Students’ difficulties with stoichiometry

Moles link the substances represented in a chemical equation to the amounts needed in practice. Moles are an abstract idea - we cannot “see” Avogadro’s number of particles, so the best we can do is to present an idea of how big this is. To use the mole meaningfully requires mathematical skills, which present an additional challenge.

One cause of the difficulties: defining “the mole”

Students’ difficulties with “the mole concept” have been known for a long period (Lazonby et al 1982). Given that particle ideas are often poor or inconsistent among teenage chemists, difficulties are unsurprising. Dierks (1981) notes that the mole has only been adopted as a unit in chemistry in relatively recent years. He says that discussion of “the mole problem” began in 1953 (p 146) and that thereafter chemists spent a number of years agreeing on a definition. The word “mole” acquired three meanings: an individual unit of mass; a portion of substance; and a number (p 150). Chemistry teachers frequently adopt the simplistic standpoint of the mole as a “counting unit”. Nelson (1991) disagrees with this approach on the grounds that in fact the mole is not strictly defined as a number, but rather as:

“…the amount of substance corresponding to the number of atoms in 0.012 kg of carbon-12.” (p 103).

Dierks suggests that problems also arise when moles are introduced to students who are not being prepared to become professional chemists. He reports that early work on students’ difficulties centred on the vital connection between chemical formulae / equations and mathematical expressions representing amounts of substance. He states:-

“It is generally argued .. that pupils need a clear conception of what is meant by amount of substance if they are to work successfully with this concept. This concept can apparently only be developed when amount of substance is interpreted as a numerical quantity.” (p 152)

Adopting the Ausubelian argument that “meaningful learning occurs when new information is linked with existing concepts” (p 153), Dierks advocates beginning to teach the mole as a “number”. This contrasts directly with Nelson (1991) who suggests strongly that the mole should be taught as an “amount”, suggesting use of the term “chemical amount” rather than “amount of substance”. This difference may be at the centre of problems associated with the mole - in teaching this concept, we may use “amount of substance” and “number of particles” synonymously, contributing unwittingly to students’ difficulties by never really explaining what we mean in either case.

More recent work by BouJaoude and Barakat (2000) makes three suggestions about teaching the mole. They developed a stoichiometry test and carried out unstructured interviews with forty 16-17 year olds revealing misunderstandings about molar quantities, limiting reagent, conservation of matter, molar volume of gases at STP and coefficients in a chemical equation. The authors suggest that teachers should help students develop clear relationships between these ideas before numerical problems are presented. They point out that teachers should also analyse students’ approaches to problem solving, suggesting that this will prevent students from continuing to use incorrect strategies. A third suggestion points to use of problems which stimulate thinking, rather than application of an algorithm. In this study, these authors found this helped to build students’ problem-solving abilities.

Students’ mathematical skills

As BouJaoude and Barakat implied above, students’ mathematical expertise also contributes to their difficulties. A student who cannot manipulate numbers readily is unlikely to be successful in learning about moles. Shayer (1970, cited in Rowell and Dawson, 1980) explains students’ difficulties in terms of their lack of the cognitive skills “necessary to deal with the concept” (p 693). Shayer believes that students who have not reached Piaget’s formal operational stage of thinking cannot learn about moles, because cognitive skills such as proportional and ratio reasoning are undeveloped. This is in broad agreement with Dierks’ suggestion, since formal operational thinking involves:-

“the ability to … see the need to control variables in making inferences from data and to impose quantitative models on observations, specifically that of proportionality.” (Driver, 1983 p 61)

Rowell and Dawson and Nelson (1991) dispute this, suggesting that students require an appropriate step-wise scheme leading towards using moles in an accepted way.

