tag:theconversation.com,2011:/institutions/carthage-college-3443/articlesCarthage College2018-08-29T19:20:59Ztag:theconversation.com,2011:article/1022782018-08-29T19:20:59Z2018-08-29T19:20:59ZTémoignage : Pourquoi j’utilise le tricot pour enseigner les mathématiques<figure><img src="https://images.theconversation.com/files/233975/original/file-20180828-86132-yf0dfm.jpg?ixlib=rb-1.1.0&rect=45%2C4%2C985%2C616&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">En tricotant eux-mêmes écharpes et bonnets, les élèves découvrent la géométrie sous un autre angle. </span> <span class="attribution"><span class="source">Carthage College</span>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span></figcaption></figure><p>Un jour enneigé de janvier, j’ai demandé à des étudiants de me dire quel était le premier mot qui leur venait à l’esprit lorsqu’ils pensent aux mathématiques. Les deux mots les plus cités ont été « calcul » et « équation ». Lorsque j’ai posé la même question à une salle remplie de mathématiciens professionnels, aucun de ces deux mots n’a été prononcé ; ils m’ont parlé « pensée critique » et « résolution de problèmes ».</p>
<p>L’expérience n’est pas très étonnante. Ce que les mathématiciens professionnels considèrent comme des mathématiques n’a rien à voir avec l’image que s’en fait la population générale. Alors que tant de gens évoquent les mathématiques comme une discipline cantonnée au calcul, il n’est pas étonnant que nous entendions si souvent dire : « Je déteste les mathématiques ».</p>
<p>J’ai donc décidé de m’emparer de cette question d’une manière peu conventionnelle, en proposant un cours intitulé « Les mathématiques du tricot » dans mon établissement, le <a href="https://www.carthage.edu/about/">Carthage College</a>. Durant les séances, j’ai choisi de supprimer tout recours aux crayons, papiers, calculatrices (gloups !) et manuels scolaires. À la place, nous avons discuté, fait des travaux manuels, dessiné et joué avec toutes sortes d’objets – ballons de plage ou mètres de couture, par exemple. En guise de devoirs personnels, nous avons nourri notre réflexion en tenant un blog commun. Et bien sûr, nous avons tricoté.</p>
<h2>Équations en question</h2>
<p>L’équation est au cœur des mathématiques ; et dans une équation, le signe égal joue un rôle crucial. Une équation comme x = 5 nous dit que le x tant redouté, qui représente une certaine quantité, a la même valeur que 5. Le nombre 5 et la valeur de x doivent donc être exactement les mêmes.</p>
<p>Le sens du signe égal est très strict. Qu’il y ait la moindre petite entorse au terme « exactement », et cela signifie que les deux choses mises en équation ne sont pas égales. Cependant, il arrive souvent, dans la vie de tous les jours, que deux quantités, sans être <em>exactement</em> les mêmes, soient équivalentes au regard d’autres critères.</p>
<p>Imaginez, par exemple, que vous disposiez de deux coussins carrés. Le premier est rouge sur le haut, jaune sur le côté droit, vert en bas et bleu à gauche. Le second est jaune en haut, vert sur son côté droit, bleu en bas et rouge à gauche. Ces coussins ne sont pas exactement les mêmes. L’un a un sommet rouge, l’autre en a un jaune. Mais ils sont similaires, sans l’ombre d’un doute. En fait, ils seraient <em>exactement</em> les mêmes si vous appliquiez au coussin avec le sommet rouge une simple rotation dans le sens contraire des aiguilles d’une montre.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/233965/original/file-20180828-86150-1fm8c2e.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/233965/original/file-20180828-86150-1fm8c2e.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=281&fit=crop&dpr=1 600w, https://images.theconversation.com/files/233965/original/file-20180828-86150-1fm8c2e.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=281&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/233965/original/file-20180828-86150-1fm8c2e.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=281&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/233965/original/file-20180828-86150-1fm8c2e.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=353&fit=crop&dpr=1 754w, https://images.theconversation.com/files/233965/original/file-20180828-86150-1fm8c2e.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=353&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/233965/original/file-20180828-86150-1fm8c2e.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=353&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Rotation de deux coussins carrés.</span>
<span class="attribution"><span class="source">Sara Jensen</span>, <span class="license">Author provided</span></span>
</figcaption>
</figure>
<p>De combien de façons différentes pourrais-je disposer le même coussin coloré sur un lit de façon à ce qu’il semble chaque fois différent ? Un petit exercice permet de démontrer qu’il existe 24 configurations de motifs possibles avec un même jeu de couleurs, dont huit sont observables en déplaçant le coussin.</p>
