paragraph, choose $\pi,\pi^{\prime}$ to begin with their largest
elements so that $\operatorname{cdes}[\pi]=\operatorname{des}\pi$ and
similarly for $\pi^{\prime}$. Since
$\operatorname{des}\pi=\operatorname{des}\pi^{\prime}$ there is a bijection
$\theta:\operatorname{Des}\pi\rightarrow\operatorname{Des}\pi^{\prime}$.
Extend this map to $\theta:[m]\rightarrow[m]$ by using any bijection between
the complements of $\operatorname{Des}\pi$ and
$\operatorname{Des}\pi^{\prime}$. The proof that $\theta$ has the desired
property is now similar to that for $\operatorname{Des}$ and so is left to the
reader. ∎

######  Theorem 3.3.

The statistic $\operatorname{cPk}$ is cyclic shuffle compatible.

###### Proof.

This proof parallels the one for $\operatorname{cDes}$, so we will only
mention the highlights. For (a) one sees that
$\operatorname{Pk}\tau=\operatorname{Pk}M[\tau]+1$ since $\tau_{1}=\max\tau$
is not a peak in $\tau$ and is removed in $M[\tau]$. But this still implies
that $\operatorname{Pk}\tau=\operatorname{Pk}\tau^{\prime}$ and the rest of
this part of the demonstration goes through.

For (b), the map $\theta$ is constructed in exactly the same way using
$A=\operatorname{cPk}\pi=\operatorname{cPk}\pi^{\prime}$. The only difference
with the remaining part of the proof is that there are two subcases when
$k-1\not\in A$ depending on whether $k-2\in A$ or not. If $k-2\in A$ then one
loses the peak which was at $\pi_{k-2}$ in $S_{i}[\pi]$. On the other hand,
the peak set stays the same modulo rotation if $k-2\not\in A$. But since the
analogous statements hold for $\pi^{\prime}$, the manipulations in these
subcases are as before. ∎

The proof of the next result is based on the proof of Theorem 3.3 in much the
same way that the demonstrations of Theorems 3.1 and 3.2 are related. So the
details are left to the reader.

######  Theorem 3.4.

The statistic $\operatorname{cpk}$ is cyclic shuffle compatible. ∎

Lest it appear that cyclic shuffle compatibility follows exactly the same
lines as the linear case, let us point out a place where they differ. Define a
factor of $\pi$ to be a subsequence of consecutive elements. A birun of $\pi$
is a maximal monotone factor. Let $\operatorname{bru}\pi$ be the number of
biruns of $\pi$. For example, $\operatorname{bru}125346=3$ because of the
biruns $125$, $53$, and $346$. It is easy to see [GZ18] that the birun
statistic is not linearly shuffle compatible. A cyclic birun of $[\pi]$ is
defined in the obvious manner and denoted $\operatorname{cbru}[\pi]$.
Returning to our example, $\operatorname{cbru}[125346]=4$ because of the
biruns above and $61$.

######  Theorem 3.5.

The statistic $\operatorname{cbru}$ is cyclic shuffle compatible.

###### Proof.

Clearly cyclic biruns always begin at a cyclic peak and end at a cyclic
valley, or vice-versa. So $\operatorname{cbru}[\pi]=2\operatorname{cpk}[\pi]$
and the result follows from Theorem 3.4. ∎

##  4 Remarks and an open question

###  4.1 Cyclic pattern avoidance and statistics

We say that $[\sigma]\in C[n]$ contains the pattern $[\pi]\in C[m]$ if there
is some subsequence $[\sigma^{\prime}]$ of $[\sigma]$ such that
$\operatorname{std}[\sigma^{\prime}]=[\pi]$. Otherwise $[\sigma]$ avoids
$[\pi]$. For any set $[\Pi]$ of cyclic permutations we let

$\operatorname{Av}_{n}[\Pi]=\\{[\sigma]\in C[n]\mid\text{$[\sigma]$ avoids
$[\pi]$ for all $[\pi]\in[\Pi]$}\\}.$

Callan calculated $\\#\operatorname{Av}_{n}[\Pi]$ when $[\Pi]$ consists of a
single element of $C[4]$. Gray, Lanning, and Wang then contributed work on
cyclic packing of patterns [GLW18] and patterns in colored cyclic permutations
[GLW19]. Most recently, the authors of this article determined
$\\#\operatorname{Av}_{n}[\Pi]$ for all $[\Pi]\subseteq C[4]$. They also
calculated the generating function

$D_{n}([\Pi];q)=\sum_{[\sigma]\in\operatorname{Av}_{n}[\Pi]}q^{\operatorname{cdes}[\sigma]}.$

for the same sets of patterns.

###  4.2 Cyclic maj?

One famous linear statistic which we have not discussed so far is the major
index of $\pi$ which is

$\operatorname{maj}\pi=\sum_{i\in\operatorname{Des}\pi}i.$

Stanley’s theory of $P$-partitions [Sta72] shows that if $\\#\pi=m$ and
$\\#\sigma=n$ then

$\sum_{\tau\in\pi\shuffle\sigma}q^{\operatorname{maj}\tau}=q^{\operatorname{maj}\pi+\operatorname{maj}\sigma}\genfrac{[}{]}{0.0pt}{}{m+n}{m}_{q}$

where the last factor is a $q$ binomial coefficient. Note that this is a
$q$-analogue of the fact that $\\#(\pi\shuffle\sigma)=\binom{m+n}{m}$.

It is easy to see that

$\\#([\pi]\shuffle[\sigma])=(m+n-1)\binom{m+n-2}{m-1}.$ (7)

Indeed, write each $[\tau]$ in the shuffle so that it starts with $\pi_{1}$.
Then there are $m+n-1$ more slots to be filled and $m-1$ of them must be for
the elements $\pi_{2}\pi_{3}\ldots\pi_{m}$ in that order. The remaining slots
can be filled with any of the $n$ rotations of $\sigma$ for a total of
$n\binom{m+n-1}{m-1}$ choices. This is equivalent to the formula above. In
[AGRR20] the following question was raised.

######  Question 4.1.

Is there a cyclic analogue of $\operatorname{maj}$ which has nice properties
such as giving a $q$-analogue of equation (7)?

More generally, one could try to find shuffle compatible cyclic analogues of
other linear statistics.

