Katrina’s progress with learning mathematics
Jan McConnochie and Greg Sneath
Katrina is 10 years old and has Down syndrome. She is making good progress with learning and numbers and mathematics. We describe how Katrina has learned number concepts and arithmetic skills over several years. We highlight the influence of early learning habits, visual supports, motivation and practice, and the uses made of different number teaching schemes.
McConnochie J, Sneath G. Katrina’s progress with learning mathematics. Down Syndrome Research and Practice. 2007;12(1);34-37.
1 | Katrina completing mathematics exercises.
We are amazed to see how well our 10 year old daughter with Down syndrome is doing with her
mathematics. Yesterday, when her dad came home, Katrina was able to tell him why 5/4 and 9/8
were improper fractions and she could sit down with her Stern fraction pieces and work out what
they were as mixed numbers (1¼ and 1⅛). Looking back, it's very interesting to see what the
steps were that helped her to arrive at this point. Like the bike riding it seemed
to be a distant and impossible goal for a long time (years) and then suddenly it all seemed to
Many years of practising mathematical skills such as addition, subtraction, multiplication and
division have helped Katrina reach the stage that processes such as long multiplication and long
subtraction are becoming so automatic that Katrina doesn't even have to consciously think about
them. We believe that in this way the processing power that is available to Katrina can be
concentrated on other aspects of maths such as problem solving. As an analogy, it is easier to
understand the meaning of text if you don't have to sound out the words while you are reading.
We have lived a transient life and in quite a few of the schools we have been to, teachers have
commented that Katrina works harder and more consistently than any other child in the class. We
think a lot of that can be traced back to the habits and expectations set by pre-school early
education classes attended at The Sarah Duffen Centre and by Portage services. Katrina has never
really known any different and for the most part she does seem happy to sit down and finish her
set tasks, particularly if she feels she is successful. These good learning habits have been
Stern Structural Arithmetic
A programme for learning number concepts that is
based on reasoning and insight into mathematical
relationships, rather than rote learning and
A home-visiting educational service for pre-school
children with additional support needs and their
families. First developed in Portage, Wisconsin, USA
in the early 1970s, the service is now widely
available across the UK.
A teaching programme that emphasises a step-by-step
approach, building on success and learning through
A multi-sensory approach to arithmetic teaching that
uses patterns that are structured to encourage the
understanding of number and number relationships.
We started with early counting up to 20 at pre-school, age 3. Our biggest hurdles have been
achieving number bonds and then times tables. Once this foundation was in place everything else
has followed more easily.
Looking back at her progress in maths, Katrina started Kumon in 2001, aged 5, but stopped again
because of the time it took. Initially, she really found the dot patterns of the teen numbers
impossible to correctly identify. It was about that time that we started to use Numicon more
consistently with the pattern shapes to learn odd and even, and to learn the addition of numbers
to 10. At that time, using Numicon we also introduced 'doubles' and 2 times tables, which
Katrina seemed to learn. Her sister was a good role model, as she was also learning her times
tables. Katrina's school was pleased with Numicon using it to support school numeracy classes
and to demonstrate some of the confusing language of maths (e.g. minus, subtract and
Practice and persistence
In 2003, when Katrina was 7, we started Kumon again. This time it was far more successful
because Katrina had developed a lot of confidence using the Numicon templates to support her
number bonds. We were very lucky that we had Marie Gibb's Andover Kumon class nearby. Marie is
very dedicated to all her pupils and also very flexible and supportive. I think Katrina enjoyed
going along to Kumon classes – certainly she enjoyed the fabulous Kumon reward days, when the
children could exchange their earned Kumon stickers for sweets. Kumon practice on number bonds
to 10, then eventually to 20, continued every day for years. When we left the UK, Marie allowed
us to continue by correspondence still with quite detailed support. (Marie even posted the
sweets on the reward days!) Looking back, Katrina was doing at least 100 number bonds every day
for about 3 years. Even in Marie's Kumon classes, Numicon supported each new stage of progress
until Katrina developed the confidence to succeed without the props. As with Numicon or Stern
Maths, a key factor in Kumon was the structure it provided, making gradual steps forward by
consolidating from a baseline level where Katrina could be successful. Most of all it gave us an
appreciation of the progress that can be made when maths is practised for a short amount of time
every day. At every stage the biggest single obstacle has been when Katrina will not try because
she feels something is too difficult for her. The structure also gave us the confidence to
Figure 2 | Example completed mathematics exercises. Further examples are
available in SUPPLEMENTARY ONLINE FIGURES.
Five times tables were introduced along with Vikki Horner's 'Charlotte' clock in 2003. Soon
after some of the other times tables were introduced as well. We had some limited success with
the tables, however they were all quickly forgotten again. We maintained our focus on number
bonds to 20 (and subtractions) with the Kumon practice every day.
We continued by correspondence with Marie Gibb right up until 2005. We could no longer afford
Kumon, but with the very strong foundation in the principles of Kumon and the established daily
routines for practice, it was a successful transition when Katrina's mum found several web sites
which allowed us to start printing off our own series of worksheets.[2,3]
We are quite convinced that the daily practice with Kumon was essential. It probably helped that
Katrina could see that her brother and sister and other children had to do their Kumon as well
(it worked well for them also).
