Actually, Printful also falls into the category of dropshipper, or even a supplier that stores the goods for you, packages themand ships them out to your clients after a purchase. So, essentially, all you have to do is upload your designs and start posting them onto your website to begin selling. In this Printful review, we will outline the advantages of the support and reveal the reason why many users select Printful over the alternatives.
The short story Is that Printful does not have as big of selection of products compared to the competition. If I wanted to print and dropship some layouts on coasters, I would have to look elsewhere. The same goes for more unique items such as yoga mats and jewellery. However, there is an onslaught of user reviews out there that state Printful is that their go-to printing press because of the quality of products. And that is a big thing.
You Might Have to Cut back on the selection of items that you sell on your site, but you know that the tops are hardy, the printing is not going to wash off fast, and customers are not going to have ticked off because of a late delivery.
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1 thing to Remember is that it often depends on the sort of product that you're printing on, the supplier you choose, and the standard of graphic you upload. So, 1 type of shirt might look fantastic from Printful, however the tote bags might not look exactly right. The main way to figure out whether Printful is for you will be to upload a few of your layouts and order samples. In this manner, you can compare pricing, quality, and shipping speed to additional printing companies and see which one is for you.
For This Printful inspection, we will prevent the caliber of its products, because that requires you to order some samples. However, we'll talk about pricing, ease-of-use, customer service, as well as the choice of products. That way, you'll have a clearer idea of just how Printful can get the job done for you and when the design and integration tools are enough for your business.
Printful supplies the products, Prints on the products, packages and fulfills the order and sends it directly to your customers, all you need to do is send orders on them and the sequence details and they will handle the rest. To find out more about how their process works, have a look at the Printful Printing Order Lifecycle.
They are in the business Of assisting you to create products to your business, and they supply all the necessary tools and resources that will help you easily integrate the dropshipping process into your ecommerce site so you can offer a seamless experience to your customers. They offer plenty of ecommerce platform integrations so you're able to utilize their service whichever platform you decide to market, they are transparent about their goods and shipping pricing so it is easy to know their prices and employ it to your product markups, they provide design, photography and branding solutions so your products can appear their best and they're located in North America as well as Europe to be able to easily reach the biggest consumer markets.
Should you order a product that comes out of a European Printful center, the shipping is free for all locations in Europe.
This may cause Problems for those trying to test new products on a regular basis. However, the firm does have some techniques to acquire additional sample orders–like by integrating having an ecommerce platform or creating over $300 in the prior month.
Another large Question most retailers have is whether it's quick and easy to walk throughout the print layout center. Also, what's it like trying to get those products on your site?
I'd choose The dimensions that I'd love to sell, then upload the print document that is going to go onto that picture.
In general, the Guide from Printful is very clear about what you might want to change with your upload. For example, most arbitrary JPGs you upload in the computer will be too small for printing onto a canvas. You are going to want a massive JPG or PNG which you exported from a RAW file in something like Lightroom or even Photoshop.
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Printful Is for ecommerce entrepreneurs who want the simplicity of selling dropshipped products without sacrificing quality or their branding at the process. Whether you're looking for a way to sell your designs to your clients be it on habit t-shirts, canvases, phone cases, cushions and much more, or if you're considering starting any sort of product-based business without the annoyance of owning inventory or production stuff, then Printful is for you.
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There are several Different printing methods but the ones that Printful sticks to are Direct-to-Garment (DTG) printing as it is a price and labor efficient way to print single order things (and the main reason why they do not require minimum orders), but in addition they offer all-over sublimation printing for a few clothing items, screen printing in some cases (for large purchases only) and embroidery for hats and coming shortly for attire.
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For more information About their products and product pricing, check out Printful's Custom Products page. If you're interested in dropshipping something which you don't find offered on Printful however, have a look at their Upcoming Product & Features page to see if it's coming soon. They listen to their vendors and they add new products frequently, so if there's a product that you need to see only inform them via their New Feature Requests page.
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As Printful supplies a Dropshipping service, there's no need for one to handle or handle any of the solution or inventory, and you don't manage the packaging, transport or fulfillment aspects either. Printful stores, prints, picks, packs, ships and fulfills every item, so you never need to.
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That can, of course, Mean that you're paying for Printful's services and also for providing warehousing, shipping, and fulfillment in order to make it a low-risk business model for you. While you do not pay for their support in monthly fees or minimum orders, your profit margins will be affected and will probably be smaller than in case you were manufacturing products yourself. This drawback is not exclusive to Printful, it applies to the dropshipping business design as a whole. It is, however, an important facet of dropshipping to keep in mind so that you can prepare your margins so.
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This is an ideal service For the ecommerce retailer who has an notion of what designs they would like to publish in their merchandise but are not certain how to make it life or the store owner who would like to outsource the artwork creation in order that they could free up some time to handle different aspects of building their enterprise. Together with Printful's dedicated design team you'll be in great hands, so if that is a service you might need, check out Printful's Design Services page here for more information.
As one of those many Services Printful supplies beyond just offering excellent products and a satisfaction service, they also offer design and branding solutions to help you take your business to the next level.
These choices help Detract from the impersonal belief that dropshipping can provide and helps to keep your brand consistency from your website to the products your clients get. To learn more about the branding services Printful offers, check out this page.
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· Printing Tips: They provide lots of posts that are readily available through their site to help vendors understand the different printing advantages and disadvantages like DTG print document transparency, cloth details, sublimation printing facts, garment printing details, and information on how to make embroidery files.
· Mockup Generator: Use their mockup generator to use your layouts digitally to their merchandise to either get a sense of what the finished product will appear to be to create product images for your ecommerce store.
· Gain Calculator: As we've said, the dropshipping business model inherently means you are going to be working with slender profit margins, so use their profit calculator to determine just what your margins will be for each item so that you can budget your markup so.
· Blog: Printful's site is a valuable resource to go to if you are interested in learning more about their merchandise, services, integrations or how to actually build your shop the way you desire it. They are experts on their products and services, so in the event that you want any more info about what they do, check out their blog posts.
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Even though Printful offers many different services, resources and tools which are valuable for their vendors, there are still some aspects of their services they are working to enhance. Let us dive into the advantages and disadvantages of using Printful as a dropshipping service supplier.
Finally, Printful Offers the best dropshipping service when it comes to apparel, accessories and homeware distributing and printing. They may not have a massive product offering but what they do offer is top quality, and they strive to get it right every time. They also use their sellers to continuously improve the products and services they offer whether that is getting new types of merchandise in or adding more choice to what they currently offer. Their drive to offer high-quality products and services and to continuously improve their products, tools, and resources is the reason so many sellers decide to work together.
However, the Main ones which also provide dropshipping include Printify (easy to get confused with Printful,) Printing Aura, and Gooten. A few different ones exist, however, there aren't many dropshipper/printers from the video game.
Printful makes Cash when you make a purchase. The same goes for the supplier you're purchasing from. What happens is you choose a product to market, and Printful lists a starting retail price for your product.
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I'm convinced That you can definitely turn a profit for nearly all of your earnings through Printful, so long as you are not pricing items over what people would usually pay. People know that habit printing is more expensive, so they are typically ready to pay a premium. Additionally, you get to determine the possible gain before publishing it into your site, so if a product doesn't yield a good return, simply look for a cheaper model or hike up the cost a little.


text:
\section{Introduction}

Let $W=W(x),\ x\in [0,\pi]$ be a standard Brownian motion on a
complete probability space $(\Omega, \mathcal{F},\mathbb{P})$.
With no loss of generality, we assume that all realizations  of $W$
are in $\mathcal{C}^{1/2-}((0,\pi))$, that is,
H\"{o}lder continuous of every order less than $1/2$.

