GeoGebra is an awesome mathematical tool! It even allowed me to discover something new. Well, at least new to me! For an insight to my ‘new discovery’, take a peek at the dynamic image (gif) above.
GeoGebra Can Open Doors To New Discoveries
For the record, I am no geek. I’m a feet-on-the-ground math person, fascinated by the craft of teaching mathematics. For many years I’ve been driven by the question “How do we stop boring kids to death with mathematics?” Let’s face it, at least 70% of people today over the age of 12 will state they were bored by mathematics at school. But, I digress…
A Pythagoras Investigation Using GeoGebra
One day, I was playing around with GeoGebra and wanted to create a file which visually demonstrated the workings of Pythagoras’ Theorem. I had junior high school students in mind. I wanted the file to cause students to respond with “OK, I see that the sum of the two smaller areas is equal to the larger area is true.”
The first file I created was the file dynamically represented below.
NOTE: This is different to the one above—in accordance with the theorem it contained squares and only squares on the three sides of the triangle.
Geogebra inspires mathematical discoveries!
So Began the Pondering
I was happy with the file. However, not long after creating it I began pondering the following:
- I wonder what would happen if, rather than having squares on the three sides, I created equilateral triangles?
- As I set about this task I had no idea what the outcome would be. (Heck, I was truly investigating mathematics!)
- I discovered that the principle worked for equilateral triangles.
- Well then, what about regular pentagons?
- Yes, it works for regular pentagons.
- Wait… maybe I can use a slider with ‘n’ varying from three sides, in intervals of one, and set the polygon to ’n’ sides. Wow, what might happen then?
- “Well look at that … this Pythagorean principle (as it relates to sums of areas) seems to apply to all sets of regular polygons on the 3 sides of a right angled triangle.”
The file I ended up with is represented below as well as by the one at the top of this article (two slightly different aspects of the same file).
Geogebra inspires mathematical discoveries!
GeoGebra Causes Curiosity
Now I’m not for a minute thinking I discovered anything new here—I’m sure I’m not the first! Anyone wanting to create some (real) mathematics out of this is welcome to do so. The point of the article is not about my ‘discovery’ nor whether the discovery is or isn’t unique or of value. The point of the article is that GeoGebra, alone, caused me to be curious.
It was GeoGebra which armed me with the ability to pursue the ‘what if’ question. It was GeoGebra that caused the ‘what if’ question to arise in the first place. Prior to this discovery I certainly had not been walking the streets of Sydney pondering the nature of the areas of regular polygons sitting on the sides of right angled triangles!
It was the act of using GeoGebra that caused me to inquire, to play, and then to discover.
If Me…Then Why Not For Students?
My point is this: If I can discover something mathematically new simply by investigating with GeoGebra then what’s stopping students benefitting from the same experience? In the classroom students would need to be guided by a well-planned investigation—some tips are provided here—but this is all very doable. Wouldn’t it be great to have our students doing some ‘real’ mathematics?
The Important Question
This leads me to an important question: Why isn’t GeoGebra (or any similar tool) an integral part of every mathematics teacher’s toolkit? Let me rephrase that:
Why doesn’t every middle and high school mathematics teacher who has access to a computer and data projector utilize GeoGebra on a regular basis—across multiple year levels and topics?
The article ‘Are You Utilizing GeoGebra?‘ contains a detailed example of using a file to impart conceptual understanding (with a downloadable file). The article also argues the case for a wider adoption of GeoGebra in maths classrooms globally.
In ‘Developing a Conceptual Understanding of Maths with GeoGebra’ I show that simply projecting quality files and allowing higher order questioning to occur requires no change in pedagogy from the teacher yet can powerfully improve conceptual understanding.
Learn Implement Share
Learn Implement Share provides an engaging path for teachers to become proficient with GeoGebra. Ultimately, professional learning is best when undertaken in TEAMs. This allows school leaders to better guide the pedagogy of their staff.
