Digging Deeper into … Representing and Interpreting Data (3rd-6th)

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For practical suggestions for families, and links to useful digital resources, to support children learning about the topic of data, please check out the following post: Dear Family, your Operation Maths Guide to Data

Data Analysis Process

Data analysis, whether at lower primary, upper primary, or even at a more specialised level of statistics, is essentially the same process:

  • It starts with a question, that doesn’t have an obvious and/or immediate answer. Information is then collected relevant to the question.
  • This collected information or data is represented in a structured way that makes it easier to read.
  • This represented data is then examined and compared (interpreted) in such a way as to be able to make statements about what it reveals and, in turn, to possibly answer the initial question (if the question remains unanswered, it may be necessary to re-start the process again, perhaps using different methods).

Thus, every data activity should start with a question, for example:

  • What is the most common eye/hair colour in our class?
  • Which fruit/pet do we prefer?
  • How did we come to school today?
  • What candidate would we vote for?
  • What is the temperature each day?
  • How many children are absent from school every day?

When choosing a question it is worth appreciating that some questions might not lend themselves to rich answers. Take, for example, the first question above; once the data is collected, and represented, there is not that much scope for interpretation of results other than identifying the most common eye/hair colour and comparing the number of children with one colour as being more/less than another colour. However, other questions might lead to richer answers, with more possibilities to collect further information, to make predictions and to create connections with learning in other areas. Take, for example, the question above about travel; the children could be asked to suggest reasons for the results e.g. can they suggest why they think most children walked/came by car on the day in question, whether weather/season/distance from school was a factor and to suggest how the results might be different on another day/time of year. Thus, the children are beginning to appreciate that data analysis has a purpose i.e. to collect, represent and interpret information, so as to answer a question. And, in the case of the questions about temperature and number of absences, the children may begin to appreciate that it is too much to give the specific details for each individual day and that a figure to represent a larger set of numbers (eg the average) is preferable in some situations.

Content overview

A quick glance at the curriculum content for representing and interpreting data for these classes, reveals the following:
3rd class: pictograms, block graphs, bar charts
4th class: pictograms, block graphs, bar charts and bar-line graphs incorporating the scales 1:2, 1:5, 1:10, and 1:100
5th class: pictograms, single and multiple bar charts and simple pie charts; calculating averages
6th class: pie charts and trend graphs; calculating averages

In Operation Maths for 3rd and 4th classes, representing and interpreting data is specifically taught in September, at the beginning of the school year, so that the children are enabled to incorporate these skills into other subject areas where possible e.g. reading and interpreting tables and graphs, collecting and displaying data in science, geography etc.

In 5th and 6th classes, representing and interpreting data is taught later in the school year, after the children have encountered degrees in lines and angles and the circle in 2D shapes, as this content is necessary prerequisite knowledge for creating pie charts. In these classes, representing and interpreting data is also taught as a double chapter (two week block), to allow for the extra time required to explore averages.

CPA

As with every topic in Operation Maths, a CPA approach is also recommended for representing and interpreting data:

Concrete: Using real objects to sort and classify eg the number of different colour crayons in a box, the different type of PE equipment in the hall etc; using unifix cubes, blocks, cuisinere rods etc to represent data; using cubes to introduce and explore the calculation of averages.
Pictorial: using multiple copies of identical images to make pictograms; using identical cut out squares/rectangles to make block graphs etc, using folded circles to make pie charts, using bar models to calculate averages.
Abstract: the final stage, where the focus is primarily on numbers and/or digits eg reading and interpreting the scale on a graph where all the scale intervals are not given; calculating averages without pictorial or concrete supports.

Interpreting data

For children to become comfortable interpreting tables and graphs it is vital that they have plenty of opportunities to look at and read a variety of tables and graphs. This shouldn’t be limited to just the tables and graphs in their maths books. In particular, data sets that are relevant to them, such as soccer league tables can be a great way to encourage the children to appreciate how relevant this strand units is to them.

Utilize every opportunity to expose them to real-life examples of data from print and digital media and use purposeful questions to highlight the features of the graph:

  • What is the title of this graph/chart?
  • How is the information displayed? Horizontally or vertically?
  • What type of chart/graph was used?
  • Why do you think this graph type was chosen? What other types would have been suitable?
  • What key information is required to interpret the data (eg scale intervals, labels on the axes, a key for piecharts)?
  • Is there information missing that would have been useful to get a better insight into the data?

The children can be asked to create questions based on the graph/chart and swap with a partner to answer. When they become adept at producing charts themselves (see next section) they can also be asked to represent the data using a different chart type.

One of the most common mistakes that children make when interpreting graphs is misreading the scale. Always draw children’s attention to this first, and ask them to identify the scale interval and what it means for the bars/blocks/points etc on the graph. The graph quiz on That Quiz provides lots of extra practice for this skill. The quizzes are also very customisable, with options to show pictograms, bar charts, trend graphs (line) and pie charts (circle), easier or normal content, and a variety of question types. Another similar activity is this one from MathsFrame which offers three different levels of questions on bar charts.

A very interesting  and very different way to explore interpreting data is to show the children graphs where much of the key information is missing initially, but is then slowly revealed as the children share their thoughts and ideas. Following on from Brian Bushart’s work on numberless word problems, many teachers have used graphs to create “slow reveal” activities or “notice and wonder graphs”, and have very generously shared these online for other teachers to use. Some of these include:

 

Representing data

As mentioned previously, where suitable children should begin to represent data themselves using concrete materials. They can build block graphs using cubes or blocks, laid flat on a piece of paper or their Operation Maths MWBs. These should all start from the same baseline and the children should also write in labels for the axes and a title.

As a development, they can then trace around the stacks of cubes and remove the cubes to have a pictorial representation of the concrete. Using cubes like this to represent 1:1 quantities can in turn lead children to see a need for one cube to represent more than one, ie scales of 1:10, 1:5 etc, especially if there are not enough cubes to represent the data or there is not enough space.

The next step could be to have small squares or rectangles of identical pieces of paper which can then be pasted onto a page to display the information. This can work particularly well for pie charts; cut out a circle of paper and divide it by folding into eighths; the circle can be left whole and the folds outlined in pencil/marker or the eighths can be cut up. A groups of eight children can then use either of these to show data like their favourite ice-cream flavour or TV programme. In this case, because the amount of data gathered is limited, the choices/categories should be limited, also, to three or four.

      

If the children are also collecting the data to make a graph or chart, they will need to come up with a system to accurately collect and record this data. This will usually involve compiling a type of table with three columns; the first column to list the categories, the second to record tally marks and the third to total the tally marks. When introducing tally systems the children could use lollipop sticks to explore and make tally marks.

For children, drawing their own graphs can present many difficulties. Some common mistakes that can be made include:

  • Incorrectly transferring the data from the table to the graph.
  • Omitting the graph title and/or category titles on the axes.
  • Using an inappropriate scale for a specific graph.
  • Not setting the scale at regular, even intervals
  • Zero being incorrectly located somewhere other than at the base line/axis.

And in other cases, it can just be a lack of neatness and exactness that reduces the quality, and readability, of a hand-draw graph. To overcome the difficulties associated with hand-draw graphs, the children could use either an online or offline computer application, all of which can produce very impressive results. Listed below are a small sample of those available; click on any of the links to access a tutorial or the application itself.

 

Calculating averages

Averages are introduced for the first time in 5th class and the children should have ample opportunities to explore this concept concretely and pictorially, before being given the formula to calculate the average of a set of numbers. Initially, the concept should be introduced as sharing amounts out to be fair/balanced:

Through plenty of concrete and pictorial opportunities to balance these separate quantities, it is hoped that the children begin to see a connection between the total number of items and the balanced quantity or average:

Bar models, one of the key problem-solving strategies used in Operation Maths, are very useful here, where comparison models can be used to compare the total of the averaged quantities with the total of the individual quantities. They are also used in Operation Maths 6 to calculate the extra number(s) when the average increases or decreases, a concept which can be very difficult to reason if no pictorial structures are used to help visualise the relationships.

You can also check out this video to see how bar models can be used to solve averages:

Further Reading and Resources

Digging Deeper into … Lines and Angles

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For practical suggestions for families, and links to useful digital resources, to support children learning about the topic of place value, please check out the following post: Dear Family, your Operation Maths Guide to Lines and Angles

Overview:

As can be seen from the overview table below, this topic is initially introduced to the children in 2nd class via “turns” and “square corners” and then develops with increasing complexity in 3rd- 6th classes.