Consider the equations
\bel{eq:main-W}
\begin{split}
\frac{\partial u_{\diamond}(t,x;\varepsilon)}{\partial t} &=
\frac{\partial^2 u_{\diamond}(t,x;\varepsilon)}{\partial x^2} +
\varepsilon u_{\diamond}(t,x;\varepsilon)\diamond \dot{W}(x),\ t>0, \
0<x<\pi,\\
u_{\diamond}(t,0;\varepsilon)&=u_{\diamond}(t,\pi;\varepsilon)=0, \
u_{\diamond}(0,x;\varepsilon)=\varphi(x),
\end{split}
\ee
and
\bel{eq:main-S}
\begin{split}
\frac{\partial u_{\circ}(t,x;\varepsilon)}{\partial t} &=
\frac{\partial^2 u_{\circ}(t,x;\varepsilon)}{\partial x^2} +
\varepsilon u_{\circ}(t,x;\varepsilon)\circ \dot{W}(x),\ t>0, \
0<x<\pi,\\
u_{\circ}(t,0;\varepsilon)&=u_{\circ}(t,\pi;\varepsilon)=0, \
u_{\circ}(0,x;\varepsilon)=\varphi(x).
\end{split}
\ee
Equation \eqref{eq:main-W} is the Wick-It\^{o}-Skorokhod formulation of the
parabolic Anderson model with potential $\varepsilon\dot{W}$; equation \eqref{eq:main-S}
is the corresponding Stratonovich (or geometric rough path) formulation.
These equations, with $\varepsilon=1$,
 are studied in \cite{Kim-Lot17} and \cite{Kim-Lot-GRP},
respectively.

The objective of the paper is to show that
\begin{itemize}
\item The solutions of \eqref{eq:main-W} and
\eqref{eq:main-S} are real-analytic functions of
$\varepsilon$: with suitable functions $u_{\diamond}^{(n)}$,
and $u_{\circ}^{(n)}$, the equalities
\begin{align}
\label{ps-W}
u_{\diamond}(t,x;\varepsilon)&=u_{\diamond}(t,x;0)+
\sum_{n=1}^{\infty} \varepsilon^n u_{\diamond}^{(n)}(t,x)\\
\label{ps-S}
u_{\circ}(t,x;\varepsilon)&=u_{\circ}(t,x;0)+
\sum_{n=1}^{\infty} \varepsilon^n u_{\circ}^{(n)}(t,x)
\end{align}
hold for all $t>0$, $x\in [0,\pi]$, $\varepsilon>0$, and every
realization of $W$;
\item The first two terms in \eqref{ps-W} and \eqref{ps-S} are the
same so that
\bel{order2}
|u_{\diamond}(t,x;\varepsilon)-u_{\circ}(t,x;\varepsilon)|=O(\varepsilon^2),
 \ \varepsilon\to 0,
\ee
 for all $t>0$ and $x\in [0,\pi]$,  and every
realization of $W$.
\end{itemize}
Equalities \eqref{ps-W} and \eqref{ps-S} are in the spirit of \cite{Lot2002}.
Equality \eqref{order2} is similar to \cite[Proposition 4.1]{Wan-Wick12};
see also \cite{Wan-Wick10}.

The precise statement of the main result is in Section \ref{sec:MR}, and
the proof is in Sections \ref{sec:W}, \ref{sec:S}, and \ref{sec:Cr}.

\section{The Main Result}
\label{sec:MR}

Denote by $p=p(t,x,y)$ the heat semigroup on $[0,\pi]$ with
zero boundary conditions:
\bel{HS}
p(t,x,y)=\frac{2}{\pi}\sum_{k=1}^{\infty} e^{-k^2t}\sin(kx)\, \sin(ky),\
t>0,\ x,y\in [0,\pi].
\ee
Let $\varphi=\varphi(x)$ be a continuous function on $[0,\pi]$,
 and let $u=u(t,x)$ be the solution of the heat equation
\bel{u0}
\begin{split}
\frac{\partial u(t,x)}{\partial t} &=
\frac{\partial^2 u(t,x)}{\partial x^2},\ t>0, \
0<x<\pi,\\
u(t,0)&=u(t,\pi)=0, \
u(0,x)=\varphi(x),
\end{split}
\ee
that is,
\bel{u0-expl}
u(t,x)=\int_0^{\pi}p(t,x,y)\varphi(y)dy.
\ee
Next, define the function $\mfu=\mfu(t,x)$ by
\bel{u1-expl}
\mfu(t,x)=\int_0^t\int_0^{\pi} p(t-s,x,y)u(s,y)\,dW(y)\, ds.
\ee
That is, $\mfu$ is the mild solution of
\bel{u1}
\begin{split}
\frac{\partial \mfu(t,x)}{\partial t} &=
\frac{\partial^2 \mfu(t,x)}{\partial x^2}+u(t,x)\dot{W}(x),\ t>0, \
0<x<\pi,\\
\mfu(t,0)&=\mfu(t,\pi)=\mfu(0,x)=0.
\end{split}
\ee
Because $u$ is non-random, no stochastic integral is required to define $\mfu$.

\begin{proposition}
If $\varphi\in \mathcal{C}((0,\pi))$, then $\mfu$ is a continuous function of
$t$ and $x$ for all $t>0$ and $x\in [0,\pi]$.
\end{proposition}

\begin{proof}
This follows by the Kolmogorov continuity criterion: $\mfu$ is a Gaussian
random field and direct computations show
$$
\bE \big(\mfu(t+\tau,x+h)-\mfu(t,x)\big)^2\leq
C(t)\big(\tau^2+h^2\big)^{1/4}\,
\max_{x\in [0,\pi]}|\varphi(x)|;
$$
cf. \cite[Sections 6 and 7]{Kim-Lot17}.
\end{proof}

Next, define the functions $u_{\diamond}^{(n)}=u_{\diamond}^{(n)}(t,x)$,
$n=0,1,2\ldots$, $t\geq 0$, $x\in [0,\pi]$, by $u_{\diamond}^{(0)}(t,x)=
u(t,x)$, and, for $n\geq 1$, $u_{\diamond}^{(n)}$ is the mild
solution of
\bel{eq:derivative-W}
\begin{split}
\frac{\partial {u}^{(n)}_{\diamond}(t,x)}{\partial t} &=
\frac{\partial^2 {u}^{(n)}_{\diamond}(t,x)}{\partial x^2} +
 {u}^{(n-1)}_{\diamond}(t,x)\diamond \dot{W}(x),\ t>0, \ 0<x<\pi,\\
{u}^{(n)}_{\diamond}(t,0)&={u}^{(n)}_{\diamond}(t,\pi)
= {u}^{(n)}_{\diamond}(0,x)=0.
\end{split}
\ee
In other words,
\bel{eq:derivative-W-expl}
u_{\diamond}^{(n)}(t,x)=\int_0^t \int_0^{\pi} p(t-s,x,y)
u_{\diamond}^{(n-1)}(s,y)\diamond d{W}(y) \,  ds,\ \ n\geq 1,
\ee
and, in particular, $u_{\diamond}^{(1)}=\mfu$.

Similarly, define the functions $u_{\circ}^{(n)}=u_{\circ}^{(n)}(t,x)$,
$n=0,1,2\ldots$, $t\geq 0$, $x\in [0,\pi]$, by $u_{\circ}^{(0)}(t,x)=
u(t,x)$, and, for $n\geq 1$, $u_{\circ}^{(n)}$ is the mild
solution of
\bel{eq:derivative-S}
\begin{split}
\frac{\partial {u}^{(n)}_{\circ}(t,x)}{\partial t} &=
\frac{\partial^2 {u}^{(n)}_{\circ}(t,x)}{\partial x^2} +
 {u}^{(n-1)}_{\circ}(t,x)\circ \dot{W}(x),\ t>0, \ 0<x<\pi,\\
{u}^{(n)}_{\circ}(t,0)&={u}^{(n)}_{\circ}(t,\pi)
= {u}^{(n)}_{\circ}(0,x)=0.
\end{split}
\ee
In other words,
\bel{eq:derivative-S-expl}
u_{\circ}^{(n)}(t,x)=\int_0^t \int_0^{\pi} p(t-s,x,y)
u_{\circ}^{(n-1)}(s,y)\circ d{W}(y) \,  ds,\ \ n\geq 1,
\ee
and, in particular, $u_{\circ}^{(1)}=\mfu$.

