Promoting interdisciplinarity through mathematical modelling

Abstract

This article presents one approach to addressing the important issue of interdisciplinarity in the primary school mathematics curriculum, namely, through realistic mathematical modelling problems. Such problems draw upon other disciplines for their contexts and data. The article initially considers the nature of modelling with complex systems and discusses how such experiences differ from existing problem-solving activities in the primary mathematics curriculum. Principles for designing interdisciplinary modelling problems are then addressed, with reference to two mathematical modelling problems—one based in the scientific domain and the other in the literary domain. Examples of the models children have created in solving these problems follow. A reflection on the differences in the diversity and sophistication of these models raises issues regarding the design of interdisciplinary modelling problems. The article concludes with suggested opportunities for generating multidisciplinary projects within the regular mathematics curriculum.

Appendices

Appendix 1: The creek watch problem*

 

Increased urban development in the Western Suburbs of Brisbane is threatening the quality of the local environment. In particular, runoff is impacting on water quality and visible signs such as rising salinity and blue-green algal blooms are becoming more prevalent.

Since 1995, scientists from the Queensland environmental protection agency (EPA) and Brisbane city council (BCC) have monitored the water quality in many creeks in South East Queensland.

In 2004 Year 5 students from Indooroopilly state school (ISS) assisted these authorities to collect important data about Moggill Creek. The Creek flows into the Brisbane River in the Western Suburbs. The members of the class collected water samples from the creek to test for chemicals. Jack Simpson, a student from ISS said, “The class also took samples of different species of fish and macro-invertebrates. I loved looking at the macro-invertebrates the best, they are great water bugs.”

Mrs. Jones from the EPA said the students’ assistance had been invaluable and the information they provided helpful. “By monitoring the waterways we can gain a picture of catchment health. Monitoring over time can provide information on the state of the catchment which can assist with the maintenance and rehabilitation of our waterways.”

When deciding how healthy Moggill Creek is, many factors have to be investigated and recorded over a period of time. A healthy river has high amounts of dissolved oxygen and low amounts of phosphorous and nitrogen. It also has relatively low salinity. “We have to combine all of these different chemical and biological measurements to come up with an indication” said Mrs. Jones.

A stretch of Moggill Creek where students from ISS collected valuable data for the study.

 

Apart from the water quality components, other factors help to determine how healthy the creek is. “Putting the chemical data together with the information that we gather about the fish and macro-invertebrates tells us how healthy the river is at that particular moment” said Mrs. Jones.

Mrs Jones said “We know that certain macro-invertebrates are highly sensitive to pollution whereas others have low sensitivity and survive in polluted water. By counting the different types of macro-invertebrates and knowing their sensitivity we can work out the pollution index of a creek.”

The river is considered clean and healthy when the river has lots of different species of water bugs, particularly highly pollution sensitive macro-invertebrates.

Obtaining samples is a time-consuming job. Having students gather data from different sites along Moggill Creek helps the department identify which areas are healthy and which sites need the most help in getting cleaned up.

While the children are supporting the local community through their investigations, they are also learning about river ecology and why it is important to keep pollution down in waterways.

 

A creek highly polluted showing an algal bloom

 

• 1. What are macro-invertebrates? (draw a picture of an example)

• 2. Why does the EPA want students to help collect data for them about the conditions of Moggill Creek?

• 3. When scientists want information on “dissolved oxygen, phosphorous and nitrogen” what are they looking for?

• 4. What levels of dissolved oxygen, phosphorus, nitrogen, and total salinity does a healthy river have?

• 5. Why is it important to know both the number of organisms and the amount of each species that a creek site has?

• 6. Why is the students’ involvement in the creek monitoring task a good idea?

• 7. What is the pollution index of a creek and how do you calculate it?

 

Jack Simpson’s class is presenting their information at a community meeting where other interested groups are presenting their conclusions. The meeting organisers are offering a prize for the group that develops the best system that describes the most important criteria in establishing the total water quality of a creek.

