Proof-related reasoning in upper secondary school: characteristics of Swedish and Finnish textbooks

Figures & data

Table 1. Distribution of analysed textbook material.

Textbook series	Justifications	Worked examples	Students’ tasks
Sw1	9	43	342
Sw2	9	41	427
Fi1	15	44	223
Total	33	128	992

Table 2. Framework for justifications in expository sections.

Code	Type of justification	Description
G	General proof	The statement is justified with a proof.
S	Specific case or other non-proof justification	The statement is justified using a deductive argument based on a specific case, or that has other flaws that makes it a non-proof justification.
L	Left to the student	A justification of the statement is explicitly left to the student to complete, typically with a problem in the exercises for which a justification of some type is required.
N	No justification	No justification is provided, and no explicit mention is made of leaving the justification to the student.

Table 3. Framework for nature of reasoning in textbook tasks.

Code	Nature of reasoning	Task in which student is asked to …
M	Make a conjecture	make a conjecture, formulate a true mathematical statement, or find the precise conditions for a certain statement to be true
I	Investigate a conjecture	investigate whether a given conjecture or statement is true or false
D	Develop an argument	justify or explain why a certain statement holds
E	Evaluate an argument	evaluate whether a certain justification or solution is correct
C	Correct or identify a mistake	find and/or correct an error in an argument or a solution
X	Counterexample	find a counterexample to a false mathematical statement
P	Outline a proof	outline an argument without the details of a full proof
O	Other	use some other element of proof-related reasoning
N	Not proof-related	do something not proof-related

Table 4. Examples of tasks in the Swedish textbooks illustrating different natures and types of reasoning

Ex	Task	Code
1	Investigate with your calculator and write down the following values: lg 2, lg 20, lg 200, lg 2000. (a) What pattern can you see? Explain. (b) What should lg 20000 and lg 0.2 be? (Sw1, Book 2c, p. 120)	GM/SM
2	(a) Give an example of a positive number which also has a positive ten-logarithm. (b) Give an example of a positive number which has a negative ten-logarithm. (c) Explain when the ten-logarithm of a positive number is positive and when it is negative. (Sw2, Book 2c, p. 100)	N/N/GM
3	The number of permutations of a certain sample is always greater than the number of combinations. True or false? Motivate your answer! (Sw1, Book 5, p. 57)	GI
4	Gina says that lg 400 must be greater than 2 but less than 3. Is she right? Answer the question without using a calculator, and justify your answer. (Sw1, Book 2c, p. 120)	SI
5	Explain why one cannot calculate ${\log }_2\left(-5\right)$. (Sw2, Book 2c, p. 113)	SD
6	Using the power laws, show that $\lg A^y=y\cdot \lg A$. (Sw1, Book 2c, p. 122)	GD
7	In the EU parliament, there are 754 people from 27 states. Show that at least 28 people are from the same state. (Sw1, Book 5, p. 10)	SD
8	Ludvig and Philip are discussing the box principle. ‘The box principle says that if I have 13 t-shirts to dye green, blue or red, I will always get at least 4 t-shirts of every colour’, Ludvig says. ‘No’, Philip says, ‘the box principle only says that you always get at least one t-shirt of every colour’. Is either of them correct? Justify. (Sw2, Book 5, p. 99)	SE
9	What errors do they make? (a) Pierre simplifies $\lg 37-\lg 8$ and gets $\lg 37\text{/}\lg 8$. (b) Fia simplifies $\lg 5x^2$ and gets $2\lg 5x$. (Sw1, Book 2c, p. 122)	SC/GC

Table 5. Examples of proof-related tasks in the Swedish textbook series.

Ex	Task	Code
10	Let’s say that $m$ objects should be placed in $n$ boxes. What should be written in the square for the statement to be true? ‘If $m>⊡$ then at least one of the boxes will contain more than $k$ objects’. (Sw2, Book 5, p. 63)	GM
11	Per says that a doubling of sound intensity implies an increase in the sound level by 3 dB. Investigate whether this is true. Choose several different intensities. (Sw2, Book 2c, p. 114)	GI
12	Is there any number $k$ that solves the equation $C\left(8,5\right)=P\left(8,k\right)$? Justify your answer. (Sw2, Book 5, p. 69)	SI
13	Show that if the pH value decreases by 1, then the concentration of hydrogen ions increases by a factor of 10. (Sw2, Book 2c, p. 110)	GD
14	Kalle solves the equation $\lg x^2=4$ like this: $\lg x^2=4$, $2\lg x=4$, $\lg x=2$, $x={10}^2$. (a) Kalle finds it strange that the equation does not have two solutions when ‘$x$ is squared’. Show by testing that the equation has the roots $x=-100$ and $x=100$. (b) Solve the equation in such a way that you get both roots. Explain what is wrong with Kalle’s solution. (Sw2, Book 2c, p. 123)	SD/GC
15	Lise and Erik solve the equation $3\cdot 5^x=12$ correctly, but in two different ways. Lise gets $x=\lg 4\text{/}\lg 5$ and Erik gets $x=(\lg 12-\lg 3)\text{/}\lg 5$. Give suggestions for how Lise and Erik might have reached their answers. (Sw2, Book 2c, p. 110)	SO
16	At a Nordic conference there were 31 students from Sweden, Norway, Denmark, Finland, and Iceland. (a) Which number is $n$ (the number of ‘boxes’)? (b) Show that some country is represented by at least 7 students. (Sw1, Book 5, p. 10)	SO/SD

Table 6. Examples of proof-related tasks in the Finnish textbook series.