## References

  * [AGRR20] Ron M. Adin, Ira M. Gessel, Victor Reiner, and Yuval Roichman.  Cyclic quasi-symmetric functions.  Israel J. Math., 243(1):437–500, 2021.
  * [BJS20] Duff Baker-Jarvis and Bruce E. Sagan.  Bijective proofs of shuffle compatibility results.  Adv. in Appl. Math., 113:101973, 29, 2020.
  * [Cal] David Callan.  Pattern avoidance in circular permutations.  See preprint at arXiv:0210014.
  * [DLM+] Rachel Domagalski, Jinting Liang, Quinn Minnich, Bruce Sagan, Jamie Schmidt, and Alexander Sietsema.  Cyclc pattern containment and avoidance.  See preprint at arXiv:2106.02534.
  * [GLW18] Daniel Gray, Charles Lanning, and Hua Wang.  Pattern containment in circular permutations.  Integers, 18B:Paper No. A4, 13, 2018.
  * [GLW19] Daniel Gray, Charles Lanning, and Hua Wang.  Patterns in colored circular permutations.  Involve, 12(1):157–169, 2019.
  * [Gri18] Darij Grinberg.  Shuffle-compatible permutation statistics II: the exterior peak set.  Electron. J. Combin., 25(4):Paper No. 4.17, 61, 2018.
  * [GZ18] Ira M. Gessel and Yan Zhuang.  Shuffle-compatible permutation statistics.  Adv. Math., 332:85–141, 2018.
  * [LMvW13] Kurt Luoto, Stefan Mykytiuk, and Stephanie van Willigenburg.  An introduction to quasisymmetric Schur functions.  SpringerBriefs in Mathematics. Springer, New York, 2013.  Hopf algebras, quasisymmetric functions, and Young composition tableaux.
  * [Oğu] Ezgi Kantarcı Oğuz.  A counter example to the shuffle compatiblity conjecture.  Preprint arXiv:1807.01398.
  * [Sag20] Bruce E. Sagan.  Combinatorics: The art of counting, volume 210 of Graduate Studies in Mathematics.  American Mathematical Society, Providence, RI, [2020] ©2020.
  * [Sta72] Richard P. Stanley.  Ordered structures and partitions.  American Mathematical Society, Providence, R.I., 1972.  Memoirs of the American Mathematical Society, No. 119.
  * [Sta99] Richard P. Stanley.  Enumerative combinatorics. Vol. 2, volume 62 of Cambridge Studies in Advanced Mathematics.  Cambridge University Press, Cambridge, 1999.  With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin.


longtable

## Maths for Einstein’s Universe -
Tools for Understanding Modern Reality.

Anastasia Popkova a, David Blair a and David Treagust b

Corresponding author: Anastasia Popkova<EMAIL_ADDRESS>
a Department of Physics, The University of Western Australia, Perth WA 6009,
Australia

b STEM Education Research Group, School of Education, Curtin University, Perth
WA 6102, Australia

Keywords: mathematics and science education, curriculum development,
Einsteinian physics

### Abstract

Aversion to mathematics is a recognised and widespread problem. Following a
review of the literature on this subject, this paper presents an education
program which has been developed to test the hypothesis that transferring
attention from traditional school arithmetic to a broad range of mathematical
skills relevant to modern science at an early age (ages 7-12) will improve
students’ attitudes to mathematics, reduce the incidence of maths anxiety and
prepare students for topics normally introduced at more senior levels. The
program entitled _Maths for Einstein’s Universe_ includes five modules
covering extreme numbers, estimation, probability, vectors and curved space
geometry taught through group activities, games and plays. The modules
complement appropriate early learning of modern physical concepts from the
subatomic world to cosmology. While connected to science, the program aims to
provide meaning and comprehension for socially relevant topics from national
budgets to pandemics and opinion polls. The program has been trialled in
multiple short workshops and extended learning programs as well as training
programs for school teachers. Analysis of knowledge and attitude tests and
questionnaires from about 170 participants demonstrate strong student
enthusiasm and positive learning outcomes in areas normally considered beyond
the ability of students in this age group. Trial results were used to identify
strategies for enhancing school mathematics based on creation of stronger
links between mathematics and science. We summarise results of pilot trials.
In the paper we present the results of learning powers of ten and vectors. In
total, around 700 participants have trialled Maths Einstein’s Universe with
nearly 200 hours of teaching for students and teachers.

Introduction

_Einstein-First project._ In the early 20th century discoveries in physics
created a new understanding of space, time, matter and radiation, described as
Einsteinian physics. It has given us knowledge of the smallest-scale phenomena
in the universe through to the vast distances of the visible universe, and
with it, the modern technologies on which our lives depend, and with which we
perceive the universe. The Einstein-First project is an educational response
to this revolutionary change in human understanding. It aims to develop a
modernised school science curriculum so that all students by year 10, their
final year of compulsory science education in Australia, achieve a basic
understanding of our current best understanding of physical reality. An
essential component of this understanding is mathematics.

The distinguished Harvard and Oxford educator Jerome Bruner stated, “Any
subject can be taught effectively in some intellectually honest form to any
child at any stage of development” (Bruner, 1976). In the spirit of Bruner,
this project is designed to determine an optimum approach and sequence for
meeting our goal. By necessity, the program must be powerful, fun and mind-
expanding. It is based on numerous peer-reviewed published results.

Many studies have demonstrated that Einsteinian Physics can be successfully
integrated into a curriculum at any level of schooling [Pitts, 2014; Kaur
(Part 1-3, 2017; Choudhary, 2018; Choudhary, 2019; Foppoli, 2019; Choudhary,
2021] when taught at the appropriate level using modern approaches including
activity-based learning with toys and models.

One component that has not been emphasised in the above research is the
development of mathematical concepts that need to be developed in parallel
with the concepts of Einsteinian physics. The need for modernised mathematics
is partly motivated by physics trials where limitations and contradictions
arose because of inadequate mathematical knowledge, but also by the fact that
the same mathematical concepts are highly relevant to modern life like
pandemic prognosis or financial risk. For this reason, in parallel with the
fundamental modernisation of science education, Einstein-First introduced a
goal of re-thinking mathematics education. The program is entitled _Maths for
Einstein’s Universe (MEU)._

Aiming to make the mathematics needed for describing physical reality more
intuitive, more relevant to students’ experience, and less dependent on rote
learning, we ask whether Fermi estimation [Philip] might be more important
than times tables, and whether the conceptual understanding and representation
of scale might be more important than arithmetic.