Students’ thinking about reacting mass reasoning

Barker (1995) reports the responses of 250 16-17 year olds to a question about the reaction between iron and sulphur, adapted from Briggs and Holding (1986). They were told that 56 g of iron reacts with 32 g sulphur to give 88 g iron sulphide and were asked to predict what would be produced when 112 g iron and 80 g sulphur react. At the start of the two-year study, about 50% gave the correct answer, that 176 g iron sulphide would be produced with some sulphur remaining. The most common incorrect response, given by 32%, was to add the two figures generating 192 g. These students had not realised the need to apply reacting mass reasoning. At the end of their two year course of study, about 72% gave the correct answer, while about 16% gave 192 g.

BouJaoude and Barakat (2000) report that about 40% of their sample of forty 16-17 year olds calculated molar mass by dividing or mulitplying the total of atomic masses by the coefficient shown in the chemical equation.

Researchers’ suggestions for learning about moles

Modelling a chemical reaction

Rowell and Dawson (1980) begin teaching moles to 16 year old students by using a model of a simple chemical reaction such as 2Na + S -> Na₂S represented in small coins. Next, the idea of proportionality is introduced by showing a reaction in which “2As” make “1C” Students are asked what would be produced if only “1A” was available. Once the idea that reactions occur in proportion was developed, Rowell and Dawson introduce the idea that the number of particles involved might be very large. At this point, they return to their original reaction and ask students to imagine that these are atoms of chemical elements. The conservation of number of atoms and masses are emphasised at each point. The authors carried out a six-week teaching strategy using this stepwise approach and tested students before, immediately afterwards and two months later. They found that twenty-one out of the twenty-four students gave error-free responses in the final test. This refutes the Shayer suggestion, since the students were not pre-selected for their ability to think in a formal operational way. The authors conclude:-

“Teaching the mole concept is not an easy task but it need not be the mountain that some have made it.” (p 707)

Using algorithms

Kean et al (1988) advocate algorithms to help teach and learn mole ideas. They note that a useful algorithm “allows students to solve problems with meaning rather than by rote” (p987). They suggest an eight-step strategy to help students devise an algorithm for converting mass into volume measurements and vice versa. Similarly, students can be taught an algorithm for solving proportionality problems and, eventually, calculation of reacting masses. This strategy may help develop students’ confidence in handling numerical data, but requires careful instruction to ensure appropriate application. Finley et al (1992) sound a warning note:-

“Recent research has indicated that the ability to solve numerical problems does not guarantee conceptual understanding of the molecular basis of the problem.” (p 254)

Although Kean et al’s proposals may provide a means to an end, the students may learn the algorithm and not its chemical meaning. Rowell and Dawson’s approach, rooted firmly in the chemical principles of stoichiometry, has much to recommend it.

Summary of key difficulties

1. Chemists do not agree on how the ”mole” should be defined

Chemists have discussed what is meant by ”one mole” over the last fifty years. The mole has three meanings: an individual unit of mass, a portion of substance and a number. Chemistry teachers frequently adopt the simplistic standpoint of a ”counting unit”, which fits with none of these. Regardless of the experts’ philosophical discussions, students need a clear approach based on making the connection between the amount of substance and a numerical quantity.

2. The mole is taught as an abstract mathematical idea

The mole is often taught in a mathematical way causing the chemical meaning to be obscured. Students who struggle to manipulate numbers and symbols will find this approach towards learning the mole very difficult to understand.

3. Students lack secure understanding of preliminary concepts

The mole is an idea which connects basic principles about chemical reactions to the more advanced concepts involving controlling reactions. Thus, prior to learning about the mole, students should understand that chemical reactions produce new substances; that matter is made from tiny particles invisible to the naked eye; and that chemists need to be able to measure amounts of substance accurately in order to be able to control chemical reactions.

4. Avogadro’s number cannot be ”seen”

The size of Avogadro’s number is too large to be readily comprehended. Students can be given an impression of its size by the use of powerful visual images such as one mole of sand grains stretching for one mile (1.6km); one mole of marbles forming a layer 1500 km deep over the UK and Eire.