<p>Les élèves en ont fait l’expérience en tricotant des coussins, composés de deux couleurs, à partir de diagrammes de tricot.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/233967/original/file-20180828-86120-10pt48i.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/233967/original/file-20180828-86120-10pt48i.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=271&fit=crop&dpr=1 600w, https://images.theconversation.com/files/233967/original/file-20180828-86120-10pt48i.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=271&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/233967/original/file-20180828-86120-10pt48i.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=271&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/233967/original/file-20180828-86120-10pt48i.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=340&fit=crop&dpr=1 754w, https://images.theconversation.com/files/233967/original/file-20180828-86120-10pt48i.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=340&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/233967/original/file-20180828-86120-10pt48i.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=340&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Diagrammes pour tricoter un coussin.</span>
<span class="attribution"><span class="source">Sara Jensen</span>, <span class="license">Author provided</span></span>
</figcaption>
</figure>
<p>Les élèves ont créé des tableaux à tricoter carrés dans lesquels les huit mouvements du tableau ont donné à chaque fois une image différente. Ces images ont ensuite été tricotées dans un coussin où leur équivalence pouvait être démontrée en déplaçant réellement le coussin.</p>
<h2>Géométrie et caoutchouc</h2>
<p>Un autre sujet que nous avons abordé est ce que l’on appelle la « géométrie du caoutchouc ». Il s’agit d’imaginer que le monde entier est fait de caoutchouc, puis de transposer cette idée sous la forme de figures géométriques.</p>
<p>Essayons de comprendre le concept avec le tricot. Pour tricoter des objets ronds – comme des chapeaux ou des gants – on peut utiliser des aiguilles à tricoter spéciales, appelées aiguilles à double pointe. En cours de fabrication, le chapeau est mis en forme par trois aiguilles, ce qui lui donne une forme triangulaire. Puis, une fois les aiguilles retirées, le fil extensible se détend pour former un cercle, et l’on reconnaît la forme caractéristique du chapeau.</p>
<figure class="align-right ">
<img alt="" src="https://images.theconversation.com/files/233968/original/file-20180828-86132-1hih22i.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/233968/original/file-20180828-86132-1hih22i.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=899&fit=crop&dpr=1 600w, https://images.theconversation.com/files/233968/original/file-20180828-86132-1hih22i.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=899&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/233968/original/file-20180828-86132-1hih22i.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=899&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/233968/original/file-20180828-86132-1hih22i.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1129&fit=crop&dpr=1 754w, https://images.theconversation.com/files/233968/original/file-20180828-86132-1hih22i.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1129&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/233968/original/file-20180828-86132-1hih22i.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1129&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Tricoter pour apprendre.</span>
<span class="attribution"><span class="source">Carthage College</span>, <a class="license" href="http://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA</a></span>
</figcaption>
</figure>
<p>C’est le concept que la « géométrie du caoutchouc » tente de saisir. D’une certaine manière, un triangle et un cercle <em>peuvent être identiques</em> s’ils sont faits d’un matériau flexible. En fait, dans ce domaine d’étude, tous les polygones deviennent des cercles.</p>
<p>Mais si tous les polygones sont des cercles, quelles sont les formes qui permettent de les différencier ? Il y a quelques traits qui se distinguent, même lorsque les objets sont flexibles – par exemple, si une forme a des bords ou non, des trous ou pas, comprend des torsions ou pas.</p>
<p>L’écharpe infinie, par exemple, n’est pas équivalente à un cercle. Pour créer une « écharpe infinie » en papier à la maison, prenez une longue bande de papier et collez ses bords les plus courts courts en associant le coin supérieur gauche au coin inférieur droit et le coin inférieur gauche au coin supérieur droit. Puis dessinez des flèches pointant tout le long de l’objet. Vous devriez observer un phénomène étonnant et sympathique.</p>
<p>Les élèves de mon cours ont passé du temps à tricoter des objets, comme des écharpes et des bandeaux infinis, qui présentaient des différences, même lorsqu’ils étaient faits d’un matériau souple. L’ajout de marqueurs – comme des flèches – a permis de visualiser précisément comment les objets se différenciaient.</p>
<h2>Saveurs multiples</h2>
<figure class="align-left ">
<img alt="" src="https://images.theconversation.com/files/233970/original/file-20180828-86126-5xvnuo.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/233970/original/file-20180828-86126-5xvnuo.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=899&fit=crop&dpr=1 600w, https://images.theconversation.com/files/233970/original/file-20180828-86126-5xvnuo.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=899&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/233970/original/file-20180828-86126-5xvnuo.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=899&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/233970/original/file-20180828-86126-5xvnuo.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1129&fit=crop&dpr=1 754w, https://images.theconversation.com/files/233970/original/file-20180828-86126-5xvnuo.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1129&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/233970/original/file-20180828-86126-5xvnuo.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1129&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Une écharpe infinie.</span>