_Maths aversion and Maths anxiety._ Math anxiety (Commodari, 2016) is
recognised as a widespread issue for all age groups. For example,
approximately 93% of adult US Americans indicated that they experience some
level of math anxiety (Luttebberger, 2018). Maths aversion and anxiety are
widely investigated psychological problems (Stodolsky, 1985), which are not
only affecting the psychological well-being of children (Winarso, 2019) but
also affect students’ performance at school (Passolunghi, 2017). The Programme
for International Student Assessment (PISA) studies in 2012 investigated Maths
anxiety amongst 15–16-year-old students in parallel with their maths
performance. Researchers found a strong negative correlation between students’
achievements and reported math anxiety (Luttebberger, 2018). Furthermore, this
correlation remained stable for several assessment periods (Pisa 2012; Lee,
2009). The consequences of math anxiety among students include avoiding math
and math-related subjects in school, as well as aversion to pursuing STEM
professions in the future. (Ashcraft, 2007).

One of the effective strategies to confront maths anxiety was reported as
hands-on and whole-body learning (Thuneberg, 2016). Researchers argued that
human cognitive processes depend on physical perception and consequently, body
movement (Valentini, 2019). A well-known historical example is fingers-
counting. Using the body as an effective tool for understanding reality makes
advanced mathematical concepts less abstract. Whole-body learning has been
recognized as a powerful method for the comprehension of some mathematical
concepts such as fractions (Isbister, 2018). Einstein-First uses whole-body
learning to help minimise maths anxiety.

Another successful strategy (Maass, 2023) for alleviating Maths anxiety is
connecting maths to real-life situations. This makes maths relevant to
students' interest so that it seen as a useful tool, leading to a build-up of
confidence. Examples include estimation of pocket money, mapping, and maths
for sporting competitions.

The approach presented here emphasises maths as a tool for the world, avoiding
abstraction by using toys and activities, as well as connecting it to physics
concepts as well as questions of students’ interests which could be the number
of galaxies in the Universe, the size of atoms, or the total mass of humans.

One of the negative factors which contribute to maths anxiety is putting
students in embarrassing situations when they cannot find the answer and feel
unable to solve a problem. Einstein-First uses toys for teaching. Children are
familiar with toys and every child naturally knows how to play and what to do.
It makes every student confident and emotionally positive about learning. Once
a student has improved maths performance, the positive outcome provides
encouragement for future learning. Einstein-First has shown that learning with
toys and activities, low-achieving students have similar improvements as
higher-achieving students (Kaur, Part 1, 2017). We hypothesise that the
improvement will also apply to the maths component.

According to (Balazer, 2020). female students are more influenced by maths
anxiety than male students. The success of the Einstein-First program
regarding girls' performance (Kaur, 2020) indicates that the same approach may
be beneficial for alleviating female maths anxiety.

The Einstein-First starts its learning progression in Year 3 (7-8 years of
age), a year before Balazar suggests that maths anxiety begins to be
manifested. We hypothesise that activity-based mathematics learning analogous
to the Einstein-First program (Kaur, Part 2, 2017) may help ensure that
students develop a positive attitude to maths, preventing maths anxiety before
it starts.

The structure of the program

To meet the goal of introducing all students to the core understandings of
modern physics by Year 10 we identified five areas in which mathematical
concepts need to be introduced early in schooling. They are summarised below,
first in brief, then each is expanded to explain the links and the sequence.
Finally, we summarise the module content in table 1. The five areas of
understanding are:

1) The development of logarithmical thinking (Mahajan, S. 2018) to facilitate
understanding of scale. 2) The development of estimation skills to allow
students to understand magnitudes without detailed arithmetic.

3) Development of vector thinking to allow students to develop skills in
symbolic representation for understanding the addition of forces, momentum,
waves and quantum probability.

4) The development of probabilistic thinking (Johnson-Laird, 1994) to
facilitate understanding of the intrinsic probabilistic nature of the quantum
world.

5) Geometry of curved space to allow students to understand Einsteinian
gravity and comprehend gravitational lensing images and time dilation.

Table 1: Five modules for Maths for Einstein Universe provide a mathematical tool for better understanding of Einsteinian physics concepts from theory of relativity and cosmology to the quantum world. ()  Module name |  Child-friendly name |  Module Description
---|---|---
() |  |
Module 1. Extreme numbers |  Powers of the Universe |  This module uses activities to introduce students to the world of huge and tiny numbers from the size of atoms to the age of the universe. Powers of two activities provide a step from linear thinking to logarithmic thinking. The transition to powers of ten reinforces and extends students’ logarithmic mental number line, allowing the powers of ten notational tool to be used to conceptualise the extreme scale range of the universe. Students develop comparison and estimation skills for extreme numbers, as well as learning how to multiply and divide these numbers, making extreme numbers easy manageable with simple arithmetic. They are empowered by new-found ability to estimate the vast scale of the universe.
Module 2. Estimation |  Roughly right is better than exactly wrong |  Fermi’s statement _Roughly right is better than exactly wrong_ defines the theme of this module that builds on module 1, emphasising powers of ten notation as tools for obtaining “roughly right” answers to estimation problems. combining it with statistical sampling concepts. Students learn how to estimate extreme numbers, such as the number of _stars in the universe_ or _the number of solar photons hitting their palms._ Through experiment they discover the $\sqrt{N}$ rule for sampling which is relevant to both opinion polls and photon imaging in telescopes.
Module 3. Vectors |  Maths of arrows |  Students learn to connect physical concepts with symbolic representation in the form of arrows. They discover experimentally that the addition of arrows is commutative. Through whole-body force activities and the graphical addition of arrows, they recognise that both scalars and vectors are needed to describe the world. Vector understanding is applied to navigation, acceleration, forces, magnetism, wind and waves, and classical and quantum interference, all represented graphically. They are introduced to the phasor wheel, a rolling wheel toy used to connect wave motion to a rotating vector.
Module 4. Probability |  Maths of chance |  From Newtonian certainty to Einsteinian uncertainty: the intrinsic probabilistic nature of our world is revealed by considering photons reflecting off windows. We see quantum uncertainty in beam splitters, light interference and radioactivity – they are all _chance machines_! This is contrasted with uncertainty for coin tossing and dice where randomness comes from human errors. Students play games that reveal the concept of statistical distributions and do experiments to reveal the $\sqrt{N}$ rule. Applying this to extreme numbers, students learn how chance connects to _statistical certainty_.
Module 5. Curved space |  Curved Space |  This module generalises geometry to the real non-Euclidean geometry of spacetime. Students study geometry on curved surfaces to learn that geometry reveals the shape of space. They study the properties of straight lines in curved space, discover that _straight is curved if space is curved,_ and discover the violation of Euclid’s third postulate and the general breakdown of Euclidean geometric formulae. A role-play asks _what is the space we live in?_ and reveals how concepts of space and gravity changed over the years, culminating in the modern understanding that gravity is curved spacetime. The tiny magnitudes of curved space errors on Earth are contrasted with their dramatic effects in astronomical images.
() |  |