  2nd 3rd 4th 5th 6th
Lines   vertical horizontal parallel perpendicular

 

oblique

    
Angles Full, half, quarter turns, square corner Right angle

 

Greater/less than a right angle

 Acute angle

 

Obtuse angle

 Straight angle

 

Reflex angle

Measuring and constructing in degrees

  
      Sum of angles in triangle  = 180° Sum of angles in quadrilateral  = 360°

As with every topic in Operation Maths, a CPA approach is also recommended for lines and angles:

  • Concrete: allow sufficient time for the children to explore making turns, lines and angles with suitable concrete materials (e.g. the children themselves, lollipop sticks, straws, geo strips, construction materials, real-life examples from the school and home environment)
  • Pictorial: activities where the focus is on drawing angles or lines on paper, MWBs etc
  • Abstract: the final stage, where the focus is primarily on numbers,  digits  and or letters to represent variables eg given the measure of two angles in a triangle calculate the third angle.

Lines:

Through exploration and activities, it is important that the children realise that:

  • A line can be classified and identified according to its position and its relationship to another line.
  • A single line can be horizontal, vertical or oblique but a single line cannot on its own be parallel or perpendicular; there must be two or more lines.
  • Parallel lines do not all have to be the same length to be parallel.
  • Parallel lines do not have to be horizontal or vertical, they can also be oblique.
  • Perpendicular lines do not have to have a horizontal and vertical line, (again they can be oblique) but there must be at least one right angle where the two lines meet.

Lines can be drawn on the Operation Maths MWBs and then rotated to reinforce this point.
It is also worth noting that in maths, when we use the word “line”, it should be assumed that this is always straight; only if the word curved is given should it be assumed otherwise.

Angles

In order for the children to recognise angles in terms of rotation, it is preferable initially, for the children to investigate the angles in their environment that are dynamic, (where the angle can be easily made bigger or smaller by increasing or decreasing the distance between the two lines) e.g. a door opening and closing, a scissors cutting paper, the angles made by the hands of a clock. The children can then proceed to examine static angles (where the angle is fixed) e.g. in 2-D shapes or 3D objects.

Operation Maths, Pupils’ Book 3: on the digital book, click on the icon on the bottom right to access a “Ready to go” activity.

In second class (and revised at the start of third class), the concept of rotation of an angle is taught through the terms quarter-turn, half-turn and full turn. Ideally, this should be introduced concretely by getting the children themselves to do half-turns and quarter-turns, and to turn in clockwise and anticlockwise directions:

  • In the classroom, the children start facing the board/front of room and make half/full/quarter turns to left/right as directed by the teacher.
  • Repeat, but this time with different starting points
  • Repeat, but this time after the teacher gives the directions the children must say where they will be facing, before they do the actual turn. The children could also record their predictions quickly on their Operation Maths MWBs

The children will be also be asked to identify 90º angles as square corners (2nd class) or right angles (3rd class). This will also be reinforced as part of the 2D shapes chapter. The children can be asked how they might decide if a corner/vertex of a shape is a square corner/right angle. Prior to the introduction of the protractor, something as simple as a corner torn from a piece of paper would suffice as an instrument with which to measure these angles.

The children should be enabled to classify angles according to the criteria appropriate to their class level (see table above). In particular, the ability to identify angles as acute (or less than a right angle), obtuse (or greater than a right angle) or reflex will greatly help the children, to later, estimate the measure of the angle in degrees, and to accurately measure and construct angles when they encounter this in 5th and 6th classes.

Operation Maths, Pupils’ Book 5

It is important to constantly reinforce the children’s understanding of what an angle actually is, i.e. an amount of turn and that this can be represented by two adjoining lines, one showing the starting position, the other showing the point after the turn. Return to concrete examples if necessary; the children stretch out two arms in front and, leaving one arm in original position, they move other arm a certain amount (90 º, 180º etc). This could also be repeated using geostrips, connected at one end using a brass clip, so as to be able to move one of the ‘arms’. Such concrete experiences also link well to measuring using a protractor; the original arm is the ‘base’ line.

Measuring and constructing angles (5th & 6th classes)

Operation Maths, Pupils’ Book 5

Using degrees to describe angles is introduced in 5th class, which develops to include measuring and constructing angles using degrees. This necessitates the use of a protractor for the first time, which in itself can lead to difficulties. The child may be unsure where exactly to place the protractor; this can come from a lack of understanding of what an angle actually is and where the angle actually is. Also, a child can be uncertain of which scale to use to measure the size of the angle eg for an acute angle measuring 45º, the child writes down 135º.

To reduce the likelihood of this arising you can ask some/all of the following questions:

  • What important tips would you give to a person about using a protractor?
  • How do you know which scale to use on the protractor?
  • What type of angle is this? How do you know? (To save time, they can write A, O or Re for Acute, Obtuse or Reflex).
  • Estimate the measure of the angle to the nearest 10º. Is your estimate/measurement sensible? Why?
  • How can you use what you know about acute and obtuse angles to check your measurement?

You can also watch some of the tutorial videos for using a protractor on the internet, such as the one below, for example. These videos can be a very visual way of demonstrating this skill. See also the list of digital resources for 5th and 6th classes on the following post: Dear Family, your Operation Maths Guide to Lines and Angles.

 

Reasonableness of answer

Angles is another area where it is important for the child to check the reasonableness of the answer. First, the child needs to identify whether the angle is acute or obtuse. Then, if measuring an acute angle, the measurement must be less than 90º. If it isn’t, then the correct scale wasn’t used.

 

Lines and Angles all around us

It is a given that lines and angles are all around us, although children may often be oblivious to the examples! Again, appropriate to each class level, the children should be encouraged to identify different types of lines and angles in their classroom, school and home. Enrich your own classroom space with lines and angles by labeling the line types in the room and the measure of angles of the open door. Make it personal by relating this topic to the children themselves and to the geometry in their names. Incorporate lines and angles into your visual art lessons (see also image below). Operation Maths 4 and 5 users can show the Maths Around Us video to their class, accessible on www.edcolearning.ie. For more ideas, check out this Lines & Angles board on Pinterest.

Operation Maths, Pupils’ Book 4

Further Reading and Resources

Digging Deeper into …. Place Value

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For practical suggestions for families, and links to useful digital resources, to support children learning about the topic of place value, please check out the following post: Dear Family, your Operation Maths Guide to Place Value

Place Value: A Fundamental Concept

When the new school year starts in September, for nearly every pupil from third to sixth class, the first mathematical topic they encounter is place value. This placement is logical; place value is the strand unit from which nearly all of the subsequent number, and measure, strand units build.

On the surface, place value may seem like it is one of the easiest topics to teach; the traditional activities simply involve counting dots on a notation board and/or beads on a place value abacus and, because of this, it is often viewed as an easy topic to kick-start the school year. And, it may appear to the teacher that the children have “got” it…..especially when they are getting all the correct answers in their books. However, it’s usually only later, when difficulties start to arise, often with operations or measures, that the teacher might start wondering “did they really get it?”

Place value is one of THE most important topics in primary mathematics, in that a child’s understanding of the fundamental concepts of place value will greatly impact on their understanding of almost all the other strand units, especially in operations, decimals and measures. Therefore, it is vital that teachers allow sufficient time for the children to explore this topic, moving from experiences with suitable concrete materials (e.g. base ten blocks) to pictorial activities (e.g. drawing base ten materials to represent a given number) and finally to abstract exercises, where the focus is primarily on numbers and/or digits.

That is why Operation Maths has a dedicated block of two weeks devoted to place value in third to sixth classes, and four weeks across the school year in first and second classes, so that there is sufficient time to explore the topic concretely and pictorially. This approach of moving from concrete to pictorial to abstract experiences is generally referred to as a CPA approach.

Indeed, spending sufficient time on meaningful activities now, may reduce potential hurdles later on. Furthermore, revisiting place value activities throughout the year, will allow the children to have ample opportunities to continuously revise and reinforce their understanding. One way to do this is to explore a Number of the Day on a regular basis; use the templates towards the back of the children’s Discovery Books or use the Number of the Day photocopiable in from the Teacher’s Resource Book (TRB). Indeed, a user of Operation Maths reported back to us “I found doing the ‘Number of the day activity’ as often as possible in September is crucial to the ‘Place Value’ chapter”.