Jack’s group needs your help to construct a model or set of guidelines that indicates the health of the creek. Your system should make use of all the data collected over the year in five locations along the creek. These data are shown in the Table attached. Jack’s class started collecting near the source of the Creek (Site E) and took samples all the way to the mouth of the Creek where it entered the Brisbane River (Site A).

Write a letter to Jack’s group that describes how you developed your system so that it can be used by others in determining the health of any creek.

*The activities displayed in each appendix were developed by Lyn English and James Watters with assistance from Jo Macri.

Appendix 1.1: Explanation of pollution index

Pollution index¹

There are a number of different things that can pollute water and consequently affect the distribution of macro-invertebrates. Pollutants include domestic waste and animal wastes (e.g. from paddocks, dairies, horse stables and yards.) These wastes can contribute to the development of toxins, bacteria, and viruses. They enter water courses through run off, or seep in through ground water. The quality of the water can be determined by calculating a pollution index.

1.1 Pollution index

Macro-invertebrates can be divided into three groups according to how sensitive they are to pollution and assigned a number related to their group:

 

Sensitive	5–10
Tolerant	3–4
Very tolerant	1–2

Each animal has a number or score next to it in the water bug table.

When you have completed the collection and identification, add the numbers assigned to each animal. For the index, only count each type of animal once. Clean water will have a high total score because it can support a lot of pollution sensitive bugs.

High abundance of only a few species might indicate poorer conditions.

Appendix 1.2: Water bugs identification chart

1.1 Water bugs identification chart

The health of a stream is given in terms of the diversity of species and the relative tolerance level of various species.

 

Species	Size	Features	Drawing	Score
High sensitivity
Caddisfly larvae	Up to 20 mm	They live in a wide range of environments from fast flowing streams to freshwater ponds. Their soft bodies are usually covered in a protective silky case		6
Dragonfly nymph	18–50 mm	Dragonfly nymphs are short and chunky with wing pads and internal gills. Their six legs are all located near the head		6
Mayfly Nymph	10–20 mm	These are only found in very clean water containing lots of oxygen. They absorb oxygen from the water through their gills.		7
Stonefly nymph	Up to 50 mm	They have three segmented tarsi and long antennae. Require lots of oxygen		8
Water mite	5 mm	Adults are free-swimming or crawling. Most common in heavily vegetated wetlands—often parasitic on other insects		5
Medium sensivity
Fairy shrimp	10–30 mm	Related to brine shrimp, copepods and Daphnia		3
Freshwater mussel		Mussels are soft bodied animals enclosed in two hinged shells		3
Leech	3–15 mm	Leeches are segmented worms with a sucker on one or both ends. They are found free swimming in water as well as on plants or on the bottom.		3
Planarian	2–5 mm	These are free living flat worms which possess a remarkable ability to regenerate their lost body parts		3
Pond snail	10–20 mm	Aquatic snails are similar in form to land snails but smaller		3
Water boatman		Water boatmen and water striders are bugs. These tend to be shield shaped when viewed from above. Their soft front wings are folded and overlap to leave a small triangle on their back.		3
Water strider		Water boatmen and water striders are bugs. These tend to be shield shaped when viewed from above. Their soft front wings are folded and overlap to leave a small triangle on their back.		3
Water tiger beetle		Beetle larvae are segmented, have three distinct pairs of legs. They are usually active with large mouth parts.		3
Waterflea	1 mm	Also known as daphnia these are related to crabs and prawns.		3
Whirligig beetle	3–35 mm	They congregate in large numbers and scurry about the water surface in a random pattern. Shiny to dull black, often with a bronzy sheen.		3
Shrimp		Shrimp are small crustations that look similar to prawns.		5
Yabby		Freshwater crayfish that are commonly found in ponds and streams.		5
Low sensitivity (tolerant)
Water scorpion	30 mm	Noted for the first pair of legs which are modified into prehensile organs for grasping prey. They are carnivorous and feed on smaller insects. The prey is held securely between their first pair of legs while the water scorpion sucks up its body fluids. Tends to be found on the muddy bottom of creeks.		2
Midge		These are small pesky biting insects as adults but are slender worm-like creatures, sometimes red, with no legs.		2
Mosquito lavae		These animals twist and squirm just below the surface of the water. The larvae look like hairy maggots with siphons.		2

Appendix 1.3: Moggill creek data

Moggill creek data collected by Indooroopilly state school creek watch students.