The overarching aim of MEU is to teach students that maths is more than just
numbers and create anticipation of future learning. The modules are described
using child-friendly names (Table 1). Each can be delivered at higher or lower
levels of complexity according to the student age group. To develop a learning
sequence for the five core topics, we took into account a) the current school
mathematical curriculum, b) results from trials in which we assessed students’
ability to conceptualize the relevant mathematical ideas, c) the mathematics
requirements needed to match and enrich the already developed Einstein-First
year 3 to 10 science learning progression, and d) the progression of
mathematical learning from module to module. The five modules form an
interrelated set which naturally forms a sequence while allowing for
repetition, extension and reinforcement of concepts throughout the program.
The connections between the five modules are illustrated schematically in
Figure 1.

Figure 1: Maths beyond arithmetic; teaching sequence and interconnections
between five modules of Maths of Einstein’s Universe. The grey arrows in-
between areas indicate topics, that can be developed only with pre-knowledge
from other modules.

We start with _Module 1: Extreme Numbers_. All modern areas of physics involve
enormous numbers whether it be cosmological distances or the number of atoms
in a cup of water, as well as the tiny numbers needed to describe the quantum
world. All of these are beyond our direct perception but are accessible
through mathematical reasoning if we first introduce powers of ten and the
logarithmic number line, as described in Table 1. The first module leads into
_Module 2: Estimations_ because powers of ten notation always involves
approximation in the form of rounding. Approximation is a challenging topic
for students for whom mathematics has focussed on finding single correct
answers but powers of ten exercises in which numbers are approximated to the
nearest power of ten and placed on a logarithmic scale emphasises the
connection between powers of ten and estimation. Once students grasp powers of
ten thinking for understanding the scale of the Universe (_Module 1_), this
skill contributes to Module 2 which is designed to deepen their understanding
of estimations. Students practice power of ten arithmetic and solve problems
with numbers that are inaccessible for exact calculations or experiments. One
example of this sort of problem is to estimate the number of galaxies in the
visible Universe from the Hubble Deep Field image that shows thousands of
galaxies in a tiny patch of sky (Williams, 2000).

_Modules 1_ and _2_ focus on numbers, but many quantities are much more
powerfully represented by arrows in the form of vectors, which is the subject
of _Module 3_. Vectors are a higher level representation where direction and
magnitude are equally important. Their use is familiar in weather forecasting,
but many aspects of physical reality, from forces to magnetism are naturally
vector quantities. Also, vectors can represent the spin of objects, whether
they be planets or the extremely tiny spin of single electrons, which add up
to create macroscopic magnetic forces.

_Module 4_ : Probability is focused on the uncertainty of events (processes)
in our universe and connects with _Module 2._ We identify two types of
probability: a) the intrinsic statistical nature of all quantum processes
(quantum probability) such as the chance that a photon will arrive at a
certain location, and b) probability associated with imprecise knowledge and
complex dynamics (classical probability), such as knowledge of life elsewhere
in the universe or the motion of a rolling dice. As an illustrative and
interesting example of mostly classical probability, we use the Drake equation
for estimating the number of civilisations in the Milky Way galaxy. This
connects probability to the other module topics: estimation and extreme
numbers. We also use videos of single photon interference (Rueckner, 1995) to
allow students to visualise the transition from quantum probability to
_statistical certainty_ when the N-value becomes large enough.

_Module 5_ : Curved Space extends geometry from the idealised geometry of
Euclid to the actual geometry of curved space, as observed throughout the
universe. The module links both to extreme numbers and to the understanding of
vectors. Extreme numbers connect to the mass range of observed black holes
where spacetime curvature has dramatic and surprising effects, to the enormous
rigidity of space and to the tiny curvature of space and stretching of time in
the weak gravity of the Earth. The displacement of vector directions when they
are _parallel transported_ on curved surfaces allows curvature to be
identified.

Maths for Einstein’s Universe: activity-based learning.

The goal of Maths for Einstein’s Universe is to give students awareness of the
breadth and scope of mathematics, to open their minds to the power of
mathematics to enhance their understanding of interesting things, and to
discover maths that does not stress the exactness of traditional arithmetic.
Activities with toys and models focus on tangible physics concepts. For
example, as discussed further below, an activity about photons and beam
splitters exposes the statistical nature of quantum reality while motivating
the learning of probability. Activity-based group learning is free of the
stress of obtaining exact answers and instead creates an enjoyable and
memorable experience.

Each lesson begins with an activity that, where possible, involves multi-
sensory learning. By placing the activity first, strong student engagement is
secured, creating the ideal platform for exploring the mathematical and
physical concepts that the activity was designed to illustrate.

The activities used in MEU can be classified in terms of the dominant learning
instrument involved. We divide them into four categories: a) the use of toys
and models, b) mathematical games, c) role-plays with songs and poems, and d)
the history of discovery. Each of these approaches provides complementary
student experiences involving tactile, auditory, social and story-based
learning. Table 2 gives some examples in each area.