Here’s what one teacher had to say after completing the online guided journey:
“[The course] allowed me to achieve what I had set out to do, and that was to increase my confidence with my use of GeoGebra. Richard set the course out in such a way that not only did you learn the mechanisms of GeoGebra, but it allowed you to think of other uses for the same applications. The reactions of the students at my school to the [files] have been great, and it has given them more insight into some concepts that they always find difficult.”
—Susan Hoy, 26/05/2015
Do your students make genuine mathematical discoveries? Can you see how well-constructed GeoGebra investigations can help? We’d love your thoughts below!
GeoGebra ni chombo cha ajabu cha hisabati! Hata iliniruhusu nijue jambo jipya. Kwa vyovyote vile, ni mpya kwangu! Kwa ufahamu wa <unk>gundua yangu mpya<unk>, kuchukua peek katika picha ya nguvu (gif) juu. GeoGebra inaweza kufungua milango kwa uvumbuzi mpya Kwa rekodi, mimi si geek. Mimi ni mtu wa hisabati mwenye miguu juu ya ardhi, mwenye kuvutiwa na ufundi wa kufundisha hisabati. "Kwa miaka mingi nimeongozwa na swali: ""Tunawezaje kuacha kuwachosha watoto kwa kufa na hisabati?"" Hebu tuone ukweli, angalau 70% ya watu wa leo wenye umri wa zaidi ya miaka 12 watasema kwamba walichoshwa na hisabati shuleni." Lakini, mimi digress... A Pythagoras Uchunguzi Kutumia GeoGebra Siku moja, nilikuwa kucheza karibu na GeoGebra na alitaka kuunda faili ambayo kioo alionyesha kazi ya Pythagoras <unk> Theorem. Nilikuwa na wanafunzi wa shule ya sekondari akilini. Nilitaka faili kusababisha wanafunzi kujibu na <unk>OK, naona kwamba jumla ya maeneo mawili madogo ni sawa na eneo kubwa ni kweli.<unk> Faili ya kwanza niliunda ilikuwa faili dynamically kuwakilishwa chini. KUMBUKA: Hii ni tofauti na moja juu <unk> kwa mujibu wa theorem ni zilizomo mraba na mraba tu juu ya pande tatu za pembetatu. Geogebra huchochea uvumbuzi wa hisabati! Hivyo alianza kutafakari Nilikuwa na furaha na faili. Hata hivyo, muda mfupi baada ya kuibuni nilianza kufikiria mambo yafuatayo: - Najiuliza ni nini kingeweza kutokea ikiwa, badala ya kuwa na mraba kwenye pande tatu, ningeunda pembetatu zenye pande sawa? - Nilipokuwa nikifanya kazi hiyo sikuwa na wazo lolote kuhusu matokeo yake. (Heck, mimi alikuwa kweli kuchunguza hisabati!) - Niligundua kwamba kanuni hiyo ilifanya kazi kwa pembetatu zenye pande sawa. - Basi, vipi kuhusu pentagoni za kawaida? - Ndiyo, inafanya kazi kwa pentagons kawaida. - Kusubiri... labda naweza kutumia slider na <unk>n<unk> kutofautiana kutoka pande tatu, katika vipindi vya moja, na kuweka polygon kwa <unk>n<unk> pande. Wow, ni nini kinachoweza kutokea basi? "Kama ilivyoelezwa hapo juu, kanuni ya Pythagoras (kwa kuwa inahusiana na jumla ya maeneo) inaonekana kutumika kwa seti zote za poligoni za kawaida kwenye pande tatu za pembetatu ya pembe moja. "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" Geogebra huchochea uvumbuzi wa hisabati! GeoGebra husababisha udadisi