Ex	Task	Code
17	Show that the value of the expression $\left({\log }_{a}b\text{/}{\log }_{ac}b\right)-\left({\log }_{a}d\text{/}{\log }_{c}d\right)$ is independent of the numbers $a$, $b$, $c$, and $d$, which all are greater than one.	GD
	(Fi1, Book 8, p. 126)
18	Prove. $\left(\begin{array}{c}n\\ k\end{array}\right)=\left(\begin{array}{c}n\\ n-k\end{array}\right)$	GD
	(Fi1, Book 6, p. 74)

Table 7. Type of reasoning in justifications in expository sections

		No justification (N)	Left to student (L)	Specific case (S)	General proof (G)	Total no. of main results
Sw1	Logarithms & combinatorics	3		5	1	9
	Primitive functions & integrals	13		6	1	20
	Total	16		11	2	29
Sw2	Logarithms & combinatorics	3	1	5	3	9
	Primitive functions & integrals	5		11	2	18
	Total	8	1	16	5	27
Sw1 + Sw2	Logarithms & combinatorics	6	1	10	4	18
	Primitive functions & integrals	18		17	3	38
	Total	24	1	27	7	56
Fi1	Logarithms & combinatorics	6		2	8	15
	Primitive functions & integrals	10	2	3	12	27
	Total	16	2	5	20	42

Remark on double coding: Two logarithm justifications in Sw2 were coded SG, one combinatorics justification in Sw2 was coded LS, and one combinatorics justification in Fi1 was coded SG.

Table 8. Labelling and placement of main results and justifications in expository sections.

		Main results				Justifications
			Labelling				Labelling			Placement
		Total no.	Theorem	Rule, law, etc.	No/other	Total no. of justified main results	Proof	Check, etc.	No/other	Before	After
Sw1	Logarithms & combinatorics	9		6	3	6			6	6
	Primitive functions & integrals	20	1	6	13	7			7	7
	Total	29	1	12	16	13			13	13
Sw2	Logarithms & Combinatorics	9		7	2	6			6	3	5
	Primitive functions & integrals	18	1	4	13	13			13	13
	Total	27	1	11	15	19			19	16	5
Sw1 + Sw2	Logarithms & Combinatorics	18		13	5	12			12	9	5
	Primitive functions & integrals	38	2	10	26	20			20	20
	Total	56	2	23	31	32			32	29	5
Fi1	Logarithms & combinatorics	15		12	3	9			9	7	2
	Primitive functions & integrals	27	6	17	4	15	7		8	7	8
	Total	42	6	29	7	24	7		17	14	10

Remark on double coding: Two logarithm statements in Sw2 had specific justifications placed before and general justifications placed after the main results.

Table 9. Type and nature of reasoning in proof-related students’ exercises.

				Type of reasoning		Nature of reasoning
		Total no. of tasks	PR tasks	S	G	M	I	D	E	C	X	P	O
Sw1	Logarithms & Combinatorics	342	44	27	17	2	12	26	1	2			2
	Primitive functions & Integrals	418	74	51	23	11	35	26		2
	Total	760	118	78	40	13	47	52	1	4			2
Sw2	Logarithms & Combinatorics	427	65	25	40	9	9	41	1	2			4
	Primitive functions & Integrals	449	46	18	28	3	19	21	1	2
	Total	876	111	43	68	12	28	62	2	4			4
Sw1 + Sw2	Logarithms & Combinatorics	769	109	52	57	11	21	67	2	4			6
	Primitive functions & Integrals	867	120	69	51	14	54	47	1	4			0
	Total	1636	229	121	108	25	75	114	3	8			6
Fi1	Logarithms & Combinatorics	223	12	1	11		1	11
	Primitive functions & Integrals	524	38	16	26	1	8	29
	Total	747	50	17	37	1	9	40

Remark on double coding: One combinatorics task in Sw1 was coded EO and one logarithms task in Sw2 was coded MI.

Table 10. Type of reasoning in justifications in expository sections by country.

	No justification	Left to student	Specific case	General proof	Total
Sweden	41%	2%	47%	12%	58
Finland	38%	5%	12%	48%	42
United States	39%	11%	22%	31%	383

Remark: Percentages do not sum to 100% due to occasional double coding and round-off errors.

Table 11. Type and nature of reasoning in proof-related students’ exercises by country.

			Type of reasoning		Nature of reasoning
	Total no. of tasks	PR tasks	S	G	M	I	D	E	C, X, P, O
Sweden	1636	14.0%	53%	47%	11%	33%	50%	1.3%	6,1%
Finland	747	6.7%	34%	74%	2.0%	18%	80%	0%	0%
United States	9742	5.4%	46%	48%	15%	31%	44%	1.9%	11%

Remark: Types and natures are reported in relation to the total number of proof-related tasks. Percentages do not sum to 100% due to occasional double coding, X tasks without type classification, and round-off errors.