<span class="attribution"><span class="source">Carthage College</span>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>Si ce dont je vous parle ici vous semble sans rapport avec les mathématiques, j’insiste : c’est pourtant le cas. Les points abordés dans cet article – l’algèbre abstraite et la topologie – sont des sujets que l’on aborde à l’université, quand on étudie les mathématiques. Or leurs enjeux sont accessibles à tous, si tant est que l’on dispose des bons outils de médiation.</p>
<p>D’après moi, ces différentes saveurs des mathématiques ne doivent pas être cachées au grand public ou moins mises en avant que les mathématiques conventionnelles. En outre, des études ont montré que l’utilisation de matériel pouvant être physiquement manipulé peut améliorer l’apprentissage mathématique, quel que soit le niveau d’études.</p>
<p>Si davantage de mathématiciens étaient capables de faire un pas de côté par apport aux techniques classiques d’enseignement, il serait possible de battre en brèche cette idée fausse mais très répandue selon laquelle les mathématiques se réduisent au calcul. Et peut-être que plus de gens seraient séduits par le raisonnement mathématique.</p>
<hr>
<p><em>Traduction : Monique Tedeschi, auteure de <a href="https://www.laplage.fr/auteurs/monique-tedeschi/">La Pédagogie Steiner Waldorf à la maison</a> et du blog <a href="https://www.chantdesfees.fr/">Chant des fées</a>.</em></p><img src="https://counter.theconversation.com/content/102278/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Sara Jensen works for Carthage College. She is a member of the Mathematical Association of America, and is a Project NExT red dot ('15).</span></em></p>Oubliez les calculatrices ! Dans la classe de cette enseignante américaine, c’est par le travail manuel qu’on s’initie aux théories mathématiques et qu’on surmonte les idées reçues.Sara Jensen, Assistant Professor of Mathematics, Carthage CollegeLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/958962018-07-16T10:39:19Z2018-07-16T10:39:19ZWhy I teach math through knitting<figure><img src="https://images.theconversation.com/files/225197/original/file-20180627-112628-1tr48e8.jpg?ixlib=rb-1.1.0&rect=0%2C145%2C752%2C598&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Math in yarn.</span> <span class="attribution"><span class="source">Carthage College</span>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span></figcaption></figure><p>One snowy January day, I asked a classroom of college students to tell me the first word that came to mind when they thought about mathematics. The top two words were “calculation” and “equation.” </p>
<p>When I asked a room of professional mathematicians the same question, neither of those words were mentioned; instead, they offered phrases like “critical thinking” and “problem-solving.”</p>
<p>This is unfortunately common. What professional mathematicians think of as mathematics is entirely different from what the general population thinks of as mathematics. When so many describe mathematics as synonymous with calculation, it’s no wonder we hear “I hate math” so often. </p>
<p>So I set out to solve this problem in a somewhat unconventional way. I decided to offer a class called “The Mathematics of Knitting” at my institution, Carthage College. In it, I chose to eliminate pencil, paper, calculator (gasp) and textbook from the classroom completely. Instead, we talked, used our hands, drew pictures and played with everything from beach balls to measuring tapes. For homework, we reflected by blogging. And of course, we knit.</p>
<h2>Same but different</h2>
<p>One crux of mathematical content is the equation, and crucial to this is the equal sign. An equation like x = 5 tells us that the dreaded x, which represents some quantity, has the same value as 5. The number 5 and the value of x must be exactly the same. </p>
<p>A typical equal sign is very strict. Any small deviation from “exactly” means that two things are not equal. However, there are many times in life where two quantities are not exactly the same, but are essentially the same by some meaningful criteria.</p>
<p>Imagine, for example, that you have two square pillows. The first is red on top, yellow on the right, green on bottom and blue on the left. The second is yellow on the top, green on the right, blue on bottom, and red on the left.</p>
<p>The pillows aren’t exactly the same. One has a red top, while one has a yellow top. But they’re certainly similar. In fact, they would be exactly the same if you turned the pillow with the red top once counterclockwise.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/224747/original/file-20180625-19396-1ychj4o.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/224747/original/file-20180625-19396-1ychj4o.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/224747/original/file-20180625-19396-1ychj4o.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=281&fit=crop&dpr=1 600w, https://images.theconversation.com/files/224747/original/file-20180625-19396-1ychj4o.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=281&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/224747/original/file-20180625-19396-1ychj4o.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=281&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/224747/original/file-20180625-19396-1ychj4o.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=353&fit=crop&dpr=1 754w, https://images.theconversation.com/files/224747/original/file-20180625-19396-1ychj4o.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=353&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/224747/original/file-20180625-19396-1ychj4o.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=353&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Rotating two square pillows.</span>