CPA

Concrete materials are key to the children developing a good conceptual understanding of place value. The children need lots of opportunities, in all classes to explore and manipulate a variety of base ten materials. Where suitable/available, these should be introduced in the following order:

  • Groupable materials that the children can physically put together in collections of tens and physically take apart. These include lollipop/bundling sticks, straws (counting straws or ordinary drinking straws), unifix/multilink cubes, ten frames and counters etc.

Example of groupable materials: bundling sticks. Example of grouped materials: base ten blocks

  • Grouped materials are those already pre-grouped as tens, hundreds, etc., e.g. base ten blocks (also known as Dienes blocks) and/or ten frame flash cards with a pre-set number of dots/counters. These can’t be physically taken apart or combined, instead exchanging/swapping is required.
  • Lastly, non-proportional base ten materials.  These are materials that still operate on a base-ten system, but are not proportional to their value i.e. the piece that represents a ten is not ten times the size of the piece representing the unit. Examples include money and place value discs. Money in particular has the advantage that it can also be used to represent decimals i.e. 10c is one tenth and 1c is one hundredth of the unit (euro).

However, money is also limited in that it can only be used to represent numbers up to 999.99. This is where the Operation Maths place value discs become extremely useful. Inspired by similar discs used with Singapore Maths, these cut-outs are included in the free ancillary resources that accompany the scheme and can be used in conjunction with the place value mats on the inside back cover of the Discovery Books. With these discs, it is possible to concretely represent numbers up to 99,999, which had not been possible, using resources available in Ireland, prior to the publication of Operation Maths.

When exploring concrete or pictorial representations of numbers, most children will not have much difficulty interpreting the number once it is presented in the typical, canonical arrangement, i.e. 345 as 3H 4T 5U. However, many children may struggle
to interpret the number correctly if it is presented in a non-canonical arrangement, e.g. 345 as 3H 3T 15U or as 2H 14T 5U. Including activities based on these less common, non-canonical arrangements can encourage children to better understand the relationship between the places and can allow you to better assess the depth of the children’s conceptual understanding, while also preparing them for regrouping.

Operation Maths users can also use the excellent Place Value e-Manipulative, accessible on www.edcolearning.ie. This manipulative can be used to show blocks, straws, money or discs to represent a variety of numbers up to 99,999 and to two decimal places.

  1. Log into your edcolearning account
  2. Click on the Pupil Book icon for your class level.
  3. Click on the Edco Resources icon (on book cover image on left-hand side)
  4. Select e-Manipulatives from list of categories and then Place Value e-Manipulative.

Read also this post from Beyond Tradition Math showing children representing three-digit numbers in various ways. And watch this video from Origo One on using Numeral Expanders to show expanded form.

Whole Numbers and Decimal Numbers

Since whole number place value and decimal place value are inherently linked, in Operation Maths for 5th and 6th classes, in the topic of place value the children will explore both whole number and decimal place value together in a very holistic way, thus reinforcing their connectedness, within this strand unit.

While place value understanding includes both whole and decimal numbers, it is important that the children appreciate the differences between them. For example, whole numbers and decimal numbers differ in the variety of correct ways in which they can be written. One ten in standard form is usually written as 10; however, one tenth can be written as 0.1, .1, 0.10, 0.100, etc. For decimals, many teachers often only use one form, usually 0.1, fearing that a variety of ways may confuse children. Conversely, using a variety of ways can actually help reinforce children’s understanding that all of the above forms show one tenth (i.e. a 1 in the tenth place immediately to the right of the decimal point), with most forms (excluding .1) having unnecessary zeros (i.e. in 0.3 the zero is unnecessary; without it the value is still 3 tenths). In other numbers, zero acts as a necessary placeholder between the digits in the neighbouring places; in 30 and .304 the zeros are necessary: without them the values would be 3 units and .34 respectively.

Verbalising Numbers

For most of our number system, we read numbers in the order that we see the digits, e.g. 345 is three hundred and forty-five. However, the numbers from 11 to 19 are an exception, and as such can present extra difficulties for struggling children. Even for the child who begins to appreciate the meaning of ‘-teen’ as ‘and ten’, the numbers 11 and 12 are additional exceptions to this pattern. Some children may also have difficulties with the -ty numbers (e.g. 120, 130, 140) and in particular may confuse them with the similar -teen number, especially in its verbal form, e.g. fifty vs. fifteen. Even children in the middle and senior classes can struggle to distinguish between the -teen and -ty numbers and it is worth being aware of them as potential hurdles, during this topic.

The children should be given ample opportunities to say numbers out loud, with the emphasis being on the use of correct language to reinforce the concept, and the place value of each digit. One way to do this is to allow individual children to call out the numbers/answers when getting feedback or when correcting. Even when using the Operation Maths MWBs, ask a child each time to say what is written on the board. Encourage all adults supporting the children, including other teachers, assistants and parents, to use the correct word forms when reading out numerals. For the number 2,150, adults may say ‘twenty one fifty’, or ‘two one five oh’ instead of two thousand, one hundred and fifty. When verbalising zero make sure that zero is said instead of ‘oh’: O is a letter of the alphabet and not a digit (In your Operation Maths TRBs see also the Home–School Links section and the ‘Dear Family’ letters in the photocopiables section).

Regarding decimal numbers, the children can use both decimal language and/or fractional language, i.e. expressing 7.381 as seven point three eight one and also seven and three hundred and eighty one thousandths. Using fractional language to read decimals reinforces the value of the digit(s) in the decimal place(s). However, when using decimal language, it is incorrect to say ‘seven point three hundred and eighty one’, as this refers to hundreds and tens (–ty) which are both whole numbers.

Number Sense and Visualisation Skills

What is bigger; 12.352 or 12.952? When ordering or comparing, children may use a procedure of comparing digits, which involves examining the digits and realising that both numbers have 1 ten, both have 2 units and the first has only 3 tenths, while the second has 9 tenths, so it is bigger. While this procedure may work successfully, it does not encourage the child to visualise the quantities involved. It would be better for the child to recognise that 12.952 is almost 13 and 12.352 is only a little more than 12 and is therefore smaller. Similarly, with younger classes, it is better for the child to recognise that 52 is just a little more than 50 whereas 58 is almost 60. Asking the children to place the numbers on an empty number line (see below) is an ideal way to promote the development of these visualisation skills. Empty number lines are also a great visual strategy to use when rounding; when rounding 12.352 to the nearest unit we can see that it is between 12.3 and 12.4 so it is closer to 12. The children can use the partial number lines in their Discovery Books as an introduction to this method and then be encouraged to solve the rounding activities in their Pupils’ Books by drawing their own number lines, either on their MWBs or in their copies.

When considering rounding, it it worth noting that it is preferable to use the phrase ‘round(s) to’ as opposed to ‘round(s) up’ and ‘round(s) down’ as these can confuse many children. For example, some people may say the number 69 rounds up to 70, which makes sense since the digit in the tens place has gone up from 6 to 7. However, if a child realises that 43 should be rounded down, they might change it to 30 instead of 40, since that looks more correct to them, i.e. the tens digit has gone down to 3.

For another idea on how to use number lines to aid rounding, please check out this short video from Origo One

Some traditional tasks and activities for place value regularly seen in maths books can give an incorrect picture of a child’s understanding of the concepts of place value. For example, tasks that involve no more than the children identifying the number of identical dots on a notation board, or the number of identical beads on a place value abacus (see opposite), are not good indicators of a child’s understanding of place value, as they are simply demonstrating their number knowledge of numbers to 9. Therefore, these types of tasks have not been included in the Operation Maths series.

Bigger Numbers

It is vital the children realise that the digits in larger numbers are organised in groups of three with commas as the digit group separators. Insist that the children also use commas in this way, and reiterate that if you can read a three-digit number you can read any size number as long as there are commas present and that you understand the role of the different commas (i.e. one comma means thousands, two shows millions etc.). Interestingly, using commas as digit group separators is a convention largely in English-speaking countries , and other European countries tend to use stops/point or spaces. It is a good idea to highlight this to the older classes, as is done in Operation Maths 6.

It is also worth noting that the concept of the size of a thousand or the size of a million is in itself quite abstract for children. When tackling the large numbers, use all available opportunities to make them real and relevant to children. There is a Maths Around Us video available in the digital resources of Operation Maths 6 made for this purpose; just click on the hyperlink when accessing the digital book.