The main result of the paper can now be stated as follows.

\begin{theorem}
\label{th:main}
 Let $\varphi\in \mathcal{C}((0,\pi))$. Then
\begin{enumerate}
\item Equality \eqref{ps-W} holds with
$u_{\diamond}^{(n)}$ from \eqref{eq:derivative-W-expl}.
\item Equality \eqref{ps-S} holds with
$u_{\circ}^{(n)}$ from \eqref{eq:derivative-S-expl}.
\item Equality \eqref{order2} holds and
\bel{order2-d}
\lim_{\varepsilon\to 0}\varepsilon^{-2}
\Big(u_{\circ}(t,x;\varepsilon)-u_{\diamond}(t,x;\varepsilon)\Big)=
\int_0^{\pi} p^{(3)}(t,x,z)\varphi(z)dz,
\ee
where
$$
p^{(3)}(t,x,z)=\int_0^{\pi}\int_0^t\int_0^s
p(t-s,x,y)p(s-r,y,y)p(r,y,z)\,dr\,ds\,dy.
$$
\end{enumerate}
\end{theorem}
The proof is carried out in the following three sections.

\section{The Wick-It\^{o}-Skorokhod Case}
\label{sec:W}

The objective of this section is the proof of \eqref{ps-W}.

The  solution of \eqref{eq:main-W} is defined as a chaos solution
(cf. \cite[Theorems 3.10]{LR-spn}). It
 is a continuous in $(t,x)$ function
 (cf. \cite[Sections 6 and 7]{Kim-Lot17})
 and has a  representation as a series
\bel{W-sol}
u_{\diamond}(t,x;\varepsilon)=
\sum_{\ba\in \cJ}u_{\ba}(t,x;\varepsilon)\xi_{\ba},
\ee
where
\begin{align*}
\cJ&=\left\{\ba=(\alpha_k,\, k\geq 1): \alpha_k\in \{0,1,2,\ldots\},\
\sum_{k}\alpha_k<\infty\right\}, \\
\xi_{\ba}& = \prod_{k}
 \left(
 \frac{\Hep_{\alpha_{k}}(\xi_{k})}{\sqrt{\alpha_{k}!}}\right),\
\Hep_{n}(x) = (-1)^{n} e^{x^{2}/2}\frac{d^{n}}{dx^{n}
e^{-x^{2}/2},\\
\xi_k&=\int_0^{\pi}\mfk{m}_k(x)\,dW(x),\
\mfk{m}_k(x)=\sqrt{2/\pi}\,\sin(kx),
\end{align*}
and, with $|\ba|=\sum_k\alpha_k$, $|(\boldsymbol{0})|=0$,
$ |\bep(k)|=1$,
\begin{align}
\notag
u_{(\boldsymbol{0})}(t,x;\varepsilon)&=u(t,x),\\
\notag
u_{\ba}(t,x;\varepsilon)&=
\varepsilon \sum_{k} \sqrt{\alpha_k}\,
\int\limits_0^t\int\limits_0^{\pi} p(t-s,x,y)
u_{\ba-\bep(k)}(s,y;\varepsilon)\mfk{m}_k(y)\,dyds;
\end{align}
see \cite[Section 3]{Kim-Lot17} for details.
In particular,
\bel{uba-ep}
\sum_{|\ba|=n}u_{\ba}(t,x;1)\xi_{\ba}=
\sum_{|\ba|=n-1}\int_0^t\int_0^{\pi}
p(t-s,x,y)u_{\ba}(s,y;1)\xi_{\ba}\diamond dW(y)\, ds.
\ee
 Comparing \eqref{eq:derivative-W-expl} with  \eqref{uba-ep}
 shows that
\bel{n-der-W}
u_{\diamond}^{(n)}(t,x)=\sum_{|\ba|=n}u_{\ba}(t,x;1)\xi_{\ba}.
\ee
In other words, \eqref{ps-W} is equivalent  to \eqref{W-sol}.

Next,
\begin{equation*}
\begin{split}
\bE|u_{\diamond}^{(n)}(t,x)|=&
\bE\left|\sum_{|\ba|=n}u_{\ba}(t,x;1)\xi_{\ba}\right|
\leq \left(\bE\left(\sum_{|\ba|=n}
u_{\ba}(t,x;1)\xi_{\ba}\right)^2\right)^{1/2}\\
&= \left(\sum_{|\ba|=n}|u_{\ba}(t,x;1)|^2\right)^{1/2}
\leq C^n(t) {n^{-n/4}}\, \sup_{x\in (0,\pi)}|\varphi(x)|^{1/2},
\end{split}
\end{equation*}
where the last inequality follows by \cite[Theorem 4.1]{Kim-Lot17}.
As a result,
$$
\sum_{n\geq 0}\varepsilon^n \bE \big|u_{\diamond}^{(n)}(t,x)\big|<\infty,
$$
that is,  the series converges absolutely with probability one for all
$t>0,\ x\in [0,\pi]$, and $\varepsilon \in \bR$.

{\em This concludes the proof of  \eqref{ps-W}.}

\section{The Stratonovich Case}
\label{sec:S}

The objective of this section is to prove  \eqref{ps-S}.
To simplify the presentation,  we use the following notations:
 \begin{align*}
\Lambda&=(-\boldsymbol{\Delta})^{1/2}, \ \ H^{\theta}=
\Lambda^{-\theta}\big(L_2((0,\pi)\big),\ \
\|\cdot\|_{\theta}=\|\Lambda^{\theta}\cdot\|_{L_2((0,\pi))},\ \
\theta\in \bR,\\
 \ p*g(t,s,x)&=\int_0^{\pi} p(t-s,x,y)g(s,y)\,dy,
\end{align*}
where $\boldsymbol{\Delta}$ is the Laplace operator on $(0,\pi)$ with
zero boundary conditions and  $p$ is the heat kernel \eqref{HS}.

By direct computation,
\bel{PosNorm}
\|p*g\|_{\gamma}(t,s)\leq C_{T,\theta}\,
(t-s)^{-\theta/2}\|g\|_{\gamma-\theta}(s),\ \theta>0,\ \
\gamma\in \bR,\ \ t\in (s,T];
\ee
cf. \cite[Lemma 7.3]{Kr_Lp1}.

Consider the equation
\bel{ghe-1}
\frac{\partial v(t,x)}{\partial t}=\frac{\partial^2 v(t,x)}{\partial x^2}+
v(t,x)\circ \dot{W}(x)+f(t,x)\circ \dot{W}(x),\ t>0,
\ee
which includes \eqref{eq:main-S} as a particular case. By definition,
a solution (classical, mild, generalized, etc.) of  \eqref{ghe-1} is a suitable
limit, as $\epsilon\to 0$,  of the corresponding solutions of
\bel{ghe-1-ep}
\frac{\partial v_{\epsilon}(t,x)}{\partial t}
=\frac{\partial^2 v_{\epsilon}(t,x)}{\partial x^2}+
v_{\epsilon}(t,x)V_{\epsilon}(x)+f(t,x)V_{\epsilon}(x) ,\ t>0,
\ee
where $V_{\epsilon}, \ \epsilon>0$ are smooth functions on $[0,\pi]$
such that
$$
\sup_{\epsilon}
\left\| \int V_{\epsilon}\right\|_{\mathcal{C}^{1/2}}<\infty,\
\lim_{\epsilon\to 0} \sup_{x\in [0,\pi]}
\left| \int_0^xV_{\epsilon}(y)dy-W(x)\right|=0.
$$

By \cite[Theorem 3.5]{Kim-Lot-GRP},
\begin{itemize}
\item The generalized solution of \eqref{ghe-1} is the
same as the generalized solution of the equation
\bel{Str-gen}
v_t=\Big(v_{x}+W(x)v+W(x)f\Big)_x-W(x)v_x-W(x)f_x;
\ee
the subscripts $t$ and $x$, as in $f_x$,  denote the corresponding
partial derivatives;
\item The mild solution of \eqref{ghe-1} is the solution of the
integral equation
 \bel{Str-mild}
 \begin{split}
v(t,x)&= \int_0^t p*\big((f+v)W\big)_x(t,s,x)\, ds-
\int_0^t p*\big((f+v)_xW\big)(t,s,x)\,
ds\\
&+\int_0^{\pi}p(t,x,y)\varphi(y)dy.
\end{split}
\ee
\end{itemize}

On the one hand,   mild and generalized  solutions of \eqref{ghe-1}
are the same: just use $\mfk{m}_k$ as the test functions. On the other
hand, different definitions of the solution lead to different regularity results.