<span class="attribution"><span class="source">Sara Jensen</span></span>
</figcaption>
</figure>
<p>How many different ways could I put the same pillow down on a bed, but make it look like a different one? A little homework shows there are 24 possible colored throw pillow configurations, though only eight of them can be obtained from moving a given pillow. </p>
<p>Students demonstrated this by knitting throw pillows, consisting of two colors, from knitting charts.</p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/224748/original/file-20180625-19390-jal506.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/224748/original/file-20180625-19390-jal506.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/224748/original/file-20180625-19390-jal506.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=271&fit=crop&dpr=1 600w, https://images.theconversation.com/files/224748/original/file-20180625-19390-jal506.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=271&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/224748/original/file-20180625-19390-jal506.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=271&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/224748/original/file-20180625-19390-jal506.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=341&fit=crop&dpr=1 754w, https://images.theconversation.com/files/224748/original/file-20180625-19390-jal506.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=341&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/224748/original/file-20180625-19390-jal506.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=341&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">A knitting chart for a throw pillow.</span>
<span class="attribution"><span class="source">Sara Jensen</span></span>
</figcaption>
</figure>
<p>The students created square knitting charts where all eight motions of the chart resulted in a different-looking picture. These were then knit into a throw pillow where the equivalence of the pictures could be demonstrated by actually moving the pillow.</p>
<h2>Rubber sheet geometry</h2>
<p>Another topic we covered is a subject sometimes referred to as “rubber sheet geometry.” The idea is to imagine the whole world is made of rubber, then reimagine what shapes would look like. </p>
<p>Let’s try to understand the concept with knitting. One way of knitting objects that are round – like hats or gloves – is with special knitting needles called double pointed needles. While being made, the hat is shaped by three needles, making it look triangular. Then, once it comes off the needles, the stretchy yarn relaxes into a circle, making a much more typical hat. </p>
<figure class="align-right ">
<img alt="" src="https://images.theconversation.com/files/225196/original/file-20180627-112601-d7vuv0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/225196/original/file-20180627-112601-d7vuv0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=899&fit=crop&dpr=1 600w, https://images.theconversation.com/files/225196/original/file-20180627-112601-d7vuv0.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=899&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/225196/original/file-20180627-112601-d7vuv0.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=899&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/225196/original/file-20180627-112601-d7vuv0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1129&fit=crop&dpr=1 754w, https://images.theconversation.com/files/225196/original/file-20180627-112601-d7vuv0.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1129&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/225196/original/file-20180627-112601-d7vuv0.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1129&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Knitting to learn.</span>
<span class="attribution"><span class="source">Carthage College</span>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>This is the concept that “rubber sheet geometry” is trying to capture. Somehow, a triangle and a circle can be the same if they’re made out of a flexible material. In fact, all polygons become circles in this field of study. </p>
<p>If all polygons are circles, then what shapes are left? There are a few traits that are distinguishable even when objects are flexible – for example, if a shape has edges or no edges, holes or no holes, twists or no twists. </p>
<p>One example from knitting of something that is not equivalent to a circle is an infinity scarf. If you want to make a paper infinity scarf at home, take a long strip of paper and glue the short edges together by attaching the top left corner to the bottom right corner, and the bottom left corner to the top right corner. Then draw arrows pointing up the whole way around the object. Something cool should happen. </p>