Also worth noting, is that Operation Maths explores numbers up to 5 digits (whole numbers) in 5th class and through millions in 6th class. However, in the curriculum there is no number limits in 5th and 6th class, therefore the children should not necessarily be limited to these numbers, particularly if they encounter bigger numbers in their environment, books, media, etc. The content in Operation Maths for these classes is deliberately presented in such a way as to encourage children to address numbers above millions where appropriate.

Place Value in the Environment

As mentioned mentioned previously, it is very important that the children can relate their understanding of place value to numbers around them. To reinforce the relevance of place value, ask the children to collect examples of numbers from the environment. This could include photographs of numbers in the school grounds or locality e.g. car registration numbers, distances on road signs (for Operation Maths 3 users, check out the Maths Trail in the car park, on page 4 of the Discovery Book) . It could also include examples of numbers from print media e.g. newspapers, magazines. In the older classes, you could challenge the children to find an example of a very large number and/or one with the most places of decimals (Operation Maths 6 users, check out the Maths Trail on the internet, on page 6 of the Discovery Book).

What to do with the numbers that the children locate:

  • Make a display for the classroom with the examples, in order of size, and giving information for the fact it relates to e.g. the distance to the nearest town etc
  • For each example the children find, they must write out the number in word and expanded form (it will likely be in standard form)
  • Round the number to the biggest place i.e. if the number is 312 round it to the hundreds, if it’s 0.012 round it to the hundredths etc.

Further Reading and Resources

This was the first in a series of “Digging Deeper into …” posts, which will take a more in-depth look at the various topics in primary maths. To ensure you don’t miss out on any future posts, please subscribe to the blog via email, on the top right hand of this page.

Operation Maths Quick-start Guide

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Category : About Operation Maths

Using Operation Maths for the first time? Here is a general quick-start guide for teaching the topics.

Operation Maths for Junior Infants to Second Class:

  1. Start with whole class warm-up and oral; see topic-specific suggestions in the Teacher Resource Book (TRB) or choose your own. Follow this with discussion questions (also in TRB).
  2. Pair work, an activity based in the At School book.
  3. Exploration of concrete materials via Maths Stations; again see topic-specific suggestions in the TRB.
  4. Complete relevant activities in At School and At Home books. The introduction of the TRB includes a year plan that lists the relevant pages of each book per topic, along with other details such as strand and strand units. Bearing in mind that Operation Maths is based on a CPA approach, it is envisaged that the child would engage in all the concrete and pictorial activities for the topic before doing the pages in their books.

For more detailed information on managing the content with Junior Infants to Second Class please read on here.

For a quick-start guide to the digital resources, please read on here.

 

Operation Maths for Third Class to Sixth Class:

  • The Teacher’s Resource Book (TRB) for Operation Maths 3-6 is divided into daily sections, each dedicated to a specific learning outcome.
  • Each topic has material for for either five or ten days, depending on whether it is a single/one week topic or double/two week topic. Double topics are indicated on the contents page of each book using an asterisk (*).

Teachers should start with the daily lesson suggestions in the Teacher’s Resource Book (TRB) as follows:

  • Oral and mental starter
  • Discuss and teach provides suggestions on how to achieve the learning outcome.
  • Pupils’ book and/or discovery book: gives the details for the location of the specific questions that reinforce and consolidate the learning outcome(s) covered in the discuss and teach section.
  • Digital Resources will briefly list any relevant digital activities that can be used from the comprehensive suite on edcolearning.ie . These are also referenced in the Pupils’ books as well and, if accessing the digital books, clicking on the hyperlinks in the Pupils Book will open the resource directly from the book (this is actually the easiest way to access the digital resources).
  • Extra exploration: Suggested activity for early finishers.

Pupils’ Book & Discovery Book: Since Operation Maths is based on a CPA approach, the children’s experience of Operation Maths should not be a purely book-based one. That said, when navigating the children’s books, it will follow this general pattern:

  • The topic (be it single or double), starts in the Discovery Book with the Starting Point activity (see example below), which revises familiar topics or sets the scene for new ones. There may often be no other book-based activities for Day 1.

  • Subsequent “days” (excluding the last day of each topic ie day 5 or day 10) may focus on the Pupils book only, or move between the Pupils’ Book and the Discovery Book. On the days when both books are in use, icons are used to indicate when it would be most appropriate to move to the other book (see below)

The icon on the extreme bottom right indicates that the child should complete the companion activities on page 16 of the Discovery Book next. The icon to the left indicates that there is also a linked digital activity for this learning outcome.

When the child has completed the activities in the Discovery Book there is often a similar icon there, redirecting the child back to the Pupils’ Book.

  • Consolidation is the focus of the last day of each topic i.e. day 5 or day 10. The children can complete the Learning Log activity in their Discovery Book either on this day or on the previous evening as a homework activity. They can also complete the topic assessment in
  • their Pupil Assessment book.

For more detailed information on managing the content with Third to Sixth Class please read on here.

For a quick-start guide to the digital resources, please read on here.

 

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Operation Maths Jr Inf-2nd: Managing the content

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Category : About Operation Maths

As outlined in a previous post, Operation Maths 3 – 6 provides fulsome content for the senior classes. The complaints heard about other schemes – that there is simply not enough to do in the senior class books – is definitely not one heard about Operation Maths! At the junior end of Operation Maths – which is the focus of this post – the Teacher Resource Books (TRBs) are the jewels in the crown: the most comprehensive available and jam packed with the ‘how to’ of setting up maths’ stations, differentiation, oral maths, discussion topics, early finisher activities and a comprehensive stand alone problem solving section. And, since this programme is also based on a CPA approach, the TRBs are full of suggestions on how to promote those methodologies in a classroom.

Familiarity with any new programme takes time and time is a very precious commodity for all us teachers. Therefore, in this post I will give you some tips on how to successfully implement the programme in the junior classes – what in today’s game parlance our students might call ‘cheats’!

1. Start from the Teacher Resource Book

Start with the weekly lesson suggestions in the Teachers’ Resource Book (TRB). Typically these will be laid out as follows:

  • A whole class warm-up and oral, designed to consolidate prior learning and lead logically into the lesson that follows. It is suggested that this lasts for 5-10 minutes  each day of the week, depending on content. While there are typically many suggestions here, it is not necessary to do all of them. If you find a starter that works particularly well, you could note this alongside the margin of your TRB, or in the notes section, to highlight it for future use.

The mini whiteboards are invaluable for this part of the lesson. Look out for children who lack the confidence or know how and are hesitant to write their answer or copy others. Encourage a growth mindset:
1. It’s okay to make mistakes, everyone does! We learn from them.
2. Often there is more than one correct approach; eg 17+19 can be modeled/thought of as move one to 19 to become 16 + 20, move 3 to 17 to become 20 +16, move one to 17 to become 18 + 18

  • Discussion questions that stimulate talk and discussion in a relevant and meaningful way. Again, only do as many as suits your circumstance.
  • Pair work, a book based activity to encourage co-operative learning. Modelling, especially when the concept of pair of group work is relatively new to a class, really sets the tone and promotes success. Choose a child to work with. Start the conversation:
    • I went first the last time, would you like to go first today?
    • Do you remember the first thing to do?
    • I think we roll the dice twice and add the numbers, do you agree?
    • Oh dear! Neither of us can remember what to do, will we quietly ask Tom?
    • Will you watch me while I’m taking my turn just in case I go wrong? I’ll help you too!
  • Stations: the organisation of these maths stations will depend on teaching style, the number of children, the ability level of the class and the assistance available from other staff members (SNAs, support teachers, etc.). And as with Pair Work it can take a little practice before the children approach stations successfully and productively – but it is well worth persevering! Station work promotes problem solving skills, group think and independence.The suggested stations can adapted in a number of ways:
    • use with similar ability groups or mixed ability
    • set up the activities at designated maths stations (tables or areas) which the class can rotate around eg 4 groups with 7 or 8 children per group; each group does two stations for 15 mins each for one class (30 mins total) and does the other two stations on the following day.
    • Each group does a station for one class, with each group working at each station over the course of the week.
    • Use the stations as a whole class activity e.g. on Monday all the class do the activities for station 1, on Tuesday do the activities for stations 2 etc. This does depend on there being enough of the required materials for the whole class to use them at the same time.
  • Books: Bearing in mind that Operation Maths is based on a CPA approach, it is envisaged that the child would engage in all the concrete and pictorial activities for the topic before doing the pages in their At School and At Home books. Furthermore, sometimes it is envisaged that the concrete activities for the topic at hand will take place during one week, followed by the book activities in the subsequent week (this will be explained in a paragraph under the Activities heading in the weekly plan in the TRB). If you are teaching in a multi-class situation, it would be better to stagger/alternate these weeks among the classes eg Week 1, first class do the concrete activities while second class are mainly book based; week 2, second class do the concrete activities while first class are mainly book based.