 By standard parabolic regularity, if
$\varphi\in H^{0}$ and $f\in L_2\big((0,T);H^{\gamma}\big)$,
$\gamma\in (1/2,1]$,
 then there is a unique generalized solution of \eqref{Str-gen}
 in the normal triple $(H^1,H^0,H^{-1})$ and
\bel{Str-gen-reg}
v\in  L_2\big((0,T);H^{1}\big)\bigcap
\mathcal{C}\big((0,T);H^{0}\big)
\ee
for every realization of $W$; cf. \cite[Theorem 3.4.1]{LM}.
Note that we cannot claim
$
v\in \mathcal{C}\big((0,T);H^{\gamma}\big)
$
even if $\varphi\in H^{\gamma}$.
 In fact,
because $W\in \mathcal{C}^{1/2-}$ is a point-wise multiplier in
$H^{\gamma}$ for $\gamma\in (-1/2,1/2)$
\cite[Lemma 5.2]{Kr_Lp1}, an attempt to find
 a traditional  regularity result for equation
\eqref{Str-gen} in a normal triple $(H^{r+1}, H^r, H^{r-1})$
leads to an irreconcilable pair of restrictions on $r$:
to have $Wf\in L_2((0,T); H^{r})$ we need $r<1/2$, whereas
to have $Wf_x\in L_2((0,T); H^{r-1})$ we need $r-1>-1/2$ or
$r>1/2$.

Accordingly, to derive a bound on $\|v\|_{\gamma}(t)$ for $t>0$,
 we use the mild formulation \eqref{Str-mild}.

\begin{proposition}
\label{pr:BasicEstimate-Str}
Let $\gamma\in (1/2,1)$, $f\in L_2\big((0,T); H^{\gamma}\big)$,
$\varphi\in H^{0}$,  and
let $v$ be the mild solution of \eqref{ghe-1} with
$v|_{t=0}=\varphi$.
Then, for every $T>0$ and every realization of $W$, there exists
a number $C_{\circ}$ such that
\bel{eq:stab0}
\|v\|_{\gamma}(t) \leq C_{\circ}\left(t^{-\gamma/2}\|\varphi\|_{0}+
\int_0^t (t-s)^{-\gamma}\|f\|_{\gamma}(s)\, ds\right).
\ee

\end{proposition}

\begin{proof}
Throughout the proof, $C$ denotes a number depending
on $\gamma$, $T$, and the
 norm of $W$ in the space $\mathcal{C}^{1-\gamma}$.
 The value of $C$ can change from one instance to another.
With no loss of generality, we assume that $\varphi$ and
$f$ are smooth functions with compact support.

To begin, let us show that if $V$ is the mild solution of
$$
\frac{\partial V(t,x)}{\partial t}=\frac{\partial^2 V(t,x)}{\partial x^2}+
f(t,x)\circ \dot{W}(x),\ t>0,
$$
$V(t,0)=V(t,\pi)=0$, $V|_{t=0}=\varphi$,
then
\bel{eq:stab1}
\|V\|_{\gamma}(t) \leq Ct^{-\gamma/2}\|\varphi\|_{0}+C
\int_0^t (t-s)^{-\gamma}\|f\|_{\gamma}(s)\, ds.
\ee
Indeed, by \eqref{Str-mild},
$$
V(t,x)= \int_0^t p*(fW)_x(t,s,x)\, ds-
\int_0^t p*\big(f_xW\big)(t,s,x)\, ds+\int_0^{\pi}p(t,x,y)\varphi(y)dy.
$$
Using  \eqref{PosNorm}  with $\theta=\gamma$,
$$
\left\|\int_0^{\pi}p(t,\cdot,y)\varphi(y)dy\right\|_{\gamma}\leq Ct^{-\gamma/2}\|\varphi\|_0.
$$
Then
\bel{str-sol-f}
\|V\|_{\gamma}(t)\leq \int_0^t \|p*(fW)_x\|_{\gamma}(t,s)\, ds+
\int_0^t \|p*\big(f_xW\big)\|_{\gamma}(t,s)\, ds
+Ct^{-\gamma/2}\|\varphi\|_{0}.
\ee

To estimate the first term on the right hand side of \eqref{str-sol-f},
 we use  \eqref{PosNorm}  with $\theta=2\gamma$. Then
$$
\|p*(fW)_x\|_{\gamma}(t,s)\leq
C(t-s)^{-\gamma}\|(fW)_x\|_{-\gamma}(s)
\leq C(t-s)^{-\gamma}\|fW\|_{1-\gamma}(s),
$$
and, because $W\in \mathcal{C}^{1/2-}((0,\pi))$
 is a (point-wise) multiplier
in  $H^{1-\gamma}$,
$$
\|fW\|_{1-\gamma}(s)\leq C_W\|f\|_{1-\gamma}(s);
$$
 recall that $0<1-\gamma<1/2$.
Finally, as $1-\gamma<\gamma$,
\bel{str-est1-f}
\|p*(fW)_x\|_{\gamma}(t,s)\leq
C(t-s)^{-\gamma}\|f\|_{\gamma}(s).
\ee

To estimate the second term on the right hand side of \eqref{str-sol-f},
 we use  \eqref{PosNorm}  with $\theta=1$. Then
$$
\|p*\big(f_xW)\|_{\gamma}(t,s)
\leq \frac{C}{\sqrt{t-s}}\,\|f_xW\|_{\gamma-1}(s)
\leq \frac{C}{\sqrt{t-s}}\,\|f_x\|_{\gamma-1}(s),
$$
that is,
\bel{str-est2-f}
\|p*\big(f_xW\big)\|_{\gamma}(t,s)
\leq C (t-s)^{-1/2}\|f\|_{\gamma}(s).
\ee
To establish \eqref{eq:stab1}, we now combine \eqref{str-sol-f},
 \eqref{str-est1-f}, and \eqref{str-est2-f},
  keeping in mind that
 $(t-s)^{-1/2}\leq C(t-s)^{-\gamma}$ because $\gamma>1/2$.