Then, over Christmas 2005, there was a concerted effort on times tables. The holiday was 6 weeks
long and each week one or two sets of times tables were concentrated on, starting with 2x and 5x
tables because Katrina had some previous confidence with them. The times tables were
demonstrated using Stern materials, and also with the ever popular raisins (which could be
eaten) and then lots of written examples using the printouts from the web pages.
Success and speed
All last year Katrina practiced several hundred maths examples every day before breakfast in
order to allow her to pass the weekly maths tests at school (addition, subtraction,
multiplication and division). To pass each level and move on to the next, the children had to be
able to answer 60 questions correctly in 4 minutes. Because of her daily practice, Katrina
managed relatively well and was able to move on to a new level most weeks. Kumon principles
insist on improving your personal times, and seeing her 'time score' remains an important
motivation and reward – as does the declaration of 'now 100%' after completing corrections (no
matter how many corrections there are). On the example work sheets you can see the times that
Katrina has recorded for each page, 4 minutes 35 seconds, 3:06 etc. This daily practice has led
to the stage where Katrina is now very confident (and very quick) with multiplication and
division up to the 12 times tables and addition and subtraction number bonds up to 20.
Figure 3 | Example completed mathematics exercises. Further examples are
available in SUPPLEMENTARY ONLINE FIGURES.
Small steps and visual support
Having confidence in number bonds and times tables has recently brought other maths operations
within her grasp. Squares and square roots were in the Year 6 tests and have been introduced
with quite good success, as have long addition / subtraction and long multiplication. We found
the Stern number board particularly useful for teaching long addition and subtraction. This
visual prop makes it obvious to children why it is necessary to carry and rename tens when
adding and subtracting larger numbers.
Katrina couldn't do it at all without the practical, visual props at each new stage of progress.
We used Numicon to start with and Stern materials when they became available later on. There
have been lots of ups and downs and periods of going backwards and periods of thinking we
weren't making much progress, but gradually Katrina developed speed and confidence with the
daily practice over several years.
We have found it is extremely important to break each mathematical process down into very small
steps to ensure constant success for Katrina. For example in long multiplication, Katrina spent
several weeks becoming very quick at long multiplication using single digit numbers (e.g. 972 x
7) before we introduced two digits (e.g. 972 x 17). We have recently introduced long division
using the Stern materials and games. Once Katrina developed a good understanding with the
concrete material we have started to introduce pages and pages of written practice of simple
divisions with remainders e.g.
4 r 3
5 | 23
Racing the clock is a great motivator. It also provides a measure of when Katrina has become
proficient enough to ensure that the long division method has become automatic (this will make
it easier to move on to more difficult calculations). It is equally important, we believe, to
teach Katrina all the correct terms for each aspect of mathematics e.g. numerator, denominator,
remainder, quotient etc. Recently when Katrina was practicing simple long division, we heard her
say for each example "and the quotient is …" It seemed that having a label for each of the terms
helped her remember each step that she needed to take.
"It seemed that having a label for each of the terms helped her remember each step that she
needed to take"
Katrina's mum has been very methodical and analytical in her approach to selecting the maths
targets, particularly when doing Kumon by correspondence. She and Marie exchanged ideas,
thoughts and recommendations about when to go back and consolidate and when to move up a level.
But equally, I think the process has been made much easier by the good habits and work skills
that were introduced during the pre-school years.
Maths skills in everyday life
We have not spent much time on problem solving and Katrina still needs a lot more practice in
applying the mathematical skills she has to real-life situations. However, over the last few
months with her improved competency in the times tables and number bonds we have noticed Katrina
effortlessly using these skills in simple daily tasks, such as calculating hours remaining on
the clock, how many places to set at the table when we have visitors, dividing up chocolates and
sweets and also in working out the amount of change she should get for simple transactions.
"Now she participates happily in family board games and some card games on an equal,
From 2003 onwards, games were fairly slow and tedious, as we helped Katrina work out the score in
simple card games, or add the numbers on the dice. Now she participates happily in family board
games and some card games on an equal, independent basis. Indeed, Katrina takes a lot of pride
and enjoyment in keeping the score for many different games and, as is the case for all of us,
there is that much more fun and enjoyment when she feels she is successful.
- Sneath G. Learning to ride a bicycle. Down
Syndrome News and Update. 2003;3(3):94-95. doi:10.3104/practice/244.
- Miller M. Free math worksheets. Homeschool Math. [Online] 2007 [cited 2007 Feb 12].
Available from: http://www.homeschoolmath.net/worksheets/
- Schoolhouse Technologies. Basic facts worksheet factory. [Electronic Download]. Seattle,
WA, USA: Schoolhouse Technologies; 2007 [cited 2007 Feb 12]. Available from:
Received: 1 February 2007; Accepted 14 February 2007; Published online: 30
DSE's Reading and Language Intervention for Children with Down Syndrome (RLI) is an evidence-based programme designed to teach reading and language skills to children with Down syndrome.
RLI incorporates best practice in structured activities delivered in fast-paced daily teaching sessions. It was evaluated in a randomised controlled trial and found to improve rates of progress compared to ordinary teaching.
Find out more...