2. You don’t have to do it all!

In the junior end TRBs,  the plans are laid out in fortnights which then break-down into weekly suggested activities. The important word here is “suggested”; you are not expected to do everything, so pick and choose the activities that are most suitable for you, your children, the physical limitations of your class and/or equipment, the availability of support personnel. For example there are Aistear-linked themes and activities in the infant TRBs, but if these don’t appeal to you, or are not practical in your specific situation, ignore them.

As explained earlier, there are regular suggestions for stations in the first and second TRBs and in places in the infant TRBs, but again if you don’t have available colleagues (eg L/S Resource teachers, SNAs etc) to help with the running of these stations, then they probably are not for you. However, you could take one or two of the station activities and instead do it with the whole class as the same time. The choice is up to you.

3. MWBs! MWBs! MWBs!

I can’t stress how fabulously adaptable are the free mini-whiteboards or how they can make getting through content so much easier. I was using them for many years before the inception of Operation Maths and found them to be an invaluable tool in the classroom. Some of the ways in which they can be used:

Give Doodle Time! The temptation to doodle is overwhelming so spare a couple of minutes for a quick doodle or two! Signal the end of doodle time with a fun rhyme such as “Rub, a dub, dub! Give your whiteboard a scrub!”

Display the ebook on your IWB for Write-Hide-Show: This works very well as the children are not looking down at their own books, only up at the board, so it’s easier for teacher to check that they are focused on the task. Highlight a specific calculation on the ebook eg 16 + 5 and ask the children to write the answer on their MWBs, hide it (place it face down on the desk, or hold it face in, to their chest) while the other pupils are afforded thinking time and finally on a specific signal (eg aon, dó, trí, taispeán dom) all the answers are revealed simultaneously. Thus, the teacher can quickly assess the accuracy of the answers and allow this feedback to inform whether the class are ready to move on, or need more reinforcement.

“Show your thinking” The children can use quick jottings to explain how they arrived at a certain answer. The MWBs are less structured and easier to use than maths copies and easier to change if you want to amend your ideas. Interesting responses or approaches could easily be brought up to the top of the class for further discussion and display. Again, encourage the growth mindset; mistakes and multiple correct answers are opportunities to learn more.

More maths done in less time. Rooting in bags, finding their book, pencil, rubber… this all leads to a delay in actually getting down to the maths at hand. Whereas, just writing on the MWBs is much quicker and gets more done. And don’t worry if the associated page in the pupils book is not completed; remember the teacher’s aim should be to enable the children to achieve a certain objective/learning outcome and however that is achieved still counts, book or otherwise.

Step-by-step to show algorithms: if you are teaching some of the standard algorithms (eg column method addition or subtraction in first and second class) the MWBs can be handy to allow the teacher and class to do it together, step-by-step, with the children holding up their MWBs at every suitable juncture to check what they have done to that point. This way, potential mistakes may be picked up quicker and addressed before they begin to occur repeatedly.

4. Reduce your preparation

The plans are all done for you, the stations are all explained, the ideas are all there! This should significantly reduce the amount of time you were spending on maths preparation. However, it is still recommended to take the time at the beginning of each fortnight to go through the TRB and familiarise yourself with the content and the activities; this is time well spend that will translate into smooth running maths classes during the fortnight. But also be flexible, and don’t stick rigidly to everything.

One of the sections in the TRB where flexibility is advantageous is the photocopiables. There is a fantastic suite of resources here with great ideas, but don’t feel that if you don’t have 30 copies done in advance that you can’t use them. One example of this are the Yahtzee photocopiables in the TRB of Operation Maths 1. The children could simply write the target numbers ( eg 2-10, 2-20 or 0-5) on their whiteboards and cross them off when rolled. This also allows the game to be played repeatedly without needing other photocopies.

The one set of photocopies to have ready in advance are the Early Finishers and the Problem-Solving photocopiables. Initially, at the beginning of the school year, try to gauge how many copies you will need;you will probably not require 1 per child. As time goes on the number of copies of each can be adjusted, as necessary. These can then be kept near at hand to distribute to children in need of a more challenging or stimulating task.

5. Go digital!

The excellent suite of  digital resources available on Edco Learning can also aid efficient progress through content. The resources are very visual and help the child grasp a solid understanding of the concepts at hand quicker than might have occurred  otherwise. The resources can all be accessed directly via the hyperlinks in the digital books and it can be beneficial to have these tabs open in advance so as to save time during maths class. For more information on the extensive range of digital resources read on here

Teaching 3rd to 6th class? Read on to find out how to manage the content for those classes. 

Operation Maths – Improving standardised test scores?

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Category : About Operation Maths

Two days after the maths standardised tests were done in our school, a teacher on my staff came to me to let me know that 18 children in her room had gone up by a STen of 1 or more, a fact she was attributing to Operation Maths, which was being used throughout the school for the first time, since the previous September.

This information made me curious to see were there similar results in other class levels; below is a summary table of my findings:

Average Percentile for each class level, current and previous years in Drumcondra Primary Mathematics Test – Revised (DPMT-R):

Current Class 2012 2013 2014 2015 2016 2017 Difference 2016 to 2017
2nd 55 61 +6 PR
3rd 63 61 77 +16 PR
4th 66 61 70 79 +9 PR
5th 63 61 58 68 68 +0 PR
6th 42 53 73 75 75 75 +0 PR

These results only include the classes that had a previous DPMT-R to which a comparison could be made. Also, they are the average of all the children’s results that completed the tests in each year, therefore other variables like children of different ability joining or leaving the school hasn’t been accounted for. However, they do make for interesting reading, while also raising interesting questions:

  • 2nd, 3rd and 4th have made significant jumps, (3rd class in particular); could this be accredited to the Operation Maths programme (there were no other new initiatives this past year due to the freeze on the SIP for numeracy as directed by our union)?
  • 5th and 6th classes stayed the same; why wasn’t the programme as effective for these classes? In the case of this school, perhaps the scores being already quite high in those classes meant there was little room for improvement. Or perhaps, because Operation Maths is a radically different programme, one that requires an openness to change the way we think about maths, it has more impact on younger classes where the children are more malleable and less rigid in their way of thinking than some of the older students. If this is the case, could we then expect to see improved results also for 5th and 6th class students in the future when they have been using the programme from 2nd and 3rd class?

 

Of course, this is only a small insight into one school’s experiences, and to have more conclusive results, data would need to be collected from a wider range of schools and this data would need to be monitored over time to see if these results were maintained. However, it does raise some interesting questions, and does indeed appear to indicate tentative evidence that Operation Maths can have a positive impact on the standardised test scores of all the children in a class. That said, improving test scores was never the main goal of Operation Maths, rather the aim is for the children to understand maths, not just do maths.  And if standardised test scores increase simultaneously, then that indeed is a positive bonus!

Did you use Operation Maths for the first time this year? Have you seen any similar trends with classes in your school? Please share with us!

Post script: Some may also suggest that Operation Maths has question items that better prepare the children for those in the test (i.e. teaching to the test). Having deliberately tried to be as unfamiliar as possible with the  DPMT-R test when authoring Operation Maths, means I can’t comment either way as I just don’t know if the question items resemble test items. Personally, I have no experience of the SigmaT at all, and at the time of authoring Operation Maths,  the only DPMT-R that I had administered in the previous 6-8 years was the DPMT-R for 5th class.

 

Number Talks & Operation Maths

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Category : About Operation Maths

“The practice of number talks is one of the most powerful vehicles I know for helping students learn to reason with numbers and make mathematically convincing arguments, for building a solid foundation for algebraic reasoning, and for teaching mathematics as a sensemaking process. If all teachers make this shift in their practice, it would represent a profound advancement in mathematics education.”
Ruth Parker, co-author of  Making Number Talks Matter

As mentioned in a previous post, one of the mathematical pedagogies currently generating significant excitement is that of number talks. The buzz in maths education circles is all about developing number sense and number talks is being seen as one of the most powerful ways to enable this.