 

Data type	Site A	Site B	Site C	Site D	Site E
Macro-invertebrates	No	No	No	No	No
Dragonfly nymphs	15	2	6	1	1
Freshwater snails	5	1	15	14	3
Stonefly nymph	4	0	0	0	3
Fly larvae	4	0	17	15	6
Cadisfly larvae	5	1	0	3	6
Mayfly nymphs	4	2	0	14	3
Aquatic segmented worms	1	0	0	7	2
Back swimmers	0	0	0	3	2
Water strider	3	0	0	0	12
Whirligig beetle larvae	2	1	0	0	3
Non-biting midge larva	3	2	1	1	3
Leeches	3	0	3	0	17
Yabbie	5	5	0	1	16
Bloodworms	11	2	19	4	6
Mosquito larvae	8	0	0	1	2
Fish
Swordtail	5	2	1	1	1
Eel	0	0	0	0	1
Mosquito fish	1	0	0	0	2
Black mangrove	1	1	0	0	0
Guppy	2	1	1	1	1
Carp	9	3	2	1	5
Platy	1	2	2	0	0
Purple-spotted Gudgeon	10	3	0	0	8
Weeds
Alligator weed	2	3	4	5	0
Chinese elm	3	4	4	5	0
Camphor Laurel	1	1	2	2	0
Chemical analysis	Units	Units	Units	Units	Units
Dissolved oxygen (DO)	105	65	77	92	98
Turbidity	7	12	10	5	2
Salinity (total dissolved solids) (TDS)	335	349	368	430	300
Total phosphorus	20	36	40	25	20
Nitrogen (Nitrate, NO3)	25	50	35	30	19
Nitrogen (Ammonia, NH3)	7	12	8	9	7
pH	6.7	6.6	6.0	6.3	7.5

Appendix 2: The summer reading problem (Michelle Heger, Purdue University, 2002)

1.1 Summer excitement strikes the library, prizes galore for local reader

Brisbane—While a long hot summer may be ahead of us, the Brisbane city council library (BCCL) is offering a chance for patrons to stay cool this year.

The annual “Reading is Cool” summer reading program will officially start at noon, June 1, in the Indooroopilly Room. Students from St. Peters will receive a free library card that will let them participate in the program.

Students can choose from an approved collection of books that the library has placed on reserve. The books have been classified by grade level (according to difficulty of the book), to help the students choose which books to read. However, students may read any of the books, regardless of their current grade level.

St. Peters students who participate will have the chance to not only earn prizes from the library, but also prizes from their school. The St. Peters School and the BCCL have teamed up to provide prizes for overall winners and classroom winners.

Some prizes that the students can win, based on a point system, include bookmarks, books, T-shirts, hats, meals from local restaurants, and compact discs. Classroom winners will also be eligible for a chance to win a $300 savings account.

To register, simply stop by the Brisbane City Council Library, 318 Moggill Rd, between 9 a.m. and 9 p.m. The contest ends Aug. 12, with the final day to collect prizes Aug. 25.

The library is accepting suggestions for this year’s reading contest. To give your input, please send a letter to Lynn, the.

Indooroopilly reading coordinator. All suggestions must be received by May 1.

 

Ready to Go: The books are all shelved at the Brisbane City Council Library in Indooroopilly. Participating students can choose from over 250 books for extra bonus points in this year’s summer reading program.

1.2 Summer reading program readiness questions

Answer the following questions using the journal article and the tables given below.