<p>Students in the course spent some time knitting objects, like infinity scarves and headbands, that were different even when made out of flexible material. Adding markings like arrows helped visualize exactly how the objects were different. </p>
<h2>Different flavors</h2>
<figure class="align-left zoomable">
<a href="https://images.theconversation.com/files/224776/original/file-20180625-19399-y5ti0b.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/224776/original/file-20180625-19399-y5ti0b.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/224776/original/file-20180625-19399-y5ti0b.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=900&fit=crop&dpr=1 600w, https://images.theconversation.com/files/224776/original/file-20180625-19399-y5ti0b.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=900&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/224776/original/file-20180625-19399-y5ti0b.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=900&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/224776/original/file-20180625-19399-y5ti0b.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1131&fit=crop&dpr=1 754w, https://images.theconversation.com/files/224776/original/file-20180625-19399-y5ti0b.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1131&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/224776/original/file-20180625-19399-y5ti0b.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1131&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">An infinity scarf.</span>
<span class="attribution"><span class="source">Carthage College</span></span>
</figcaption>
</figure>
<p>If the things described in this article don’t sound like math to you, I want to reinforce that they very much are. The subjects discussed here – abstract algebra and topology – are typically reserved for math majors in their junior and senior years of college. Yet the philosophies of these subjects are very accessible, given the right mediums. </p>
<p>In my view, there’s no reason these different flavors of math should be hidden from the public or emphasized less than conventional mathematics. Further, <a href="https://files.eric.ed.gov/fulltext/ED321967.pdf">studies have shown</a> that using materials that can be physically manipulated can improve mathematical learning at all levels of study. </p>
<p>If more mathematicians were able to set aside classical techniques, it seems possible the world could overcome the prevailing misconception that computation is the same as mathematics. And just maybe, a few more people out there could embrace mathematical thought; if not figuratively, then literally, with a throw pillow.</p><img src="https://counter.theconversation.com/content/95896/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Sara Jensen works for Carthage College. She is a member of the Mathematical Association of America, and is a Project NExT red dot ('15).</span></em></p>In this professor’s class, there are no calculators. Instead, students learn advanced math by talking, drawing pictures, playing with beach balls – and knitting.Sara Jensen, Assistant Professor of Mathematics, Carthage CollegeLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/960862018-07-13T10:24:32Z2018-07-13T10:24:32ZScientist at work: Identifying individual gray wolves by their howls<figure><img src="https://images.theconversation.com/files/227222/original/file-20180711-27015-1dr73z1.jpg?ixlib=rb-1.1.0&rect=164%2C226%2C2079%2C1483&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Each wolf calls with its own 'voice.'</span> <span class="attribution"><span class="source">Angela Dassow</span>, <a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span></figcaption></figure><p>Love them or hate them, wolves are vital members of natural ecosystems and the health of a wolf population can be an important factor in <a href="https://www.livingwithwolves.org/about-wolves/why-wolves-matter/">maintaining balance</a> among species. Wolf populations are growing in North America – the Great Lakes region in particular now supports over <a href="https://www.fws.gov/midwest/wolf/aboutwolves/wolfpopus.htm">3,700 individuals</a>. Keeping track of wolf pack movements is important for reducing human-wolf conflicts which can arise when packs move too close to ranches.</p>
<p>The traditional way to track wolves involves setting traps, sedating and then radio-collaring individual animals. While effective, this approach is time intensive and expensive, and entails risks for the animals. </p>
<p>I was fortunate to participate in this entire process firsthand as an undergraduate student. During the summer trapping seasons, I became familiar with each of the wolves in the central forest region of Wisconsin. This experience led to several conversations with the wildlife biologists in the area about whether wolf howls could be used to help identifying non-radio-collared pack members.</p>
<p><audio preload="metadata" controls="controls" data-duration="5" data-image="" data-title="Howl from a wild adult wolf, recorded in central Wisconsin by author Angela Dassow and Carthage College biology students, Cara Hull and Caitlin McCombe." data-size="119400" data-source="" data-source-url="" data-license="Author provided" data-license-url="">
<source src="https://cdn.theconversation.com/audio/1199/south-bluff-filtered-howl.mp3" type="audio/mpeg">
</audio>
<div class="audio-player-caption">
Howl from a wild adult wolf, recorded in central Wisconsin by author Angela Dassow and Carthage College biology students, Cara Hull and Caitlin McCombe.