Next,   \eqref{eq:stab1} applied to \eqref{ghe-1} implies
$$
\|v\|_{\gamma}(t)\leq C\int_0^t (t-s)^{-\gamma}
\|v\|_{\gamma}(s)\,ds+C\int_0^t (t-s)^{-\gamma}
\|f\|_{\gamma}(s)\,ds+Ct^{-\gamma/2}\|\varphi\|_{0},
$$
and then a generalization  of the Gronwall inequality
 (e.g. \cite[Corollary 2]{GG}) leads to \eqref{eq:stab0}.
\end{proof}

\begin{corollary}
If  $\varphi\in H^{0}$ and $\gamma\in (1/2,1)$, then, for
every $T>0$, $a>0$, and every realization of $W$, there
 exists a number $\tilde{C}_{\circ}$ such that the mild
solution of \eqref{eq:main-S} satisfies
\bel{norm-u}
\sup_{|\varepsilon|<a}\|u_{\circ}(t,\cdot,\varepsilon)\|_{\gamma}
\leq \tilde{C}_{\circ}\,t^{-\gamma/2}\|\varphi\|_0,\ t\in (0,T].
\ee
\end{corollary}

Next, define the functions $u_{\circ}^{(n),\varepsilon}=
u_{\circ}^{(n),\varepsilon}(t,x)$,\ $n=0,1,2\ldots,$
$t\geq 0$, $x\in [0,1]$, $\varepsilon\in \bR$,
by $u_{\circ}^{(0),\varepsilon}(t,x)=u_{\circ}(t,x;\varepsilon)$ and, for
$n\geq 1$,
\bel{eq:main-dk-ep}
\begin{split}
\frac{\partial u_{\circ}^{(n),\varepsilon}(t,x)}{\partial t} &=
\frac{\partial^2 u_{\circ}^{(n),\varepsilon}(t,x)}{\partial x^2} +
\varepsilon u_{\circ}^{(n),\varepsilon}(t,x)\circ \dot{W}(x)+
 u_{\circ}^{(n-1),\varepsilon}(t,x)\circ \dot{W}(x),\\
u_{\circ}^{(n),\varepsilon}(t,0)
&=u_{\circ}^{(n),\varepsilon}(t,\pi)=0, \
u_{\circ}^{(n),\varepsilon}(0,x)=0.
\end{split}
\ee
 In particular,
 $$
 u_{\circ}^{(n),0}(t,x)=u_{\circ}^{(n)}(t,x).
 $$
 Note that all equations in \eqref{eq:main-dk-ep}
  are of the form \eqref{ghe-1}.

\begin{proposition}
\label{prop:St}
If $\varphi\in H^{0}$, then, for every $\gamma\in (1/2,2/3)$
 and every realization of $W$,
\bel{gen-der}
\lim_{\varepsilon\to \varepsilon_0}
\frac{1}{(\varepsilon-\varepsilon_0)^{n}}
\Big\|u_{\circ}(t,\cdot;\varepsilon)-\sum_{k=0}^n
(\varepsilon-\varepsilon_0)^ku_{\circ}^{(k),\varepsilon_0}(t,\cdot)
\Big\|_{\gamma}=0,
\ee
$n\geq 0$, $\varepsilon_0\in \bR$,\ $t\geq 0$.
\end{proposition}

\begin{proof}
Throughout the proof, $C$ denotes a number depending
on $\gamma$, $T$, $\varepsilon_0$, and the
 norm of $W$ in the space $\mathcal{C}^{1-\gamma}$.
Define
\bel{dffrence-gen}
\mathfrak{v}^{(n)}_{\varepsilon}(t,x)=
\frac{1}{(\varepsilon-\varepsilon_0)^{n}}
\left(u_{\circ}(t,x;\varepsilon)-\sum_{k=0}^n
(\varepsilon-\varepsilon_0)^ku_{\circ}^{(k),\varepsilon_0}(t,x)\right).
\ee
By \eqref{eq:main-S},
\bel{eq:main-Vd1}
\begin{split}
\sum_{k=0}^n(\varepsilon-\varepsilon_0)^k
\frac{\partial u_{\circ}^{(k),\varepsilon_0}(t,x)}{\partial t}
&= \sum_{k=0}^n
(\varepsilon-\varepsilon_0)^k
\frac{\partial^2 u_{\circ}^{(k),\varepsilon_0}(t,x)}{\partial x^2}\\
&+\varepsilon_0\sum_{k=0}^n(\varepsilon-\varepsilon_0)^k
u_{\circ}^{(k),\varepsilon_0}(t,x)\circ \dot{W}(x)\\
 &+ (\varepsilon-\varepsilon_0)\sum_{k=1}^n
 (\varepsilon-\varepsilon_0)^{k-1}
  u_{\circ}^{(k-1),\varepsilon_0}(t,x) \circ \dot{W}(x),
\end{split}
\ee
so that
\bel{eq:main-Vd2-gen}
\begin{split}
\frac{\partial \mathfrak{v}^{(0)}_{\varepsilon}(t,x)}{\partial t} &=
\frac{\partial^2 \mathfrak{v}^{(0)}_{\varepsilon}(t,x)}{\partial x^2}
+\varepsilon_0\mathfrak{v}^{(0)}_{\varepsilon}(t,x)
\circ \dot{W}(x)
 + (\varepsilon-\varepsilon_0) u_{\circ}(t,x;\varepsilon)\circ
 \dot{W}(x),\\
\frac{\partial \mathfrak{v}^{(n)}_{\varepsilon}(t,x)}{\partial t} &=
\frac{\partial^2 \mathfrak{v}^{(n)}_{\varepsilon}(t,x)}{\partial x^2}
+\varepsilon_0\mathfrak{v}^{(n)}_{\varepsilon}(t,x)\circ \dot{W}(x)
 +  \mathfrak{v}^{(n-1)}_{\varepsilon}(t,x)\circ \dot{W}(x),
 \ n\geq 1,
\end{split}
\ee
 $\mathfrak{v}^{(n)}_{\varepsilon}(0,x)=0$, $n\geq 0$, and
\eqref{gen-der} becomes
\bel{gen-der-v}
\lim_{\varepsilon\to\varepsilon_0}
\|\mathfrak{v}^{(n)}_{\varepsilon}\|_{\gamma}(t)=0,\
\ n\geq 0,\ t\geq 0,\ \varepsilon_0\in \bR.
\ee
 Note that all equations in
\eqref{eq:main-Vd2-gen} are of the form \eqref{ghe-1},
and \eqref{gen-der-v} trivially holds for $t=0$.
Accordingly, combining the second equation in \eqref{eq:main-Vd2-gen}
with \eqref{eq:stab0},
$$
\|\mathfrak{v}^{(n)}_{\varepsilon}\|_{\gamma}(t) \leq
C\int_0^t
(t-s)^{-\gamma}
\|\mathfrak{v}^{(n-1)}_{\varepsilon}\|_{\gamma}(s)\,ds,
$$
$n\geq 1$,
and then, for $t>0$, \eqref{gen-der-v} follows by induction: for  $n=0$,
 \eqref{norm-u} yields
\begin{align*}
&\|\mathfrak{v}^{(0)}_{\varepsilon}\|_{\gamma}(t)
\leq |\varepsilon-\varepsilon_0|\,
C\int_0^t (t-s)^{-\gamma}\|u_{\circ}(s,\cdot,\varepsilon)\|_{\gamma}\,ds
\\
& \leq  C  |\varepsilon-\varepsilon_0|\, \|\varphi\|_0
\int_0^t (t-s)^{-\gamma}s^{-\gamma/2}\,ds
\leq C  |\varepsilon-\varepsilon_0|\, \|\varphi\|_0 t^{1-(3/2)\gamma}
\to 0,\ \varepsilon \to \varepsilon_0;
\end{align*}
similarly,  for $n\geq 1$,
$$
\|\mathfrak{v}^{(n)}_{\varepsilon}\|_{\gamma}(t) \leq
C^{(n)}|\varepsilon-\varepsilon_0|\, \|\varphi\|_0,
$$
because $1-(3/2)\gamma>0$.
\end{proof}

\begin{proposition}
\label{prop:Sa}
If $\varphi\in H^{0}$ and  $\gamma\in (1/2,1)$, then
 \bel{n-der-S}
\lim_{n\to \infty} c^n\sup_{|\varepsilon|<a}
\|u_{\circ}^{(n),\varepsilon}\|_{\gamma}(t)=0,\ t\geq 0,
\ee
for all $c>0$, $a>0$, and every realization of $W$.
\end{proposition}

\begin{proof}
Throughout this proof,  $C$ denotes a number depending
on $\gamma$, $T$, $a$, and the
 norm of $W$ in the space $\mathcal{C}^{1-\gamma}$.