• 1. Drew read The Tell-Tale Heart and Roll of Thunder, Hear My Cry. Should he receive the same number of points for each book? Why or why not?

  

   

• 2. If a sixth grader and a ninth grader both read A Tale of Two Cities, should they both earn the same number of points? Why or why not?

  

   

• 3. If Shelly reads Jurassic Park and Much Ado About Nothing, should she get the same number of points for each?

  

   

1.2.1 Examples of approved books

 

Title	Author	Reading level (by grade)	Pages
Sarah, Plain and Tall	Patricia MacLachlan	4	58
Are You There God? It’s Me Margaret.	Judy Blume	4	149
Awesome Athletes	Multiple Authors	5	288
Encyclopedia Brown and the case of Pablo’s Nose	Donald J. Sobol	5	80
Get Real (Sweet Valley Jr. High, No.1)	Francine Pascal, Jamie Suzanne	6	144
Roll of Thunder, Hear My Cry	Mildred Taylor	6	276
The Tell-Tale Heart	Edgar Allan Poe	6	64
Little Women	Louisa Mae Alcott	7	388
The Scarlet Letter	Nathaniel Hawthorne	7	202
Aftershock (Sweet Valley High)	Kate Williams, Francine Pascal	8	208
Jurassic Park	Michael Crichton	8	400
A Tale of two cities	Charles Dickens	9	384
Lord of the flies	William Golding	9	184
Much Ado about nothing	William Shakespeare	10	75

Title	Brief description of book
Sarah, Plain and tall	When their father invites a mail-order bride to come and live with them in their prairie home, Caleb and Anna are captivated by their new mother and hope that she will stay.
Are you there god? It’s me Margaret	Faced with the difficulties of growing up and choosing a religion, a twelve-year-old girl talks over her problems with her own private God.
Awesome athletes	Sports illustrated for kids
Encyclopedia Brown and the case of Pablo’s Nose	America’s Sherlock Holmes in sneakers continues his war on crime in ten more cases.
Get real (sweet valley Jr. high, no. 1)	Describes the trials and tribulations of twins that moved to a new junior high school.
Roll of thunder, hear my cry	A black family living in the South during the 1930s is faced with prejudice and discrimination that its children do not understand.
The tell-tale heart	The murder of an old man is revealed by the continuous beating of his heart.
Little women	A story of family, of hope, of dreams, and of growing up as four devoted sisters search for romance and find maturity in civil-war era 19th century New England.
The Scarlet letter	Hawthorne’s masterpiece about Hester Prynne, hapless victim of sin, guilt and hypocrisy in Puritan New England.
Aftershock	Twins deal with the pain and shock of an earthquake.
Jurassic park	A modern-day scientist brings to life a horde of prehistoric animals and dinosaurs.
A tale of two cities	A highly charged examination of human suffering and human sacrifice, private experience and public history, during the French revolution.
Lord of the flies	The classic tale of a group of English school-boys who are left stranded on an unpopulated island.
Much Ado about nothing	Shakespeare comedy.

1.2.2 Summer reading problem

 

Information: The Brisbane City Council Library and St. Peters School are sponsoring a summer reading program. Students in grades 6–9 will read books and prepare written reports about each book to collect points and win prizes. The winner in each class will be the student who has earned the most reading points. The overall winner will be the student that earns the most points. A collection of approved books has already been selected and put on reserve. See the previous page for a sample of this collection.

Students who enroll in the program often read between ten and twenty books over the summer. The contest committee is trying to figure out a fair way to assign points to each student. Margaret Scott, the program director, said, “Whatever procedure is used, we want to take into account: (a) the number of books, (b) the variety of the books, (c) the difficulty of the books, (d) the lengths of the books, and (e) the quality of the written reports.

Note: The students are given grades of A + , A, A-, B + , B, B-, C + , C, C-, D, or F for the quality of their written reports.

1.2.3 Your mission…

Write a letter to Margaret Scott explaining how to assign points to each student for all of the books that the student reads and writes about during the summer reading program.