<span class="attribution"><span class="license">Author provided</span><span class="download"><span>117 KB</span> <a target="_blank" href="https://cdn.theconversation.com/audio/1199/south-bluff-filtered-howl.mp3">(download)</a></span></span>
</div></p>
<p>This question remained a fun thought experiment for many years. Now <a href="https://www.carthage.edu/live/profiles/1488-angela-dassow">as a biology professor</a> who specializes in <a href="http://ocr.org/learn/bioacoustics/">bioacoustics</a>, I’ve been able to turn that thought experiment into a full research question: Can we use acoustic features to identify individual wolves in the wild? </p>
<h2>Downsides of radio collaring</h2>
<p>Because of the many challenges involved in radio collaring an animal, it would be useful to have a new way to identify and track wild wolves.</p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/227472/original/file-20180712-27024-1pztro3.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/227472/original/file-20180712-27024-1pztro3.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/227472/original/file-20180712-27024-1pztro3.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=568&fit=crop&dpr=1 600w, https://images.theconversation.com/files/227472/original/file-20180712-27024-1pztro3.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=568&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/227472/original/file-20180712-27024-1pztro3.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=568&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/227472/original/file-20180712-27024-1pztro3.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=714&fit=crop&dpr=1 754w, https://images.theconversation.com/files/227472/original/file-20180712-27024-1pztro3.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=714&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/227472/original/file-20180712-27024-1pztro3.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=714&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">A U.S. Fish and Wildlife Service employee fastens a radio collar onto a sedated female gray wolf.</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/usfwsmtnprairie/8488974469">Lori Iverson/USFWS</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>To successfully set a trap, wildlife managers must first spend days, if not weeks, scouting for signs of wolves. Once they’ve identified a suitable area, they set traps that must be checked every 24 hours. If successful, the animal needs to be sedated before it can be removed from the trap – which can be stressful both for the wolf and the researchers involved.</p>
<p>A sedated wolf cannot regulate its body temperature and overheating can become an issue on hot days. Human handling of a sedated wolf can also be stressful on the pack members that are often nearby, observing the scene. Even after an animal is successfully radio-collared and released, it’s still vulnerable to predators while the sedative wears off.</p>
<p>In spite of these risks, radio-collaring has been the standard way to track populations because each collar’s radio-transmitter frequency acts as a unique identifier of an individual. Researchers can then use aerial surveys where a pilot searches for the collared animal or ground surveys where a field crew drives throughout a pack territory searching for feedback from the radio signal. This method is used to track a wide array of animals, including turtles, birds, bats, whales, fish, snakes and more.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/223229/original/file-20180614-32319-1juwm5r.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/223229/original/file-20180614-32319-1juwm5r.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/223229/original/file-20180614-32319-1juwm5r.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=450&fit=crop&dpr=1 600w, https://images.theconversation.com/files/223229/original/file-20180614-32319-1juwm5r.JPG?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=450&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/223229/original/file-20180614-32319-1juwm5r.JPG?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=450&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/223229/original/file-20180614-32319-1juwm5r.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=566&fit=crop&dpr=1 754w, https://images.theconversation.com/files/223229/original/file-20180614-32319-1juwm5r.JPG?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=566&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/223229/original/file-20180614-32319-1juwm5r.JPG?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=566&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Angela Dassow and Cara Hull survey a road in central Wisconsin for signs of wolves.</span>
<span class="attribution"><span class="source">Caitlin McCombe</span>, <a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
</figcaption>
</figure>
<h2>Listening to learn who’s who</h2>
<p>In 2013, behavioral ecologist <a href="https://scholar.google.com/citations?user=CZyS1lMAAAAJ&hl=en&oi=sra">Holly Root-Gutteridge</a> and her colleagues successfully demonstrated that they could <a href="https://doi.org/10.1080/09524622.2013.817318">identify individual wolves in captivity using acoustic features</a>. Their research provided evidence that it made sense to test whether vocal identification in wild animals is possible.</p>