Suggested activities^[1]

This is a tried and tested teaching sequence. With patience and care, the mole can be taught to most chemistry students with little difficulty.

1. Show students elements in a whole-number mass ratio

Have ready pre-weighed samples of familiar chemical elements and compounds clearly labelled with symbols, formulae and Ar / Mr values. Include two elements with Ar values in a simple whole number ratio – copper (assume Ar = 64) and sulfur (Ar= 32)are good examples. Start two columns on a board, one for each element, writing the symbol at the top of each and the Ar value then the ratio (2:1) underneath. Ask students to imagine one atom of each element and to state what the ratio of the masses will be (2:1). This is easy! Next ask if it is possible to weigh one atom conventionally. The answer is no. Explain that chemists need to be able to compare masses that can be measured easily.

2. Show that the ratio remains fixed, regardless of the number of atoms

Extend the columns by writing increasingly large numbers of atoms - the number could be written once for both columns, or separately in each. The numbers should go up to 1 000 000. Point out that even one million atoms cannot be measured out conventionally because atoms are just too small. Each time a number is added, refer to the mass ratio – note that this is always the same, 2:1. Reinforce the fact that the ratio is fixed.

3. Introduce the masses in grammes which chemists use: ask about the number of atoms present

Next, refer to the labels on the jars – it is possible to measure out amounts of chemicals, but how have chemists done this? What is special about the amounts? Find the jars for the two chosen elements and write these masses in grammes on the lists. Ask students to state the ratio of the masses – this is unchanged, 2:1. Then comes the crucial question, ”What can they say about the numbers of atoms in the two jars?” Wait patiently for the response, repeating the discussion if necessary. Eventually someone will make the connection between the ratio of masses of atoms and the ratio of masses weighed out in the jars, saying that therefore the number of atoms in the jars must be the same.

4. Introduce Avogadro’s number, reinforce atom size

When satisfied that this has been understood by most of the group, introduce these points: that atoms are extremely small; the number of atoms present is extremely large; the number of atoms is called Avogadro’s number; that we call this number and amount of material ”one mole”. Reinforce the discussion with visual images of Avogadro’s number, such as that one mole of the sweet called ”marshmallows” would make a layer 1000 km deep over the entire USA (see also 9.5.4 above).

5. Use formula cards to reinforce the ideas

Make a set of ”formula cards” (see Barker 2002 for examples). Each card in the pack will have the symbol and Ar value for one chemical element. Make elements with one ”combining power” square-shaped, then double and triple the length for elements with combining powers of 2 and 3. This will enable students to make formulae of simple compounds such as water by aligning two hydrogen cards with one oxygen card, making a two-by-two square, for example. Ensure there are sufficient cards for students to be able to construct the reagents for a chemical reaction. For example, the reaction between hydrogen and oxygen will require four hydrogen cards and two oxygen cards. Use the cards every time students meet a new chemical reaction. Reinforce the notions that new substances are made and that mass is conserved. Encourage students to write chemical equations accurately, based on the formulae they make with the cards. Students can then see easily that the Mr values of elements and compounds are the sums of the Ar values of the component elements and be introduced to the fact that the large numbers in front of formulae represent the number of moles present. These cards are invaluable when working with students of all abilities.

6. Introduce mathematics later

With sufficient practice, students will see the relationship between moles, mass and Ar / Mr values for themselves. When introducing this, ensure that all examples fit simple ratios and use familiar substances. Support learning using chemical reactions which can be demonstrated in front of students, measuring amounts explicitly and showing what happens when excess is used. The reaction between iron and sulfur is a good example. Then, when and not before students are secure using formula cards, for example, seeming not to need them for simple reactions and showing signs of memorising the information they contain, take them away. At this point, mathematics can take their place. The embedding process may take weeks, not just a few lessons. If necessary, revisit the original discussion and provide support for students for whom progress is less rapid. Being patient while gradually building students’ confidence will pay dividends later.