Combining  \eqref{eq:main-dk-ep} and \eqref{eq:stab0},
$$
\|u_{\circ}^{(n),\varepsilon}]\|_{\gamma}(t) \leq
C\int_0^t (t-s)^{r-1}\|u_{\circ}^{(n-1),\varepsilon}\|_{\gamma}(s)\,ds.
$$
 By iteration and \eqref{norm-u}, with $r=1-\gamma>0$,
\begin{equation*}
\begin{split}
\sup_{|\varepsilon|<a}
&\|u_{\circ}^{(n),\varepsilon}\|_{\gamma}(t)
\leq C^n\|\varphi\|_0\\
&\times\int_0^t\int_0^{s_{n-1}}\!\!\!\!\!\ldots\int_0^{s_2}
(t-s_n)^{r-1}(s_n-s_{n-1})^{r-1}
\cdots (s_2-s_1)^{r-1}s_1^{-\gamma/2}ds_1\cdots ds_n\\
&=C^n \|\varphi\|_0 \,
 \frac{\big(\Gamma(r)\big)^n\Gamma(1-(\gamma/2))}
 {\Gamma\big(nr+1\big)} \,t^{nr-(\gamma/2)},
\end{split}
\end{equation*}
 where $\Gamma$  is the Gamma function
$$
\Gamma(y)=\int_0^{\infty} t^{y-1}e^{-t}dt.
$$
Then \eqref{n-der-S} follows by the Stirling formula.
\end{proof}

Equality  \eqref{ps-S} now follows:
\begin{itemize}
\item By the Sobolev embedding theorem, every element,
 or equivalence, class from $H^{\gamma}$, $\gamma>1/2$,
  has a representative
 that is a continuous function on $[0,\pi]$;
\item By Proposition \ref{prop:St} and the Taylor formula,
$$
u_{\circ}(t,x)=u(t,x)+\sum_{k=1}^n
u_{\circ}^{(k),0}(t,x)\varepsilon^k+R_n(t,x);
$$
\item By Proposition \ref{prop:Sa},
$$
\lim_{n\to \infty} R_n(t,x)=0.
$$
\end{itemize}

{\em This concludes the proof of  \eqref{ps-S}.}

\section{The Correction Term}
\label{sec:Cr}

The objective of this section is the proof of \eqref{order2-d}.

Using \eqref{ps-W} and \eqref{ps-S}, and remembering that
$u_{\circ}^{(1)}=u_{\diamond}^{(1)}=\mathfrak{u}$,
\begin{equation}
\label{WS-diff0}
\begin{split}
\lim_{\varepsilon\to 0}\varepsilon^{-2}&
\Big(u_{\circ}(t,x;\varepsilon)-u_{\diamond}(t,x;\varepsilon)\Big)=
u_{\circ}^{(2)}(t,x)-u_{\diamond}^{(2)}(t,x)\\
&=
\int_0^t \int_0^{\pi} p(t-s,x,y)\big(\mathfrak{u}(s,y)\circ d{W}(y)-
\mathfrak{u}(s,y)\diamond d{W}(y)\big)\,ds.
\end{split}
\end{equation}
By definition,
$$
\xi_k\diamond\xi_n=
\begin{cases}
\xi_k\xi_n,& k\not=n,\\
\xi_n^2-1,& k=n,
\end{cases}
$$
and therefore
\bel{WDif}
\xi_k\xi_n-\xi_k\diamond\xi_n=
\begin{cases}
0& k\not=n,\\
1,& k=n,
\end{cases}
\ee
Then  \eqref{WDif} and \cite[Theorem 3.1.2]{Nualart} imply that,
for  a function $f=f(x)$ of the form
 $$
 f(x)=\sum_{k=1}^{\infty} f_k(x)\xi_k,
 $$
 with $f_k$ non-random and satisfying
 \bel{int-cond}
 \sum_k\int_0^{\pi}|f_k(x)|\,dx <\infty,
 \ee
  the following equality holds:
 \bel{WS-dif1}
 \int_0^{\pi} f(x)\circ dW(x)-\int_0^{\pi}
 f(x)\diamond dW(x)=
 \sum_{k=1}^{\infty} \left(\int_0^{\pi} f_k(x)\mathfrak{m}_k(x)\,
 dx\right).
 \ee
 Condition \eqref{int-cond} ensures that the sum on the
 right-hand side of \eqref{WS-dif1} converges absolutely.

 Next, recall that, by \eqref{u1-expl},
$$
\mathfrak{u}(s,y)=\sum_{k=1}^{\infty}
\left(\int_0^{\pi} \int_0^s p(s-r,y,z)u(r,z)\mathfrak{m}_k(z)\, dr\, dz
\right)\,\xi_k.
$$
For fixed $s\in [0,T]$ and $y\in[0,\pi]$, define
$$
g(z)=\int_0^s p(s-r,y,z)u(r,z)dr,\ g_k=\int_0^{\pi}g(z)\mfm_k(z)\,dz.
$$
Then
$$
\mathfrak{u}=\sum_{k=1}^{\infty} g_k\xi_k,
$$
and \eqref{int-cond} in this case will follow from uniform,
in $(s,y)$ convergence of
$$
\sum_{k=1}^{\infty} |g_k|,
$$
which, by Bernstein's theorem \cite[Theorem VI.3-1]{Zygmund},
will, in turn,  follow from
\bel{HoldCont}
|g(z+h)-g(z)|\leq Ch^{\delta}
\ee
with $\delta\in (1/2,1)$ and $C$ independent of $s,y,z$.

Recall that
$$
u(r,z)=\sum_{k=1}^{\infty} \varphi_k e^{-k^2r} \, \mfm_k(z),\
\varphi_k=\int_0^{\pi} \varphi(x)\mfm_k(x)\, dx,
$$
and, by integral comparison,
$$
\sum_{k=1}^{\infty} k^p e^{-k^2\,t}\leq \frac{C(p)}{t^{(1+p)/2}},\
p\geq 0.
$$
Also,
$$
|\sin(k(z+h))-\sin(kz)|\leq k^\delta h^{\delta},\ \delta\in (0,1),
$$
and the maximum principle implies $|u(r,z)|\leq C$.
Then
\begin{align*}
&p(s-r,y,z)\leq \frac{C}{\sqrt{s-r}},\ \
|p(s-r,y,z+h)-p(s-r,y,z)|\leq \frac{Ch^{\delta}}{(s-r)^{(1+\delta)/2}},\\
& |u(r,z+h)-u(r,z)|\leq \frac{Ch^{\delta}}{r^{(1+\delta)/2}},
\end{align*}
and \eqref{HoldCont} follows because
$$
\int_0^s\frac{dr}{\big(r(s-r)\big)^{(1+\delta)/2}}<\infty
$$
for $\delta\in (1/2,1)$.

We  now apply \eqref{WS-dif1} to \eqref{WS-diff0}:
 \begin{align*}
&\int_0^t \int_0^{\pi} p(t-s,x,y)\big(\mathfrak{u}(s,y)\circ d{W}(y)-
\mathfrak{u}(s,y)\diamond d{W}(y)\big)\,ds\\
&=\int_0^t \int_0^{\pi} \sum_{n=1}^{\infty}
\left(\int_0^{\pi} \!\left(\int_0^s p(s-r,y,z) {u}(r,z)\,dr\right)
\mathfrak{m}_n(z)\, dz\right)\mathfrak{m}_n(y)p(t-s,x,y)\, dy\,ds\\
&= \int_0^{\pi}\int_0^t\int_0^s p(t-s,x,y)p(s-r,y,y)u(r,y)\, dr\, ds\, dy,
\end{align*}
which, in view of \eqref{u0-expl} and the Fubini theorem, is the same as
\eqref{order2-d}.