<p>So with the support of the <a href="https://www.carthage.edu/sure/">Summer Undergraduate Research Experience</a> at <a href="https://www.carthage.edu/">Carthage College</a>, volunteers from the <a href="http://www.timberwolfinformation.org/">Timber Wolf Information Network</a>, and wildlife managers at <a href="https://dnr.wi.gov/topic/lands/wildlifeareas/sandhill/">Sandhill Wildlife Area</a> in Babcock, Wisconsin, my undergraduate students Cara Hull and Caitlin McCombe and I began to record wolves in the wild.</p>
<p>It would be an understatement to say fieldwork can be challenging. On any given day, there can be daunting weather fluctuations. Biting insects, especially mosquitoes and deer flies, are abundant in wolf habitat. We had to constantly check ourselves for ticks. And then of course comes the actual fieldwork. </p>
<p>Wolves naturally avoid coming near people, but the best quality recordings are made up close to where the animals are producing the sounds. To get close with our audio equipment, we had to track the wolves every day to learn where they’d most recently been within their large territories. That’s how we’d establish a starting point for our nightly recording sessions.</p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/223228/original/file-20180614-32304-mgakkr.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/223228/original/file-20180614-32304-mgakkr.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/223228/original/file-20180614-32304-mgakkr.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=765&fit=crop&dpr=1 600w, https://images.theconversation.com/files/223228/original/file-20180614-32304-mgakkr.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=765&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/223228/original/file-20180614-32304-mgakkr.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=765&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/223228/original/file-20180614-32304-mgakkr.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=962&fit=crop&dpr=1 754w, https://images.theconversation.com/files/223228/original/file-20180614-32304-mgakkr.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=962&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/223228/original/file-20180614-32304-mgakkr.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=962&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Fresh track from an adult gray wolf.</span>
<span class="attribution"><span class="source">Angela Dassow</span>, <a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
</figcaption>
</figure>
<p>Conducting a daily survey of wolf habitat requires driving or walking down every possible path within a wolf’s territory. Signs of activity could include fresh footprints or tracks. This can tell us how many animals were in the area and what direction they were heading.</p>
<p>Large dogs can produce footprints that are similar in size to those of wolves; but the pattern of tracks can be distinguished based on the placement of their feet and the directness of the chosen route. Dogs have a tendency to wander more, while wolves will walk in a more efficient straight line.</p>
<p>In addition to tracks, we conduct a survey of fresh scat. It’s not glamorous, but examining their feces provides valuable information about what the wolves have been eating and how recently they walked along a trail.</p>
<figure class="align-left zoomable">
<a href="https://images.theconversation.com/files/223050/original/file-20180613-32323-8i1dwh.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/223050/original/file-20180613-32323-8i1dwh.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/223050/original/file-20180613-32323-8i1dwh.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=800&fit=crop&dpr=1 600w, https://images.theconversation.com/files/223050/original/file-20180613-32323-8i1dwh.JPG?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=800&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/223050/original/file-20180613-32323-8i1dwh.JPG?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=800&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/223050/original/file-20180613-32323-8i1dwh.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1005&fit=crop&dpr=1 754w, https://images.theconversation.com/files/223050/original/file-20180613-32323-8i1dwh.JPG?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1005&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/223050/original/file-20180613-32323-8i1dwh.JPG?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1005&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Carthage College biology students Cara Hull and Caitlin McCombe conduct a howl survey in central Wisconsin.</span>
<span class="attribution"><span class="source">Angela Dassow</span>, <a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
</figcaption>
</figure>