Table 2: Four main components are employed to construct activity-based lessons for Maths for Einstein’s Universe in a non-abstract and age-appropriate way by using toys, games, drama, and history. ()  Method |  Examples
---|---
() |
1. Toys and Models |  Lycra fabric models of orbits in curved spacetime, spinning tops and gyroscopes for understanding spin vectors, toy atoms and molecules as tangible representations of extremely small sizes, and toy cars to define straight line trajectories on curved two-dimensional spaces, and learning the right-hand rule to find spin vector.
2. Mathematical games |  Exponential dice, a game in which dice sides represent powers of ten, and scores move the player up or down a logarithmic number line that extends from -10 (atomic dimensions in meters) to +26 (scale of the visible universe). Rules allow students to develop and practice their skills and match dimensions to known objects. A rice grain game on a chess board gives students a tangible understanding of doublings and powers of two. The tug of war game explicitly introduces the concept of vector addition and the idea of the resultant vector.
3. Role plays and songs |  Discovery of Zero is a role-play about a birthday party set 4000 years ago when zero was invented as a placeholder to prevent errors in arithmetic (Blair, 2023). Ten Times Alice is a role-play based on Alice in Wonderland in which bites of magical food cause Alice to change size ten times. Songs and imagination reinforce the rules of division and multiplication and allow students to practice logarithmic thinking (Blair, 2023).
4. Historical content |  The history of zero as an operator, which enables logarithmic thinking and allows us to imagine the entire universe. The millennia-long quest to determine the value of $\pi~{}$to greater and greater precision includes feats of memory and supercomputing. The history of the understanding of space from Euclid to the precise observation of curved space at Wallal (Blair, 2022) including Gauss’s mountain top triangle experiment where he explored the understanding of geometry, the testing of Euclid’s geometry and the concept that light trajectories define straight lines.
() |

To further illustrate how activities designed with these learning instruments
can elucidate the mathematical concepts in an age-appropriate and engaging
way, we give three examples. First an exponential growth activity relating to
_Modules 1_ and _2_. Second an activity about vectors in an abstract context
(_Module 3_), and finally an activity on probability for a quantum world
(_Module 4_).

1._The mathematician tricks the emperor_ : The rice-on-the-chessboard activity
is a realisation of the well-known story of the mathematician who tricks the
emperor (Demi,1997). Following the mathematician’s demand, students place one
grain on the first square and double the number for each square. Students
typically manage to fill about 10 squares (210 =1024 grains). Having
physically realised the first ten squares, the activity progresses using
estimation and imagination. Students have previously learnt through other
activities, and repeated here, that 210 ~ 103. They apply this approximation
to the chess board, recognising that 220 ~ 106, 230 ~109 etc, ultimately
determining the number of rice grains for square number 64 to be ~16 x 1018
grains.

This physical activity connects powers of two to powers of ten, and utilises
both estimation and extreme numbers, all connected to the concept of
exponential growth. It provides a framework for developing abstract
representation and logarithmic understanding of other domains such as time and
distance. If one rice grain represents one second, the age of the universe is
represented by the square number 60 ~1018 seconds. If one grain represents 1
meter, then square number 40 takes you to the edge of the Solar system and
square number 54 represents the distance to the nearest star. The combination
of activity, imagination, estimation, and simple arithmetic has transported
the students from pure numbers to astronomical scales of time and space.

2\. _Phasor wheel vectors_. In Module 3, vectors are first introduced for
tangible representations such as forces as described in Table 2, item 3 (Tug
of war game). We also introduce two abstract representations. First, we used
phasor vectors as a tool for understanding the interference of light. Students
observe the striking light interference patterns when a laser pointer is
reflected off a soap film. The patterns of light and dark are taught as a
vector addition process. A toy called a phasor wheel is introduced, which uses
a rolling wheel with a phasor vector and a cam bar attached. (Choudhary ,
2021) It shows the connection between a rotation, wave motion (the
displacement of the cam) and the phasor (a radial arrow on the wheel). Bright
spots arise when the phasor arrows from two paths are aligned.

Secondly, we used vectors to describe spin. We use the term spin, rather than
angular momentum because we wish to introduce the concept of quantum spin
using the analogy with classical spin. To introduce the spin vector, we simply
introduce the right-hand rule and use simple toy tops to study the physical
concept of spin, including the addition and cancellation of spin and the
precession of spin vectors when forces are applied.

3\. _Quantum probability and partial reflections_ : Students investigate the
reflection of a laser pointer from a window or else they photograph coloured
objects seen directly, in a mirror and in a partial reflection from a window.
Pre-knowledge is introduced, that light comes as photons and their energy
depends on their colour. The key observation made by students is that partial
reflections do not change colour – they only change their intensity. This
means that all the photons are reflected “whole”. How does each photon "know"
where to go? It's a matter of quantum chance. For normal window glass, the
probability of a single photon being reflected very roughly equates to the
probability of rolling a double six with a pair of dice. By rolling dice and
tabulating double-sixes versus all other scores, students obtain an
experiential connection between photon reflection, the statistical nature of
quantum physics, and the concept that image brightness represents the number
of photons. Further activities are presented in (Popkova, 2020), (Blair, 2022)
and the accompanying paper (Popkova, 2023)."

Trials of Maths for Einstein’s Universe

The MEU program was developed iteratively following the Model of Educational
Reconstruction in which trial results were used to identify the most effective
learning instruments at different ages. For initial trials we offered
approximately 40 diverse 45 minute out-of-school workshops covering components
from all five modules. They attracted the participation of over 700 primary
and secondary school students. Following the initial trials, formal lessons
were carefully designed and trialled in schools as both long-term and short-
term programs.These trials are presented in Table 3.

Table 3: Trials of Maths for Einstein’s Universe in 2021-2023: a) after-school workshops in the Einstein-First facility b), d), g), h) government schools, c) PEAC program e) long-term program for a stable multi-age class, taught in three different age groups f) remote aboriginal school. () |  Modules |  Dates |  Number of sessions x Duration |  Number of participants |  Year level
---|---|---|---|---|---
() |  |  |  |  |
a) |  1 and 5 |  May 2021. Nov. 2022 4 weeks |  3 x 1 hr 1 x hr |  28 |  5-9
|  |  |  |  |
b) |  3 |  Jul. 2021 Two days |  2 x 2 hr |  27 |  4
|  |  |  |  |
c) |  1 and 3 |  Jul. 2021 Two days |  3 x 2 hr |  9 |  5-6
|  |  |  |  |
d) |  1 |  Sept 2021 4 weeks |  4 x 1.5 hr, |  27 |  5-6
|  |  |  |  |
e) |  1- 5 |  Feb. 2022 -May 2023 |  53 x 2hr |  30 |  2 - 9
f) |  1, 2, 3, 5 |  Jul. 2022. Sept 2022 Two days |  2 x 1hr 1 x 1hr |  26 |  5-9
|  |  |  |  |
g) |  1 |  Aug. 2022 4 weeks |  4 x 1.5 hr |  30 |  5-6
|  |  |  |  |
h) |  3 |  May 2023 4 weeks |  16 x 1 hr |  100 |  5-6
|  |  |  |  |
Totals |  1-5 |  2021-2023 |  151 hr |  277 |
() |  |  |  |  |

Feedback from participants was collected and analysed for the purpose of
developing and testing each module. We collected data from 277 students and 63
teachers. We also collected data on teacher’s responses to our teacher
training courses that consisted of micro-credential courses (micro-
credentials, 2023) and training workshops. Table 4 presents a summary of the
teaching training and micro-credential courses for teachers.