Here in Ireland, although the Professional Development Service for Teachers (PDST) has advocated the use of number talks in the PDST Mental Maths workshops and supporting manuals, and the more recent PDST Number Sense workshops, number talks is still relatively unknown. Similarly, there is very little in most of the maths text books available here, which explicitly promotes the development of specific mental maths strategies.

Not so Operation Maths. The promotion of the development of number sense is a key principle of the Operation Maths programme, as is the explicit exposure to a wide range of mental calculation strategies, most of which are also specified in the number talks approaches.

In this post, the connections between both number talks and Operations Maths will be shown, while also outlining how Operation Maths is the best programme to support the introduction and use of number talks in Irish classrooms. To read more about number talks generally, and access a whole suite of supporting resources  for all classes across the school,  please click here. To find out more about how Operation Maths works so well with number talks, please read on.

What does a Number Talk look like?

One of the definitive number talks texts is Sherry Parrish’s book Number Talks: Helping Children Build Mental Math and Computation Strategies, Grades K-5. In this book, she recommends the following structure:

Number Talks Approach

Operation Maths & Number Talks
1. The teacher presents a number sentence to the class; the students are given thinking time to mentally solve it. The horizontal number sentences in Operation Maths can in themselves inspire or be used as the basis for a number talk. For example, similar number sentences to the ones shown below this table were used to encourage the children to use compensation to solve calculations.
2. The students start with one fist to their chest;  they make a “thumbs-up” on their chest to show that they have found an answer. They then use the remaining time to try to think of another way/strategy which they then indicate by putting up a thumb and a finger, and so on. While I initially used this “fist and thumbs-up” system when collecting answers, after multiple times hearing “I had the same answer as Jack/Jill”, I returned to my preferred tool of using the Operation Maths mini-whiteboards, (to maximise on participation and honesty regarding answers).  It is important to insist that the MWBs are not to be used at all for working out, all of which is to happen in the heads, rather they should only be used to record the answer(s).
3. The teacher asks a number of children to volunteer their answers and all given answers are recorded on the board.
4. The teacher asks a child to “defend their answer”/”explain their strategy”. For the children to explain clearly, they need to have the correct mathematical language so that all listeners can follow their thinking. Thus, children who have been using the Operation Maths programme are typically better able to express their thinking using the correct mathematical language and terminology that is being emphasised throughout these books.
5. All strategies are recorded on board by teacher, using visuals where possible to make the strategy less abstract for the other listeners. Many of the visual strategies that are specifically recommended to be used are ones that already used extensively throughout Operation Maths eg frames, empty number lines, bar models (referred to as part/whole models), arrays and  area models. Branching is another visual way to demonstrate strategies particularly when partitioning (breaking into place value parts) /or compensation is involved.
6. The children agree on the “real” answer. Depending on the range of possible answers given, the children can also be asked to identify any unreasonable answer from those suggested and explain why they think so. This in turn encourages them to apply the variety of estimation strategies taught in the Operation Maths programme

These actual number sentences or similar ones could be used as the basis for a number talks session (from Operation Maths 1)

Other ways in which Operation Maths and Number Talks work so well together:

  • In the junior end of the school, number talks is very much about the children developing their ability to conceptually subitise  (i.e. to recognise that there is 8 counters because there is a group of  5 and a group of 3) using a variety of images, including five and ten frames. Operation Maths also recognises the value of using frames throughout the programme in Junior Infants to Second class and provides these frames as part of the pupils’ book packs in these classes, as well as having digital eManipulatives  (i.e. the Sorting eManipulative) to support their use.
  • In Sherry Parrish’s book Number Talks: Helping Children Build Mental Math and Computation Strategies, Grades K-5, she lists a whole range of specific strategies for the four operations, almost all of which are also explicitly taught or emphasized in the Operation Maths programme, including the strategy of compensation. To see an overview of the number talks strategies and where they overlap with Operation Maths click this link: Strategies in Number Talks & Operation Maths
  • For those teachers using Operation Maths, they are already familiar with the structure of having an oral and mental starter at the beginning of each maths lesson. Number talks can be used interchangeable with the starters in the Operation Maths starters bank so as to add further variety to lessons.
  • The strong emphasis on talk and discussion ( eg Talk Time in the pupils books, discussion and questions given in the TRBs) in Operation Maths further supports number talks as it prepares the children for situations in which they will be asked to explain their reasoning.

So there you have it, Number Talks & Operation Maths: a perfect partnership for each other!

Are you compensating?!

Tags :

Category : About Operation Maths

A key recurrent theme in Operation Maths is the teaching of specific strategies to promote the development of flexible and fluent mathematical learners. In a similar way to the Building Bridges approach to reading, which advocates explicitly teaching specific reading comprehension skills, Operation Maths explicitly explores a range of specific strategies in a spiral and progressive way, in order to equip the children with the necessary skills for them to become capable and confident at problem-solving and computing mentally. Particular to mental computation, Operation Maths introduces the children to a range of of mental calculation skills, one of which is compensation.

 

Compensation

 

Compensation is primarily an addition strategy where the aim is to to adjust one addend to become an easier number to add with.  This involves moving the quantity required to do this  from one addend to the other. In Operation Maths, these easier numbers are usually referred to as  friendly or compatible numbers and can include doubles, multiples of ten (10, 20, 30…) or, in the older classes, multiples of the powers of ten (100, 200, 300…..; 4,000,  5,000,  6,000 etc).

 

Concrete

As with all new concepts and strategies, Operation Maths advocates a CPA approach. An ideal introduction to compensation is with the Operation Maths frames in first class when the children first begin to notice how adding onto 9 can be made easier by moving a counter from the other quantity to the 9 to make it become a ten. When ready, the children can also begin to explore how they can also make tens when adding to 8 and 7 by moving 2 and 3 counters respectively.

This can progress to using cubes  for bigger numbers; again, this should start with addends ending in 9 eg 19, 29, 39 etc. Encourage the children to see ways to make the calculations become easier, and encourage them to use the language of moving (not adding or subtracting) a cube from one number to the other, to make a friendly number. When ready, they should then develop this strategy to use with addends ending in 8 and 7, by moving 2 and 3 from the other number. In this way, the children can also begin to start doing addition with renaming, without having to grapple with the traditional written algorithm ( or column method).

Pictorial

With first and second classes, it can be helpful also to show what is happening to the actual numbers in the calculation by using an arrow to highlight the quantity moving from one addend to the other. Notice how the calculation is being presented horizontally; this encourages children to consider the whole number and how it relates to the other number in the calculation. It also encourages the child to consider alternatives to the written column method, on which many children can be over-reliant.

In the senior end books for Operation Maths, branching (see red figures below) is used  to show the process of compensation and this can be particularly useful when the numbers involved are bigger than what might practically be shown using concrete materials. Never-the-less, it is always recommended to return to examples that can be demonstrated concretely, if the child finds the intermediary branching stage difficult to understand.

Abstract

The ultimate aim is, that when presented with a random calculation, that the children will recognize and use compensation if it is an appropriate and efficient strategy. The suitability of compensation as an efficient strategy will depend on the numbers involved, which in turn requires flexibility on the child’s part. In most cases, this will only be likely, if they have previously encountered compensation, and a variety of other mental computation strategies, in structured  and meaningful lessons, like those provided by Operation Maths.

 

Further reading:

 

Fostering the development of correct mathematical language and terminology

Tags :

Category : About Operation Maths

Last Friday, I was working with a group of first class children who were completing some first grade activities on Splash Math, an American maths site. While, on the plus side, the activities on this site are very visual and promote a CPA approach to mathematical instruction, on the down side, the first grade in the US isn’t aligned exactly to the maths curriculum for first class in Ireland, and so we have regularly encountered activities that might have unfamiliar language and terminology.

This was one of those days. We were looking at 2D shapes in the geometry section when a child said quizzically to me, “I’m stuck, miss”. The question was “How many vertices has the shape (a circle): 1, 3, 0, 2?”. I asked the class could anybody remember, from the previous day, what vertices meant? A flurry of hands went up to tell me “corners” at which point the child had no difficulty identifying 0 as the correct answer. Then I asked the children to remind me of all the other American words to do with geometry that we had come across the previous day, which I then recorded on the board for the benefit of all the children (see image below).