Sasa mimi si kwa dakika kufikiri mimi aligundua kitu chochote kipya hapa mimi nina uhakika mimi si wa kwanza! Mtu yeyote anayetaka kuunda baadhi ya (halisi) hisabati nje ya hii ni kukaribishwa kufanya hivyo. Jambo la makala si kuhusu "ugunduzi" wangu wala kama ugunduzi ni au si ya kipekee au ya thamani. Jambo kuu la makala hiyo ni kwamba GeoGebra, peke yake, ilinifanya niwe na udadisi. Ilikuwa GeoGebra ambayo silaha mimi na uwezo wa kufuatilia <unk> nini kama<unk> swali. Ilikuwa GeoGebra ambayo ilisababisha swali la 'nini kama' kutokea kwa mara ya kwanza. Kabla ya ugunduzi huu mimi hakika hakuwa na kutembea katika barabara za Sydney kutafakari asili ya maeneo ya polygons mara kwa mara kukaa juu ya pande za pembe tatu haki! Ilikuwa ni kitendo cha kutumia GeoGebra ambayo ilinifanya kuuliza, kucheza, na kisha kugundua. Ikiwa Mimi... Basi Kwa Nini Wasicheze? Jambo langu ni hili: Kama naweza kugundua kitu kisayansi mpya kwa kuchunguza tu na GeoGebra basi nini ni kuacha wanafunzi kufaidika na uzoefu huo huo? Katika darasa wanafunzi ingekuwa haja ya kuongozwa na uchunguzi vizuri iliyopangwa - baadhi ya vidokezo hutolewa hapa - lakini hii ni yote sana doable. Je, si ingekuwa kubwa kuwa na wanafunzi wetu kufanya baadhi ya "halisi" hisabati? Swali muhimu Hii inaniongoza kwa swali muhimu: Kwa nini GeoGebra (au chombo chochote sawa) si sehemu muhimu ya kila mwalimu wa hisabati chombo? Hebu nikubadili maneno: Kwa nini si kila mwalimu wa hisabati wa shule ya sekondari na ya sekondari ambaye ana upatikanaji wa kompyuta na projector ya data hutumia GeoGebra kwa ukawaida katika ngazi nyingi za mwaka na mada? Makala ‘Je, Unatumia GeoGebra? ina mfano wa kina wa kutumia faili ya kutoa uelewa wa dhana (na faili downloadable). Makala pia inadai kesi kwa ajili ya kupitishwa kwa GeoGebra katika madarasa ya hisabati ulimwenguni kote. Katika <unk>Kujenga uelewa wa dhana ya hisabati na GeoGebra<unk> Ninaonyesha kwamba tu kuingiza faili za ubora na kuruhusu maswali ya utaratibu wa juu kutokea haihitaji mabadiliko katika ufundishaji kutoka kwa mwalimu lakini inaweza kuboresha uelewa wa dhana. Kujifunza kutekeleza kushiriki Kujifunza kutekeleza kushiriki hutoa njia ya kuvutia kwa walimu kuwa na ujuzi na GeoGebra. Hatimaye, kujifunza kitaaluma ni bora wakati uliofanywa katika TEAMs. Hii inaruhusu viongozi wa shule kuongoza vizuri pedagogy ya wafanyakazi wao. Hapa ni nini mwalimu mmoja alikuwa na kusema baada ya kukamilisha online kuongozwa safari: [Mtaala] kuruhusiwa mimi kufikia kile nilikuwa kuweka nje ya kufanya, na kwamba ilikuwa kuongeza imani yangu na matumizi yangu ya GeoGebra. Richard alianzisha kozi hiyo kwa njia ambayo si tu kwamba ulijifunza utaratibu wa GeoGebra, lakini ilikuruhusu kufikiria matumizi mengine kwa maombi yaleyale. "Mfano wa wanafunzi wa shule ya ""Susana"" ni kwamba, wanafunzi wa shule ya ""Susana"" wana uwezo wa kugundua mambo mengi ya hisabati." Je, unaweza kuona jinsi vizuri kujengwa GeoGebra uchunguzi inaweza kusaidia? Tungependa mawazo yako hapa chini!