Observations and Qualitative Outcomes

The trials described in Table 3 included students across an age range from 7
to 14. Because the program covered material normally outside the school
curriculum, we had no benchmarks or prior data to calibrate our expectations.
For this reason, observations and qualitative measures were essential for
testing and refining lessons, evaluation of the optimum age for introducing
learning content, and evaluation of student responses.

Table 4: MEU teacher training courses: a) and d) workshops; b), c) and e) part of a 72-hour micro-credential course. () |  Name of course/workshop |  Dates |  Duration |  Audience |  Number of participants
---|---|---|---|---|---
() |  |  |  |  |
a) |  Maths of arrows |  Oct. 2021 |  1 hr |  Pre-service teachers |  20
b) |  Maths for Einstein’s Universe (micro-credential courses) |  Jan. 2022- Feb. 2022 |  18 hr |  Primary and secondary teachers |  9
c) |  Maths for Einstein’s Universe (micro-credential courses) |  Aug. 2022- Oct. 2022 |  18 hr |  Primary and secondary teachers |  8
d) |  How small is an atom? Numbers in the Universe on a Chessboard |  Oct. 2022 |  1.5 hr |  Primary teachers |  12
e) |  Maths for Einstein’s Universe (micro-credential courses) |  May. 2023- Jul. 2023 |  9 hr |  Primary and secondary teachers |  15
Totals |  2021-2023 |  46.5 hr |  |  63
() |  |  |  |  |

The qualitative observations are drawn from post-lesson notes for each
workshop as well as some video and photographic data. The MEU activities were
differentiated to suit the various age groups and ability levels, while in
some cases mixed ages allowed direct comparison of age-related engagement.

The observations have allowed us to obtain the following general description
of student performance and age-related aptitude, providing data used to define
the next iteration of the MEU program. We categorized our observations using
toys, mathematical games, role-plays, concept engagement duration, language
development, age-dependent learning styles, and learning retention across
three main groups of children: a) Year 3-4, b) Year 5-6, and c) Year 7-9:

  1. 1.

_Engagement with toys:_ All students regardless of age were observed to relate
readily to activities with toys, such as toy atoms and molecules, toy photons
and toy spacetime, as well as related games. The toys clearly provided visual
and tactile meaning for the relevant mathematical and physical concepts, such
as scale, speed, geometry and statistical properties. Students accepted the
toy models, without appearing to need to question their adequacy as accurate
representations.

  2. 2.

_Mathematical games._ From youngest to oldest, independent on gender, all
students enthusiastically participated in mathematical games. It was observed
that students in years 3-4 showed a higher level of engagement and preference
for games that involved physical and tactile experiences, compared to older
year groups.

  3. 3.

_Concept engagement duration_. The introduction of new concepts and language
requires time for assimilation. The activities with toys extended the duration
that students across all the age groups engage with key concepts, expediting
the formation of their cognitive concepts on a deeper level. In relation to
this we asked what the optimum duration of activities is, and how does the
optimum duration depend on age. We observed that both Year 5-6 students and
Year 3-4 students showed an equal amount of time being involved in playing.
However, Year 5-6 students were more engaged with advanced toys such as phasor
wheels, whereas year 3-4 students tended to spend more time with simpler toys
like toy atoms. Students in Year 7-9, while engaging with toys were quickly
ready to move on to an analytical and formal explanation, balancing playing
with mathematical reasoning.

We were surprised to observe that students in Year 3-6 not only achieved
similar scores but sometimes also demonstrated a deeper understanding of
certain concepts, thanks to their prolonged engagement with educational toys.
Older students who spent less time engaging with toys sometimes missed
important detail. For instance, in working with vector addition, older
students who spent less time engaged with toy magnetic arrows were more likely
to confuse the direction of a resultant vector.

It is important to note that an excessive duration of unstructured play can
lose the connection between the toy and the concept. The best learning
outcomes occurred when the connection between the toy and the concept was
repeatedly reinforced by the teacher, and by sequential activity tasks.

A fraction of students, especially those from lower socio-economic
backgrounds, found it difficult to establish a strong link between activities
and concepts. In a low SES school, one third of students appeared to
experience this difficulty, while in extension classes, only 1-2 students in a
typical class (a few percent) faced this challenge.

Despite these learning challenges, the universally high engagement with the
activities, and positive student responses (see below) leads us to believe
that the pleasurable and memorable tactile associations are likely to
contribute positively to future learning.

  1. 4.

_Role Plays_ : Role-playing combines historical knowledge with imagination and
creativity. It was particularly effective for Year 3-5 school students, who
were highly engaged. A common response at the end of a role-play session was
‘can we do it again?’. Songs in the role plays were clearly remembered, and
very useful for cementing concepts such the role of zero, as well as difficult
new words such as photon. One weakness of role-play learning is that drama can
take precedence over conceptual understanding. We found that it was very
important for the teacher to draw-out meaning through classroom discussion and
examination of the concepts.

  1. 5.

_Learning styles._ Students in years 3-4 showed impressive ability to connect
concepts to everyday experiences and imagination. For example, students in
this group were easily able to recognize the trajectory of a cam-rod on a
phasor wheel (Choudhary, 2021) as an approximate wave trajectory: _‘It is like
an ocean wave’_ , stated an 8-year-old during program e) (table 3). Year 5-6
students demonstrated a remarkable ability to embrace scientific notation by
only establishing connections between the number of zeros in a billion and
index notation. Formal mathematical conceptualization of exponentials is
clearly not needed to create the basic skills for logarithmic thinking. Year
7-9 students exhibited a keen interest in applying powers of ten notation for
tackling large number estimation problems.