<p>Using the information from our daytime survey, we plan a shorter nighttime howling route. Howling is a natural behavior during the evenings, when wolves call to signal that a territory is occupied. At each stopping point on our route, a researcher must get out of the vehicle and howl while another researcher records with a microphone any wolf responses, announcing their presence or defending territory. If we are successful in eliciting a response, we continue in its direction until we get as close as possible.</p>
<p>Use of lights is discouraged since it can deter the wolves from calling again, so we needed to feel our way through the forest at night. Personally, I think it is incredibly exciting to be walking down a trail in the dark and have a wolf walk within feet of where I am. It may sound scary, but we are not in any danger since wolves prefer to avoid contact with humans. During our month-long survey, we were fortunate to experience two close wolf encounters. </p>
<h2>Back in the lab, analyzing the calls</h2>
<p>With the howls recorded, we can return to the lab to analyze our findings using audio software.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/223230/original/file-20180614-32316-16wyrn2.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/223230/original/file-20180614-32316-16wyrn2.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/223230/original/file-20180614-32316-16wyrn2.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=450&fit=crop&dpr=1 600w, https://images.theconversation.com/files/223230/original/file-20180614-32316-16wyrn2.JPG?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=450&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/223230/original/file-20180614-32316-16wyrn2.JPG?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=450&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/223230/original/file-20180614-32316-16wyrn2.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=566&fit=crop&dpr=1 754w, https://images.theconversation.com/files/223230/original/file-20180614-32316-16wyrn2.JPG?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=566&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/223230/original/file-20180614-32316-16wyrn2.JPG?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=566&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Acoustic properties are measured using Adobe Audition.</span>
<span class="attribution"><span class="source">Angela Dassow</span>, <a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
</figcaption>
</figure>
<p>We were able to isolate 21 howls from two adult wolves over two evenings. For each howl, we made six types of frequency measurements and two types of duration measurements. Frequency is how high or low the pitch of the howl sounds and duration is the length of time the howl lasted.</p>
<p>For wild gray wolves, we found that the maximum frequency – that is, the highest sound an animal produced – and the frequency at the end of the howl were the two variables that were most individualistic. For captive wolves, it was different. The lowest frequency an individual produced – what in acoustics is called their fundamental frequency – and the loudness of its calls were the factors that best differentiated among the captive individuals.</p>
<p>The differences in useful identification information between wild and captive howls are likely a reflection of signal quality. The captive recordings are much clearer than what we were able to record in the wild, where we were typically at least half a mile away from the wolves; the signal degrades with distance. As signal quality declines, maximum frequency and end frequency become more useful in individual identification.</p>
<p>Based on our findings and previous research, it is possible to monitor gray wolf populations using non-invasive methods. To do so effectively, researchers would need to record known individuals in a particular area. Once they’ve built up a database of known individuals’ howls, they can conduct nightly surveys. Comparing new recordings to those in the audio library would let them determine which individuals are in an area. </p>
<p>While radio-collaring procedures may still be useful in some cases, vocal identification is a promising alternative for monitoring individuals. Acoustic surveys are still a time-consuming process, but they eliminate the time needed to trap individuals and remove any possibility of accidentally injuring an animal in a trap. Additionally, once researchers gather a database of positively identified individuals, they can use remote monitoring stations to record howls, thus reducing the amount of time spent conducting nightly surveys. Acoustic monitoring could potentially track all the wolves in multiple packs whereas radio-collaring is typically used to track a single member in select packs.</p><img src="https://counter.theconversation.com/content/96086/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Angela Dassow does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Tracking wild animals can provide lots of valuable data. New research suggests audio recordings of wild wolves can replace the typical radio collars, which can be expensive and intrusive.Angela Dassow, Assistant Professor of Biology, Carthage CollegeLicensed as Creative Commons – attribution, no derivatives.