It brought home to me how correct mathematical language and terminology is much more prevalent in the primary maths curricula and texts of other countries, and how it is often even introduced much earlier, when compared to Ireland. And, how much of a disservice we do to children in Ireland if we try to shield them from this language in primary school, only to have it all thrust at them in secondary, where some children might wonder if it is the same subject they are doing at all!

It also reminded me of an RSE inservice I attended years ago, which stressed the importance of the children being introduced to the correct terminology for the body parts, so they might be able to properly communicate and report any incidences that might occur. In a similar way, should we not introduce children to the correct mathematical terminology, so as to enable them to communicate their thinking more clearly and to explain the approaches they took and the strategies they used?

That is why Operation Maths has been written as a programme which does not shy away from the correct mathematical language and terminology, rather it specifically uses words like commutative, distributive, associative, dividend, product etc when explaining concepts. Furthermore, when introducing new terminology it is done via concrete and pictorial activities with the back-up of  a range of images that enable the children to not just know the word, but to be able to picture it also, and in that way to truly understand the concept it describes.

As can be seen from the example above, new terminology and language is typically introduced as part of the teaching panels (yellow-coloured sections) and is often in a blue bold font to highlight it as being new/significant. The new term is then explained in simpler words and using visual examples to reinforce its meaning for the children. Since it is envisaged that these teaching panels would be presented/mediated by the teacher, this ensures that the teacher can help explain the vocabulary and that the child is not meeting the new term  in a random section of text.

The questions/exercises for the children that follow these teaching panels have also been specifically chosen to help reinforce the new term and consolidate the concept that it entails. These typically incorporate the use of concrete materials or pictorials representations (as in the case of the 100 dots grid/sheet mentioned above) for further exploration and reinforcement.

With all new terminology, when met again, there is typically some supporting text to remind the child and/or revise the meaning. Furthermore, the child can always consult the colourful glossary at the back of his/her pupil’s book if necessary.

Some of the advantages of using correct mathematical terminology in primary mathematics:

Preparation for second level: The NCCA has published a number of Bridging materials for maths, which encourage continuity between mathematics in primary and post-primary schools. Included in these materials, there is a glossary of terminology that teachers of 5th and 6th classes are encouraged to incorporate, where possible, so that children will be better prepared for second level maths, thus easing the transition from primary. This terminology was deliberately included in the Operation Maths books for 5th and 6th. Furthermore, where useful, some terminology was also incorporated in a simpler way in the Operation Maths books for 3rd and 4th so as to make the introduction more gradual.

Number Sense & Number Talks: The buzz in maths education circles is all about developing number sense. One approach that is being encouraged to support this is to have regular Number Talks to encourage the children to communicate how they mentally solved a calculation and to explore and discuss the various strategies that could  be used. The promotion of the development of number sense is a key principle of the Operation Maths programme, from the use of frames in the junior classes, right up to the use of thinking strategies, bar models and other pictorial structures in the senior classes. Similarly, the strong emphasis on talk and discussion ( eg Talk Time in the pupils books, discussion and questions given in the TRBs) in Operation Maths further supports this process. Ultimately however, this is all dependent on the children having a well developed range of mathematical terminology, by which they can clearly communicate and express their ideas and approaches.

Maths on the internet: Most of the maths we access on the net is american-based, be it You Tube videos, teaching sites, games, drill and practice sites. In the case of the latter, in many schools and homes, the children are encouraged to access teaching, drill and practice sites such as Khan Academy, Manga High, Splash Math, IXL.com etc to complement their core mathematical texts. As a result, Irish children will likely encounter, initially, terminology that is unfamiliar.  However, if they have encountered this terminology in their Operation Maths books, this will better prepare them for these sites. Indeed for those children and classes who have regularly accessed these non-Irish sites, they will probably have developed an understanding of this terminology already and its inclusion in Operation Maths will be unlikely to faze them at all.

 

Some FAQs:

Is this mathematical terminology in-line with the Irish Primary Mathematics Curriculum?

This is taken direct from the curriculum:
Third Class > Number > Operations >
The child should be enabled to explore, understand and apply the zero, commutative and distributive properties of multiplication.

Thus, not only is the terminology in-line with the curriculum, it raises the question how a child could previously have been enabled to “apply the commutative property” without being able to explain what he/she was doing and why, and furthermore how he/she could explain this without using the word “commutative” or “turn-around fact”?

Is is worth noting that the Teacher Guidelines, that accompanies the mathematics curriculum here in Ireland, includes a limited list of symbols, numerals, fractions and certain terminology for each class level (p. 70). However, other more generic terminology (eg product, factor, dividend etc) has not been categorised according to class levels, which contrasts with the curricula of other countries where specific terminology is typically specified for each year level/grade. Therefore, in writing Operation Maths, the authors categorised terminology into certain class levels based on evidence and practice in other countries.

Are the children expected to learn off and define this terminology? 

Of course not. In the same way as a teacher might use such terminology as simile, metaphor, alliteration etc to explain writing concepts in English, it is hoped that the teacher would use and reinforce specific terminology when appropriate, and in this way some of the children might also pick up this vocabulary and use it themselves when communicating their ideas. But it is not suggested or encouraged that these terms be drilled and “learnt off”.

We have a high number of children with dyslexia/English as a second language; should we avoid Operation Maths because of the language?

Actually, quite the opposite. While the teaching panels of Operation Maths may have more mathematical vocabulary that the competitor texts, they also have many more visual images that explain and demonstrate the concepts, and both the teaching panels and the exercises that follow are more concrete-based and pictorial in nature. This will in fact be better for children with limited language or language difficulties, as opposed to texts which are largely just digits and symbols, which themselves can be too abstract, particularly for senior classes. Plus, deliberately avoiding this language in primary only moves the issue on to becoming a bigger one when those children go to second level.
As mentioned previously, all of the Operation Maths programme is based on a CPA approach,  from the Pupils’ Book to the Discovery book, which is dominated by visual, rather than text, activities, to the free place value materials and frames, to the digital resources, eManipulatives and videos all of which place the emphasis on visual representations of content. This makes Operation Maths the most suitable programme for any child who is more of a visual learner.

Further suggestions, hints and tips:

Repetition, repetition, repetition! Whenever a new term is encountered don’t expect the children to know it,  understand it and use it straight away; research suggests that a child will typically need to encounter a word 15-20 times before they will start to use it. This is why it is important to use the term at every suitable opportunity and why in Operation Maths the term will be used repeatedly in various contexts to help this.

Use glossaries: As well as the Operation Maths glossary, use Jenny Eather’s, Maths Dictionary for Kids to look up new terminology and explore the visual and interactive activities that typically accompany each term. Another useful resource are the Math Vocabulary Cards from the Math Learning Centre, available to use online or download as a  free app. However, bear in mind that, while a definition in a glossary is useful, new terms must be also understood from meaningful examples and contexts relevant to the child.

Maths Word Wall: Whenever you encounter new terminology display it on your maths wall for future reference. This can be printed out vocabulary posters from the internet or small flash cards/A4 posters created by the children themselves. Aim to always include a pictorial representation and not just text. There are also lots of printable charts and posters available to download free from Jenny Eather’s, Maths Dictionary for Kids .

Start a personal maths dictionary: This allows children to keep a personal record of the vocabulary they encounter. Operation Maths users can use the vocabulary sections in the Discovery Book, where the children in 3rd and 4th must match the term to a definition and to an example. In Operation Maths 5, the children must provide the term to match the definition and, in Operation Maths 6, the children must provide the definition to match the term, as well as drawing an example in both cases. Thus the activities are getting slightly more difficult at each class level while continuing to emphasise the visual representations.

Use Number Talks: Through the regular use of Number Talks the children will begin to appreciate how having a good grasp of the correct mathematical language can help them explain their thinking in a more accurate and efficient way during number talks. Furthermore, he/she will realise that it is easier to understand the approach of a peer when they use terminology that he/she recognises and understands.

Make it fun: Play games such as matching games or “Just a Minute” word games.

Use matching activities, true or false, always, sometimes, never true etc: These type of language activities are included in the Operations Maths books to reinforce and consolidate the language acquisition. Also included are  oral discussion activities and “Talk Time” activities, to further promote discussion and exploration.