  1. 6.

_Learning retention._ In many trials younger (Year 3-4) and older children
(Year 7-9) demonstrated similar learning performance. However, by testing
after a few weeks, we found that younger students forget more quickly. This
reduced level of retention is not surprising (Piaget, 1983). In a spiral
curriculum where concepts are revisited in later years, their early
comprehension is still likely to assist in developing deeper understanding in
later years, but in general this observation implies that younger students
need ongoing reinforcement.

Results and Evaluations

We shall now present general results obtained from our trials, which provide a
comprehensive expression of the program's effectiveness. In addition to formal
questionaries, parents and students were asked simple evaluation questions.
Recorded answers were transcribed and approved for correctness. Teachers
provided in-depth written responses to evaluation questions.

First, we will present some examples of student’s attitude results from across
the MEU program, followed by knowledge test results from individual programs.
Then we will go on to present evaluation results from across the MEU program.

_Attitude questions._ One of our primary objectives of MEU is to enhance
children's attitudes towards mathematics and science. In program f, as
outlined in Table 3, students engaged in a chessboard activity as part of the
program _The mathematician tricks the emperor_ (see above), a paper cutting
activity (Popkova, 2020), maths of arrows linked to hair interference
experiments and eclipse activities (Blair, 2022). In the post-test, two
statements were designed to assess students' perceptions of learning through
activities, using a Likert scale. More than 90 percent of children taking part
in Maths for Einstein’s Universe agreed or strongly agreed with the statement,
"I like learning things with activities” and “Science with activities is fun”
while 73 percent agreed or strongly agreed with the statement, "I like doing
activities to learn about numbers."

_Knowledge questions._

  1. a)

The results of pilot trials of _Powers of the Universe_ module from pre-test
and post-test comparisons indicate that nearly 100% of students were able to
use powers of two for describing numbers in processes such as bacteria
splitting or numbers of ancestors. In the post-test, 78% of students preferred
to use powers of ten notation over multiple zeros for big and small numbers.
According to post-test results, 98% of students expanded their understanding
of the scale of the Universe by demonstrating knowledge of cosmological
objects and subatomic particles in response to questions about the largest and
smallest objects or magnitudes they were familiar with.

  2. b)

The results of the _Maths of Arrows_ module from the post-test demonstrated
that nearly 100% of students were able to understand the idea of vector and
vectors addition as well as use it for solving problems such as the addition
of forces when many people push a car.

  3. c)

Understanding phasors and quantum spin vectors: In post-testing nearly 90% of
students were able to correctly use the right-hand rule for defining spin
vectors of spinning objects. More than 83% of students could add phasors shown
by a phasor wheel and proximately 90 % could correctly correlate the resultant
with light intensity in an interference pattern.

Detailed outcomes of the program can be found in the papers (Popkova, 2020),
(Blair, 2022) and forthcoming papers.

_Feedback responses from interviews with teachers, parents, students_

To assess MEU programs, we collected written feedback from each teacher (T)
who delivered the program or assisted in the teaching, including a STEM
coordinator and a teaching principal. We also collected verbal feedback
(recorded and transcribed) from parents (P), and students (S). We observed
that all responses shared four common parameters given in table 5.
Furthermore, we conducted an analysis of teachers' confidence. We give some
typical responses in Table 5, followed by a more detailed analysis.

Table 5: The key feedback citations collected from teachers, students, and parents reflect common impressions across all three categories and teachers’ confidence. ()  Feedback parameters |  Examples of feedback (exact quotations)
---|---
() |
1) Program structure and way of learning |  T: _“A good blend between concepts, activities, games and assessment.”_ P: _“I am truly impressed by the wide range of activities.”_
|
2) Enthusiasm and engagement |  T: _“Students' attendance remained high even though it was an after-school program.”_ P: _“At home, he always talks about the activities.”_ S: _“It was so cool for all of us to push the car together!”_
|
3) Effectiveness |  T: _“The review questions asked at the beginning of the sessions were a further indicator that students were attending and understanding what had been previously covered.”_ P: _My son had no problems with advanced scientific problems in school extension classes……._
|
4) Teachers’ confidence |  T: _“I did need to ask her_ [Einstein-First team member] _a few times whether what I was teaching was correct.”_
|
5) Future collaboration |  T: _“I look forward to teaching it again.”_ P: _“She wants to learn science and maths in this way.”_ S: _“I want to play with toys to learn maths and science.”_
() |

  1. a)

Program structure and way of learning. The teachers reported back that the
program is _“well-sequenced”_ , _“well supported with practical
demonstrations”_ and _“rigorous but achievable”_ and _“a good blend between
concepts, activities, games and assessment”_ , which helps to understand
modern science in an interesting and meaningful way.

Teachers were enthusiastic about the approach, which effectively prevent the
lessons _“…to get either superficial or too heavy”._ One teacher remarked
_“Excellent; exposed the students to new concepts in an active, hands-on
way”._ The emphasis on outdoor time was particularly well appreciated, with
one teacher noting _“The outdoor activities were especially well received”._
On group activities, a teacher commented that the lessons _“incorporate
discussion and presentation skills through an assumption of different roles in
their groups. Students get a choice to work collaboratively with each other or
individually, or even at times to take on the role of an observer.”_

One parent mentioned, "_I am truly impressed by the wide range of activities
my child has been exposed to through this program_."

  1. b)

Enthusiasm and engagement _._ Teacher responses emphasise a high level of
student engagement and enthusiasm. Teachers reported that students appeared to
be “ _highly engaged and motivated_ ”, noting _“the unprecedented growth of
students numbers”._ :

  1. a.

_“Students' attendance remained high even though it was an after-school
program “_

  2. b.

_"Students completed the homework which was not compulsory in the context”_

  3. c.

_“Students carry over their discussions from the classroom to their informal
conversations outside the class._

  4. d.

_“Students felt privileged to be included”_ in the MEU program.