{\em This concludes the proof of  \eqref{order2-d}.}

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\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]
  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}

redpajama_set_name: RedPajamaArXiv


There are many approaches our teachers use to personalize the learning experience for students. One way is by giving students different ways to demonstrate their learning. Edwin Bryson, Greenwood's Vice-Principal of Teacher and Staff Development, shares an example from his Grade 10 Introduction to Business class, in which differentiated assessments were used to personalize for student readiness, interest and learning profile.
I began by identifying what each student should know, understand and be able to do as a result of a particular chunk of learning. In this case, students needed to "demonstrate financial planning skills and produce a...personal financial plan (e.g. monthly plan, budget)."
Create a board game that illustrates income, expenses and savings, etc.
The goal is to determine financial planning skills, but the teacher has the flexibility to create more than one type of assessment for this skill.
The student understands the relationship between types of income, fixed and flexible expenses.
The student demonstrates the use of planning skills (gathering information, organizing a budget/project).
The student uses critical/creative thinking processes (evaluation of spending and saving goals, actual versus planned budgeting).
The student makes connections between the financial planning process and future career and life goals.
The fourth step was to select a few assessments that would meet the different levels of student readiness, interests and learning profile.
Complete a monthly budget using a template: This option would suit students who are still gaining confidence with financial planning, liked working individually and benefited from concrete and sequential tasks rather than abstract and non-sequential tasks.
Work in pairs and create a board game: The game should demonstrate the key concepts of income, personal income tax, expenses, savings and investment. This option would suit students who had attained a conceptual understanding of financial planning, liked working collaboratively and enjoyed abstract and non-sequential thinking.
Create a role-playing game: This option was created by a few students who wanted to modify the board game assessment to create a role-playing game. Since the rubric focused on learning outcomes, rather product specifications, it was very easy to accommodate this request.
Below is a summary of the differentiation found in each assessment.
Create a board game Strong conceptual understanding Wants to expand and extend their learning of this topic Is stimulated by creative challenges and conceptual thinking. Prefers working with peers.
Create a role-playing game Strong conceptual understanding Wants to expand and extend their learning of this topic Is stimulated by open-ended challenges and enjoys abstract thinking. Prefers working with peers.
Regardless of which assignment was chosen, students were engaged in an authentic learning experience and making meaning by linking the concepts to their own lives. Each assignment allowed students to reach the top level of achievement and provided different approaches to suit their learning styles. The end result was increased engagement and more accurate evaluation of each student's progress toward meeting the course's learning goals.
Note: Many of the concepts discussed here are borrowed from one of the foremost authors on the subject, Carol Ann Tomlinson, in her book How to Differentiate Instruction in Mixed-Ability Classrooms (2nd Edition).
Anthony Costa, Subject Team Leader for Health and Physical Education at Greenwood, explores how practical skill acquisition is one method for helping students to develop life-long transferable skills that support an active, healthy lifestyle.
Healthy active living is a life-long endeavour and the improvement of one's own healthy practices requires continuous pruning. Whether it is keeping current about nutrition, improving personal fitness techniques, learning how to be aware of mental health issues, or acquiring new skills, Health and Physical Education is continually evolving.
One way that the Health and Physical Education team at Greenwood College School is working to have students continually challenge themselves is through improvement of Physical Literacy. This video gives a brief explanation of why physical literacy is so important. We endeavour to work on this in many ways at Greenwood. However, I would like to highlight one example for now: on-going skill acquisition.
Teachers create an on-going skills continuum across various grades and ability levels and in each sport or activity, in order to ensure continuous challenge and growth for students. This continuum also allows teachers to use professional judgement and have students practice a skill set based on readiness. For example, currently in soccer activity units, students can practice shooting, passing and dribbling skills, body control, and team movement patterns. The way that these are practiced can be effectively individualized, in order to have students indirectly practice and learn how to continually grow. The idea is to give students a constant challenge within their readiness zone in order to help them grow.
Focusing on the skill acquisition of dribbling as a specific example, students with self identified "novice" skills can practice and develop the skill of moving the ball to various areas of the pitch while maintaining proper technique. If students consider themselves to be at an "intermediate" level they can work to improve dribbling skills with their non-dominant foot in order to acquire a more complete set of skills in the sport. If students consider themselves at an advanced or "expert" level for the skill, they will have the opportunity to practice advanced techniques of dribbling such as the "crossover," change of direction between the legs, different spin moves, dragging the ball and other creative techniques. Furthermore, if students are interested, they may access videos highlighting the extraordinary skills of the best players in the world and practice these elite skills to ensure continuous challenge.
Once introduced, students utilize the skill on a daily basis in order to refine technique and increase their speed in mini game activities and modified game situations. If students can continually challenge themselves in a skill like dribbling, then they are developing a framework for how to improve upon other skills as well - not only physically, but also cognitively and academically.
To improve requires basic knowledge, background and a mindset of growth. If students can experience a successful step-by-step process of improvement in physical literacy, they can potentially transfer the same mindset towards many other aspects of life. Perseverance through a set of math problems, determination through science labs, focus when working on French verb tenses and practice routines in the arts and music are just a few examples. It is the hope that these transferable skills create well rounded and capable young problem solvers.
Through collaboration and the creation of on-going skill continua and unit plans, Health and Physical Education teachers at Greenwood can modify and adjust daily lesson plans to meet the needs of the varied ability levels of students with input from colleagues. After all, two heads are better than one! From "novice" to "expert," teachers are working to ensure that all students will have the opportunity to improve upon their skills, and ultimately, develop a framework for practice in whatever aspect of life they choose to pursue with the most rigour.
An overall goal of the Health and Physical Education curriculum is to have students become healthy and active people for life. By participating in these habits regularly in soccer, basketball, volleyball and other activities, Greenwood students can practice transferable skills and apply them to continual growth and improvement in their individual mental and physical health. Health and Physical Education equips students with valuable decision-making skills for a positive and healthy lifestyle. Skill acquisition is just one piece of the puzzle but it can be a very effective one.
It is very easy to rationalize a personalized approach to learning. Given the choice between an educational program that is tailored to individual needs and a program that is the same for everybody, the choice seems obvious. The difficulty with the former lies in implementation. Personalized learning gives students more control and ownership over their learning but control can be very daunting for some students. To successfully transition into a personalized program it is essential that students possess the proper skills. It is the teacher's role to help students develop these skills.
The first is a technological element. In order to facilitate a flexible learning environment, technology is generally utilized, which requires students to have a certain skill set.
The other essential element is a collection of personal skills that learners need to possess. For example, in order to thrive in a personalized learning environment, students need to be organized, self-motivated, good communicators and able to persevere. Fortunately, all of these skills can be developed with time and practice.
One course that has been redeveloped at Greenwood with the goal of better preparing students for personalized learning is the Information and Communication Technology in Business Course, more commonly known as BTT. A large component of this course focuses on the development of important IT skills. Students are exposed to a variety of software and become skilled with word processing, spreadsheets, databases, presentation software, developing web pages and using search engines. With the strong IT skills developed in this course, any technological barriers preventing students from being successful in a personalized program are removed.
Developing the personal skills to be successful in a student-centred program is a considerably more challenging task. In a traditional classroom where teachers control the pace and content, it is very easy for students to become passive learners. Passive learners are accustomed to being told what, how and when they need to learn.
BTT, on the other hand, gives students complete control over their learning in an environment that is flexible and accommodating, and allows students to develop the necessary skills for a personalized program over time. Although BTT is a grade 9 course, at Greenwood, the course is generally started in grade 7 and completed by students over 2 years. By extending the program over 2 years, students have a lot of time to develop the required skills. This course is also available to new grade 9 students so that they can reach the same level as returning students, but in this case it is completed in 1 year.
BTT is run as an independent course, with only extra help scheduled, and is integrated with other subjects. Integrating with other subjects allows students to complete assignments for BTT and other courses simultaneously. For example, one unit in BTT requires students to use software to create a presentation. Since the assessment for BTT is primarily skills-based, students can easily create a presentation with content for another class, ensure that the appropriate skills are demonstrated, and then submit the same assignment for BTT.
The format of this course requires students to look ahead and determine the most appropriate assignments to complete, to organize their time, to coordinate meetings for extra help and to collaborate with each other. The flexibility of the course and the 2-year completion time allows students to improve their work and hone their skills without worrying about rigid deadlines.
While it is very challenging for many grade 7, 8, and 9 students to take ownership of their learning, this course aims to promote and help develop the skills needed for a personalized learning program in a safe and supportive environment.