 

Further reading:

Developing mathematical vocabulary

Operation Maths 3-6: Managing the content

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Category : About Operation Maths

It can be very difficult to strike the correct balance of content in a maths programme;  a more able class might fly through the activities and conversely a less able group of children may work through content at a much slower pace. In a multi-class situation, the teacher may prefer to have more, rather than less, content so that one or more groups in the room can be kept occupied while the teacher is instructing a different group. Therefore, the volume of content required varies greatly from class to class and from school to school.

During the research and development phase of Operation Maths, the message from teachers was very clear: they wanted a maths programme with sufficient content and ideas, with no need to have to go sourcing extra material. Because of this feedback, the Operation Maths authors decided to err on the side of more, rather than less, content and designed a comprehensive maths programme that has considered everything a teacher may require, while also being able to be pared back to suits the needs of students and classes where a slower pace is preferable.

And not only is the Operation Maths programme highly adaptable to each unique teaching and learning situation, it is also based on the current, most forward-thinking approaches to maths education.

This post will provide some tips on how to best manage the programme in the senior classes, from third to sixth.

1. Start from the Teacher Resource Book

As always, when taking on any new programme it can sometimes take a while to discover the best ways to utilise it in order to maximise on its full potential for both you, as the teacher, and the children. Furthermore, since Operation Maths is based on many, very new and different approaches to the teaching of maths, this may leave teachers feeling a bit adrift initially.
That is why we recommend that those using Operation Maths for the first time should always start with the daily lesson suggestions in the Teachers Resource Book (TRB).  Typically, these will be laid out as follows:

  • A recommended oral and mental starter, designed to consolidate prior learning and lead logically into the lesson that follows. It is suggested that this lasts for 5-10 minutes.
  • The objective/learning  outcome for that day. This will also be given in the pupil book and/or discovery book
  • Discuss and teach is the most important section. This will give suggestions on how to achieve the objective learning outcome. The suggestions will differ depending on the specific learning outcome(s); for example there may be probing questions given or suggestions for a concrete, pictorial or digital activity which may lead the children to deduce the new learning outcome(s) for themselves. It may involve reading and discussing a teaching panel (yellow panel) in the pupils’ book.
  • Pupils’ book and/or discovery book: This gives the details for the location of the specific questions that reinforce and consolidate the learning outcome(s) covered in the discuss and teach section. It is not expected that all questions would be completed by all children and this is the main place where the teacher needs to decide what question activities are a priority for his/her pupils. Typically, the question sets are arranged to start with easier tasks and then graduate towards more difficult ones. There is often a section towards the end entitled “Work it Out” (blue panels in the pupils’ book) which contain the most difficult tasks and might be most suitable for the higher attainers (HAs) in the class.
  • Digital Resources will list any relevant digital activities that can be used from the comprehensive suite on edcolearning.ie . These are given in the Pupils’ books as well and may also have been referenced previously in the discuss and teach section.
  • Extra exploration: this is typically a suggestion of a game from the games bank that could be played by early finishers to reinforce the learning outcome of the day.

2. You don’t have to do it all!

As explained above, in the senior end TRBs , each topic is broken down into day-by-day plans which have a specific objective(s)/learning outcome(s) eg 5th class, Division, “I am learning to divide using chunking” or 3rd class, place value, “I am learning to identify the value of each digit in a number”. The discuss and teach sections lay out how to explore and teach each specific concept and what activities, either concrete, pictorial, digital or book-based can be used to reinforce the understanding.

However, it is not necessary that the class would do every single book-based exercise before they can move on. Rather, the teacher can select which ones they think most suitable for the ability level of their class. As explained previously, the initial question sets in each “day” are easier and then they progress in difficulty, often culminating in a  “Work it Out!” section. There are many different ways that a teacher could direct a class to answer these questions so as to facilitate differentiation:

  • The children progress through the questions themselves at their own pace, as individuals or as pairs perhaps, for support.
  • The children do a certain number in each question set eg first three in each; every second question etc
  • The teacher could allow the children to choose what questions to answer eg ” I want you to do five questions, you pick which ones” or “I want you to select two questions out of each set, you choose”. The children could discuss at the end the reasoning behind their choices thus providing a great insight into their understanding of a topic and their concept of themselves as learners.
  • The teacher could assign a number of incomplete questions as homework for that evening.

At the end of that “day”s maths lesson, it is likely that the children will have achieved the learning outcome, albeit to a variety of different depths, eg the child can identify the value of a digit in a number, even if not with 100% success rate. Irrespective of the content covered that day, in the next maths class, the teaching should move on to focus on the the next “day” and the next learning outcome(s), as envisaged in the day-by-day plans, thus ensuring that the children get a broad and balanced experience of the maths curriculum.

3. MWBs! MWBs! MWBs!

The free mini-whiteboards (MWBs) that accompany the Operation Maths programme are very adaptable  and can make covering content, in a meaningful way, so much more efficient. Some of the ways in which they can be used:

  • Display  the ebook on your IWB (this is often preferable to the children looking at their own books  as they are looking straight up at the board, and therefore easier to check that the children are focused on the teacher and the task). Then, using  a selection of suitable questions from the book, you can use the MWBs for some fast-paced answering. This can be a great way to get through all/most of that day’s content, while also revealing any problematic questions/misconceptions that can then be focused on again as part of class-based reinforcement or practiced as part of homework.
  • “Show your thinking”. The children can use quick jottings to explain how they arrived at a certain answer. The MWBs are less structured and easier to use than maths copies and are quicker and easier to change if you want to amend your ideas. Interesting responses or approaches could easily be brought up to the top of the class for further discussion and display.
  • More maths done in less time. Rooting in bags, finding a copy, ruler, pencil, pen…ruling the copy, asking what date it is….this all leads to a delay in actually getting down to the maths at hand. Whereas, just writing on the MWBs is much quicker and gets more done.
  • Bar models: This is one of the key problem-solving strategies used in Singapore Maths and a key strategy also in Operation Maths. If your pupils are not very familiar or comfortable with bar model drawing (for example if the children are using an Operation Maths book and didn’t have Operation Maths the previous year) it can be a great idea to draw the bar models step-by-step with the children i.e. the teacher draws on the classroom board and the children draw on their MWBs. Alternatively, the teacher can use the Bar Modelling eManipulative, available on Edco Learning to model the problems on the main IWB.
  • Quick fire estimations: estimating should be quick responses and not take as long to produce as a full calculation would; otherwise they are not efficient (see this post for more on this). To practice these quick fire responses, you could quickly display a calculation on the class board from the Operation Maths ebook and then hide the calculation (eg use the no-show button on your projector remote) while the children quickly jot down estimated answers. These should then be compared and discussed, with reasons given as to why some estimates are more reasonable than others, before then agreeing on the most reasonable estimate(s).
  • Step-by-step to show algorithms: if you are teaching some of the standard algorithms (eg the long division or long multiplication method) the MWBs can be handy to allow the teacher and class to do it together, step-by-step, with the children holding up their MWBs at every suitable juncture to check what they have done to that point. This way potential mistakes may be picked up quicker and addressed before they begin to occur repeatedly.

For other ideas on how you can use your Operation Maths MWBs across the curriculum, read on here.

4. Bar model drawing

As mentioned above, bar model drawing can be a difficult concept for both teachers and children to grasp when they’ve never come across them before. That said, they are an invaluable strategy and worth the investment; already feedback from teachers using the programme for the first time have revealed that topics the children previously found very problematic (eg fractions in all classes, cost price and selling price in 6th class), have now become so much easier and clearer, thanks to the structure of the bar models.

A way to make your collective introduction to bar models much easier, is to display the Thinking Blocks site on the class IWB and to get the children to respond by drawing the bar models and/or giving answers on their MWBs. Such an activity would also work well as an oral and mental starter that could be used regularly throughout the year.

5. Go digital!

The excellent suite of  digital resources available on Edco Learning can also aid efficient progress through content. The resources are very visual and help the child grasp a solid understanding of the concepts at hand quicker than might have occurred  otherwise. The resources can all be accessed directly via the hyperlinks in the digital books and it can be beneficial to have these tabs open in advance so as to save time during maths class. For more information on the extensive range of digital resources read on here

Teaching Junior Infants to 2nd class? Read on to find out how to manage the content for those classes. 

 

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