One teacher also noted, _“We even have parents come back to Anastasia_ [MEU
external teacher] _with physics questions, as children share what they have
learnt with them at home”._

Parents also indicate a high level of enthusiasm: “ _At home, he always talks
about the activities he had during the lessons such as counting rice on a
chessboard and rolling balls on the elastic space-time”_ and also the impact
on the attitude to science in general: _“Whenever the topic of science arises,
he eagerly raises his hand to share the ideas he has learned with MEU”_.
Parents have also noticed an increase in STEM-related discussions at home due
to MEU _“One of the topics my son finds most interesting is the population of
the world. He was impressed by this unimaginable number and asked lots of
questions.”_

Most of the students expressed verbal excitement about the new way of
learning: _“I like the maths of arrows. It was so cool for all of us to push
the car together! Wow! My favourite one.”_

  1. c)

Effectiveness. The teachers suggest that the program was very effective in
achieving its learning outcomes: “ _The review questions asked at the
beginning of the sessions was a further indicator that students were attending
and understanding what had been previously covered.”_ Parents described how
the program benefitted their children: _“My daughter doesn’t feel confident
learning abstract concepts at school, but she has very positive self-esteem
with Maths for Einstein’s Universe, providing an easy and enjoyable of
understanding science”._ _“Thanks to Maths for Einstein’s Universe, my son
…demonstrated an excellent performance_ ” _._

  1. d)

Teachers’ confidence. The initial trials of the program revealed a need for
additional support and assistance. A first-time teacher commented _“the
external presenter who had a greater depth of knowledge”_ and referring to the
Einstein-First person stated _“I did need to ask her a few times whether what
I was teaching was correct. Without her there I would have felt less
confident.”_ However, as the teachers gained experience through these trials,
in all cases, they expressed a desire to continue implementing the program
independently. The positive response of the teachers who participated in
initial trials led to the creation of micro-credential courses (micro-
credentials, 2023)as well as specialised workshops organized to offer ongoing
support and assistance.

  1. d)

Future collaboration. The schools and teachers involved in this program were
asked about future collaboration. “ _I look forward to teaching it again”; “We
are very pleased to have tried the Einstein-First approach at our centre”_.
All the trial schools indicate that they want teach MEU on a regular basis.
Students also found the program more engaging and enjoyable than their regular
classes at school and emphasize the willingness to learn MEU. One Parent
reported: “ _My daughter declared that Maths for Einstein’s Universe is her
favourited subject. She wants to learn science and maths in this way”_ ; One
student said:_“I want to play with toys to learn maths and science. Spinning
tops are my favourite. At school, the teacher talks and talks about science,
and then we fill in boring worksheets”_

Based on the feedback, _Maths for Einstein’s Universe_ appears to be a
successful and engaging program for students promoting interest in science and
maths. The positive feedback on every level suggests that the "Maths for
Einstein's Universe" program was a positive and valuable educational
experience for students with remarkable outcomes. A school principal who
observed the program carefully stated _“I would have not believed that it
could be taught to the target age group (7- 14-year-old students) if I had not
witnessed the astonishing comprehension that even middle primary students have
demonstrated”._

Conclusion

We have presented a hypothesis that the modernisation of mathematics content
to make it more relevant to student interests and to science, combined with
less focus on arithmetic is likely to help reduce maths anxiety. We have
suggested that maths anxiety could be reduced by emulating the approach used
in the Einstein-First program which has shown the significant benefit of an
activity-based modern curriculum in improving teenage girls’ attitudes to
science (Kaur, 2020)

We have presented five modules for the _Maths for Einstein's Universe_
program. They focus on key mathematical areas that a) enhance understanding of
the physical universe, and b) change the emphasise of mathematics from
exactness to a broader conceptual foundation. It is suggested that by
enhancing the understanding of modern physics concepts and associated
mathematics from the early primary school level, we will be providing students
with intellectual tools that are of value throughout our modern world while
also preparing them for exciting future learning.

We have presented evidence that MEU does indeed provide students with an
intuitive grasp of the mathematical foundations necessary for a deeper
comprehension of physical reality. Evaluations indicate that it strongly
improves student attitudes to mathematics. It is too early to determine how it
could affect future learning and maths anxiety.

We have outlined the learning progression for these modules, from Powers to
the Universe to curved space geometry along with trial results, learning
outcomes, attitudes, in-class observations, and feedback analysis.

The initial success of the roughly 200 hours of delivered programs indicates
that Maths for Einstein’s Universe effectively achieves its teaching
objectives, fosters interest and engagement in mathematics and related science
and among young students. Further quantitative results will be presented
through case studies of individual modules in forthcoming papers.

In general, based on these preliminary observations, it is fruitful to start
learning modern physics concepts in middle primary school through engagement
with simple toys, capitalising on young learners’ willingness to play, while
slowly building up their formal scientific language and advanced abstract
concepts. In order to present stronger and statistically significant
conclusions, it is necessary to conduct observations with a larger sample size
of students.

Test results for individual modules using quantitative analysis of pre and
post-tests will be published in forthcoming papers.

Funding

This research was supported by the Australian Research Council Linkage Grant
LP 180100859 led by Professor David Blair at the University of Western
Australia. The third author David Treagust, from Curtin University, is a
member of the Einstein-First project.

Acknowledgements

The authors would like to acknowledge the contribution of all the Einstein-
First collaboration members: Marjan Zadnik, David Wood, Kyla Adams, Jessse
Santos, Tejinder Kaur,Ju Li, Elaine Horne. We would like to thank Peter
Rossdeutscher and Howard Golden, who have enabled us to raise donation funds
to supplement our ARC Linkage funding (LP180100859) that allowed us to develop
our on-line training programs. We also thank the ARC Centre of Excellence for
Gravitational-Wave Discovery (OzGrav) for their continual support, especially
in enabling us the develop school kits for our activities. We also thank our
Einsteinian Physics Education Research (EPER) collaborators for their
enthusiastic support. We are very grateful to the West Australian Department
of Education for their support, the Independent Schools Association of Western
Australia who have facilitated many of our trials, and the Science Teachers
Association of Western Australia who have provided essential and continuous
support. We are grateful to our participating schools principals, teachers,
and students for allowing us to run the program and for granting us permission
to use their photographs and data for research purposes.

Declaration of interest statement

The authors have no relevant financial or non-financial interests to disclose
in relation to the publication of this paper. The authors also have no
conflicts of interest to declare that are relevant to the content of this
article. Ethics declarations The participants involved in this study gave
their informed consent for this publication. The research was carried out
under the University of Western Australia Ethics approval number
2019/RA/4/20/5875.

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