text:
Labour law 'reforms'
July 26, 2019 ThePatriot
The Working Journalists Acts of 1955 and 1958 are among 12 labour laws to be subsumed in proposed labour codes that will reduce legal protections for journalists at their workplaces
A wave of layoffs is washing through Indian newsrooms. In June, Newslaundry had reported that digital news portal Scroll.in laid off more than 16 employees from its 40-member editorial team, including several state correspondents and senior editors. Just days later, Tiranga TV laid off more than 200 employees leading to a spate of public protests by retrenched journalists demanding transparency and better severance pay.
Earlier, last month, journalist Hartosh Singh Bal, now political editor of Caravan magazine, won against his former employer Open magazine in an industrial dispute he had filed in a Delhi labour court for wrongful termination from work six years back. The court awarded him compensation of Rs 10 lakh for harassment by his then employer in violating the law, back-wages amounting to six months' notice pay, and provided for the cost of litigation to be reimbursed by the employer.
What legal protections do journalists enjoy at their workplace? What rights do they have in case of retrenchment?
Bal moved a labour court to seek redress because The Working Journalists and other Newspaper Employees (Conditions of Service) and Miscellaneous Provisions Act of 1955 says the Industrial Disputes Act of 1947's provisions apply to journalists. In case of retrenchment, however, lawyers state that the Working Journalists Act enhances the protections for journalists above and beyond the protections the Industrial Disputes Act offers to workers in other sectors.
"While the Industrial Disputes Act Section 25F gives only one month notice period in case of retrenchment of a workman, the current legal protections are that if you wish to retrench a journalist, you have to give three months notice, and if the person is at the level of an editor, then you have to give six months notice," says Bal's lawyer Raghav Awasthi, explaining Section 3(2) of the Working Journalists Act. Awasthi adds: "The law provides journalists greater protection than a worker in any other private service as a measure for protecting free speech and the independence of journalists, insulating them from political pressure on media owners." These protections apply even if journalists sign contracts with employers that do not provide for such a notice period.
Section 16(2) of the Working Journalists Act states:
Nothing contained in this Act shall be construed to preclude any newspaper employee from entering into an agreement with an employer for granting him rights or privileges in respect of any matter which are more favourable to him than those to which he would be entitled under this Act.
Thus, the Working Journalists Act provides crucial protections for journalists against arbitrary termination, and even under the contract system, journalists continue to be protected by the Working Journalists Act.
But a major limitation is that the Act, framed in the 1950s, applies only to print journalists. It is yet to be extended to TV, radio and digital news journalists who do similar editorial work in new media. It covers only journalists in newspaper establishments which it defines as a "printed periodical work" of public news and comments notified by the central government. It defines "working journalists" as those who do editorial work, including writers, editors, sub-editors, cartoonists and news-photographers, excluding those employed mainly in a managerial or administrative capacity such as financial officers or HR managers.
Now, even the protections that only print journalists get are being diluted in the new labour codes being considered by the National Democratic Alliance government. On July 10, the Union cabinet chaired by Prime Minister Narendra Modi approved a Code on Occupational Safety, Health and Working Conditions Bill, 2019. This labour code amalgamates 12 labour laws into one legislation. These labour laws include the Working Journalists Act, 1955, and the Working Journalists (Fixation of Rates of Wages) Act, 1958, along with sector-specific labour laws applying to workers in mines, factories, plantations, and building and construction.
The Ministry of Labour and Employment in a statement said the new labour code, which will now be laid before Parliament, will streamline and modernise labour laws and "enhance the coverage of the safety, health and working conditions provisions manifold". But the draft drops Section 3(2) of the Working Journalists Act which offers enhanced protections against retrenchment to working journalists and reduces this to a one-month notice period, like workers in any other factory or private establishment. Instead of expanding the definition of "working journalists" to include those doing editorial work in TV, digital, and radio media, it replicates the definition of "working journalists" from the 1955 law by limiting it only those working in newspaper establishments.
A few journalists' unions expressed concerns over this. The Delhi Union of Journalists in a statement expressed "its incredulity over how these laws are now sought to be repealed by placing them innocuously in a Labour Code on Occupational Health and Safety". The statement said: "It was honourable members of Parliament who, shortly after Independence, realised the need to protect journalists and enable them to earn a decent livelihood by setting basic standards for the newspaper industry through these two Acts … Without such legal protection the independence of journalists shall be further weakened."
Sanjay Upadhyay, vice president of Working Journalists of India—a union affiliated with the Bharatiya Mazdoor Sangh, the trade union wing of the Rashtriya Swayamsewak Sangh—told Newslaundry he and other members of National Union of Journalists (India), and Delhi Journalists Association had met with the Minister for Labour and Employment, Santosh Gangwar, on July 11 to reconsider these provisions in draft labour code.
"The Second Press Commission report of 1982 came before the advent of private electronic media and radio channels. We asked the minister to incorporate new categories of jobs in electronic and digital media in the labour codes," says Upadhyay. "We asked for six months' retrenchment pay for all categories of journalists. The minister told us he would look into it."
On July 23, when tabling the labour codes, the Lok Sabha included union demands to expand the definition of working journalist. The definition now says "in relation to, one or more newspaper establishment, or other establishment relating to any electronic media such as newspaper or radio or like other media". However, the notice period of three to six months is absent.
Caravan magazine's political editor Hartosh Singh Bal says he believes the Working Journalists Act continues to be relevant as it provides crucial protections for journalists against arbitrary termination recognising the vital role that journalists play in a democracy. "Journalists themselves started assuming after the 1990s that if they have signed contracts, they are outside the Working Journalists Act protections. But the contract system cannot displace the Working Journalists Act is what I found out in my case," he says.
Bal thinks it's essential that TV and digital journalists are included in the Working Journalists Act as these are categories of journalists doing the same editorial work as print journalists, and similar equal legal protections must apply to them. He adds: "Our failure to report on labour laws and conditions is almost suicidal. It is the same conditions that we work in. It is worrisome that these protections are being diluted. There should be far more outrage about it."
Senior journalist and readers' editor of The Wire, Pamela Philipose, says journalists need to oppose the shrinking of their legal protections. "Many journalists fought for years for these legislative protections and the labour Codes in their current form are going to invalidate them. Over a period of time, journalists lost interest in protecting the law and instead looked for better deals for themselves personally. But in this process, they have let their side down. Journalists need to understand the value of collective action at a time when their rights are under attack."
Sevanti Ninan, founder-editor of The Hoot, a South Asian media watch website, says: "The advent of TV and digital has both increased media employment and reduced job security. Media has increased hugely in India, the advertising available to newspapers and others has shrunk, and even former exemplary employers such as The Hindu Group have reduced or done away with jobs covered by the Working Journalists Act."
Ninan adds: "With job security being a thing of the past, journalists have to be far more watchful of the contracts they sign with employers, and ensure that they get proper compensation and notice periods when laid off. And be prepared to go to court to fight layoffs in unjust conditions. Every journalist who goes to court and wins as Hartosh Bal did recently, helps to ensure that there is a precedent created for his fellow professionals."
She expresses concern that the "regional press offers practically no security to parttime workers who do news collection in the country's hinterland".
Shekhar Gupta, president of the Editors' Guild of India, who is also the founder and editor-in-chief of the news portal The Print, says he is not aware of the proposals in the new proposed code but he plans to examine them.
RK Gupta, who is joint secretary, labour reforms, at the Ministry of Labour and Employment, did not respond to phone calls and an email questionnaire. The story will be updated if the ministry responds.
The story has been updated with Lok Sabha proceedings of July 23
www.newslaundry.com
ThePatriot
http://thepatriot.in/author/thepatriot/
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